Production Economics

Barry T. Coyle

This is a draft: please do not distribute c Copyright; Barry T. Coyle, 2010

October 9, 2010

University of Manitoba

Foreword

This booklet is typed based on professor Barry Coly’s lecture notes for ABIZ 7940 Production Economics in Winter 2008. I am responsible for all the errors and typos.

Winnipeg, October 2010 Ning Ma

Contents

1 Static Cost Minimization ...... 1 1.1 Properties of c.w; y/ ...... 1 1.2 Corresponding properties of x.w; y/ solving problem (1.1) ...... 3 1.3 Second order relations between c.w; y/ and f .x/ ...... 4 1.4 Additional properties of c.w; y/ ...... 6 1.5 Applications of dual cost function in econometrics ...... 7 1.6 Conclusion ...... 8 References ...... 9

2 Static Competitive Profit Maximization ...... 11 2.1 Properties of .w; p/ ...... 11 2.2 Corresponding properties of y.w; p/ and x.w; p/ ...... 13 2.3 Second order relations between .w; p/ and f .x/, c.w; y/ ...... 14 2.4 Additional properties of .w; p/ ...... 16 2.5 Le Chatelier principles and restricted profit functions ...... 17 2.6 Application of dual profit functions in econometrics: I ...... 19 2.7 Industry profit functions and entry and exit of firms ...... 21 2.8 Application of dual profit functions in econometrics: II ...... 23 References ...... 25

3 Static Utility Maximization and Expenditure Constraints ...... 27 3.1 Properties of V .p; y/ ...... 29 3.2 Corresponding properties of x.p; y/ solving problem (3.1) ...... 31 3.3 Application of dual indirect utility functions in econometrics ...... 32 3.4 Proﬁt maximization subject to budget constraints ...... 34 References ...... 38

4 Nonlinear Static Duality Theory (for a single agent) ...... 39 4.1 The primal-dual characterization of optimizing behavior ...... 39 4.2 Producer behavior ...... 41 4.3 Consumer behavior ...... 43

vii viii Contents

References ...... 45

5 Functional Forms for Static Optimizing Models ...... 47 5.1 Difficulties with simple linear and log-linear Models ...... 47 5.2 Second order flexible functional forms ...... 49 5.3 Examples of second order flexible functional forms ...... 51 5.4 Almost ideal demand system (AIDS) ...... 57 5.5 Functional forms for short-run cost functions ...... 58 5.5.1 Normalized quadratic: c c=w0, w w=w0...... 59 5.5.2 Generalized Leontief:. . . .D ...... D ...... 59 5.5.3 Translog: ...... 59 References ...... 59

6 Aggregation Across Agents in Static Models ...... 61 6.1 General properties of market demand functions ...... 61 6.2 Condition for exact linear aggregation over agents ...... 64 6.3 Linear aggregation over agents using restrictions on the distribution of output or expenditure ...... 72 6.4 Condition for exact nonlinear aggregation over agents ...... 75 References ...... 76

7 Aggregation Across Commodities: Non-index Number Approaches . . 77 7.1 Composite commodity theorem ...... 78 7.2 Homothetic weak separability and two-stage budgeting ...... 79 7.3 Implicit separability and two-stage budgeting ...... 83 References ...... 85

8 Index Numbers and Flexible Functional Forms ...... 87 8.1 Laspeyres index numbers and linear functional forms ...... 87 8.2 Exact indexes for Translog functional forms ...... 89 8.3 Exact indexes for Generalized Leontief functional forms ...... 96 8.4 Two-stage aggregation with superlative index numbers ...... 97 8.5 Conclusion ...... 100 References ...... 100 8.6 Laspeyres and Paasche cost of living indexes ...... 102 8.7 Fisher indexes (for inputs) ...... 103

9 Measuring Technical Change ...... 105 9.1 Dual cost and proﬁt functions with technical change ...... 106 9.2 Non-Parametric index number calculations of changes in productivity ...... 110 9.3 Parametric index number calculations of changes in productivity . . . 113 9.4 Incorporating variable utilization rates for capital ...... 117 9.5 Conclusion ...... 120

Index ...... 123 Chapter 1 Static Cost Minimization

Consider a firm producing a single output y using N inputs x .x1; ; xN / according to a production function y f .x/. The firm is a priceD taker in its N D factor markets, i.e. the firm treats factor prices w .w1; ; wN / as given. We assume that all inputs are freely adjustable and perfectD rental markets exist for all capital goods, i.e. for now we ignore all costs of adjustment that can lead to dynamic behavior. A necessary condition for static profit maximization is static cost minimization, i.e. at the profit maximizing level of output y and input prices w the firm necessarily solves the following cost minimization problem:

N X min wi xi wx x 0 D (1.1) i 1 D s.t. f .x/ y The minimum cost c wx to problem (1.1) depends on the levels of input prices w and output y, andÁ of course on the production function y f .x/. By solving (1.1) using different value of .w; y/ we can in principle trace outD the relation between minimum cost c and parameters .w; y/, conditional on the ﬁrm’s particular production function y f .x/. This relation c c.w; y/ between minimum cost and parameters .w; y/ isD called the ﬁrm’s dual costD function.

1.1 Properties of c.w; y/

Property 1.1. a) c.w; y/ is increasing (or, more precisely, non-decreasing) in all parameters .w; y/.

1 2 1 Static Cost Minimization

Fig. 1.1 c.w; y/ is concave c(wA, wB, y) in w

b) c.w; y/ is linear homogeneous in w. i.e. c.w; y/ c.w; y/ for all scaler > 0 (if all factor prices w increase by the same proportion.D e.g. 10%, then the minimum cost of attaining the same level of output y also increases by this proportion) c) c.w; y/ is concave in w. i.e. c.wA .1 /wB ; y/ c.wA; y/ .1 C C /c.wB ; y/ for all 0 1. See Figure 1.1 d) If c.w; y/ is differentiableÄ Ä in w, then

@c.w; y/ xi .w; y/ D @wi

(Shephard’s Lemma)1

Proof. Property 1.1.a is obvious from Equation (1.1), and 1.1.b follows from the fact that an equiproportional change in all factor prices w does not change relative factor prices and hence does not change the cost-minimizing level of inputs x for problem (1.1). 1.1.c is not so obvious. In order to prove it simply note that, for wC wA .1 /wB and x solving (1.1) for .wC ; y/, Á C C

c.wC ; y/ wC x Á C wAx .1 /wB x (1.2) D C C C c.wA; y/ .1 /c.wB ; y/ C since wAx c.wA; y/ and wB x c.wB ; y/ (e.g. wAx cannot be less than C C C the minimum cost for problem (1.1) given prices wA wC : wAx > c.wA; y/ ¤ C unless xC solves (1.1) for prices wA as well as for prices wC ). Numerous proofs of Shephard’s Lemma 1.1.d are available. Here we simply consider the most obvious method of proof (see Varian 1992 for alternative methods). Expressing (1.1) in Lagrange form

1 Note that c.w; y/ can be differentiable in w even if, e.g. the production function y f .x/ is Leontief (ﬁxed proportions). In general differentiability of c.w; y/ is a weaker assumptionD than differntiability of y. 1.2 Corresponding properties of x.w; y/ solving problem (1.1) 3 min wx .f .x/ y/ wx f .x/ y c.w; y/ (1.1’) x; Q D Q D Q with ﬁrst order condition @f.x/ wi 0 Q @xi D

f .x/ y 0 D Then @c.w;y/ can be calculated by total differentiation as follows: @wi

N @x Â Ã @c.w; y/ X j @f.x/ @Q xi wj f .x/ y @wi D C @wi Q @xj @wi j 1 (1.3) D x D i by the first order conditions to problem (1.1’). ut It is important to note that Shephard’s Lemma 1.1.d is simply an application of the envelope theorem (Samuelson 1947). The lemma states that, for an infinitesimal change in factor price wi (all other factor prices and output remaining constant), the change in minimum cost divided by the change in wi is equal to the equilibrium level of input i in the absence of any change in .w; y/. In other words, in the limit, zero changes in equilibrium x in response to a change in wi are optimal. Obviously such a lemma has no economic content, i.e. does not describe optimal response to finite changes in wi . Nevertheless Shephard’s and analogous envelope theorems are critical to the empirical and theoretical application of duality theory. This distinction is easily missed in more complex models.

1.2 Corresponding properties of x.w; y/ solving problem (1.1)

Property 1.2. a) x.w; y/ is homogeneous of degree 0 in w. i.e. x.w; y/ x.w; y/ for all scalar 0. D h i b) @x.w;y/ is symmetric negative semideﬁnite. @w N N

Proof. 1.2.a simply states that the cost minimizing solution x to problem (1.1) depends only on relative prices. In order to prove 1.2.b, note that the Hessian matrix h 2 i @ c.w;y/ is symmetric negative semideﬁnite by concavity and twice differen- @w@w N N @c.w;y/ tiability of c.w; y/ in w, and then note that @w x.w; y/ for all w (Shephard’s h 2 i h i D Lemma) @ c.w;y/ @x.w;y/ . ) @w@w N N D @w N N ut 4 1 Static Cost Minimization

In order to test economics theories it is important to know all of the restrictions that are placed on observable behavior by particular theories. This is known as the integrability problem in economics. It can easily be shown that 1.2.a-b exhausted the (local) properties that are placed on factor demands x.w; y/ by the hypothesis of cost minimization (1.1).

Proof. It has already been shown that the properties 1.1 of the cost function imply 1.2. In order to show that 1.2 exhausts the the implications of cost minimization (1.1) for local properties of x.w; y/, ﬁrst total differentiate c.w; y/ wx.w; y/ Á with respect to wi ,

N @c.w; y/ X @xj .w; y/ xi .w; y/ wj i 1; ;N: (1.4) @wi Á C @wi D j 1 D

2 PN @xj .w;y/ 1.2.a implies (by Euler’s theorem ) wj 0 (i 1; ;N ) and to- j 1 @wi D D @c.w;y/D gether with symmetry 1.2.b this reduces the identity (1.4) to @w xi .w; y/ h i i D 0,(i 1; ;N ) (Shephard’s Lemma). In addition @x.w;y/ 1.2.b implies D @w N N @c.w;y/ that the system of differential equations xi .w; y/ .i 1; ;N/ inte- D @wi D grates up to an underlying cost function c.w; y/ (Frobenius theorem3). Shephard’s Lemma also implies (by simple differentiation of x.w; y/ @c.w; y/=@w with h i h 2 i D respect to w) @x.w;y/ @ c.w;y/ which is negative semidefinite @w N N D @w@w N N h 2 i by 1.2.b. It can then be shown that @ c.w;y/ negative semidefinite implies @w@w N N y f .x/ is quasiconcave at x x.w; y/ (see (1.6) below). This establishes theD second order conditions on theD production function y f .x/ for competitive cost minimization. The first order condition follow from theD fact that 1.2 establishes Shephard’s Lemma for all @x.w;y/ , @.w;y/ , satisfying 1.2 (see (1.3)). @w @w ut

1.3 Second order relations between c.w; y/ and f .x/

It is sometimes interesting to ask whether or not the firm’s production function y f .x/ can be recovered from knowledge of the firm’s cost function c.w; y/, i.e.D can we construct y f .x/ directly from knowledge of c.w; y/? The answer is essentially yes (the onlyD qualification is that we cannot recover f .x/ at levels of x that cannot be solutions to a cost minimization problem (1.1) for some .w; y/, i.e. at levels of associated with locally non-convex isoquants). For example, given knowl-

2 Euler’s theorem states that ,if g.x/ r g.x/ for all scaler > 0 (i.e. the function g.x/ is DP @g.x/ @g.x/ P @2g.x/ homogenous of degree r), then rg.x/ i xi and .r 1/ i xi D @xi @xj D @xi @xj 3 @ .v/ The Frobenius theorem states that a system of differential equations gi .v/ .i @vi @g .v/ D D 1; ;T/ has a solution .v/ if and only if @gi .v/ j .i; j 1; ;T/. @vj D @vi D 1.3 Second order relations between c.w; y/ and f .x/ 5 edge of c.w; y/ and differentiability of c.w; y/, application of Shephard’s Lemma @c @c x1; ; xN immediately gives the cost-minimizing levels of inputs @w1 @wN x correspondingD to .w;D y/ (this assumes a unique solution x to problem (1.1)). By varying .w; y/ we can map out y f .x/ from c.w; y/ in this manner. A related point that will be importantD later (in a lecture on functional forms) is that the first and second derivatives of f .x/ at x.w; y/ can be calculated directly from knowledge of the first and second derivatives of c.w; y/. The first derivatives can be calculated simply as

@f .x.w; y// w i i 1; ;N (1.5) @xi D @c.w; y/=@y D

@f.x/ using the ﬁrst order conditions wi 0 .i 1; ;N/ for cost minimiza- @xi Q D D tion (1.1’), where @c.w;y/ . The procedure for calculating the second derivatives Q Á @y of f .x/ from c.w; y/ is not quite as obvious. The corresponding formula in matrix notation is

2 2 1 " @c.w;y/ @2f @f # " @ c.w;y/ @ c.w;y/ # @y @x@x @x @w@w @w@y ˝ 2 2 (1.6) @f 0 D @ c.w;y/ @ c.w;y/ @x .N 1/ .N 1/ @y@w @y@y .N 1/ .N 1/ C C C C where it is assumed without loss of generality that the above inverse exists4.

Proof. Consider the case N 2 so y f .x1; x2/ and c c.w1; w2; y/. The ﬁrst order condition for cost minimizationD (1.1’)D can be writtenD as

w1 cy .w; y/fx 0 1 D w2 cy .w; y/fx 0 (1.7) 2 D y f .x/ D @c.w;y/ @f.x/ where for now cy .w; y/ , fx , etc. Á @y 1 Á @x1 Total differentiating these conditions (1.7) with respect to .w1; w2; y/ yields (using Shephard’s Lemma):

4 PN @2c.w;y/ c.w; y/ linear homogenous in w and Euler’s theorem imply 0 wj .i j 1 @wi @wj D h 2 i D D 1; ;N/, which implies @ c.w;y/ does not have full rank. Nevertheless the above bor- @w@w N N " @2c.w;y/ @2c.w;y/ # @w@w @w@y dered matrix @2c.w;y/ @2c.w;y/ generally have full rank. @y@w @y@y .N 1/ .N 1/ C C 6 1 Static Cost Minimization 8 1 cyw fx cy fx x cw w cy fx x cw w 0 @ <ˆ 1 1 1 1 1 1 1 2 1 2 D cyw fx cy fx x cw w cy fx x cw w 0 @w ) 1 2 1 2 1 1 2 2 1 2 D 1 :ˆ fx1 cw1w1 fx2 cw1w2 0 8 D cyw fx cy fx x cw w cy fx x cw w 0 @ <ˆ 2 1 1 1 1 2 1 2 2 2 D 1 cyw fx cy fx x cw w cy fx x cw w 0 (1.8) @w ) 2 2 1 2 1 1 2 2 2 2 D 2 :ˆ fx1 cw1w2 fx2 cw2w2 0 8 C D cyy fx cy fx x cw y cy fx x cw y 0 @ <ˆ 1 1 1 1 1 2 2 D cyy fx2 cy fx1x2 cw1y cy fx2x2 cw2y 0 @y ) ˆ D : fx cw y fx cw y 0 1 1 C 2 2 D Writing (1.8) in matrix notation, 2 3 2 3 2 3 cw1w1 cw1w2 cyw1 cy fx1x1 cy fx1x2 fx1 1 0 0

4 cw1w2 cw2w2 cyw2 5 4 cy fx1x2 cy fx2x2 fx2 5 4 0 1 0 5 (1.9) D cyw1 cyw2 cyy fx1 fx2 0 0 0 1

1.4 Additional properties of c.w; y/

Property 1.3. a) y f .x/ homothetic c.w; y/ .y/ c.w; 1/ for some function . b) y D f .x/ constant returns, to scaleD c.w; y/ y c.w; 1/. c) AllD the partial elasticity of substitution, betweenD inputs i and j (output y constant) @.xi .w;y/=xj .w;y// xi =xj ij .w; y/ Á @.wi =wj / wi =wj can be calculated simply as

2 c.w; y/ @ c.w;y/ @wi @wj ij .w; y/ D @c.w;y/ @c.w;y/ @wi @wj (see Uzawa 1962, p. 291-9). d) Assuming a vector of outputs y y1; ; yM . The transformation function D f .x; y/ 0 is disjoint (i.e. y1 f1.x1/; ; yM fM .xM / where input D D @ 2c.w;y/D vector x1; ; xM do not overlap) only if 0 for all i j , all @yi @yj D ¤ .w; y/. 1.5 Applications of dual cost function in econometrics 7

1.5 Applications of dual cost function in econometrics

The above theory is usually applied by ﬁrst specifying a functional form .w; y/ for the cost function c.w; y/ and differentiating .w; y/ with respect to w in order to obtain the estimating equations

@.w; y/ xi i 1; ;N (1.10) D @wi D

2 2 (employing Shephard’s Lemma). Then the symmetry restrictions @ @ @wi @wj D @wj @wi h 2 i .i 1; ;N/ are tested and the second order condition @ negative D @w@w N N semideﬁnite is checked at all data points .w; y/. For example a cost function could be postulated as having the functional form

N N 1 1 X X 2 2 c y aij w w D i j i 1 i 1 D D (a Generalized Leontief functional form with y f .x/ showing constant returns to scale), which leads to the following equations forD estimation:

N 1 Â Ã 2 xi X wj aij i 1; ;N (1.11) y D wi D j 1 D

@xi @xj Here the symmetry restriction are expressed as aij aj i .i @wj D @wi D D 1; ;N/ which are easily tested. Equation (1.10) can be interpreted as being derived from a cost function c.w; y/ for a producer showing static, competitive cost- minimizing behavior if and only if the symmetry and second order conditions are satisfied. The major advantage of this approach is that it permits the specification of a system of factor demand equations x x.w; y/ that are consistent with cost minimization and with a very general specificationD of technology. In contrast, suppose that we wished to specify explicitly a solution to a cost minimization problem. Then we would estimate a production function directly with first order conditions for cost minimization: 8 1 Static Cost Minimization y f .x/ D @f=@xi wi @f=@x D w j j (1.12) @f=@xi wi

@f=@xk D wk

However, unless a very restrictive functional form is specified for the production function (e.g. Cobb-Douglas), then we can seldom drive the factor demand equations x x.w; y/ explicitly from (1.12). Since policy makers are usually more interestedD in demand and supply behavior than in production functions per se, this greater ease in specification of x.w; y/ is an important advantage of duality theory. Two other advantages of a duality approach rather than a primal approach (1.12) to the estimation of producer behavior are apparent. First, the hypothesis of competitive cost minimization is more readily tested in the framework of equations (1.10) than (1.12). Second, variables that are omitted from the econometric model (but are observed by producers) influence both the error terms and production decisions but do not necessarily influence factor prices to the same degree (e.g. factor prices may be exogenous to the industry). This tends to introduce greater simultaneous equations biases into the estimation of (1.12) than of (1.10). One disadvantage of (1.10) is that output y, as well as factor prices w, is treated as exogenous. Since the firm generally in effect chooses y jointly with x , this mis- specification can lead to simultaneous equations biases in the estimators. So the extent that production is constant return to scale with a single output, this difficulty can be avoided by using 1.3.b to specify a unit cost function c.w/ c.w; y/=y and applying Shephard’s Lemma to obtain estimating equations D

x @c.w/ i i 1; ;N (1.13) y D @wi D for a given functional form c.w/ (e.g. (1.11)). Under constant returns to scale xi .w; y/=y depends on w but not on y, so that (1.13) is well defined and estimation is independent of whether y is endogenous or exogenous to the firm. A second disadvantage of this duality approach to the specification of functional forms for econometric models, and a disadvantage of primal approaches such as (1.12) as well, is that it is derived from the theory of the individual firm but is usually applied to market data that is aggregated over firms. Difficulties raised by such aggregation will be discussed in a later lecture.

1.6 Conclusion

The dual cost function approach offers many advantages in the estimation of production technologies. The estimated factor demands x x.w; y/ measure factor D References 9 substitution along an isoquant and the effects of scale of output on factor demands, and first and second derivatives of the production function can be calculated. More- over the assumption of cost minimization is consistent with various broader theories of producer behavior. However effective policy making depends more on a knowledge of producer behavior than of production functions per se. By ignoring the effect of output prices on the firm’s input levels, the dual cost function approach generally is inappropriate for the modeling of economic behavior. Of course cost functions can be embedded within a broader behavioral model. For example, static competitive profit maximization implies a cost minimization model such as 1.1 together with first order conditions

@c.w; y/ p (1.14) @y D for optimal output levels (marginal cost equals output price). This equation implicitly defines the optimal level of y as y y.w; p/ provided of course that y en- @c.w;y/ D ters (1.14), i.e. @y is not independent of y or equivalently f .x/ does not show constant returns to scale. In this case equations 1.1.d, (1.10) and (1.14) can be estimated jointly, and the second order condition for profit maximization are expressed @2c.w;y/ as c.w; y/ concave and @y@y 0. Nevertheless there can be substantialÄ disadvantages to this approach to modeling competitive profit maximization. The assumption of constant returns to scale is commonly employed in empirical studies, and the assumption of profit maximization is not so easily tested here (the homogeneity and reciprocity condition for cost minimization do not imply integration up to a profit function). Therefore, for policy purposes, it is often better to model and test directly (using dual profit functions) the hypothesis of competitive profit maximization behavior.5

References

1. Samuelson, Paul A., 1947, Enlarged ed., 1983. Foundations of Economic Analysis, Harvard University Press. ISBN 0-674-31301-1 2. Uzawa, H. (October 1962). Production Function with Constant Elasticities of Substitution, Review of Economics Studies, Vol. 29, pp. 291-299. 3. Varian, H. R. (1992). Microeconomic Analysis, Third Edition. W. W. Norton & Company, 3rd edition.

5 In passing, note that this indirect approach to the modeling of proﬁt maximization (1.1.d, (1.10), (1.14)) may be superior to a direct approach (see next lecture) when there is substantially higher multicollinearity between factor prices w and output prices p than between w and output levels y.

Chapter 2 Static Competitive Proﬁt Maximization

As before, consider a firm producing a single output y using N inputs x D .x1; ; xN / according to a production function y f .x/, and assume that the firm is a price taker in all N factor markets. In addition,D we now assume that the firm takes output price p as given and chooses its levels of inputs to solve the following static competitive profit maximization problem:

( N ) X max pf .x/ wi xi pf .x/ wx: (2.1) x 0 D i 1 D

The maximum profit pf .x/ wx to problem (2.1) depends on prices .w; p/ and the firm’s productionÁ functiony f .x/. The corresponding relation .w; p/ between maximum profit andD prices is denoted as the firm’s dual profitD function.

2.1 Properties of .w; p/

Property 2.1. a) .w; p/ is decreasing in w and increasing in p. b) .w; p/ is linear homogeneous in .w; p/. i.e., .w; p/ .w; p/ for all scalar > 0. D c) .w; p/ is convex in .w; p/. i.e.,

ŒwA .1 /wB ; pA .1 /pB .wA; pA/ .1 /.wB ; pB / C C Ä C for all 0 1: See Figure 6.1. Ä Ä

11 12 2 Static Competitive Proﬁt Maximization Fig. 2.1 .w; p/ is convex π(w , p) in .w; p/

d) if .w; p/ is differentiable in .w; p/, then

@.w; p/ y.w; p/ ; D @p (Hotelling’s Lemma) @.w; p/ xi .w; p/ i 1; ;N: D @wi D

Proof. Properties 2.1.a–b follow obviously from the deﬁnition of the ﬁrm’s maximization problem (2.1). In order to prove 2.1.c simply note that, for .wC ; pC / Á .wA; pA/ .1 /.wB ; pB / and x solving 2.1 for .wC ; pC /, C C

.wC ; pC / pC f .xC / wC xC Á pAf .x / wAx .1 / pB f .x / wB x (2.2) D C C C C C .wA; pA/ .1 /.wB ; pB / Ä C since .wA; pA/ pAf .x / wAx , .wB ; pB / pB f .x / wB x . In order to C C C C prove Hotelling’s Lemma 2.1.d simply total differentiate .w; p/ pf .x/ wx with respect to .w; p/ respectively and then apply the standard ﬁrstÁ order conditions for an interior competitive proﬁt maximum:

N Ä @.w; p/ X @f.x/ @xk f .x/ p wk @p D C @xk @p k 1 D f .x/ D (2.3) N Ä @.w; p/ X @f.x/ @xk xi p wk @wi D C @xk @wi k 1 D x i 1; ;N: D i D

ut 2.2 Corresponding properties of y.w; p/ and x.w; p/ 13

Hotelling’s Lemma plays the same role in the theory of competitive profit maximization as Shephard’s Lemma plays in the theory of competitive cost minimization. Hotelling’s Lemma is an envelope theorem. The Lemma applies only for infinitesimal changes in a price and yet is critical to the empirical theoretical application of dual profit functions.

2.2 Corresponding properties of y.w; p/ and x.w; p/

Property 2.2. a) y.w; p/ and x.w; p/ are homogeneous of degree 0 in .w; p/. i.e., y.w; p/ y.w; p/ and x.w; p/ x.w; p/ for all scalar 0. " @x.w;p/ D @y.w;p/ # D @w N N @w N 1 b) @x.w;p/ @y.w;p/ is symmetric positive @p 1 N @p 1 1 .N 1/ .N 1/ semideﬁnite. C C

Proof. Property 2.2.a follows directly from the maximization problem (2.1).1 In or- h 2 i der to prove 2.2.b, note that @ .w;p/ is symmetric positive semidef- @w@p .N 1/ .N 1/ inite by 2.1.c and then apply 2.1.d to evaluateC thisC matrix. ut Moreover, 2.2.a–b exhaust the (local) properties that are placed on output supply y.w; p/ and factor demand x.w; p/ relations by the hypothesis of competitive proﬁt maximization (2.1).

Proof. First total differentiate .w; p/ pf .x.w; p// wx.w; p/ to obtain Á N @.w; p/ @y.w; p/ X @xk.w; p/ y.w; p/ p wk @p Á C @p @p k 1 D N @.w; p/ @y.w; p/ X @xk.w; p/ xi .w; p/ p wk i 1; ;N @wi Á C @wi @wi D k 1 D (2.4)

Property 2.2.a implies (by Euler’s theorem)

1 Alternatively, .w; p/ homogeneous of degree one in .w; p/ implies (by Euler’s theorem) @.w;p/ xi .w; p/ homogeneous of degree 0. @wi D 14 2 Static Competitive Proﬁt Maximization

N @y.w; p/ X @y.w; p/ p wk c @p C @wk D k 1 D N @xi .w; p/ X @xi .w; p/ p wk 0 i 1; ;N @p C @wk D D k 1 D and property 2.2.b states the reciprocity relations

@y.w; p/ @xk.w; p/

@wk D @p @x .w; p/ @x .w; p/ i k i; k 1; ;N; @wk D @wi D so properties 2.2.a–b jointly imply

N @y.w; p/ X @xk.w; p/ p wk 0 @p @p D k 1 D N @y.w; p/ X @xk.w; p/ p wk 0 i 1; ;N: @wi @wi D D k 1 D Substituting this into the identity (2.4) yields

@.w; p/ y.w; p/ 0 @p D (Hotelling’s Lemma) @.w; p/ xi .w; p/ 0 i 1; ;N: @wi D Ä D The reciprocity relations 2.2.b imply that the system of differential equations @.w;p/ @.w;p/ y.w; p/, xi .w; p/, .i 1; ;N/ integrates up to an @p D @wi D D underlying function .w; p/ (Frobenius theorem). The positive semidefiniteness restriction 2.2.b implies positive semidefiniteness of the Hessian matrix of .w; p/, and this in turn implies y f .x/ is concave at all x (see (2.6) below). This establishes the second order conditionsD on the production f .x/ for competitive profit maximization. The first order conditions follow from the fact that 2.2 establish @y.w;p/ @x.w;p/ @y.w;p/ @x.w;p/ Hotelling’s Lemma for all @p , @p , @w , @w satisfying properties 2.2 (see (2.3)). ut

2.3 Second order relations between .w; p/ and f .x/, c.w; y/

As in the case of a cost function, the ﬁrm’s production function y f .x/ can be recovered from knowledge of the proﬁt function .w; p/. Given knowledgeD of 2.3 Second order relations between .w; p/ and f .x/, c.w; y/ 15

.w; p/ and differentiability of .w; p/, application of Hotelling’s Lemma immediately gives a profit maximizing combination .x; y/ for prices .w; p/. Likewise the first and second derivatives of f .x/ at x x.w; p/ can be calculated directly from the first and second derivatives of .w;D p/. The first derivatives can be calculated simply as

@f Œx.w; p/ w i i 1; ;N (2.5) @xi D p D using the ﬁrst order conditions for an interior solution to problem (2.1). The second derivatives can be calculated from the matrix equation

2 2 1 Ä@ f .x.w; p// Ä@ .w; p/ p (2.6) ˝ @x@x N N D @w@w N N h @2.w;p/ i assuming an inverse for @w@w .

@f.x / Proof. Simply total differentiate the ﬁrst order conditions p wi 0 (i @xi 1; ;N ) with respect to w to obtain D D N 2 X @ f .x / @xk.w; p/ p 1 0 i; j 1; ;N; (2.7) @xi @xk @wj D D k 1 D 2 Substitute @xk .w;p/ @ .w;p/ (by Hotelling’s Lemma) into (2.7) and express @wj D @wk @wj the result in matrix form. ut This result (2.6) can easily be extended to the case of multiple outputs (Lau 1976). Since elasticities of substitution (holding output y constant) and scale effects are easily expressed in terms of a dual cost function c.w; y/, it is useful to note that .w; p/ also provides a second order approximation to c.w; y/. The ﬁrst derivatives of c.w; y/ can be calculated simply as

@c.w; y/ xi .w; y/ @wi D @.w; p/ i 1; ;N (2.8) D @wi D @c.w; y / p @y D where y y.w; p/ @.w;p/ , the proﬁt maximizing level of output given Á D @p .w; p/. The second derivatives of c.w; y/ at y y.w; p/ can be calculated from D h @2c.w;y/ i .w; p/ using the following matrix relations, here cww .w; y/ , Á @w@w N N h @2c.w;y/ i h @2.w;p/ i cwy .w; y/ , ww .w; p/ , wp.w; p/ Á @w@y N N Á @w@w N N Á 16 2 Static Competitive Proﬁt Maximization h 2 i @ .w;p/ , @w@p N N 1 T cww .w; y/ ww .w; p/ wp.w; p/ pp.w; p/ wp.w; p/ D C 1 cwy .w; y/ wp.w; p/ pp.w; p/ (2.9) D 1 cyy .w; y/ pp.w; p/ D

Proof. xi .w; p/ xi Œw; y.w; p/ .i 1; ;N/ implies (by Hotelling’s Lemma and Shephard’s Lemma)D D

w .w; p/ cw .w; y/ (2.10) i D i where y y.w; p/. Differentiating with respect to .w; p/, D T ww .w; p/ cww .w; y/ cwy .w; y/ wp.w; p/ D C (2.11) wp.w; p/ cwy .w; y/ pp.w; p/ D Combining (2.11),

1 T cww .w; y/ ww .w; p/ wp.w; p/ pp.w; p/ wp.w; p/ D C (2.12) 1 cwy .w; y/ wp.w; p/ pp.w; p/ D

Finally differentiating the ﬁrst order condition cy .w; y/ p (for proﬁt maximiza- 1D tion) with respect to p yields cyy .w; y / pp.w; p/ . D

2.4 Additional properties of .w; p/

Property 2.3.

a) the partial elasticity of substitution ij between inputs i and j , allowing for variation in output, can be deﬁned as

2 .w; p/ @ .w;p/ @wi @wj ij .w; p/ i; j 1; ;N D @.w;p/ @.w;p/ D @wi @wj

and, in the case of multiple outputs y .y1; ; yM /, the partial elasticity of transformation between outputs i andD j can be deﬁned as

2 .w; p/ @ .w;p/ @pi @pj tij .w; p/ i; j 1; ;N D @.w;p/ @.w;p/ D @pi @pj 2.5 Le Chatelier principles and restricted proﬁt functions 17

b) the transformation function f .x; y/ 0 is disjoint in outputs only if D @2.w; p/ 0 for all i j , all .w; p/: @pi @pj D ¤

2.5 Le Chatelier principles and restricted proﬁt functions

Samuelson proved the following Le Chatelier principle: ﬁxing an input at its initial static equilibrium level dampens own-price comparative static responses, or more precisely

@y.w; p/ @y.w; p; x1/ N 0 @p @p (2.13) @xi .w; p/ @xi .w; p; x1/ N 0 @wi Ä @wi Ä

@y.w;p/ @x.w;p/ where x1 x1.w; p/, i.e., , denote the comparative static changes N Á @p @w in .y; x/ when the level of input 1 cannot vary from its initial equilibrium level x1.w; p/ (Samuelson 1947). The following generalization of this result is easily established using duality theory:

" @x.w;p/ @y.w;p/ # " @x.w;p;x1/ @y.w;p;x1/ # @w @w @w N @w N @x.w;p/ @y.w;p/ @x.w;p;x1/ @y.w;p;x1/ N N @p @p .N 1/ .N 1/ @p @p .N 1/ .N 1/ C C C (2.14)C is a positive semideﬁnite matrix.

Proof. By the deﬁnition of competitive proﬁt maximization (2.1),

( N ) ( N ) X X A .w; p/ max pf .x/ wi xi max pf .x/ wi xi .w; p; x1 / Á x 0 x 0 Á i 1 i 1 D D A s.t. x1 x1 D (2.15) A i.e., adding a constraint x1 x1 to a maximization problem (2.1) generally de- creases (and never increases)D the maximum attainable proﬁts. Then ( 0 for all .w; p; xA/ .w; p; xA/ .w; p/ .w; p; xA/ 1 (2.16) 1 1 A Á 0 for x x1.w; p/ D 1 D A A i.e., .w; p; x1 / attains a minimum ( 0) over .w; p/ at all x1 x1.w; p/. By the second order condition for an interiorD minimum, D 18 2 Static Competitive Proﬁt Maximization

Ä@2.w; p; xA/ Ä@2.w; p/ Ä@2.w; p; xA/ 1 1 (2.17) @w@p .N 1/ .N 1/ Á @w@p @w@p C C A is positive semidefinite at x1 x1.w; p/. (2.17) and Hotelling’s Lemma establish (2.14). D ut Samuelson’s Le Chatelier Principle (2.13)/(2.14) has often been given the following dynamic interpretation: assuming that the difference between short-run, intermediate run and long-run equilibrium can be characterized in terms of the number of inputs that can be adjusted within these time frames, the magnitude of the firm’s response xi (or y) to a given change in price wi (or p) increases over time, and the sign of these responses does not vary with the time frame. However this characterization of dynamics in terms of a series of static models with a varying number of fixed inputs is unsatisfactory. In general dynamic behavior must be analyzed in terms of truly dynamic models. For example, it often appears that an increase in price for beef output leads to a short-run decrease in beef output and a long-run increase in output, which contradicts the dynamic interpretation of Samuelson’s Le Chatelier Principle (2.13). This can be explained in terms of the dual role of cattle as output and as capital input to future production of output: a long-run increase in output generally requires an increase in capital stock, and this can be achieved by a short-run decrease in output (Jarvis 1974). This illustrates the following point: a series of static models with a varying number of fixed inputs completely ignores the intertemporal decisions associated with the accumulation of capital (durable goods). Nevertheless, versions of profit functions conditional on the levels of certain inputs can be useful in applied work. The following restricted dual profit function is conditional on the level of capital stocks K:

( N ) X .w; p; K/ max pf .x; K/ wi xi : (2.18) Á x 0 i 1 D .w; p; K/ has the same properties in its price space .w; p/ as does the unrestricted proﬁt function .w; p/, which implcitly treats capital (or services from capital) as @.w;p;K/ a freely adjustable input. In addition @K measures the shadow price of capital, and twice differentiability of .w; p; K/ and Hotelling’s Lemma establish the following reciprocity conditions:

@y.w; p; K/ @ Â@.w; p; K/Ã @K D @p @K (2.19) @x .w; p; K/ @ Â@.w; p; K/Ã i i 1; ;N: @K D @wi @K D There are three major advantages to specifying a restricted dual profit function. First, a restricted profit function .w; p; K/ is consistent with short-run equilibrium for a variety of dynamic models that dichotomize inputs as being either perfectly 2.6 Application of dual profit functions in econometrics: I 19 variable or quasi-fixed in the short-run. Second, it is more realistic to assume that firms are in a short-run equilibrium rather than in a long-run equilibrium, and mis- specifying an econometric model as long-run equilibrium (e.g. using an unrestricted profit function .w; p/) implies that the resulting estimators of long-run equilibrium effects of policy are unreliable. Third, long-run equilibrium effects can sometimes be inferred correctly from the estimates of .w; p; K/ using the following first order condition for a long-run equilibrium level of capital stock K:

@.w; p; K/ @.w; p; K/ wK or .r ı/pK (2.20) @K D @K D C where wK rental price of capital, pK asset price of capital, r an appropriate discount rateÁ and ı rate of depreciationÁ for capital. After estimatingÁ .w; p; K/ Á 2 we would solve (2.20) for K.

2.6 Application of dual proﬁt functions in econometrics: I

As in the use of a cost function c.w; y/, the above theory is usually applied by ﬁrst specifying a functional form .w; p/ for a proﬁt function .w; p/ and differentiating .w; p/ with respect to .w; p/ to obtain the estimating equations

@ .w; p/ y D @p (2.21) @ .w; p/ xi i 1; ;N D @wi D

(employing Hotelling’s Lemma). Then the symmetry restrictions @y @xi , @wi @p h 2 i D @xi @xj ,(i; j 1; ;N ) are tested and @ is checked for @wj D @wi D @w@p .N 1/ .N 1/ positive semideﬁniteness. For example, a proﬁt function couldC beC postulated as

N N N X X X aij pwi pwj a0j pwj pp ap D C C i 1 j 1 j 1 D D D

2 @.w;p;K/ In general @K cannot be measured directly from the estimating equations (a) y @.w;p;K/ @.w;p;K/ P P D @p , (b) x @w , e.g. when K i j aij pvi pvj aKK K, D @.w;p;K/ D C (v .w; p/). Estimates of @K can be obtained either by estimating .w; p; K/ Á @.w;p;K/ D jointly with (a)–(b) and then calculating directly @K , or by estimating (a)–(b) and then @.w;p/ @y.w;p/ PN @xi .w;p/ calculating @K p @K i 1 wi @K . This last equation is derived as fol- D D @.w;p;K/ @.w;p;K/ @.w;p;K/ lows, .w; p; K/ .w; p; K/ @K @K @K @2.w;p;K/ PN D @2.w;p;K/ ) D ) D p wi (Euler’s theorem), and then applying (2.19). @p@K i 1 @wi @K C D 20 2 Static Competitive Proﬁt Maximization

(this is a Generalized Leontief functional form, which imposes homogeneity of degree one in prices on the proﬁt function). This leads to the following equations for estimation

N 1 Â Ã 2 X wj y a a0j D C p j 1 D (2.22) N 1 1 Â Ã 2 Â Ã 2 X wj p xi aij ai0 i 1; ;N D wi C wi D j 1 D (applying Hotelling’s Lemma). Here the symmetry restrictions are expressed as aij aj i (i; j 1; ;N ) and a0i ai0 (i 1; ;N ). If and only if the symmetryD andD second order conditionsD are satisfied,D then equations (2.21) can be interpreted as being derived from a profit function .w; p/ for a produces showing static, competitive profit maximizing behavior. Similar comments apply to the estimation of a restricted profit function .w; p; K/. Here we can note three major advantages of this approach to the modeling of producer behavior. First, it enables us to specify systems of output supply and factor demand equations that are consistent with profit maximization and with a general specification of technology. This cannot be achieved by estimating a production function directly together with first order conditions for profit maximization. Sec- ond, modeling a dual profit function explicitly allows for the endogeneity of output levels to the producer. This is in contrast to cost functions c.w; y/ where output generally is treated as exogenous. Third, aggregation problems are likely to be less severe for unrestricted dual profit functions than for dual cost functions. If all firms f 1; ;F (a constant number of firms over time) face identical prices .w; p/ thenD it is clear that a profit function can be well defined for aggregate market data:

F X f .w; p/ ˆ.w; p/ ….w; p/: (2.23) D Á f 1 D In the special case of profit maximization at identical prices, a cost function also is well defined for data aggregated over firms even though the output level yf varies over firms (we shall use this in a later lecture); but in this case we may as well estimate ….w; p/ directly. The disadvantage of estimating a profit function .w; p/ rather than a cost function c.w; y/ is that the former imposes stronger behavioral assumptions which are often very unrealistic. For example, at the time of production decisions, farmers generally have better knowledge of input prices than of output prices forthcoming at the time of marketing in the future. Thus risk aversion and errors in forecasting prices are more likely to influence the choice of output levels rather than to contradict the hypothesis of cost minimization. In addition in the case of food retail industries, hypothesis of oligopoly behavior plus competitive cost minimization may be more realistic than the hypothesis of competitive profit maximization. 2.7 Industry profit functions and entry and exit of firms 21 2.7 Industry profit functions and entry and exit of firms

As mentioned above, an industry profit function is well defined provided that all firms face identical prices .w; p/ and the composition of the industry in terms of firms does not change. In this case the industry profit function is simply the sum of PF the profit functions of the firms f 1; ;F : ….w; p/ f 1 f .w; p/ for all .w; p/. D D D A more realistic assumption is that changes in prices .w; p/ induce some established firms to exit the industry and some new firms to enter the industry. Suppose that there is free entry and exit to the industry (all firms in the industry earn non- negative profits because all firms that would earn negative profits at .w; p/ are able to exit the industry, and all firms excluded from the industry are also at long-run equilibrium). Then the industry profit function inherits essentially the some properties as the individual firm’s profit function f .w; p/. Proposition 2.1 provides a characterization of the industry profit function .w; p/, assuming (a) competitive behavior, (b) both input prices w .w1; ; wN / and output prices p are exogenous, and (c) a continuum of firmsD and free entry/exit to the industry. The role of the assumption of a continuum of firms deserves comment. As noted by Novshek and Sonnenschein (1979) in the case of marginal consumers, an infinitesimal change in price will lead to entry/exit behavior only in the case of a continuum of agents. Therefore it is necessary to assume that there exists a continuum of firms. Industry profits are calculated by integrating over the continuum of firms in the industry:

Z f m.w;p/ ….w; p/ .w; p; f /.f / df D 1 where .f / is the density of firms f . .w; p; f / denotes the individual firm’s profit function conditional on the firm being in the industry, and industry profits are obtained by integrating over those firms in the industry given prices .w; p/ and free entry/exit. Adapting the arguments of Novshek and Sonnenschein, the assumption of a continuum of firms also establishes the differentiability of the industry profit function ….w; p/, industry factor demands X.w; p/ and industry output supplies Y.w; p/ with free entry/exit. The argument can be outlined as follows. Since the individual firm’s profit function .w; p; f / is conditional on firm f remaining in the industry, it is reasonable to assume that .w; p; f / is twice differentiable in .w; p/, i.e., .w; p; f / is not kinked at 0 due to exit from the industry. .w; p; f / is dif- D ferentiable in f and the derivative f .w; p; f / < 0 assuming a continuum of firms indexed in descending order of profits. Then (by the implicit function theorem) a marginal firm f m f m.w; p/ is defined implicitly by the zero profit condition .w; p; f m/ 0, andD f m.w; p/ is differentiable. Under these assumptions it can easily be shownD that the industry profit function with free entry/exit is differentiable, and its derivatives …w .w; p/, …p.w; p/ can be calculated by applying Leibnitz’s 22 2 Static Competitive Profit Maximization rule to the above equation for industry profits (note that the zero profit condition .w; p; f m/ 0 established Hotelling’s Lemma at the industry level). A similar procedure establishesD the differentiability of industry factor demands X.w; p/ and output supplies Y.w; p/ with free entry/exit.

Proposition 2.1. Assume that the industry consists of a continuum of firms such that each individual firm’s profit function .w; p; f / is twice differentiable in .w; p/, differentiable in f , f .w; p; f / < 0, and is linear homogeneous and convex in .w; p/ and satisfies Hotelling’s Lemma. Also assume .w; p; f m/ 0 for a marginal firm f m. Then .w; p/ is linear homogeneous and convexD in .w; p/. Moreover, it also satisfies Hotelling’s Lemma, i.e.,

@….w; p/ Xi .w; p/ i 1; ;N D @w D i (2.24) @….w; p/ Yk.w; p/ k 1; ;N D @pk D and industry derived demands X.w; p/ and output supplies Y.w; p/ are differentiable.

Proof. Given prices .w; p/ and free entry/exit, index firms in the industry in descending order of profits. Industry profits can be calculated as

Z f m.w;p/ ….w; p/ .w; p; f / .f / df (2.25) D 1 where .f / is the density of firms f and a marginal firm f m satisfies the zero profit condition .w; p; f m/ 0: (2.26) D Assuming .w; p; f / differentiable in .w; p; f / and f .w; p; f / < 0, the zero profit condition (2.26) establishes (using the implicit function theorem) f m f m.w; p/ is defined and differentiable. Differentiability of .w; p; f / and f m.w;D p/ establishes (using (2.25) ….w; p/ is differentiable). Applying Leibnitz’s rule to (2.25), Pro:2.1

m @….w; p/ Z f .w;p/ @.w; p; f / @f m.w; p/ .f / df .w; p; f m.w; p// .f m.w; p// @wi D 1 @wi C @wi Xi .w; p/ i 1; ;N D D (2.27)

m @….w; p/ Z f .w;p/ @.w; p; f / @f m.w; p/ .f / df .w; p; f m.w; p// .f m.w; p// @pk D 1 @pk C @pk

Yk.w; p/ k 1; ;M D D (2.28) 2.8 Application of dual proﬁt functions in econometrics: II 23 using Hotelling’s Lemma for the individual ﬁrm in the industry and the marginal condition .w; p; f m.w; p// 0. Similarly D Z f m.w;p/ xi .w; p/ xi .w; p; f / .f / df i 1; ;N (2.29) D 1 D

Z f m.w;p/ yk.w; p/ yk.w; p; f / .f / df k 1; ;M (2.30) D 1 D Differentiability of x.w; p; f /, y.w; p; f /, f m.w; p/ implies (using (2.29)–(2.30)) X.w; p/, Y.w; p/ differentiable. ….w; p/ homogeneous of degree one in .w; p/ follows from (2.25): .w; p; f / is linear homogeneous in .w; p/ for all f , and the continuum of firms 1; ; f m.w; p/ is invariant to equiproportional changes in .w; p/. Convexity of ….w; p/ can be established as follows. Define a profit function .w; p; g/ for each firm g allowing for free entry/exit to the industry: .w; p; g/N 0 for all .w; p; g/, and .w; p; g/ 0 when the firm has exited the industryN in order to avoid negative profits.N Thus ….w;D p/ with entry/exit is the sum or integral over the continuum of profit functions .w; p; g/ for the fixed set of potential firms. .w; p; g/ is convex in .w; p/ by standardN arguments (e.g. McFadden 1978), and inN turn ….w; p/ is convex in .w; p/. ut

2.8 Application of dual proﬁt functions in econometrics: II

Our development of Hotelling’s Lemma when the number of firms is variable to the industry has important implications for empirical studies. An industry profit function ….w; p/ satisfies Hotelling’s Lemma in two extreme cases: the number of firms in the industry is fixed (the standard case) or there is free entry/exit to the industry (Proposition 2.1). In intermediate cases, where the number of firms is variable but entry/exit is not instantaneous and costless, the Lemma does not apply. This can be seen from equations (2.29) and (2.30) in the proof of Proposition 2.1: applying Leibnitz’s rule to the integral for industry profits over a continuum of firms with entry/exit Z f m.w;p/ ….w; p/ .w; p; f /.f / df D 1 yields Hotelling’s Lemma only if there exist marginal firms earning zero profits, i.e., .w; p; f m.w; p// 0 (assume for the sake of argument that f m.w; p/ is differentiable in the intermediateD case). This zero profit condition is characteristic of free entry/exit with a continuum of firms but it is not characteristic of the intermediate case. The usual assumptions that are acknowledged in standard applications of Hotelling’s Lemma to industry-level data are that each firm in the industry shows static competitive profit maximizing behavior (conditional on the firm being in the industry). In addition we must generally add the restrictive assumption that the composition 24 2 Static Competitive Profit Maximization of the industry does not change over the time period of the data or there is free entry/exit to the industry. Nevertheless there is at least in principle a simple procedure for avoiding this additional restrictive assumption: the industry profit function can be defined explicitly as conditional upon the number of firms of each different type. For example, suppose that an industry consists of two homogeneous types of firms in variable quantities F1 and F2. The industry profit function can be written as ….w; p; F1;F2/, where F1 nd F2 are specified as parameters along with .w; p/. Then X.w; p; F1;F2/ D …w .w; p; F1;F2/, Y.w; p; F1;F2/ …p.w; p; F1;F2/ by standard argument (e.g. Bliss). Of course in practice reasonableD data on the number of firms by type is not always available, but whenever possible such modifications seem likely to improve the specification of the model. Thus, if we have time series data on the number of firms F1 and F2 in the two classes as well as data on total output, total inputs and prices, then we can postulate an industry profit function ….w; p; F1;F2/ conditional on F1, F2 and apply Hotelling’s Lemma to obtain the estimating equations

@….w; p; F ;F / Y 1 2 D @p (2.31) @….w; p; F1;F2/ Xi i 1; ;N: D @wi D

If there is free entry/exit to the industry, then (by 2.1) parameters F1 and F2 drop out of the system of estimating equations (2.31)—in this manner the assumption of long-run industry equilibrium is easily tested. If there is not free entry/exit and if the number of firms F1 and F2 varies over time, then the parameters F1 and F2 are significant in equations (2.31). The stocks of firms F1 and F2 at any time t probably can be approximated as predetermined at time t (i.e., the number of firms is essentially inherited from the past, given substantial delays in entry and exit). Then F1;t and F2;t do not necessarily covary with the disturbance terms at time t for equations (2.31), so that equations (2.31) may be estimated consistently even if there is costly entry/exit to the industry. However for policy purposes it may be desirable to estimate (2.31) jointly with equations of motion for the number of firms:

F1;t 1 F1;t F1.wt ; pt ;F1;t ; / C D (2.32) F2;t 1 F2;t F2.wt ; pt ;F2;t ; / C D Equations (2.31) can indicate the short-run impact of price policies or industry output and inputs levels .Y; X/, i.e., the impact of the price policies before there is an adjustment in the number of ﬁrms. Equations (2.31) and (2.32) jointly indicate intermediate and long-run impacts of price policies. For example, at a static long-run equilibrium there is no exit/entry to the industry, so the long-run equilibrium numbers of ﬁrms F1, F2 for any .w; p/ can be calculated from (2.32) by solving the implicit equations F1.wt ; pt ;F / 1;t D References 25

0, F2.wt ; pt ;F2;t/ 0. Obvious difﬁculties here are (a) problems in specifying the dynamics of entryD and exit (equations (2.32)) correctly, and (b) dangers in using

(2.31) to extrapolate to long-run equilibrium numbers of ﬁrms F1, F2 that are outside of the data set.

References

1. Fuss, M.& McFadden, D. (1978). Production Economics: A Dual Approach to Theory and Applications: The Theory of Production, History of Economic Thought Books, McMaster University Archive for the History of Economic Thought 2. Jarvis, L. (1974). Cattle as Capital Goods and Ranchers as Portfolio Managers: An Applica- tion to the Argentine Cattle Sector, Journal of Political Economy, pp. 489–520 3. Lau, L. (1976). A Characterization of the Normalized Restricted Proﬁt Function, Journal of Economic Theory, pp. 131–163. 4. Novshek, W. & Sonnenschein, H .(1979). Supply and marginal ﬁrms in general equilibrium, Economics Letters, Elsevier, vol. 3(2), pp. 109–113. 5. Samuelson, P. A. (1947). Enlarged ed., 1983. Foundations of Economic Analysis, Harvard University Press.

Chapter 3 Static Utility Maximization and Expenditure Constraints

Here we model both consumer and producer behavior subject to expenditure constraints. We begin with the case of consumer. Consider a consumer maximizing utility u by allocating his income y among N commodities x .x1; ; xN /, and denote his utility function as u u.x/. D D Assume that the consumer takes commodity prices p .p1; ; pN / as given and solves the following static competitive utility maximizationD problem:

max u.x/ u.x/ x 0 D N (3.1) X s.t. pi xi y Ä i 1 D

The maximum utility V u.x/ to problem (3.1) depends on prices and income .p; y/ and the consumer’sÁ utility function u u.x/. The corresponding relation V V .p; y/ between maximum utility and pricesD and incomes is denoted as the consumer’sD dual indirect utility function. A necessary condition for utility maximization (3.1) is that the consumer attains the utility level u.x/ at a minimum cost. In other words, if x does not minimize PN the cost i 1 pi xi subject to u.x/ u.x/, then a higher utility level u > u.x/ D PND can be attained at the same cost i 1 pi xi y (for this result we only need to D D assume local nonsatiation of u.x/ in neighborhood of x). Thus a solution x to the utility maximization problem (3.1) also solves the following cost minimization problem when the exogenous utility level u is equal to u.x/:

N N X X min pi xi pi xi x 0 D (3.2) i 1 i 1 D D s.t. u.x/ u PN The minimum cost E i 1 pi xi to problem (3.2) depends on prices and utility level .p; u/ and the consumersD D utility function u.x/. The corresponding relation

27 28 3 Static Utility Maximization and Expenditure Constraints between minimum expenditure and prices and utility level, E E.p; u/, is denoted as the consumers dual expenditure function. Note that (3.2) isD formally equivalent to PN the producer’s cost minimization problem minx 0 i 1 wi xi s.t. f .x/ y (1.1). Thus the expenditure function E.p; u/ inherits the essentialD properties (1.1) of the producers cost function c.w; y/:

Property 3.1. a) E.p; u/ is increasing .p; u/. b) E.p; u/ is linear homogeneous in p. c) E.p; u/ is concave in p. d) If E.p; u/ is differentiable in p, then

@E.p; u/ xi .p; u/ i 1; ;N: (Shephard’s Lemma) D @pi D

Also note that a sufﬁcient condition for utility maximization (3.1) is that the PN consumer attains the utility level u.x/ at a minimum cost equals to i 1 pi xi. In other words, if xA solves (3.2) subject to u.x/ uA, then xA also solvesD (3.1) PN PN A D subject to i 1 pi xi y i 1 pi xi . D D Á D Proof. Assume the that u.x/ is continuous and xA solves (3.2) at the exogenously A A determined utility level u u . Now suppose that x rather than x solves (3.1) at Á A PN A the exogenously determined expenditure level y i 1 pi xi > 0. This would A Á PDN PN A imply u.x/ > u.x / and (given local nonsatiation) i 1 pi xi i 1 pi xi . D D D Then continuity of u.x/ would imply that there exits an x in the neighborhood of x A PN PNQ A such that u.x/ u.x/ u.x / and i 1 pi xi < i 1 pi xi , which contradicts the assumption xA solvesQ (3.2). ThereforeDxA solvesQ (3.2)D implies xA solves (3.1). ut Thus xA solves (3.1) if and only if xA solves (3.2) subject to u u.xA/. This implies that the restrictions placed on Marshallian consumer demandsÁ x x.p; y/ (corresponding to problem (3.1)) by the hypothesis of utility maximizationD (3.1) can be analyzed equivalently in terms of the restrictions placed on Hicksian consumer demands x xh.p; u/ (corresponding to problem (3.2)) by the hypothesis of cost minimizationD (3.2). In other words, the hypothesis of cost minimization (3.2) exhausts the restrictions placed on Marshallian demands x x.p; y/ by the hypothesis of utility maximization. Properties 3.1 of E.p; u/ implyD

Property 3.2. a) x.p; u/ are homogenous of degree 0 in p. i.e. x.p; u/ x.p; u/ for all scalar > 0. D 3.1 Properties of V .p; y/ 29

Ä@x.p; u/ b) is symmetric negative semideﬁnite. @p N N

And properties 3.2 exhausts the implications of cost minimization for the (local) properties of Hicksian demands xh.p; u/ (the proof is the same as in the case of cost minimization by a producer). Therefore, by the proof immediately above, 3.2 also exhausts the implications of utility maximization (3.1) for the (local) properties of Marshallian demands x.p; y/, where 3.2 is evaluated at a utility maximization u u .x.p; y//. Of course the characterization of utility maximization in terms ofD (3.2) is not immediately useful empirically in the sense that utility level u is not observed (nor is u exogenous to the consumer). This relation between utility maximization and cost minimization is very different from the relation between profit maximization and cost minimization for the producer: profit maximization implies but is not equivalent to cost minimization in any sense. The explanation is that in the consumer case, in contrast to the producer case, maximization is subject to an expenditure constraint defined over all commodities.

3.1 Properties of V.p; y/

Property 3.3. a) V .p; y/ is decreasing in p and increasing in y. b) V .p; y/ is homogenous of degree 0 in .p; y/. i.e. V .p; y/ V .p; y/ for all scalar > 0. D c) V .p; y/ is quasi-convex in p. i.e. p V .p; y/ k is a convex set for all scalar k 0. See Figure 3.1 f W Ä g d) If V .p; y/ is differentiable in .p; y/, then

@V .p; y/=@pi xi .p; y/ i 1; ;N D @V .p; y/=@y D (Roy’s Theorem)

Proof. Properties 3.3.a–b follows simply from the deﬁnition of the consumer’s maximization problem (3.1). For a proof of 3.2.c see Varian (1992, pp. 121–122). In order to prove 3.3.d note that, if u is the maximum utility for (3.1) given parameters .p; y/, then u V .p; y / where y E.p; u /, i.e. y is the minimum expen- Á Á diture necessary to attain a utility level u given prices p. Total differentiating this 30 3 Static Utility Maximization and Expenditure Constraints

Fig. 3.1 V .p; y/ is quasi- convex in p

identity u V .p; E.p; u // with respect to pi , Á @V .p; y / @V .p; y / @E.p; u / 0 i 1; ;N (3.3) D @pi C @y @pi D

h which yields 3.3.b since @E.p; u /=@pi x .p; u / xi .p; y /, i 1; ;N . D i D D ut Roy’s Theorem and V .p; y/ quasi-convex in p can also be derived as follows. Since V .p; y/ maxx u.x/ s.t. px y, Á D V .p; y/ u.x/ 0 for all .p; y; x/ such that px y (3.4) D or equivalently

PN Á V p; i 1 pi xi u.x/ 0 for all .p; x/ (3.5) D PN where y is now defined implicitly as y i 1 pi xi . By (3.5), D D PN Á G.p; x/ V p; i 1 pi xi u.x/ 0 for all p Á D PN Á (3.6) G.p; x/ V p; i 1 pi xi u.x/ 0 at p such that x x.p; y/ Á D D D PN for y i 1 pi xi Á D In other words, given that xA solves (3.1) conditional on .pA; yA/, then G.p; xA/ attains a global minimum over p at pA. This implies the following first and second order conditions for maximization: @G.pA; xA/ 0 i 1; ;N @p D D i (3.7) Ä@2G.pA; xA/ is symmetric positive semidefinit. @p@p N N PN Á Since G.p; x/ V p; i 1 pi xi u.x/, conditions (3.7) imply Á D 3.2 Corresponding properties of x.p; y/ solving problem (3.1) 31 @G.p; x/ @V .p; y/ @V .p; y/ xi 0 i; j 1; ;N (3.8a) @pi Á @pi C @y D D @2G.p; x/ @2V .p; y/ @2V .p; y/ @2V .p; y/ @2V .p; y/ xj xi xj xi @pi @pj Á @pi @pj C @pi @y C @y@pi C @y@y (3.8b)

h 2 i (3.8a) is Roy’s Theorem. @ G.p;x/ symmetric positive semideﬁnite (3.7) im- @p@p N N plies semideﬁnite—these are the second order restrictions implied by V .p; y/ quasi- convex in p.

3.2 Corresponding properties of x.p; y/ solving problem (3.1)

Property 3.4. a) x.p; y/ is homogeneous of degree 0 in .p; y/. i.e. x.p; y/ x.p; y/ for all scalar 0. D h @xi .p; y/ @xi .p; u/ @xi .p; y/ b) xj .p; y/ i; j 1; ;N @pj D @pj @y D h h i @x .p;u/ (Slutsky equation) where u V .p; y/ and is symmetric Á @p N N negative semideﬁnite.

Proof. Property 3.4.a is obviously true. In order to prove 3.4.b, let u be the max- imal utility for problem (3.1) conditional on .p; y/. Then we have proved the fol- h lowing identity (see pages 27–28): xi .p; y/ xi .p; u/ where y E.p; u/, i.e. Á Á h x .p; u/ xi p; E.p; u/ i 1; ;N (3.9) i Á D In words, the Hicksian and Marshallian demands xh.p; u/ and x.p; y/ are equal h when x .p; u/ are evaluated at a utility level u u solving (3.1) for prices p and a given income y and x.p; y/ are evaluated at pDand a level y E.p; u / solving Á (3.2) given p and the level u u . Differentiating (3.9) with respect to pj (holding D utility level u constant) yields

@xh.p; u / @x .p; y/ @x .p; y/ @E.p; u / i i i i; j 1; ;N: (3.10) @pj D @pj C @y @pj D

Since (using Shephard’s Lemma) @E.p; u/=@pj xj .p; u/ xj .p; y/ .j h i D D D 1; ;N/ and @x.p;u/ is symmetric negative semideﬁnite from 3.2.b, (3.10) @p N N establishes 3.4.b. ut 32 3 Static Utility Maximization and Expenditure Constraints

The integrability problem has traditionally been prosed as follows: given a set of Marshallian consumer demand relations x x.p; y/, what restrictions on x.p; y/ exhaust the hypothesis of competitive utilityD maximizing behavior by the consumer? We can now construct an answer as follows. Since utility maximization (3.1) and expenditure minimization subject to u.x/ u are equivalent (see pages 27–28), we can rephrase the question as: what restrictionsD on x.p; y/ exhaust the hypothesis of competitive cost minimizing behavior by the consumer? Using the Slutsky equation h h i 3.4.b we can recover the matrix of substitution effects @x .p;u/ for the cost @p N N h h minimization demands x .p; u/ from the Marshallian demands x.p; y/. x .p; u/ h h i homogeneous of degree 0 in p and symmetry of @x .p;u/ imply the set @p N N h of differential equations xi .p; u/ @E.p; u/=@pi , i 1; ;N (Shephard’s Lemma) (see page 4). By the FrobeniusD theorem, these differentialD equations can be h h i @x .p;u/ integrated up to a macrofunction E.p; u/ if and only if is sym- @p N N h h i metric. Furthermore @x .p;u/ negative semideﬁnite implies (by Shephard’s @p N N h 2 i @ E.p;u/ Lemma) negative semideﬁnite, which in turn implies E.p; u/ @p@p N N concave and u.x/ quasi-concave at x.p; y/. Thus Marshallian demand relations x x.p; y/ can be interpreted as being derived from competitive utility maximiz- ingD behavior if and only if

Ä Ä " h # @x.p; y/ @x.p; y/ @x .p; u/ Œx.p; y/1 N @p C @y Á @p N N N 1 N N is symmetric negative semideﬁnite, (3.11a) x.p; y/ homogenous of degree 0 in .p; y/.1 (3.11b)

3.3 Application of dual indirect utility functions in econometrics

The above theory is usually applied by ﬁrst specifying a functional form .p; y/ for the indirect utility function V .p; y/ and differentiating .p; y/ with respect to .p; y/ in order to obtain the estimating equations

@ .p; y/=@pi xi i 1; ;N (3.12) D @ .p; y/=@y D

1 h x.p; y/ homogenous of degree 0 in .p; y/ implies x .p; u/ homogenous of degree 0 in p, since x.p; y/ xh .p; V .p; y//. Á 3.3 Application of dual indirect utility functions in econometrics 33

(using Roy’s Theorem). .p; y/ is usually speciﬁed as being homogeneous of degree 0 in .p; y/. The symmetry conditions

@2 .p; y/ @2 .p; y/ @2 .p; y/ @2 .p; y/ ; i; j 1; ;N @pi @pj D @pj @pi @pi @y D @y@pi D (3.13) are satisﬁed if and only if corresponding Hicksian demand relations xh.p; u / i D @E.p; u /=@pi .i 1; ;N/ integrate up to a macrofunction E.p; u /. D h Proof. By the Frobenius theorem, the Hicksian demands xi .p; u/ @E.p; u/=@pi hD .i 1; ;N/ integrate up to a macrofunction if and only if xi .p; u/=@pj h D @V .p;y/=@pi D x .p; u /=@pi .i; j 1; ;N/. Differentiating xi (Roy’s Theo- j D D @V .p;y/=@y rem) with respect to .p; y/ and substituting into the Slutsky equation 3.4.b yields

h Â Ã @x .p; u / Vp p Vy Vyp Vp Vp y Vy Vyy Vp Vp i i j j i i i j (3.14) @pj D Vy Vy C Vy Vy Vy

@ @V .p;y/ Á @ @V .p;y/ Á @ @V .p;y/ Á where Vp p , Vyp , Vp p , i j Á @pj @pi j Á @pj @y j i Á @pj @pi h h etc. Inspection of (3.14) shows that @x .p; u /=@pj @x .p; u /=@pi if and only i D j if Vp p Vp p , Vyp Vp y .i; j 1; ;N/. i j D j i i D i D ut Thus the hypothesis of competitive utility maximization is veriﬁed by (a) testing for the symmetry restrictions (3.13) and (b) checking the second order conditions (3.8b) at all data points .p; y/ or (equivalently) using Slutsky relations to recover the Hicksian matrix @xh=@p and checking for negative semideﬁniteness. For example an indirect utility function could be postulated as having the func- P P 1=2 1=2 tional form V y= i j aij pi pj (a Generalized Leontief reciprocal indirect utility functionD with homotheticity). Applying Roy’s Theorem leads to the following functional form for the consumer demand equations:

PN 1=2 y j 1 aij .pj =pi / xi D i 1; ;N: (3.15) D PN PN 1=2 1=2 D j 1 k 1 ajkpj pk D D

Here the symmetry restrictions (3.13) are expressed as aij aj i .i; j 1; ;N/ which are easily tested. Equations (3.15) can be interpretedD as being derivedD from an P P 1=2 1=2 indirect utility function V .p; y/ y= i j aij pi pj for a consumer showing static competitive utility maximizingD behavior if and only if the symmetry and second order conditions are satisfied. The major advantage of this approach to modeling consumer behavior is that it permits the specification of a system of Marshallian commodity demand equations x x.p; y/ that are consistent with utility maximization and with a very gen- eralD specification of the consumer’s utility function.2 In contrast, even if an accept-

2 The high degree of flexibility of V .p; y/ in representing utility functions (consumer preferences) u.x/ follow from the fact that the first and second derivatives of V .p; y/ determine the first and 34 3 Static Utility Maximization and Expenditure Constraints

@u.x/=@xi able measure of utility is defined and the first order conditions @.x /=@x pi =pj j D .i; j 1; ;N/ are estimated, the corresponding Marshallian demand equations x x.p;D y/ can be recovered explicitly only under very restrictive functional forms forDu.x/ (e.g. Cobb-Douglas). A serious problem with this and all other consumer demand models derived from microtheory (behavior of the individual consumer) is that the data is usually aggregated over consumers. Difficulties raised by the use of such market data will be discussed in a later lecture.

3.4 Proﬁt maximization subject to budget constraints

The above model (3.1) of consumer behavior subject to a budget constraint is formally equivalent to the following model of producer behavior subject to a budget constraint ( N ) X max pf .x/ wi xi .w; p; b/ x 0 Á i 1 D (3.16) N X s.t. wi xi b D i 1 D since (3.16) and max pf .x/ R.w; p; b/ x 0 Á N (3.17) X s.t. wi xi b D i 1 D have identical solutions x assuming that the budget constraint is binding. Here purchases of all inputs are assumed to draw upon the same total budget b, and there is a single output with production function y f .x/. In this case the solution x to (3.16) also solves the cost minimization problemD

min wi xi c.w; y/ x 0 Á (3.18) s.t. f .x/ y D where y f .x/. Note that cost minimization (conditional on y) is sufﬁcient as well as necessaryD for a solution to (3.16) (see pages 27–28), in contrast to the case of proﬁt maximization without a budget constraint.

second derivatives of the corresponding utility function u.x/ at x.p; y/. This can be seen by dif- @u.x / @V .p;y/ PN ferentiating the ﬁrst order conditions pi , pi x y .i 1; ;N/ @xi @y i 1 i for utility maximization and employing Roy’sD Theorem, in a mannerD analogousD toD the derivation of (1.6). 3.4 Proﬁt maximization subject to budget constraints 35

The envelop relations and second order conditions for (3.16) and (3.17) are easily derived as follows. (3.16) implies

.w; p; b; x/ .w; p; b/ pf .x/ wx 0 Á f g for all .w; p; b; x/ such that wx b (3.19) D .w; p; b; x.w; p; b// 0 D or, substituting wx for b in . /, .w; p; x/ .w; p; wx/ pf .x/ wx 0 for all .w; p; x/ Q Á f g (3.19’) .w; p; x.w; p; b// 0 Q D Thus w; p; x.w0; p0; b0/ attains a minimum over .w; p/ at .w0; p0/. This im- Q plies the following ﬁrst order conditions for a minimum:

@ .w0; p0; x0/ @.w0; p0; b0/ Q f .x0/ 0 i 1; ;N @p Á @p D D (3.20) @ .w0; p0; x0/ @.w0; p0; b0/ @.w0; p0; b0/ Q 0 0 xi xi 0 @wi Á @wi C @b C D i.e. (3.16) implies

@.w; p; b/ y.w; p; b/ D @p (3.21) @.w; p; b/=@wi xi .w; p; b/ i 1; ;N D 1 @.w; p; b/=@b D C Note that (3.21) reduces to Hotelling’s Lemma if @.w; p; b/=@b 0 , i.e. if the budget constraint is not binding. The second order conditions forD a minimum of w; p; x.w0; p0; b0/ over prices .w; p/ are .w; p; x0/ positive semideﬁ- Q Q w;p nite. Twice differentiating the identity .w; p; x/ .w; p; wx/ pf .x/ wx Q with respect to prices .w; p/ yields Á f g

@2 .w; p; x/ @2.w; p; b/ @2.w; p; b/ @2.w; p; b/ @2.w; p; b/ Q xj xi xi xj @wi @wj Á @wi @wj C @wi @b C @b@wj C @b@b @2 .w; p; x/ @2.w; p; b/ @2.w; p; b/ Q xi @wi @p Á @wi @p C @b@p @2 .w; p; x/ @2.w; p; b/ @2.w; p; b/ Q xi @p@wi Á @p@wi C @p@b @2 .w; p; x/ @2.w; p; b/ Q i; j 1; ;N @p@p Á @p@p D (3.22) 36 3 Static Utility Maximization and Expenditure Constraints

Thus profit maximization subject to a budget constraint wx b implies that the .N 1/-dimensional matrix defined in (3.22) is symmetric positiveD semidefinite. LikewiseC (3.17) implies

.w; p; x/ R.w; p; wx/ pf .x/ 0 for all .w; p; x/ Á (3.23) .w; p; x.w; p; b// 0 D Proceeding as above we obtain the envelop relations

@R.w; p; b/ y D @p (3.24) @R.w; p; b/=@wi xi D @R.w; p; b/=@b and second order conditions analogous to (3.22) symmetric positive semideﬁnite. Nevertheless, assuming an expenditure constraint over all inputs and a single output, the simplest approach is to estimate a cost function c.w; y/ using Shephard’s Lemma to obtain estimating equations

@c.w; y/ xi i 1; ;N: (3.25) D @wi D Under the above assumptions the hypothesis of cost minimization conditional on y exhausts the implications of the hypothesis of profit maximization subject to a budget constraint, although of course this approach (3.25) still mis-specifies the maximization problem (3.16) by treating output y as exogenous. As a second and more interesting example of modeling budget constraints in production, suppose that expenditures on only a subset of inputs are subject to a budget constraint, (e.g. different inputs may be purchased at different times and cash constraints may be binding only at certain times, or alternatively credit may be available for the purchase of some but not all inputs). In this case the firm solves the profit maximization problem

( NA NB ) X A A X B B max pf .xA; xB / wi xi wi xi .w; p; b/ xA;xB 0 Á i 1 i 1 D D (3.26) NA NB XC s.t. wB xB b i i D i NA 1 D C (3.26) implies

.w; p; x/ .w; p; wB xB / pf .xA; xB / wAxA wB xB Á f g 0 for all .w; p; x/ (3.27) 0 for .w; p; x.w; p; b// D 3.4 Proﬁt maximization subject to budget constraints 37

Proceeding as before leads to the envelop relations

@.w; p; b/ y D @p

A @.w; p; b/ xi A i 1; ;N (3.28) D @wi D B B @.w; p; b/=@wi x i NA 1; ;NA NB i D 1 @.w; p; b/=@b D C C C and to the second order relations

2 2 2 2 2 @ .w; p; x/ @ .w; p; b/ @ .w; p; b/ B @ .w; p; b/ B @ .w; p; b/ B B B B B B B xj B xi xi xj @wi @wj Á @wi @wj C @wi @b C @b@wj C @b@b

i; j 1; ;NB D 2 2 2 @ .w; p; x/ @ .w; p; b/ @ .w; p; b/ B B A B A A xi i 1 ;NB j 1; ;NA @wi @wj Á @wi @wj C @b@wj D D 2 2 2 @ .w; p; x/ @ .w; p; b/ @ .w; p; b/ B B B xi i 1 ;NB j 1; ;NA @wi @p Á @wi @p C @b@b D D @2 .w; p; x/ @2.w; p; b/ A A A A i 1; ;NA @wi @wj Á @wi @wj D @2 .w; p; x/ @2.w; p; b/ A A i 1; ;NA @wi @p Á @wi @p D @2 .w; p; x/ @2.w; p; b/ @p@p Á @p@p (3.29) The .NA NB 1/-dimensional matrix defined by (3.29) should be symmetric positive semidefinite,C C assuming profit maximization subject to a binding budget constraint wB xB b. The aboveD theory of profit maximization subject to a budget constraint can be applied by first specifying a functional form .w; p; b/ for the profit function .w; p; b/ and differentiating .w; p; b/ with respect to .w; p; b/ in order to obtain the estimating equation

@ .w; p; b/ y D @p

A @ .w; p; b/ xi A i 1; ;NA (3.30) D @wi D B B @ .w; p; b/=@wi x i NA 1; ;NA NB i D 1 @ .w; p; b/=@b D C C C 38 3 Static Utility Maximization and Expenditure Constraints where only inputs xB are subject to a budget constraint wB xB b. The symmetry @2 @2 @2 @2 D conditions , , .i; j 1; ;NA NB / imply that @wi @wj D @wj @wi @wi @p D @p@wi D C the output supply and factor demand equations (3.30) integrate up to a macrofunction .w; p; b/, and satisfaction of the second order conditions (3.29) (replacing with in the derivatives) implies that .w; p; b/ can be interpreted as a proﬁt function .w; p; b/ corresponding to the behavioral model (3.26). Note that models such as (3.30) may be useful in testing whether expenditures on a particular input draw on a binding budget, i.e. in testing whether the shadow price for the budget constraint @ .w; p; b/=@b is signiﬁcant in the demand equation for a particular input. For example, the functional form for .w; p; b/ could be hypothesized as

Â 1 1 1 1 Ã X X 2 2 X 2 b aij w w a0i p 2 w (3.31) D i j i j C i i (a Generalized Leontief functional form with constant returns to scale). This leads to the following functional form for the output supply and factor demand equations:

N N 1 A B Â Ã 2 XC wi y b a0i D p i 1 D N N 1 A B Â Ã 2 A XC wj xi b aij i 1; ;NA D wi D j 1 D 1 PNA NB 2 b C a w =w B j 1 ij j i xi D 1 1 1 N N N N N N 1 D P A B P A B 2 2 P A B 2 2 1 j 1C k 1C ajkwj wk j 1C a0j p wi C D D C D i NA 1; ;NA NB D C C(3.32) Note that the particular functional form (3.31) implies @.w; p; b/=@b .w; p; b/=b, D so that the demand equations for inputs xB can be simpliﬁed to

N N 1 Â 2 Ã A B Â Ã 2 B b XC wj xi aij i NA 1; ;NA NB (3.33) D b wi D C C j 1 C D However, in the absence of constant returns to scale, the demand equations for inputs subject to a budget constraint generally will be nonlinear in the parameters to be estimated.

References

1. Varian, H. R. (1992). Microeconomic Analysis, Third Edition. W. W. Norton & Company, 3rd edition. Chapter 4 Nonlinear Static Duality Theory (for a single agent)

4.1 The primal-dual characterization of optimizing behavior

In previous lecture we have constructed primal-dual relations such as

G.w; p; x/ .w; p/ pf .x/ wx (page 33) (4.1a) Á f g G.p; y; x/ V .p; y/ u.x/ ((3.6) on page 30) (4.1b) Q Á G.p; u; x/ E.p; u/ p x (4.1c) Q Á where the first term denotes the optimal value of profits/utility/expenditure for an agent solving a profit maximization/utility maximization/cost minimization problem and the second term denotes a feasible level of profits/utility/expenditure, respectively. It is obvious that the primal-dual relations (4.1a)–(4.1b) attain a minimum value (equal to zero) at an equilibrium combination of choice variables x and parameters: .w; p; x.w; p///.p; y; x.p; y//. Likewise the primal-dual relations c for cost minimization attains a maximum value (equal to zero) at an equilibrium combination of choice variables x and parameters: .p; u; x.p; u//. This implies that, given equilibrium levels of x of the agent’s choice variables x, the primal-dual relations (4.1a)–(4.1b) attain a minimum over possible values of the parameters at the particular level of the parameters for which x solves the profit/utility maximization problem: given x0 x.w0; p0/ solving a profit maximization problem (2.1): Á .w0; p0/ solves min G.w; p; x0/ .w; p/ ˚pf .x0/ wx0« 0 (4.2) w;p Á D given x0 x.p0; y0/ solving a utility maximization problem (3.1): Á

39 40 4 Nonlinear Static Duality Theory (for a single agent)

.p0; y0/ solves min G.p; y; x0/ V .p; y/ u.x0/ 0 s.t. px0 y p;y Q Á D D (4.3) and similarly for case (4.1c), given x0 x.p0; u0/ solving a cost minimization problem (3.2): D

p0 solves max G.p; u0; x0/ E.p; u0/ p x0 0 (4.4) p Q Á D where u0 u.x0/. By analyzing the first order conditions for an interior solution to these problems,Á we easily derived Hotelling’s Lemma (page 12), Roy’s Theorem (page 29) and we can easily derive Shephard’s Lemma, respectively. To repeat, it is obvious that the hypotheses of profit maximization, utility maximization and cost minimization imply that an equilibrium combination of choice variables and parameters are obtained by solving (4.2)–(4.4), respectively. In this sense a necessary condition for x0 to solves a profit maximization problem (2.1) conditional on prices .w0; p0/ is that .w0; p0/ solve (4.2), and similarly for utility maximization and cost minimization. A further question is: does x0 solve a profit maximization problem (2.1) (e.g.) conditional on .w0; p0/ if and only if (w0; p0) solves (4.2) conditional on x0? The answer is yes and this implies that problems (4.2)–(4.4) exhaust the implications of behavioral models (2.1), (3.1) and (3.2), respectively, for local comparative static analysis of changes in prices. The intuitive explanation of this result is surprisingly simple (given the confusion that has been raised in the recent past over this matter, especially Silberberg 1974, pp. 159–72). For example, note that the profit maximizing derived demands x.w; p/ solving maxx pf .x/ wx .w; p/ also solve f g Á min G.w; p; x/ .w; p/ pf .x/ wx 0 (4.5) x Á f g D and note that any x such that G.w; p; x/ 0 also solves maxx pf .x/ wx conditional on .w; p/. Since the minimum valueD of G.w; p; x/ over .w;f p/ is also 0,g 0 0 0 0 it follows that any combination .w ; p ; x / solving minw;pG.w; p; x / also solves 0 0 0 0 0 0 0 minxG.w ; p ; x/, and conversely any .w ; p ; x / solving minxG.w ; p ; x/ also 0 solves minw;p G.w; p; x /. 0 Thus solving minw;p G.w; p; x / is equivalent to solving the profit maximiza- ˚ 0 0 « tion problems maxx p f .x/ w x . Also note that an (interior) global solution 0 0 0 .w ; p / to minw;p G.w; p; x / implies for local comparative static purposes only that @G.w0; p0; x0/ @G.w0; p0; x0/ 0; 0 i 1; ;N (4.6a) @p D @wi D D Ä@2G.w0; p0; x0/ symmetric positive semidefinite (4.6b) @p@w .N 1/ .N 1/ C C 4.2 Producer behavior 41 i.e. only the first and second order conditions for an interior minimum are relevant.1 The implication of the above argument is that (4.6) exhaust the restrictions placed on the comparative static effects of (local) changes in prices .w; p/ by the hypothesis of competitive profit maximization for the individual firm. Similar conclusions hold for the cases of utility maximization (4.3) and cost minimization (4.4). More generally, consider an optimization problem

max f .x; ˛/ .˛/ x Á (4.7) s.t. g.x; ˛/ 0 D where both the objective function f .x; ˛/ and constraint (or vector of constraint) g.x; ˛/ 0 are conditional on a vector of parameters ˛ (e.g. prices or parameters shiftingD the price schedules facing the agent). Then x0 solves (4.7) conditional on ˛0 if and only if ˛0 solves the following problem conditional on x0:

min G.x0; ˛/ .˛/ f .x0; ˛/ ˛ Á (4.8) s.t. g.x0; ˛/ 0 D Therefore the ﬁrst and second order conditions for a solution to problem (4.8) in parameter (˛) space exhaust the implications of the behavioral model (4.7) for comparative static effects of (local) changes in parameters ˛.2

4.2 Producer behavior

Consider the following general proﬁt maximization problem

max R.x; ˛A/ c.x; ˛B / .x; ˛/ .˛; b/ x f g ÁQ Á (4.9) s.t. c.x; ˛C / b D where all prices may be endogenous to the producer and there is a budget constraint (or vector of constraints) c.x; ˛C / b limiting expenditures on at least some in- D puts. Hence R.x; ˛A/ denotes total revenue as a function of the output level y (or equivalent the inputs levels x, given a single output production function y f .x/) D and parameters ˛A shifting the price schedules p.y; ˛A/ facing the ﬁrm. c.x; ˛B / denotes total costs as a function of the input levels x and the parameters ˛B shifting

1 0 0 0 0 The condition G.w ; p ; x / 0 for a global solution to minw;p G.w; p; x / implies only that .w0; p0/ p0f .x0/ wD0x0, which is not directly relevant to local comparative statics. D 2 Likewise, if we are only interested in the comparative static effects of changes in a sub- 0 0 0 set ˛B of parameters ˛, then x solves (4.7) conditional on ˛ if and only if ˛B solves 0 0 0 0 min˛ G.x ; ˛ ; ˛B / s.t. g.x ; ˛ ; ˛B / 0. In turn the first and second order conditions B A A D in the parameter ˛B subspace for this minimization problem exhaust the implications of (4.7) for the comparative static effects of changes in parameters ˛B . 42 4 Nonlinear Static Duality Theory (for a single agent) the factor supply schedules wi .x; ˛B / .i 1; ;N/ facing the firm. .˛; b/ denotes the dual profit function for (4.9), i.e.D the relation between maximization attainable profits and parameters .˛; b/. Formally (4.9) allows for traditional monopoly/ monopsony behavior, the specification of financial constraints in terms of both fixed cash constraints (at level b) and costs of borrowing the vary with the level of borrowing (and the firm’s debt-equity ratio), the endogeneity of the opportunity cost (value of forgone leisure) of farm family labor, etc. Consider the corresponding minimization problem

min G.˛; b; x/ .˛; b/ R.x; ˛A/ c.x; ˛B / ˛;b Á f g (4.10) s.t. c.x; ˛C / b D or equivalently (substituting out the budget constraint c.x; ˛C / b) D

min G.˛; x/ .˛; c.x; ˛C // R.x; ˛A/ c.x; ˛B / (4.11) ˛ Q Á f g The analysis in the previous section implies that the following ﬁrst and second order conditions for an (interior) solution to (4.11) exhaust the implications of proﬁt maximization for the effects of local changes in parameters ˛: (using obvious vector notation):

G˛.˛; x/ 0 Q D (4.12) G .˛; x / symmetric positive semideﬁnite Q ˛˛

h @G @G i h @2G i where x x.˛; b/ solves (4.9), and G˛ Q Q , G˛˛ Q . D Q Á @˛1 @˛´ 1 Z Q Á @˛@˛ Z Z Since G.˛; x / .˛; c.x ; ˛ // .x ; ˛/, restrictions (4.12) can be rewritten Q C as Á Q

G˛.˛; x/ ˛.˛; b/ b.˛; b/c˛.x; ˛/ ˛.x; ˛/ 0 (4.13a) Q Á C Q D

G˛˛.˛; x/ ˛˛.˛; b/ ˛b.˛; b/c˛.x; ˛/ b˛.˛; b/c˛.x; ˛/ Q Á C C bb.˛; b/c˛.x; ˛/c˛.x; ˛/ b.˛; b/c˛˛.x; ˛/ (4.13b) C C ˛˛.x; ˛/ symmetric positive semidefinite. Q In order to derive a system of estimating equations for a behavioral model of the type (4.9), we can begin by specifying a functional form .˛; b/ for the firm’s dual profit function .˛; b/. Then we calculate corresponding envelop relations (4.13a). Note, however, that calculation of these envelop relations also requires knowledge of 3 the derivatives c˛.x; ˛/, R˛.x; ˛/, C˛.x; ˛/, i.e. we must specify functional forms for R.x; ˛/, C.x; ˛/ and c.x; ˛/ in addition to the functional form for .˛; b/. This does not appear to cause any serious problems: although these functions are not independent of the functional form for .˛; b/, a flexible specification (see next

3 In the competitive case R.x; ˛A/ pf .x/, C.x; ˛B / wx, c.x; ˛C / wC xC ; So that Á Á Á R˛ y, C˛ x, c˛ xC . A D B D C D 4.3 Consumer behavior 43 lecture) of these functions and of .˛; b/ need not severely restrict the implicit speciﬁcation of the production technology y f .x/. After specifying the functional forms ofD.˛; b/ and R.x; ˛/, C.x; ˛/ and c.x; ˛/, we then solve the envelope relations in the form

˛.x; ˛/ b.˛; b/ c˛.x; ˛/ ˛.˛; b/ at b c.x; ˛/ (4.14) Q Q D Á we see (using the Frobenius theorem) that these estimating equations integrate up to a macrofunction .˛; c.x; ˛// if and only if the symmetry condition (4.13b) is satisﬁed. Thus we test the symmetry conditions G .˛; x / .˛; b/ symmet- Q ˛˛ ˛˛ ric and check the second order conditions G .˛; x / positive semideﬁnite, using Q ˛˛ (4.13b).

4.3 Consumer behavior

Consider the following general utility maximization problem

max u.x/ V .˛; y/ x Á (4.15) s.t. c.x; ˛/ y D where these is a nonlinear budget constraint c.x; ˛/ y, i.e. in general prices of the commodities x can depend upon the levels of the commoditiesD purchased by the consumer as well as upon the parameters ˛. Also consider the corresponding cost minimization problem min c.x; ˛/ E.˛; u/ x Á (4.16) s.t. u.x/ u D Comparative static effects for the maximization problem (4.15) can be analyzed in term of the primal-dual relation

min G.˛; y; x/ V .˛; y/ u.x/ ˛;y Á (4.17) s.t. c.x; ˛/ y D or equivalently (substituting out the budget constraint)

min G.˛; x/ V .˛; c.x; ˛// u.x/ (4.18) ˛ Q Á The ﬁrst and second order conditions for a solution to (4.18) yield

G˛.˛; x/ V˛.˛; y/ Vy .˛; y/ c˛.x; ˛/ 0 (4.19a) Q Á C D 44 4 Nonlinear Static Duality Theory (for a single agent)

G˛˛.˛; x/ V˛˛.˛; y/ V˛y .˛; y/ c˛.x; ˛/ Vy˛.˛; y/ c˛.x; ˛/ Q Á C C Vyy .˛; y/ c˛.x; ˛/ c˛.x; ˛/ Vy .˛; y/ c˛˛.x; ˛/ (4.19b) C C symmetric positive semideﬁnite

Equation (4.19a) c˛.x; ˛/ V˛.˛; y/=Vy .˛; y/ are a generalization of Roy’s Theorem. D Likewise comparative static effects for the minimization problem (4.16) can be analyzed in terms of the primal-dual relation

max H.˛; x/ E.˛; u.x// c.x; ˛/ (4.20) ˛ Á The ﬁrst and second order conditions for a solution to (4.20) yield

H˛.˛; x/ E˛.˛; u/ c˛.x; ˛/ 0 (4.21a) Á D

H˛˛.˛; x/ E˛˛.˛; u/ c˛˛.x; ˛/ symmetric negative semideﬁnite Á (4.21b)

(4.21a) c˛.x; ˛/ E˛.˛; u/ is a generalization of Shephard’s Lemma. In order to deriveD a generalization of the Slutsky equation, note that the Hick- sian demands xh.˛; u/ solving the cost minimization problem (4.16) for parameters .˛; u/ also solve the utility maximization problem (4.15) for parameters .˛; b/ .˛; E.˛; u//, i.e. D xh.˛; u/ xM .˛; E.˛; u// i 1; ;N for all ˛ (4.22) i D i D Differentiating (4.22) with respect to ˛ (holding utility level u constant),

@xh.˛; u / @xM .˛; y/ @xM .˛; y/ @E.˛; u / i i i i 1; ; N j 1; 2 @˛j D @˛j C @y @˛j D D (4.23) Substituting the generalized Shephard’s Lemma (4.21a) into (4.23),

@xM .˛; y/ @xh.˛; u / @xM .˛; y/ @c.x ; ˛/ i i i i 1; N j 1; 2 @˛j D @˛j @y @˛j D D (4.24) where @c.x; ˛/=@˛j xj .˛; y/ except in the competitive case, where c.x; ˛/ PN ¤ Á i 1 ˛i xi .˛i wi /. Also note that the second order conditions for cost minimiza- D Á h h i @x .˛;u/ tion (4.21b) do not reduce to @˛ symmetric negative semideﬁnite except in the competitive case. An alternative generalization of the Slutsky equation that directly relates Mar- shallian demands to the second order conditions for cost minimization (4.21b) can be derived as follows. The ﬁrst order conditions for cost minimization E˛.˛; u / c˛.x ; ˛/ 0 (4.21a) can be rewritten as D M E˛.˛; u/ c˛.x; ˛/ for all ˛ at x x .˛; E.˛; u// (4.25) D Á References 45 using (4.22). Differentiating (4.25) with respect to ˛,

h M M i E˛˛.˛; u/ c˛˛.x; ˛/ c˛x.x; ˛/ x .˛; y/ x .˛; y/E˛.˛; u/ (4.26) D C ˛ C y substituting (4.25) into (4.26) and rearranging,

h M M i c˛;x.x; ˛/ x .˛; y/ x .˛; y/ c˛.x; ˛/ E˛˛.˛; u/ c˛˛.x; ˛/ ˛ C y D symmetric negative semideﬁnite (4.27) by the second order conditions (4.21b) for cost minimization. In order to derive a system of estimating equations for a behavioral model of the type (4.15), we can begin by specifying a functional form for the consumer’s indirect utility function V .˛; y/ and the budget constraint c.x; ˛/ y. Then we calculate D corresponding envelope relations c˛.x ; ˛/ V˛.˛; y/=Vy .˛; y/ (4.19a) which D we then solve for consumer demand relations x x.˛; y/. Since utility maximization and cost minimization are equivalent, theD Marshallian demand equations integrate up to a macrofunction V .˛; y/ representing utility maximization behavioral if and only if the corresponding envelope relations c˛.x; ˛/ E˛.˛; u/ for cost minimization satisfy the symmetry and second order conditionsD (4.21b) for integration up to a macrofunction E.˛; u/ representing cost minimizing behavior. These symmetry and second order conditions are related to the parameters of the estimated Marshallian demand equations using the generalized Slutsky equations (4.27).

References

1. Silberberg, E. (1974). A revision of comparative statics methodology in economics. Journal of Economic Theory, pages 159–72

Chapter 5 Functional Forms for Static Optimizing Models

5.1 Difﬁculties with simple linear and log-linear Models

The main purpose of this lecture is to discuss the concept of ﬂexible functional forms and to present speciﬁc functional forms that are commonly employed with static duality theory. In order to appreciate the potential value of such functional forms, we begin with a discussion of the problem in using simpler linear and log-linear models. For simplicity we restrict this discussion to cost-minimizing behavior. First suppose that a simple linear model of a cost function c.w; y/ is formulated:

N X c a0 ai wi ay y: (5.1) D C C i 1 D If a producer minimizes his total cost of production, then Shephard’s Lemma applies to the cost function. Applying Shephard’s Lemma to (5.1),

@c.w; y/ xi ai i 1; ;N: (5.2) D @wi D D i.e. the cost-minimizing factor demands x x.w; y/ are in fact independent of factor prices and the level of output. AlternativelyD suppose that factor demands are estimated as a linear function of prices and output:

N X xi ai0 aij wj aiy y i 1; ;N: (5.3) D C C D j 1 D However if factor demands are homogeneous of degree 0 in prices then (by Euler’s PN @xi .w;y/ theorem) wj 0 .i 1; ; N:/ (see footnote 2 on page 3). So j 1 @wj D D (5.3) is consistentD with cost minimizing behavior only if (5.3) reduces to

xi ai0 aiy y i 1; ;N: (5.4) D C D

47 48 5 Functional Forms for Static Optimizing Models i.e. cost minimizing factor demands are independent of factor prices (implying a Leontief production function). Of course these difﬁculties can be circumvented by ﬁrst normalizing prices on a numeraire price:

N Â Ã X wj xi ai0 aij aiy y i 1; ;N: (5.5) D C w1 C D j 2 D Here the hypothesis that factor demands are homogenous of degree 0 in prices w is imposed a priori by linearizing on normalized prices; so homogeneity cannot place any further restrictions on the functional form (5.5) for factor demands. Thus, to the extent that factor demands are homogenous of degree 0 in prices, (5.5) clearly is a better speciﬁcation of factor demands than is (5.3). Nevertheless, note that (5.5) does impose an symmetry on the effects of the numeraire price w1 relative to the effects of other prices on factor demands. Responses @c.w;y/ where (j 1) can be calculated @wj ¤ directly form (5.5) are simply @c.w;y/ aij , but responses @c.w;y/ must be calcu- @wj D w1 @w1 @c.w;y/ PN @c.w;y/ lated indirectly from the homogeneity condition w1 wj 0 @w1 j 2 @w1 C D D @c.w;y/ PN wj Á as aij .i 1; ;N/. @w1 D j 2 w1 D Next consider simpleD log-linear models of cost minimizing behavior. A log-linear cost function (corresponding to a cobb-Douglas technology)

N X ln c a0 ai ln wi ay ln y (5.6) D C C i 1 D

@ ln c @ ln c @c.w;y/ c wi xi implies that ai , but = using Shephard’s @ ln wi @ ln wi @wi wi c Lemma. Then the shareD of each inputÁ in that costs isD independent of prices and output: wi xi si ai i 1; ;N: (5.7) Á c D D Alternatively consider log-linear factor demands:

N X ln xi ai0 aij ln wj aiy y i 1; ;N: (5.8) D C C D j 1 D

PN @xi .w;y/ 1 @ ln xi Homogeneity implies j 1 @w = w 0 .i 1; ;N/ and @ ln w D j j D D j D @xi .w;y/ = xi using Shephard’s Lemma; so homogeneity of factor demands does not @wj wj PN imply the restriction j 1 aij 0 .i 1; ;N/. On the other hand, in the case of log-linear consumer demandsD Dx x.p;D y/ conditional on prices p and income PN @xDi .p;y/ y, the adding up constraint i 1 @y pi 1 (derived by differentiating the budget constraint px y with respectD to y) is generallyD satisfied only if all income elasticities are equal toD1 (see Deaton and Muellbauer 1980, pp. 16–17). 5.2 Second order flexible functional forms 49 5.2 Second order flexible functional forms

The previous section illustrated the severe restrictions implied by linea or log-linear models of cost functions or (by extension) profit functions or indirect utility functions. These functional forms imply extremely restrictive production functions and behavioral relations. Likewise the most commonly employed production functions, i.e. Cobb-Douglas or CES, place significant restrictions on behavioral relations (all elasticities of substitution equal 1 or all elasticities of substitutions are equal, respectively). Also note that a Cobb-Douglas production function is equivalent to a Cobb-Douglas functional form for the associated cost function and profit function (e.g. Varian 1984, p. 67). The concept of second order flexible functional forms has been employed in order to generate less restrictive functional forms for behavioral models (see Diewert 1971, pp. 481–507). Suppose that the true production function, cost function, profit function or indirect utility function for an agent is represented by the functional form f .x/. Then, taking a second order Taylor series approximation of f .x/ about a point x0,

2 X @f.x0/ 1 X X @ f .x0/ f .x0 x/ f .x0/ xi xi xj : (5.9) C C i @xi C 2 i j @xi @xj

Thus for a small x, f .x0 x/ can be closely approximated in terms of the 2 C @f.x0/ @ f .x0/ Á level of f at x0 (f .x0/) and its first and second derivatives at x0 @x ; @x@x . In turn, any other function g.x/ whose level at x0 can equal the level of f at x0 .g.x0/ f .x0// and whose first and second derivatives at x0 can equal the first 2 2 D @g.x0/ @f.x0/ @ g.x0/ @ f .x0/ Á and second derivatives of f at x0 ; can closely @x D @x @x@x D @x@x approximate f .x/ for small variations in x about x0. To be more formal, g.x/ provides a second order (differential) approximation to f .x/ at x0 if and only if

g.x0/ f .x0/ (5.10a) D @g.x / @f.x / 0 0 i 1; ;N: (5.10b) @xi D @xi D @2g.x / @2f .x / 0 0 i; j 1; ;N: (5.10c) @xi @xj D @xi @xj D

In general the true functional form f .x/ is unknown. Thus g.x/ provides a second order flexible approximation to an arbitrary function f .x/ at x0 if conditions (5.10) can be satisfied at x0 for any function f .x/. In other words, g.x/ is a second order flexible functional form if, at any point x0, any combination of level g.x0/ and 2 @g.x0/ @ g.x0/ derivatives @x , @x@x can be attained. Thus a general second order flexible N.N 1/ functional form g.x/ must have at least 1 N 2C free parameters (assuming @2g C C @x@x symmetric). If g.x/ is linear homogeneous in x, then (using Euler’s theorem) 50 5 Functional Forms for Static Optimizing Models there are 1 N restrictions. C N X @g.x0/ g.x0/ xi0 D @xi i 1 D (5.11) N 2 X @ g.x0/ 0 xj 0 i 1; ;N D @xi @xj D j 1 D on g.x0/ and its first and second derivatives. Then a linear homogeneous second or- N.N 1/ der flexible functional form g.x/ has 2C free parameters. Note that the number N.N 1/ of free parameters 2C increase exponentially with the dimension N of x. Consider a functional form c.w; y/ for a producer’s cost function. c.w; y/ is a 2 @c.w0;y0/ @c.w0;y0/ h @ c.w0;y0/ i second order flexible functional form if c.w0; y0/, , , @w @y @w@y .N 1/ .N 1/ C C are not restricted a priori (except for homogeneity restrictions (5.11)) at .w0; y0/. Moreover a second order flexible approximation c.w; y/ to a true cost function implies a second order flexible approximation to a true production function (see equation (1.5), (1.6) on page (1.5)). Next consider a functional form .w; p/ for a firm’s profit function. .w; p/ @.w0;p0/ is a second order flexible functional form if the combination .w0; p0/, @w , h 2 i @.w0;p0/ , @ .w0;p0/ is not restricted a priori (except for homogene- @p @w@p .N 1/ .N 1/ ity restrictions). In additionC a secondC order flexible approximation to a true profit function implies a second order flexible approximation to a true production function (see equation (2.5), (2.6) of page (2.5)). Likewise a second order approximation V .p; y/ to a true indirect utility function implies a second order approximation to a true utility function (structure of preferences). Thus, to the extent that w, y is small over the data set .w; y/, a second order flexible functional form for a producer’s cost function c.w; y/ provides a close approximation to the true cost function and to the underlying production function. Similar comments apply to profit functions and indirect utility functions. Note that if prices w are highly co-linear over time then in effect the variation w in prices may be small. Thus second order flexible functional forms may be useful in modeling behavior over data sets with very high multicollinearity. On the other hand, for a cost function, changes in output .y/ are likely to be substantial over time and not perfectly correlated with changes in factor prices .w/. In this case it may be desirable to provide a third order or even higher order of approximation in output y to the true cost function, e.g. by specifying a cost function c.w; y/ such that any combination of the following cost and derivatives can be attained at a point .w0; y0/ (subject to homogeneity restrictions): 5.3 Examples of second order flexible functional forms 51

c.w0; y0/ (5.12a) @c.w ; y / @c.w ; y / 0 0 0 0 i 1; ;N (5.12b) @wi @y D @2c.w ; y / @2c.w ; y / @2c.w ; y / 0 0 0 0 0 0 i; j 1; ;N (5.12c) @wi @wj @wi @y @y@y D @3c.w ; y / @3c.w ; y / 0 0 0 0 i 1; ;N (5.12d) @wi @y@y @y@y@y D Notice, however, that if third derivatives (5.12d) as well as (5.12a)–(5.12c) to be free then a greater number of free parameters must be estimated in the model, which implies a lost of degrees of freedom in the estimation. In sum, there are two serious problems in the application of flexible functional forms. First, the number of parameters to be estimated increases exponentially with the number of variations (e.g. prices, outputs) included in the functional form and with the order of the Taylor series approximation. Thus flexible functional forms generally require a high level of aggregation of commodities; but consistent aggregation of commodities is possible only under strong restrictions on the underlying technology or preference structure (see next lecture). Second, when there is substantial variation in the data, the global properties of flexible functional forms (how well these forms approximate the unknown true function f .x/ over large variation in prices and output) become very important. Unfortunately the global properties of many common flexible functional forms are not clear. It is often difficult to discern whether these functional forms impose restrictions over large x that are unreason- able on a priori grounds and hence seriously bias econometric estimates (see pages 232–236 of M. Fuss, D. McFadden, Y. Mundlak, “A Survey of Functional Forms in the Economic Analysis of Production,” in M. Fuss, D. McFadden, Production Economic: A Dual Approach to Theory and Applications, 1978 for a good early discussion of this problem). Nevertheless the concept of flexible functional forms appears to be useful in practice.

5.3 Examples of second order ﬂexible functional forms

The most common flexible functional forms are the Translog and Generalized Leon- tief, so we focus on these plus the Normalized Quadratic (which is the most obvious flexible functional form). First consider dual profit functions .w; p/, which we now write as .v/ where v .w; p/. The most obvious candidateÁ for a flexible functional form for a dual profit function .v/ is the quadratic:

N N N X 1 X X .v/ a0 ai vi aij vi vj (5.13) D C C 2 i 1 i 1 j 1 D D D 52 5 Functional Forms for Static Optimizing Models

Note that this quadratic can be viewed as a second order expansion of in powers of v. In the absence of homogeneity restrictions, (5.13) obviously is a sec- 2 ond order flexible functional form: differentiating twice yields @ =@vi @vj aij D .i; j 1; ;N/ i.e. each second derivative is determined as a free parameter aij ; D PN differentiating once yields @=@vi ai j 1 aij vj .i 1; ;N/ i.e. there D C D D is a remaining free parameter ai to determine @=@vi at any level; and finally the parameter a0 is free to determine at any level. However linear homogeneity of PN @2 .v/ in v implies (by Euler’s theorem) vj 0 .i 1; ;N/ and in j 1 @vi @vj D D PN D turn (by Hotelling’s Lemma) j 1 aij vj 0. Thus the quadratic profit function (5.13) reduces to a linear profit function:D D

N X a0 ai vi (5.14) D C i 1 D which implies (using Hotelling’s Lemma) that output supplies and factor demands are independent of prices v. This problem is circumvented by deﬁning the quadratic proﬁt function in terms of normalized prices:

N N N X 1 X X .v/ a0 ai vi aij vi vj (5.15) z Q D C Q C 2 Q Q i 1 i 1 j 1 D D D where vi vi =v0 .i 1; ;N/, i.e. the inputs and outputs of the firm are indexed Q Á D i 0; ;N and v0 is chosen as the numeraire. By construction (5.15) satisfies theD homogeneity condition (i.e. only relative prices v matter) so homogeneity does Q not place any further restrictions on the functional form (5.15). Since vi vi =v0 1 Q Á implies dvi dvi (for v0 fixed), the derivatives of .v/ and the correspond- Q D v0 z Q @.v/ @.v/ @2.v/ 1 @2.v/ ing .v/ can be related simply as follows, z Q , z Q @vi @vi @vi @vj v0 @vi @vj Q D Q Q D .i; j 1; ;N/. The derivatives of .v/ with respect to v0 can then be recovered fromD (5.15) using the homogeneity conditions (5.11). Often calculating the data v Q from the data .v0; vi ; ; vN / we can assume without loss of generality (since only relative prices matter) that v0 1 and specify the estimating equations as Á N @.v/ X yi z Q ai aij vj D @vi D C Q Q j 1 D (5.16) N @.v/ X xi z Q ai aij vj i 1; ;N D @vj D Q D j 1 Q D for outputs y and inputs x. The corresponding equation for the numeraire commodity can be recovered from (5.16) using homogeneity. These equations (5.16) @2.v/ integrate up to a macrofuction .v/ if @v@Q vQ is symmetric, i.e. if aij aj i z Q Q Q D 5.3 Examples of second order flexible functional forms 53

@2.v/ .i; j 1; ;N/, and profit maximization implies that the matrix @v@z vQ aij is symmetricD positive semidefinite. Q Q D Next consider a Generalized Leontief dual profit function:

N N N X 1 X X .v/ a0 ai pvi aij pvi pvj (5.17) D C C 2 i 1 i 1 j 1 D D D P p This is a second order expansion of in powers of pv. Note that j aij pvi vj P P P D aij pvi pvj whereas ai pvi p ai pvi ; so .v/ .v/ re- j i D i D quires a0 0, ai 0 .i 1; ;N/. Thus, imposing the restriction of linear homogeneity,D the GeneralizedD D Leontief .v/ can be rewritten as

N N 1 X X .v/ aij pvi pvj (5.18) D 2 i 1 j 1 D D 1=2 @ P vj Á Differentiating .v/ (5.18) once yields @v aij j i aij v .i i D C ¤ i D 1; ;N/, and differentiating .v/ twice yields @2.v/ a ij j i @vi @vj D pvi pvj ¤ 2 (5.19) @ .v/ X pvj a i; j 1; ;N ij 3=2 @vi @vi D D j i vi ¤ This provider a second order ﬂexible form for .v/ subject to homogeneity restrictions. The corresponding estimating equations are

Â Ã1=2 @.v/ X vj yi aii aij D @vi D C vi j i ¤ (5.20) Â Ã1=2 @.v/ X vj xi aii aij i 1; ;N D @vi D vi D j i ¤ for outputs y and inputs x. These equations integrate up to a macrofunction .v/ if @2.v/ @v@v is symmetric, i.e. aij aj i .i; j 1; ;N/ (see (5.19)). Profit maximiza- @D2.v/ D tion implies that the matrix @v@v is symmetric positive semidefinite. Third consider a Translog dual profit function

N N N X 1 X X ln .v/ a0 ai ln vi aij ln vi ln vj (5.21) D C C 2 i 1 i 1 j 1 D D D 54 5 Functional Forms for Static Optimizing Models

This is a second order expansion of ln in powers of ln v. In order to determine homogeneity restrictions, note that .v/ .v/ implies ln .v/ ln ln .v/. Multiplying prices v by in (5.21).D D C X X X ln .v/ a0 ai ln.vi / aij ln.vi / ln.vj / D C i C i j X X 2 X X a0 ln ai ai ln vi .ln / aij D C i C i C i j X X X X 2 ln aij ln vi aij ln vi ln vj C i j C i j X X X Á 2 X X ln .v/ ln ai 2 ln aij ln vi .ln / aij D C i C i j C i j (5.22)

Thus the linear homogeneity condition ln .v/ ln ln .v/ is satisﬁed if D C N X ai 1 D i 1 D (5.23) N X aij 0 i 1; ;N D D j 1 D Rearranging @ ln @ = as @ @ ln and differentiating with respect to @ ln vi D @vi vi @vi D @ ln vi vi vj ,

2 2 @ .v/ @ ln @ ln vj @ ln @ 1

@vi @vj D @ ln vi @ ln vj @vj vi C @ ln vi @vj vi Ä @ ln @ ln aij i j D vi vj C @ ln vi @ ln vj ¤ 2 2 (5.24) @ .v/ @ ln @ ln vi @ ln @ 1 @ ln 2 @vi @vi D @ ln vi @ ln vi @vi vi C @ ln vi @vi vi @ ln vi .vi / " # Â @ ln Ã2 @ ln 2 aii i; j 1; ;N D .vi / C @ ln vi @ ln vi D

@2 ln @ ln vj 1 @ @ ln since aij (5.21), and . Thus, as in the @ ln vi @ ln vj D @vj D vj @vj D @ ln vj vj cost of the normalized Quadratic and Generalized Leontief, the differential equa- @.v/ @.v/ tions yi .v/ (for outputs), xi .v/ (for inputs) integrate up to a D @vi D @vi macrofunction .v/ if aij aj i .i; j 1; ;N/. Profit maximization further @2.v/D D requires that the matrix @v@v defined by (5.24) is symmetric positive semidefinite. In contrast to the normalized Quadratic and Generalized Leontief, the Translog model (5.21) is more easily estimated in terms of share equations rather than output supply and factor demand equations per se. The elasticity formula @ ln @ = @ ln vi D @vi vi 5.3 Examples of second order flexible functional forms 55

@ ln @ ln and Hotelling’s Lemma yield yi , xi ; so substituting for D @ ln vi vi D @ ln vi vi @ ln and from (5.21) yields estimating equations for y and x that are nonlinear in @ ln vi the parameters .a0; ; aN ; a11; ; aNN / which are to be estimated. On the other hand, the elasticity formula and Hotelling’s Lemma directly imply pi yi @ ln D @ ln vi and wi yi @ ln . Thus the Translog model (5.21) can be most easily estimated @ ln vi in terms ofD equations for “proﬁt shares”:

N pi yi X si ai aij ln vj Á D C j 1 D (5.25) N wi xi X si ai aij ln vj i 1; ;N Á D D j 1 D The homogeneity condition (5.23) and reciprocity conditions aij aj i .i; j @2.v/ D D 1; ;N/ can easily be tested, and the restriction @v@v positive semideﬁnite can be checked at data points .v; / using (5.24). The only complication in this procedure for estimating Translog models arises from the fact that the sum of the output shares minus the sum of the input shares P pi yi P wi xi equals 1 : i i 1. Writing these equations as si PN D D D .ai j 1 aij ln vj / ei .i 1; ;N/ where ei denotes the disturbance ˙for equationC Di, we see thatC this linearD dependence between shares implies a linear dependence between disturbances, i.e. ei for any equation must be a linear combination of the disturbances for all other equations. This in turn implies that one of the share equations must be dropped from the econometric model for the purposes of estimation, but this is not a serious problem since there are simple techniques for insuring that econometric results are invariant with respect to which equation is actually dropped. The above functional forms are easily generalized to cost functions c.w; y/, and the homogeneity, reciprocity and concavity conditions are calculated in an analogous manner. The normalized quadratic cost function can be written as

N N N N X 1 X X X 2 c.w; y/ a0 ai wi ay y aij wi wj aiy wi y ayy y (5.26) Q Q D C Q C C 2 Q Q C Q C i 1 i 1 j 1 i 1 D D D D

wi where wi , and the corresponding factor demand equations are (using Shep- w0 hard’s Lemma)Q Á @c.w; y/ xi Q Q D @wi Q N (5.27) X ai aij wj aiy y i 1; ;N D C Q C D j 1 D 56 5 Functional Forms for Static Optimizing Models

The effects of changes in the numeraire price w0 can be calculated from (5.27) using the homogeneity restrictions (5.11). Note that the parameters ay and ayy are not included in the factor demand equations, so that @c.w; y/=@y cannot be recovered directly from (5.27). Nevertheless @c.w; y/=@y canQ Q be calculated indirectly from (5.27) using the homogeneity condition @c.w; y/=@y @c.w; y/=@y and the 2 2 D reciprocity relations @ c.w; y/=@wi @y @ c.w; y/=@y@wi .i 1; ;N/ plus Shephard’s Lemma (See Footnote 2 onD page 19). AlternativelyD the cost function (5.26) can be estimated directly along with N 1 of the factor demand equations (5.27). The Generalized Leontief cost function is often written as

N N N 1 X X 2 X c.w; y/ y aij pwi pwj y aiy wi (5.28) D 2 C i 1 j 1 i 1 D D D where aiy 0 i 1; ;N/ implies that the production function is constant returns to scale.D TheD corresponding factor demand equations are

@c.w; y/ xi D @wi 1 Â Ã 2 (5.29) X wj 2 aii y aij y aiy y i 1; ;N D C wi C D j i ¤ The Translog cost function is

N N N X 1 X X ln c.w; y/ a0 ai ln wi ay ln y aij ln wi ln wj D C C C 2 i 1 i 1 j 1 D D D (5.30) N X 2 aiy ln wi ln y ayy .ln y/ C C i 1 D and the corresponding factor share equations are

N wi xi X si ai aij ln wj aiy ln y i 1; ;N (5.31) Á c D C C D j 1 D PN As in the case of the Translog proﬁt function, i 1 si 1, so that one share equation must be deleted from the estimation. D D Finally, we brieﬂy consider indirect utility functions V .p; y/ where p denotes prices for consumer goods and y now denotes consumer income. The Normalized Quadratic indirect utility function can be written as

N N N X 1 X X V.p/ a0 ai pi aij pi pj (5.32) z Q D C Q C 2 Q Q i 1 i 1 j 1 D D D 5.4 Almost ideal demand system (AIDS) 57 where pi pi =y, so that (5.32) is by construction homogeneity of degree 0 in Q Á PN @V .p;y/ .p; y/. @V .p; y/=@y can be calculated from (5.32) using the condition pi i 1 @pi @V .p;y/ D C @y y 0 implied by zero homogeneity. Then the consumer demand equations D @V .p;y/ @V .p;y/ can be calculated using Roy’s identity xi = .i 1; ;N/. @pi @y Roy’s identity generally leads to equations thatD are nonlinear in the parametersD (a) to be estimated. A simple example of a Translog indirect utility function is

N N N X 1 X X ln V.p/ a0 ai ln pi aij ln pi ln pj (5.33) Q D C Q C 2 Q Q i 1 i 1 j 1 D D D where again pi pi =y .i 1; ;N/. (see Varian 1984, pp. 184–186 for further examples of functionalQ Á formsD for indirect utility functions).

5.4 Almost ideal demand system (AIDS)

Suppose consumer expenditure function in the form

log E.p; u/ a.p/ b.p/ u p .p1; ; pM / (5.34) D C D where X 1 X X a.p/ ˛0 ˛i log pi rij log pi log pj D C i C 2 i j (5.35) Y b.p/ ˇ0 pi ˇi ˇ0 p1ˇ1 pM ˇM D i ( Since E.p; u/ E.p; u/, the following relation apply, D M X ˛i 1 D i 1 D (5.36) M M M X X X r r ˇi 0 ij D ij D D i 1 j 1 i 1 D D D By Shephard’s Lemma.

@ log E @E pi xi pi si (cost share i) (5.37) @ log pi Á @pi E D E Á So that cost share equations are attained from Shephard’s Lemma by differentiating (5.34)–(5.35): 58 5 Functional Forms for Static Optimizing Models @ log E si D @ log pi X @b (5.38) ˛i rij log pj u D C j C „ƒ‚… @ log pi E a. / „ ƒ‚ … log by (??) @a b./ @ log pi

This (5.38) simpliﬁes to

N X si ˛i rij log pi ˇi log Y=P (5.39) D C C i 1 D where Y is consumer expenditure .E/ and P is a price index given by

M M M X 1 X X log P ˛0 log pi rij log pi log pj (5.40) D C C 2 j 1 i 1 j 1 D D D and 1 rij .r r / for all i; j (5.41) Á 2 ij C j i Except for the price index P , demands (5.39) are linear is coefﬁcients. Homogeneity and symmetric imply.

M X rij 0 i 1; ;M (homogeneity) (5.42a) D D j 1 D rij rj i for all i; j 1; ;M: (5.42b) D D In practice P is usually approximated by an appropriate arbitrary pure index, e.g.

M X log P sj log pj (5.43) j 1 D and then (5.39) is estimated.

5.5 Functional forms for short-run cost functions c.w; y; K/ CRTS f .x; K/ c.x; K/ c.w; y; K/ c.w; y; K/ D ) D The issue: form for c.w; y; K/ should be 2nd-order ﬂexible functional form with and without CRTS. References 59

5.5.1 Normalized quadratic: c c=w , w w=w . D 0 D 0 e.g. Â Ã X 1 X X X Á 2 c a0 ai w aij ww y b0 bi w y D C i i C 2 i j i j C C i i X Á X Á 2 X Á p c0 ci w K d0 di w K e0 ei w Ky C C i i C C i i C C i i (under CRTS) ) Â Ã X 1 X X X Á X Á p c a0 ai w aij ww y c0 ci w K e0 ei w Ky D C i i C 2 i j i j C C i i C C i i

5.5.2 Generalized Leontief:

Â Ã 1 X X X Á 2 X Á X Á 2 c aij pwi pwj y bi wi y ci wi K di wi K D 2 i j C i C i C i X Á p ei wi Ky C i (under CRTS) ) Â Ã 1 X X X Á X Á p c aij pwi pwj y ci wi K ei wi Ky D 2 i j C i C i

5.5.3 Translog:

X 1 X X 2 log c a0 ai .log wi / aij .log wi /.log wj / b0 log y b1.log y/ D C i C 2 i j C C X 2 X bi .log wi /.log y/ c0 log K c1.log K/ ci .log wi /.log K/ e.log y/.log K/ C i C C C i C (under CRTS) ) log c.w; y; K/ log log c.w; y; K/ D C

References

1. Deaton and Muellbauer (1980). Economics and Consumer Behavior. pp. 16–17 60 5 Functional Forms for Static Optimizing Models

2. Diewert W. (1971), An Application of the Shephard Duality Theorem: A Generalized Leontief Production Function, Journal of Political Economy, 1971, pp. 481–507 3. Fuss M., McFadden D. (1978). Production Economic: A Dual Approach to Theory and Ap- plications 4. Varian H. (1984), Microeconomic Analysis, second edition,WW Norton & Co. pp. 67 Chapter 6 Aggregation Across Agents in Static Models

6.1 General properties of market demand functions

In previous lecture we characterized the behavior of an individual producer or consumer at a static equilibrium. However in practice we often have data aggregated over agents rather than data for the individual producers or consumers. This leads to the following question: do the behavioral restrictions that apply to data at the firm or consumer level also apply to data that has been aggregated over agents? In several simple behavioral models the answer to this question is “yes”. The most obvious example is the simple theory of static profit maximization, where all input are free variable and all firms face the same prices w, p. Then the market prices w, p correspond exactly to the parameter facing the individual firms so that industry output supply and factor demand relations Y Y.w; p/, X X.w; p/, PF f P f D D where Y f 1 y and X f x , do not misrepresent the parameters facing individualD firms.D More precisely,D

( N ) f f X f .w; p/ py wi x 0 for all .w; p/ f 1; ;F i D i 1 D ( N ) X f X f X f .w; p/ py wi x 0 for all .w; p/ (6.1) ) i f f i 1 D ( N ) X i.e. ….w; p/ pY wi Xi 0 for all .w; p/ i 1 D By (6.1), we can develop the properties of the industry proﬁt function ….w; p/ and industry output supplies Y Y.w; p/ and derives demands X X.w; p/ D D

61 62 6 Aggregation Across Agents in Static Models in essentially the same manners as in the case of data for the individual ﬁrm (see section 4.1 of lecture 4 on nonlinear duality).1 As a second example consider the case where each consumer maximizes utility P f f f f subject to the value pi w of his initial endowments w .w ; ; w / of i i D 1 N the N commodities rather than an income yf that is independent of commodity prices p, i.e. each ﬁrm f solves

max uf .xf / xf N N (6.2) X f X f s.t. pi x pi w : i D i i 1 i 1 D D

Then the solution xf to (6.2) is conditional on .p; wf / and in turn aggregate market demands P xf are conditional on .p; w1; ; wF /. If the en- f dowments wf of consumer f are constant over time for each consumer f P f D 1; ;F , then for purposes of estimation the market demands f x are in effect conditional only on the prices p which are common to each consumer: x x .p/ P xf .p/. Then the aggregate market demand relation x x.p/ D D f D are well deﬁned and inherit all linear restrictions on xf .p/. This includes the homogeneity restrictions xf .p; yf / xf .p; yf /, which can be expressed as xf .p; wf / xf .p; wf /, but apparentlyD excludes the nonlinear second or- Dh f f f f i h H f i der conditions x p; y x f p; y xj x p; u symmetric i pj i y Á i pj negative semideﬁnite. In general the answer to the above question is “no”, i.e. the behavioral restrictions that apply to data at the level of the individual agent generally do not apply to data that has been aggregated over agents. This question has been addressed in the context of utility maximization by consumers:

max uf .xf / xf N xf xf .p; yf / X f f ! D s.t. pi x y i D i 1 D which implies the Slutsky relations

xf .p; yf / xf .p; yf /xf .p; yf / symmetric negative semideﬁnite (6.3) p C yf (property (3.4).b on page 31). It has been shown that, in the absence of special restrictions on utility functions uf .xf / or on the distribution of income yf over consumers, the above restrictions (6.3) on demands of the individual consumer do not carry over to aggregate demands X X.p; Y / where X P xf and Y D Á f Á 1 The one exception concerns whether these relations .w; p/, y.w; p/, x.w; p/ are well de- ﬁned in cases of free entry and exit to the industry (see footnote 2 on page 19). 6.1 General properties of market demand functions 63

P f f y . Indeed, if the number of consumers is equal to or greater than the number of goods N , then any continuous function X.p; Y / that satisﬁes Walras’ Law Walras Law is a principle in general equilibrium theory asserting that when con- sidering any particular market, if all other markets in an economy are in equilibrium, then that speciﬁc market must also be in equilibrium. Walras Law hinges on the mathematical notion that excess market demands (or, conversely, excess market supplies) must sum to zero. That is, P XD P XS 0. Walras’ Law is named for the mathematically inclined economistD Leon Walras,D who taught at the University of Lausanne, although the concept was expressed earlier but in a less mathematically rigorous fashion by John Stuart Mill in his Essays on Some Unset- tled Questions of Political Economy (1844). (in particular the adding up properties PN @xi .p;y/ PN @xi .p;y/ @xi .p;y/ pi 1, pi y 0, j 1; ;N in the i 1 @y D i 1 @pj C @y D D differentiableD case) can be generatedD by some set of utility maximizing consumers with some distribution of income (see Debrew 1974, pp. 15–22; Sonnenschein 1973, pp. 345–354). There is a relatively simple way of addressing the above question. Following Diewect (1977, page 353–362) write the Slutsky relations (6.3) in the form

f f f f f f f xp .p; y / K x f .p; y / x .p; y / (6.4) N N y N N D 1 N N 1 f hf f where K xp .p; u / is the Hickscian substitution matrix which is symmetric Á h i negative semidefinite. In general xf .p; yf /xf .p; yf / is neither symmetric nor yf negative semidefinite. Choose N 1 linearly independent vector v1; v2; ; vN 1 f such that x vn 0 for n 1; ;N 1, and write v1; ; vN 1 in matrix 1 N N 1 D D form as V . Pre and post-multiplying (6.4) by V , N .N 1/ T f T f T f f T f V xp V V K V V xy x V V K V (6.5) .N 1/ N N .N 1/ D D N N f f hf f since x V 0. Since K xp .p; u / is symmetric negative semidefinite, D f T f Áf f (6.5) implies that V xp .p; y /V is symmetric negative semidefinite. Now sum (6.4) over consumers f 1; ;F to obtain the matrix of partial derivatives of aggregate demands X DP xf with respect to prices p as D f f X f X f f f f Xp.p; y / K x .p; y /x .p; y /: (6.6) D yf f f

Let v1; ; vN F be N F linearly independent vectors, each of which is orthog- onalQ to x 1 ; Q ; xF , the set of initial demand vectors of the F consumers. Define the N .N M/ matrix V Œv1; ; vN M . Pre and postmultiplying (6.6) by V , Q D Q Q Q X X f X V T X V V T Kf V V T Kf x xf V V T Kf V (6.7) Q p Q Q Q Q yf Q Q Q .N M/ .N M/ D D f f f 64 6 Aggregation Across Agents in Static Models since V T xf 0 for all f 1; ;F . Therefore Kf symmetric negative Q N F semidefinite (fD 1; ;F ) impliesD D V T X V is a symmetric negative semidefinite matrix of dimension.N F/ .N F /: Q p Q (6.8) (6.8) can be viewed as restrictions places on aggregate consumer demands X.p; y1; ; yF / by symmetry negative semidefiniteness of Hicksian demand re- hf f sponses xp .p; u / .f 1; ;F/. Moreover these are essentially the only restrictions on the first derivativesD of market demand functions (aside from adding up constraints) (Mantel 1977). Note that if the number of consumers F is equal to the number of goods N , then (6.8) places 0 restrictions on aggregate demands (the dimensions of the matrix of restrictions are .N F/ .N F/ equals 0 0 in this case). This is consistent with the result of Debreu and Sonnenschein that sym- hf f f metry negative semidefiniteness of Hicksian demand responses xp .p; u / K places no restrictions on aggregate demands when the number of consumers isÁ equal to or greater than number of goods. In sum, in general (i.e. in the absence of restrictions on production/utility functions or on the distribution of exogenous parameters across agents) the microtheory of the individual agent cannot be applied to data that has been aggregated over agents. In the next two sections we investigate how restrictions on production/utility functions and on the distribution of parameters across agents can influence this conclusion.

6.2 Condition for exact linear aggregation over agents

Here we consider restrictions on production and utility functions that imply consistent aggregation over agents for any distribution of the exogenous parameters that vary over agents. Fist, consider the conditional factor demands Á xf xf w; yf i 1; ; N f 1; ;F (6.9) i D i D D where w vector of factor prices (which are common to each ﬁrm) and yf Á P f P f Á output level of ﬁrm f . Aggregate factor demands f x X.w; f y / exist if and only if D

P f Á P f f Xi w; y x w; y i 1; ;N (6.10) f D f i D for all data .w; y1; ; yF /. Differentiating (6.10) w.r.t. yf , 6.2 Condition for exact linear aggregation over agents 65

f f @Xi .w; y/ @y @xi .w; y / @y @yf D @yf (6.11) @X .w; y/ @xf .w; yf / i.e. i i i 1; ; N f 1; ;F @y D @yf D D

f P f since @y=@y 1 in the case of linear aggregation y f y . Thus, in theD absence of restrictions on the distributionD of total output y over firms, P f f f f aggregate factor demands X.w; f y / exist if and only if @x .w; y /=@y are constant across all firms (for a given w). This condition implies that conditional factor demand equations for firms are of the following form:

f f f x ˛i .w/y ˇ .w/ i 1; ; N f 1; ;F (6.12) i D C i D D where the function ˛i .w/ is invariant over ﬁrms. (6.12) is called a “Gorman Polar Form” Gorman polar form is a functional form for indirect utility functions in economics. Imposing this form on utility allows the researcher to treat a society of utility-maximizers as if it consisted of a single individual. W. M. Gorman showed that having the function take Gorman polar form is both a necessary and sufﬁcient for this condition to hold.. Here the scale effects @xf .w; yf /=@yf are independent of the level of output, which implies that the production function yf yf .xf / is “quasi-homothetic”: D

Fig. 6.1 quasi-homothetic f production function x1

y1f y0f xf 0 2

Here, as output yf expands and factor price w remain constant, the cost minimizing level of inputs increase along a ray (straight line) in input space. Condition (6.12) implies that aggregate demands have a Gorman Polar Form2:

2 xf .w;yf / The more restrictive assumption of hornatheticity implies that the expansion path 4 yf is a ray through the origin. Given the Gorman Polar Form (6.12), homotheticity requires ˇ f4.w/ 0 (which in turn implies the stronger assumption of constant restrun to scale). D 66 6 Aggregation Across Agents in Static Models

X f f X f X f Xi x .w; y / ˛i .w/ y ˇ .w/ ˛i .w/Y ˇi .w/ D i D C i D C f f f (6.13) i 1; ;N: D Condition (6.12) implies the existence of aggregate conditional factor demands satisfying (6.10), irrespective of whether producers are minimizing cost. The next question is: under what conditions do the restrictions on cost minimizing factor P f demands at the firm level generalize to aggregate factor demands X.w; f y /? The simplest way to answer this question is as follow: a Gorman Polar Form for aggregate demands (6.13) implies that identical demand relations could be gener- P f ated by a single producer with an output level equal to f y , and by assumption this producer is a cost minimizer. Therefore aggregate demands of a Gorman Po- lar Form necessarily inherit the properties of cost minimizing factor demand for a single agent. To answer the above question more rigorously, first note that the assumption of cost minimization and condition (6.12) for existence of aggregate factor demands P f X.w; f y / are jointly equivalent to the assumption of cost minimization plus a Gorman Polar Form for the cost function of firm f :

cf .w; yf / a.w/yf bf .w/; f 1; ;F: (6.14) D C D f f Sufﬁciency is obvious: (6.14) and Shephard’s Lemma imply xf .w; yf / @c .w;y / i @wi f f D D @a.w/ yf @b .w/ where @a.w/ ˛.w/, @b .w/ ˇf .w/, and necessity follows @wi @wi @wi @wi simply byC integrating (6.12) up toD a cost functionD using Shephard’s Lemma. This implies the existence of an industry cost function which is also a Gorman Polar Form: X X Á cf .w; yf / a.w/yf bf .w/ f D f C X X a.w/ yf bf .w/ D f C f X (6.15) a.w/ yf b.w/ D f C X Á C w; yf D f

P f P f Moreover this industry cost function C.w; f y / a.w/ f y b.w/ inherits all the properties of cost minimization by a ﬁrm. ToD see this, note thatC 6.2 Condition for exact linear aggregation over agents 67

N f f f Á X f c w; y .x / wi x 0 for all w f 1; ;F i Ä D i 1 D N X f f f Á X X f c w; y .x / wi x 0 for all w ) i Ä f f i 1 D N X X X X f f a.w/ yf .xf / bf .w/ w x 0 for all n (6.16) , C i i Ä f f f i 1 D N X f X a.w/ y b.w/ wi xi 0 for all w , C Ä f i 1 D N X f Á X C w; y wi xi 0 for all w , f Ä i 1 D and likewise

N f f f Á X f c w; y .x / wi x 0 f 1; ;F i D D i 1 D (6.17) N X f Á X C w; y wi xi 0 ) f D i 1 D P f PN Thus the aggregate primal-dual relation C.w; y / wi xi has the same f i 1 f f D PN f properties as the cost minimizing ﬁrm’s primal-dual c .w; y / wi x .f i 1 i D 1; ;F/, and in turn C.w; P yf / has the same properties as the costD minimizing f cf .w; yf /:

Property 6.1. P f P f a) C.w; f y / is increasing in w, f y ; Á b) C.w; P yf / C w; P yf ; f D f P f c) C.w; f y / is concave in w; P f 3 d) C.w; f y / satisﬁes Shephard’s Lemma

P f Á @C w; f y X f Á Xi w; y i 1; ;N f D @wi D

3 P f P f Conditions Prop.6.1.a–c are satisﬁed for C.w; f y / a.w/ f y b.w/ if a.w/ > 0 and both a.w/ and b.w/ are increasing, linear homogeneousÁ and concave inCw. 68 6 Aggregation Across Agents in Static Models

In turn the aggregate demands satisfy the restrictions as the cost minimizing demands for a ﬁrm:

Property 6.2. X Á X Á a) X w; yf X w; yf f D f "@X.w; P yf /# b) f is symmetric negative semideﬁnite. @w N N

f Similar results hold for linear aggregation of consumer demand equations xi f f D xi .p; y / where p price of consumer goods (which are assumed to be identical for all consumers) and yf exogenous income of consumer f . The conditions for existence of aggregate demandsÁ are

X f Á X f f Á Xi p; y x p; y i 1; ;N (6.18) f D f i D and the conditions are equivalent to Gorman Polar Forms

f f Á f f x p; y ˛i .p/y ˇ .p/ i 1; ; N f 1; ;F (6.19) i D C i D D which imply

X f Á X f Xi p; y ˛i .p/ y ˇi .p/ i 1; ;N: (6.20) f D f C D In order to determine rigorously the restrictions that permit aggregate demands (6.20) to inherit the properties of utility maximization, ﬁrst note that utility maximization by a consumer is essentially equivalent to cost minimization (see page 28): N X f f f min pi xi E .p; u / xf D i 1 (6.21) D s.t. uf .xf / uf D where uf is the maximum attainable utility level for the consumer given his budget f f P f constraint px y . An aggregate cost function E.p; f u / exists if and only if D X Á X E p; uf Ef .p; uf / (6.22) f D f and this condition is equivalent to the existence of Gorman Polar Forms

Ef .p; uf / a.p/uf bf .p/ f 1; ;F (6.23) D C D 6.2 Condition for exact linear aggregation over agents 69 which in turn implies X Á X E p; uf a.p/ uf b.p/ (6.24) f D f C (6.23) and (6.24) imply

N f f f Á X f E p; u .x / pi x 0 for all p f 1; ;F i Ä D i 1 D N X f Á X E p; u pi xi 0 for all p ) f Ä i 1 D (6.25) N f f f Á X f E p; u .x / pi x 0 f 1; ;F i D D i 1 D N X f Á X E p; u pi xi 0: ) f D i 1 D P f Therefore E.p; f u / inherit the properties of cost minimization, and Shephard’s Lemma implies that

P f Á @E p; f u h X f Á Xi p; u f D @pi (6.26) @a.p/ X @b.p/ uf i 1; ;N D @pi f C @pi D In order to relate (6.26) to Marshallian demands, note that (6.23) and utility maximization by consumers f imply

yf a.p/uf bf .p/ D C yf bf .p/ (6.27) uf f 1; ;F ) D a.p/ D i.e. the consumer’s inherit utility function has the Gorman Polar Form

V .p; yf / a.p/yf bf .p/ (6.28) DQ C Q where a.p/ 1=a.p/, b.p/ bf .p/=a.p/. Substituting (6.27) into (6.26), Q D Q D 70 6 Aggregation Across Agents in Static Models @a.p/ X yf bf .p/ Xi D @pi a.p/ f @a.p/ 1 X @a.p/ X bf .p/ @b.p/ yf i 1; ;N D @pi a.p/ @pi a.p/ C @pi D f f (6.29)

(6.29) implies the existence of aggregate Marshallian demands of a Gorman Polar Form:

X f Á X f Xi p; y a.p/ y b.p/Q i 1; ;N (6.30) f D Q f C Q D where a.p/ @a.p/ 1 , b.p/ @a.p/ b.p/ @b.p/ . @pi a.p/ @pi a.p/ @pi Thus costD minimization byD individual consumersC and a Gorman Polar Form (6.18/6.30) that satisfy restrictions analogous to restriction on Marshallian demands by individual utility maximizing consumers. The restriction (6.1) implied by cost P f minimization are satisﬁes for E.p; f u / (6.24) if

a.p/ > 0 (6.31a) I a.p/, b.p/ increasing and linear homogeneous in P ; (6.31b) Ä@2a.p/ Ä@2b.p/ , symmetric, negative semideﬁnite. (6.31c) @p@p @p@p P f These restrictions can be tested or imposed on the Gorman Polar Form X.p; f y / (6.30) Alternatively we could try to incorporate the utility maximization restrictions P f into aggregate demands X.p; f y / directly by specifying consumers’ indirect utility functions V f .p; yf / as Gorman Polar Form:

V f .p; yf / a.p/yf bQf .p/: (6.32) D Q C Q However inverting this relation yields

uf b.p/Q yf Q (6.33) D a.p/ Q i.e. 1 bQf .p/ Ef .p; uf / uf Q f 1; ;F: (6.34) D a.p/ a.p/ D Q Q Thus an indirect utility function V f .p; yf / has a Gorman Polar Form if and only if the corresponding cost function Ef .p; uf / has a Gorman Polar Form. Therefore the above analysis in term of cost minimizing behavior exhaust the restrictions per- 6.2 Condition for exact linear aggregation over agents 71

P f mitting the existence of aggregate Marshallian demands X.p; f y / that satisfy the same restrictions as Marshallian demands xf .p; yf / for an individual utility maximizing consumer. Note that these restrictions (6.23–6.24, 6.30–6.31) on preference structure do imply mild restriction on the distribution of expenditures over consumers:

Ef a.p/uf bf .p/ a.p/ > 0 D C (6.35) yf > bf .p/ f 1; ;F ) D Aside from this restriction, Gorman Polar Forms permit aggregate demands to inherit the properties of utility maximizing demands irrespective of the distribution of expenditure over consumers. bf .p/ can be viewed as the agent’s “committed expenditure” at prices p (since it is independent of utility level uf ), and yf bf .p/ can be defined as the corresponding “uncommitted expenditure”. Aggregation problems also arise when there are variations in prices over agents, and these problems can be severe. For example, suppose that profit maximizing competitive firms in an industry are distributed across different regions of the coun- try and as a result firms face different output prices pf . Aggregate demands can P f P f P f P f be defined as f x X.w; f f p /, f y Y.w; f f p / where P f D D f f p denotes a (weighted) average output price p. Aggregate demands and supplies exist if N

X f Á X f f Xi w; f p x .w; p / i 1; ;N f f i D D (6.36) X f Á X f f Y w; f p y .w; p / f D f These conditions are satisﬁed if the aggregate demands and supplies have Gorman Polar Forms:

X f Xi ˛i .w/ f p ˇi .w/ i 1; ;N f D C D (6.37) X f Y ˛0.w/ f p ˇ0.w/ D f C Now suppose further that these aggregate relation are to inherit the properties of profit maximizing demand and supply relations for the individual firm. In the absence of any restrictions on the distribution of prices pf over firms, this requires the firm and industry profit functions to have the following Gorman Polar Forms

f .w; pf / a.w/pf bf .w/ f 1; ;F D C D X f Á X f (6.38) … w; f p a.w/ f p b.w/ f D f C However Hotelling’s Lemma now implies that output supplies are independent of output prices: 72 6 Aggregation Across Agents in Static Models

@f .w; pf / yf .w; pf / a.w/ f 1; ;F (6.39) D @pf D D using (6.38). In sum, Gorman Polar Forms (GPF) are both necessary and sufficient for exact linear aggregation over agents. Unfortunately these functional forms are fairly restrictive. For example, GPF conditional factor demands x x.w; y/ imply linear expansion paths, and GPF consumer demands x x.p; y/D imply linear Engel curves (and income elasticilties tend to unity as totalD expenditure incuaser). These assumptions may or may not be realistic over large changes in output or expenditure (see Deaton and Muellbauer 1980, pp. 144–145, 151–153). On the other hand it should be noted that Gorman Polar Form cost and indirect utility functions are flexible functional forms. These former provide second order approximations to arbitrary cost and indirect utility functions (Diewert 1980, pp. 595–601). Therefore Gorman Polar Forms provide a local first order approximation to any system of demand equations (except, of course, for cases such as (6.38)).

6.3 Linear aggregation over agents using restrictions on the distribution of output or expenditure

The analysis in the previous section was aimed at achieving consistent aggregation over agents while imposing essentially zero restrictions on the distribution of output or expenditure over agents. However these distribution are in fact highly restricted in most cases, and these restrictions may help to achieve consistent aggregation. Unfortunately there are few results on the relation between restrictions on the distribution of “exogenous” variables such as output or income and consistent aggregation. This section simply summaries several example that illustrate the effects of alternative restrictions on the distribution of such variables. First, output or expenditure may be choice variables for the agent rather than exogenous variables, and it is very important to incorporate this fact into the analysis of possibilities for consistent aggregation. For example, suppose that producers are competitive profit maximizers and face the same output price p. Then the first order condition in the output market for profit maximization is

@cf .w; yf / p f 1; ;F (6.40) @yf D D

f f PN f f f f where c .w; y / minxf i 1 wi xi s.t. y .x / y . This implies that the marginal cost is identicalD acrossD firms at all observed combinationsD of output levels 1 F .y ; ; y / for given prices w, p. Now remember from our earlier discussion that identical marginal cost across firms is the condition for consistent linear aggregation of cost functions across firms 6.3 Linear aggregation over agents using restrictions on the distribution of output or expenditure73

(irrespective of the distribution of output across firms) (see pages 64–66). Thus, the assumption of competitive profit maximization and identical price imply that P f an aggregate cost function C.w; f y / exists over all profit maximizing lev- P f P f els of output f y f y .w; p/. Moreover this aggregate cost function P f PD f P f C.w; f y /, where f y is restricted to the equilibrium levels f y .w; p/, also inherits the properties of cost minimization.4 As a second example, supposed that cost functions of individual consumers are of the form

Ef .p; uf / af .p/uf bf .p/ f 1; ;F: (6.41) D C D this is slightly more general than the Gorman Polr Form (6.23) in the sense that have the function af .p/ can vary over consumers. Nevertheless preference are still quasi-homothetic. Solving yf af .p/uf bf .p/ (6.41) for the consumer’s indirect utility function yields D C

yf bf .p/ V f .p; yf / f 1; ;F; (6.42) D af .p/ D and applying Roy’s Theorem to this result (6.42) yields

h @bf .p/ f @af .p/ f f i. f 2 @p a .p/ @p y b .p/ a .p/ xf .p; yf / i i i D 1ıaf .p/ (6.43) , @bf .p/ @af .p/ Á yf bf .p/ af .p/ D @pi C @pi

Thus, in contract to the Gorman Polar Form, , xf .p; yf / @af .p/ i af .p/ i 1; ; N f 1; ;F (6.44) f @y D @pi D D i.e. Engel curves can vary over consumers (although these curves are still linear). Now suppose that the distribution of “uncommitted expenditure” remains pro- portionally constant over consumers, i.e. consumer incomes y1; ; yF satisfy the following restrictions for all variations in commodity prices p:

4 On the other hand, endogenizing consumer expedition yf , as the wage rate w times the amount of labor supplied by the agent, is not sufﬁcient for consistent linear aggregation in the case of utility maximization. Endogenizing yf in this manner implies the additional ﬁrst order condition f f Le Le h f f f i Le @u .x ; x /=@x @V .p; y /=@y w where x le of leisure for con- D Á f f Le Le sumer f , .f 1; ;F/. Since the marginal utility of leisure @u .x ; x /=@x will D f f f generally vary over consumers, the marginal utility of income @V .p; y /=@y also varies over consumers. 74 6 Aggregation Across Agents in Static Models

ÄXF Á yf bf .p/ f yf bf .p/ > 0 f 1; ;F D f 1 D D (6.45) XF where f > 0, f 1. f 1 D D If the individual cost functions are of the form (6.41) and if the distribution of expenditure over agents satisﬁes the restriction (6.45), then aggregate Marshallian demand functions exist, have Gorman Polar Form and inherit the properties of utility maximization. Proof. Summing (6.43) over consumers and substituting in (6.45),

f f , X f X @b .p/ X @a .p/ f hX f f Ái f xi y b .p/ a .p/ f D f @pi C f @pi f f f f X @b .p/ X @a .p/=@pi X X @a .p/=@pi X f yf f bf .p/ f f D f @pi C f a .p/ f f a .p/ f Â Ã @ˇ.p/ @˛.p/=@pi X Á yf ˇ.p/ i 1; ;N D @pi C ˛.p/ f D (6.46)

P f QF f f where ˇ.p/ f b .p/, ˛.p/ f 1 a .p/ . Thus there exist aggregate Marshallian demandÁ functions of GormanÁ PolarD Form. A GPF implies that aggregate demands are identical to those that could be generated by a single consumer with P f expenditure f y , and by assumption this consumer maximizes utility. Thus the aggregate demand relations (6.46) inherit the properties of utility maximization. ut We have the following corollary to the above result.

Corollary 6.1. Suppose that the utility function of individual consumers are homothetic, so that the expenditure function of individual consumers have the form

Ef .p; uf / af .p/uf f 1; ;F (6.47) D D f P f and also suppose that each consumer’s share in total expenditure f y is P f ﬁxed over the data set. Then aggregate Marshallian demand relations f x P f D X.p; f y / exist, have the form Â Ã X f @˛.p/=@pi X x yf i 1; ;N (6.48) f i D ˛.p/ f D (using the notation of (6.46)) and inherit the properties of utility maximization. In order to see that this is a special case of the result proves above, simply note that (6.47) is a special case of (6.41) where bf .p/ 0, and that bf .p/ 0 .f 1; ;F/ reduce (6.45) to Á Á D 6.4 Condition for exact nonlinear aggregation over agents 75

F X yf f yf > 0 f 1; ;F D D f 1 D (6.49) F X where f > 0, f 1 D f 1 D In one respect the assumption of homotheticity in (6.47) is more restrictive than the quasi-homothetic Gorman Polar form (6.23) at constant price p the ratio of cost minimizing demands is independent of the level of expenditure, i.e. increases in expenditure yf lead to increases in consumption of all commodities along a ray from the origin rather than from an arbitrary point. On the other hand, (6.47) is more general than Gorman Polar Form (6.23) in the sense that the function af .p/ can vary over agent, so that (linear) Engel curves can vary over agents.

6.4 Condition for exact nonlinear aggregation over agents

The previous sections assumed that aggregate output or expenditure Y is to be con- P f f structed as a simple linear sum Y f y of the outputs of expenditure y of individual agents. More generally, theD aggregation relation can be written as

Y Y.y1; ; yF /: (6.50) D Then the conditions for existence of an aggregate cost function for producers (for example) can be written as Á X C w; Y.y1; ; yF / cf .w; yf / (6.51) D f for all .w; y1; ; yF /. Differentiating (6.51) w.r.t. yf , @C.w; Y / @Y @cf .w; yf / f 1; ;F (6.52) @Y @yf D @yf D which implies , @Y @y @cf .w; yf / @cg .w; yg / f; g 1; ;F: (6.53) @yf @yg D @yf @yg D

Condition (6.53) implies that Y Y.y1; ; yF / is strongly separable (see Deaton and Muellbauer 1980, pp. 137–142),D so that X Á Y Y hf .yf / : (6.54) D f 76 6 Aggregation Across Agents in Static Models

The corresponding aggregate cost function and aggregate demands are Á X Á C w; Y.y1; ; yf / ˛.w/ Y hf .yf / ˇ.w/ D f C 1 F Á @˛.w/ X f f Á @ˇ.w/ Xi w; Y.y ; ; y / Y h .y / i i 1; ;N: D @wi f C @w D (6.55)

The above cost function is more general than the Gorman Polar Form since the ag- P f gregate output Y in not restricted to the linear case f y . The aggregate cost function inherits the propertied of cost minimization, so the aggregate factor demands X w; Y.y1; ; yF / can be derived from the aggregate cost function using Shep- ard’s Lemma. these conditions for exact nonlinear aggregation are less restrictive than Gorman Polar Form, but it is not easy make this distinction operational (see Deaton and Muellbauer 1980, pp. 154–158, for one attempt).

References

1. Deaton A. & Muellbauer, J. (1980). Economics and Consumer Behavior: pp. 144–145, 151– 153 2. Debreu, G. (1974). Excess demand functions, Journal of Mathematical Economics 1: pp. 15– 22 3. Diewert, W. E. (1977). Generalized slutsky conditions for aggregate consumer demand functions, Journal of Economic Theory, Elsevier, vol. 15(2), pages 353-362, August. 4. Diewert, W. E. (1980). Symmetry Conditions for Market Demand Functions, Review of Eco- nomic Studies: pp. 595–601 5. Mantel, R.(1977). Implications of Microeconomic Theory for Community Excess Demand Function, pages 111–126 in M. D. Intriligator, ed., Frontiers of Quantitative Economics, Vol. III A, North-Hollard 6. Sonnenschein, H. (1973). Do Walras’ identity and continuity characterize the class of community excess demand functions?. Journal of Economic Theory 6: pp. 345–354 Chapter 7 Aggregation Across Commodities: Non-index Number Approaches

Consumers and also producers generally use a wide variety of commodities, so substantial aggregation (grouping) of commodities is necessary to make econometrics studies manageable. This is particularly the case with flexible functional forms, where the number of parameters to be estimated increases exponentially with the number of commodities that are modeled explicitly. Presumably aggregation over commodities generally misrepresent the choices faced by consumers and producers, so the microeconomic theory that applies to the true behavioral model with disaggregated commodities may not generalize to a model with highly aggregated commodities. This leads to the following questions: when does aggregation over commodities not misrepresent the agent’s choices or behavior, and how is this aggregation procedure defined? This lecture summaries two types of results related to this question: the composite commodity theorem and conditions for two stage budgeting. The composite commodity theorem of Hicks demonstrates that certain restrictions on the covariation of prices permit consistent aggregation, and the discussion of two stage budgeting shows that separability restrictions (plus other restrictions) on the structure of utility functions, production functions etc. also permit consistent aggregation over commodities.1 The next lecture provides a more satisfactory answer to the above question. Cer- tain index number formulas for aggregation over commodities can be rationalized in terms of certain functional forms for production functions, cost functions, etc., including popular second order flexible functional forms. Thus certain index number formulas for aggregation over commodities inherit the desirable properties of approximation that characterize the corresponding second order flexible functional forms.

1 For simplicity we will assume a single agent, i.e. we will abstract from problems in aggregating over agents.

77 78 7 Aggregation Across Commodities: Non-index Number Approaches 7.1 Composite commodity theorem

If the prices of several commodities are in ﬁxed propositions over a data set, then these commodities can be correctly treated as a single composite commodity with one price. For example, suppose that a consumer maximizes utility over three commodities x1; x2; x3 and that prices p2 and p3 remain in ﬁxed proportion over the data set, i.e.

p2;t t p2;0 p3;t t p3;0 for all times t (7.1) D D where p2;0 and p3;0 are base period prices of commodities 2 and 3, and t is a (positive) scalar that varies over time t. Equivalently the consumer can be viewed as solving a cost minimization problem

3 X E .p1; p2; p3; u/ min pi xi D x i 1 (7.2) D s.t. u.x/ u D Combining (7.1) and (7.2),

E .p1;t ; p2;t ; p3;t ; ut / E .p1;t ; t p2;0; t p3;0; ut / D (7.3) E .p1;t ; t ; ut / for all t D Q Cost minimizing behavior (7.2) implies that E .p ; ; u/ inherits all the properties Q 1 of a cost function, including Shephard’s Lemma:

@E.p1; ; u/ Q x1 @p1 D

@E.p1; ; u/ @ (7.4) Q .p1x1 p2;0x2 p3;0x3/ @ D @ C C p2;0x2 p3;0x3 D C by the envelope theorem. c Thus p2;0x2;t p3;0x3;t can be interpreted as the quantity x of a composite C commodity at time t, and t is the price of the corresponding composite commodity at time t. The resulting system of Hicksian demands

h x1 x1 .p1; ; u/ D (7.5) c ch x p2;0x2 p3;0x3 x .p1; ; u/ D C D inherit the properties of cost minimizing demands. Therefore the corresponding system of demands x1 x1 .p1; ; y/ c D c (7.6) x p2;0x2 p3;0x3 x .p1; ; y/ D C D 7.2 Homothetic weak separability and two-stage budgeting 79 inherit the properties of utility maximizing demands. However, this composite commodity theorem may be of limited value in justify- ing and deﬁning aggregation over commodities. Even if two commodities are per- ceived as relatively close substitutes in consumption, their relative prices may vary signiﬁcantly due to differences in the supply schedules of the two commodities.

7.2 Homothetic weak separability and two-stage budgeting

The assumption of “two-stage budgeting” has often been used to simplify studies of consumer behavior. The general utility maximization problem (3.1) can be written as

max u.xA; ; xA ; xB ; ; xB ; ; xZ; ; xZ / V .p; y/ x 1 NA 1 NB 1 NZ Á NA NB NZ X X X (7.7) s.t. pAxA pB xB pZxZ y i i C i i C C i i D i 1 i 1 i 1 D D D A A B B Z Z where x .x ; ; x ; x ; ; x ; ; x ; ; x /, i.e. there are NA D 1 NA 1 NB 1 NZ C NB NZ commodities. Two-stage budgeting can then be outlined as follows. In theC ﬁrst C stage total expenditures y are allocated among broad groups of commodities x. For example the consumer decides to allocate the expenditures yA to commodities xA .xA; ; xA /, yB to commodities xB .xB ; ; xB /, , D 1 NA D 1 NB yZ to commodities xZ .xZ; ; xZ /, where yA yB yZ y. This 1 NZ allocation of expendituresD among broad groups requiresC knowledgeC C of totalD expenditures y and of an aggregate price pA; pB ; ; pZ for each group of commodities. Thus in the ﬁrst stage a consumerQ is viewedQ asQ solving a problem of the form

max u.xA; xB ; ; xZ/ x Q Q Q Q (7.8) s.t. pAxA pB xB pZxZ y Q Q CQ Q C C Q Q D where x .xA; ; xZ/ and p .pA; ; pZ/ denote aggregate commodities and pricesQ D forQ the broad Q groupsQA;D Q;Z. Alternatively, Q utilizing the equivalence between utility maximization and cost minimization (see pages 27 to 28), in the ﬁrst stage a consumer can be viewed as solving a cost minimization problem of the form

min pAxA pB xB pZxZ x Q Q CQ Q C C Q Q Q (7.9) s.t. u.x/ u Q D In the second stage the group expenditure yA; yB ; ; yZ are allocated among the commodities within each group. Here the consumer solves maximization problems of the form 80 7 Aggregation Across Commodities: Non-index Number Approaches

A A A Z Z Z max u .x1 ; ; xNA / max u .x1 ; ; xNZ / xA xZ NA NZ (7.10) X X s.t. pAxA yA s.t. pZxZ yZ i i D i i D i 1 i 1 D D where uA.xA; ; xA /; ; uZ.xZ; ; xZ / are interpreted as “sub-utility func- 1 NA 1 NZ tion” for the commodities within each group A; ;Z. What restrictions on the consumer’s utility function u.x/ imply that two-stage budgeting is realistic, i.e. under what restrictions do the general utility maximization problem (7.7) and the two-stage procedure (7.8) and (7.10) yield the same solutions x? First consider the second stage of two-stage budgeting. Define a weakly separable utility function as follows: Definition 7.1. A utility function u.x/ is defined as “weakly separable” in commodity groups xA .xA; ; xA /; ; xZ .xZ; ; xZ / if and only if u.x/ can 1 NA 1 NZ be written as D D h i u.x/ u uA.xA; ; xA /; ; uZ.xZ; ; xZ / DQ 1 NA 1 NZ over all x for some macro-utility function u u.uA; ; uZ/ and sub-utility function uA uA.xA; ; xA /; ; uZ uZ.xDQZ; ; x Z /. D 1 NA D 1 NZ The second stage of two-stage budgeting is equivalent to the restriction that the consumer’s utility function u.x/ is weakly separable. To be more precise,

Property 7.1. The general utility maximization problem (7.7) and a series of “second stage” maximization problems (7.10) (conditional on group expen- A Z diture y ; ; y ) yield the same solution x if and only if u.x/ is weakly separable in the above manner in Deﬁnition 7.1.

Proof. First, suppose that u.x/ is weakly separable as in Definition 7.1 and that @u=@uA > 0; ;@u=@uZ > 0. Then utility maximization (7.7) requires that eachQ subutility u A; Q ; uZ be maximized conditional on its group expenditure A Z A A A y ; ; y . For example if x solving Definition 7.1 does not solve u .x / P NA A A A A A s.t. i 1 pi xi y , then y could be reallocated among commodities x so as to increaseD uA withoutD decreasing uB ; ; uZ, i.e. so as to increase the total utility level u without violating the budget constraint px y. Thus the second stage of two-stage budgeting is satisfied if u.x/ is weaklyD separable. Second, suppose A A that two-stage budgeting is satisfied. two-stage budgeting implies x x .p; y/ solving (7.7) can be written as D

A A A A x x .p ; y / i 1; ;NA (7.11) i D i D and similarly for subgroups B; ;Z. Without loss of generality xA solves 7.2 Homothetic weak separability and two-stage budgeting 81

A B Z Á max u x ; x ; ; x xA

NA (7.12) X s.t. pAxA yA i i D i 1 D where xB ; ; xZ are fixed at their equilibrium levels solving (7.7), (7.11) im- A A A B Z plies that (given p ; y ) x is independent of p ; ; p and hence indepen- B Z dent of x ; ; x . Thus (7.12) reduces to (7.10), i.e. there exists a subutility function uA .x A/ for commodity group A that is independent of the levels of other commodities xB ; ; xZ. Thus two-stage budgeting implies weak separability u uA.xA/; ; uZ.x Z /. ut Second, consider the first stage of two-stage budgeting. The critical point here is to be able to construct a price pA; ; pZ for each commodity group A; ;Z such that a first stage utility maximizationQ Q or cost minimization problem yields the optimal allocation of expenditure across subgroups A; ;Z. Given weak separability of u.x/, a sufficient condition for the first stage is homotheticity of each subutility function uA.xA/; ; uZ.xZ/. To be more precise,

Property 7.2. If u.x/ is weakly separable and each subutility function uA.xA/; ; uZ.xZ/ is homothetic, then there exists a ﬁrst stage maximization problem that obtains the same distribution of group expenditures yA; ; yZ as in the general case (7.7).

Proof. Weak separability of u.x/ implies second stage maximization (7.10) and equivalently a series of cost minimization problems.

NA X A A A A A min pi xi E .p ; u / xA D i 1 D A A A s.t. u .x / u D : : (7.13)

NZ X Z Z Z Z Z min pi xi E .p ; u / xZ D i 1 D Z Z Z s.t. u .x / u D A A A Z Z Z A Z where u u .x /; ; u u .x / for x ; ; x solving (7.7). Un- der weak separabilityD the consumer’s D general cost minimization problem E.p; u / D minxpx s.t. u.x/ u can be restated as D 82 7 Aggregation Across Commodities: Non-index Number Approaches

A A AÁ Z Z ZÁ E p; u min E p ; u E p ; u D uA; ;uZ C C (7.14) A ZÁ s.t. u u ; ; u u Q D uA.xA/ homothetic implies EA.pA; uA/ .uA/EA.pA/ (see Property 1.3.a on page 6) where without loss of generalityD we can deﬁne .uA/ uA (constant returns to scale), since the indexing of the indifference curves representingD the consumer’s preferences is arbitrary. Therefore under homotheticity (7.14) reduces to

A A AÁ Z Z ZÁ E p; u min u e p u e p D uA; ;uZ C C (7.15) A ZÁ s.t. u u ; ; u u Q D This can be interpreted as a ﬁrst stage cost minimization problem with aggregate prices cA.pA/; ; cZ.pZ/ and leading to the following optimal allocation of expenditures y over subgroups:

A A A A Z Z Z Z y u e .p /; ; y u e .p / D D Weak separability places substantial restrictions on the degree of substitution between commodities in different groups. Since weak separability implies that utility maximizing demands for a group of commodities xA depends only on prices pA and group expenditure yA (7.11), it follows that other prices pB inﬂuence demands xA only through changes in the optimal level of expenditure yA for the group. This implies the following restriction on the Hicksian substitution effects across groups:

A A A A @xB .pB ; yB / @xi .p; u/ AB @xi .p ; y / j B ' .p; y/ A B @pj D @y @y (7.16)

i 1; ;NA j 1; ;NB D D where the function 'AB .p; y/ is independent of the choice of commodities i; j from group A; B (see Deaton and Muellbauer, pp. 128–129 for a sketch of a proof), moreover, this relationship (7.16) is both necessary and sufficient for weak separability of groups A and B, i.e. restrictions (7.16) exhaust the implications of weak separability. The most obvious application of this discussion of two-stage budgeting is in the estimation of, e.g., consumer demand for food. In general the demand for food goods xA .xA; ; xA / will depend on the prices of all food and non-food final goods 1 NA andD on total expenditure y. On the other hand suppose that the consumer’s utility function u.x/ is weakly separable in food commodities xA and all other commodities xB , i.e. u.x/ u uA.xA/; uB .xB /. Then the demand equations for food A A A BDQ A A A A x x p ; p ; y .i 1; ;NA/ can be simplified to x x p ; y i D i D i D i .i 1; ;NA/ (7.11) which can be derived from a subutility maximization prob- lemD 7.3 Implicit separability and two-stage budgeting 83

max uA.xA/ V A.pA; yA/ xA D NA (7.17) X s.t. pAxA yA i i D i 1 D Here V A.pA; yA/ has the properties of an indirect utility function and the corresponding Marshallian demands xA xA.pA; yA/ inherit the properties of utility maximization. D One complication in estimating the above model (7.17) for food demand is that food expenditure yA is not exogenous to the consumer, and ignoring this fact generally leads to biases in estimation. In the absence of any further assumptions (be- yond weak separability) yA generally depends on all prices and total expenditure, i.e. yA yA.pA; pB ; y/. HoweverD if the subutility functions uA.xA/ and uB .xB / are homothetic (implying two-stage budgeting), then the corresponding expenditure functions can be written as uAeA.pA/ and uB eB .pB / and the relation yA yA.pA; pB ; y/ can be D rewritten as h i yA yA eA.pA/; eB .pB /; y (7.18) D Thus we can posit homothetic functional forms EA uAeA.pA/ and EB uB eB .pB /, derive the corresponding indirect utility functionD V A yA=eA.pAD/ and differentiate this indirect utility function to obtain the estimatingD equations xA xA.pA; yA/, and use these functional forms eA.pA/; eB .pB / as the aggre- gateD prices in (7.18). Unfortunately the expenditure equation (7.18) still depends on prices pB as well as pA, but at least the estimating equation (7.18) is more restricted than the general equation yA yA.pA; pB ; y/.2 D

7.3 Implicit separability and two-stage budgeting

An alternative two-stage budgeting procedure can be obtained if certain separability restrictions hold for the consumer’s cost function E.p; u/ rather than for his utility function u.x/. Preferences are deﬁned as “implicitly separable” in broad groups A; ;Z if the cost function can be written in the form h i E.p; u/ E eA.pA; u/; ; eZ.pZ; u/; u (7.19) D Q where pA .pA; ; pA /; ; pZ .pZ; ; pZ /. Note that total util- D 1 NA D 1 NZ ity u (rather than subutilities uA; ; uZ) appear in each of the cost function eA.pA; u/; ; eZ.pZ; u/, so there are no group subutilities in contrast to the case where u.x/is weakly separable.

2 Alternatively we can eliminate yA from the demand equations xA xA.pA; yA/ using the A PNA A A D identity y i 1 pi xi . Á D 84 7 Aggregation Across Commodities: Non-index Number Approaches

The following two-stage budgeting procedure is deﬁned by simple differentiation of the macrofunction E.eA; ; eZ/ and then the group cost function eA.pA; u/; ; eZ.pZ; u/: Q

PNA A A A Z A i 1 pi xi @ log E.e ; ; e ; u/ (ﬁrst stage) s D Q Á y D @ log eA : : (7.20a)

PNZ Z Z A Z Z i 1 pi xi @ log E.e ; ; e ; u/ s D Q Á y D @ log eZ

A A A A A pi xi @ log e .p ; u/ (second stage) si A A i 1; ;NA Á y D @ log pi D : : (7.20b) Z Z Z Z Z pi xi @ log e .p ; u/ si Z Z i 1; ;NZ Á y D @ log pi D

A Z A PNA A A where y y y (i.e. total expenditure) and y i 1 pi xi (expenditure of groupÁ A),C etc. C Thus the budget shares sA; ; sZ ofÁ groupsD A; ;Z are obtained by simple logarithmic differentiation of the macrofunction E.eA ; ; eZ; u/, A Z Q and the share si .i 1; ;NA/, , si .i 1; ;NZ/ of each commodity in group expenditure isD obtained by simple logarithmicD differentiation of the group cost functions eA.pA; u/; ; eZ.pZ; u/. A Proof. Differentiating (7.19) w.r.t. pi and applying Shephard’s Lemma,

@E.p; u/ @E. / @eA. / Q A A A @pi D @e @pi (7.21) A x .p; u/ i 1; ;NA; D i D Thus

NA X yA xApA Á i i i 1 D NA @E. / X @eA. / (7.22) Q pA D @eA @pA i i 1 i D @E. / Q eA. / by Euler’s Theorem D @eA References 85 since eA.pA; u/ is linear homogeneous in pA.3

yA sA Á y @E. /=@eA Q (7.23) D y=eA. / @ log E. / Q D @ log eA

@E. / yA @eA. / @E. / Q Q A which is (7.20a). Substituting @eA eA. / (7.22) into A @eA xi (7.21) D @pi D yields (7.20b).

Note that implicit separability is sufﬁcient for the two-stage budgeting procedure outlined in (7.20). In contrast weak separability of u.x/ was sufﬁcient only for the second stage of the budgeting procedure discussed in the previous section. Also note that implicit separability implies that the ratio of commodities within a group is independent of prices of commodities outside the group, i.e.

h A A i @ xi .p; u/=xj .p; u/ B 0 i; j 1; ;NA k 1; ;NB (7.24) @pk D D D (see (7.21)). In contrast, weak separability of u.x/ implied that the ratio of marginal rates of substitution between commodities within a group is independent of levels of commodities outside the group.

References

1. Deaton A. & Muellbauer, J. (1980). Economics and Consumer Behavior

3 E.p; u/ is linear homogeneous in p only if eA.pA; u/; ; eZ .pZ ; u/ are also linear homogeneous in prices and E.eA; ; eZ ; u/ is linear homogeneous in eA; ; eZ . How- Q ever note that eA.pA; u/ does not equal total expenditure yA on group A (see (7.22)). Thus cA.pA; u/; ; cZ .pZ ; u/ are to be interpreted as group price indexes that depend on utility level u.

Chapter 8 Index Numbers and Flexible Functional Forms

In this lecture we show that particular index number formulas for aggregating over commodities can be rationalized in terms of particular functional forms for production functions or dual cost or profit functions. Approximately correct procedures for aggregating over commodities are presented for cases of Translog and Generalized Leontief functional forms. Since these functional forms provide a second order approximation to any true form, the corresponding index number formulas can often be interpreted as approximately correct. The results obtained here should be contrasted with the previous lecture. There consistent aggregation over commodities was rationalized essentially in terms of assumptions of separability between groups of commodities. Here specific aggregation procedures are rationalized essentially in terms of specific functional forms for production functions or dual cost functions. Since assumptions of Translog or Generalized Leontief functional forms are usually considered less restrictive than assumptions of (homothetic) weak separability, this lecture presents a more useful basis for a theory of approximate aggregation over commodities.

8.1 Laspeyres index numbers and linear functional forms

Until recently most index number computations have used simple base period weighting schemes, and the most common of these are Laspeyres quantity and price indexes. The Laspeyres quantity index can be written as

PN X1 i 1 pi;0xi;1 D (8.1) X D PN 0 i 1 pi;0xi;0 D where p0 .p1;0; ; pN;0/ denotes the prices for the N commodities in the base D period .t 0/, x0 .x1;0; ; xN;0/ denotes the quantities of the N commodities D D in the base period .t 0/, and x1 .x1;1; ; xN;1/ denotes the quantities of the N commodities in anyD other periodDt 1. D

87 88 8 Index Numbers and Flexible Functional Forms

This aggregation procedure (8.1) can be deﬁned as correct if the ratio of aggregates X1, X0 provides an accurate measure of the contributions of inputs 1; ;N to the producer’s output in different time periods. Thus in the case where aggregation is deﬁned over all inputs (1; ;N is to be interpreted as all inputs) and there is a single output, the quantity index X1=X0 in (8.1) is correct if X1=X0 is equal to the ratio of outputs f .x1;1; ; xN;1/=f.x1;0; ; xN;0/ for any time periods t 0; 1. Similarly a Laspeyres price index can be written as D

PN P1 i 1 xi;0pi;1 D (8.2) P D PN 0 i 1 xi;0pi;0 D where the prices p .p1; ; pN / for any period are weighted by the base period D quantities x0 .x1;0; ; xN;0/. Interpreting commodities 1; ;N as inputs in production andD assuming cost minimizing behavior, this aggregation procedure can be defined as correct if the ratio of the aggregates P1, P0 provides an accurate measure of the contribution of inputs 1; ;N to the cost of attaining a given level of output y. In the case where aggregation is defined over all inputs (1; ;N is to be interpreted as all inputs), the price index P1=P0 is correct if P1=P0 is equal to the ratio of minimum costs C.p1;1; ; pN;1; y/=C.p1;0; ; pN;0; y/ for any two time periods and a common output level y. Are the above aggregation procedures (8.1)–(8.2) correct for some cases of production functions? The answer is yes: assuming static competitive profit maximizing or cost minimizing behavior for a firm, an aggregation procedure (8.1) or (8.2) over inputs x .x1; ; xN / can be rationalized in terms of a linear production function with a familyD of parallel straight line isoquants

0 x1 (8.3) and also in terms of a linear production function with ﬁxed coefﬁcients, i.e. right angle isoquants 8.2 Exact indexes for Translog functional forms 89

0 x1 (8.4) The ﬁrst case (8.3) assumes perfect substitution between inputs and the second case (8.4) assumes zero substitution between inputs. More formally, given Laspeyres quantity and price indexes X1=X0 (8.1), a linear production function y f .x/ satisfying either (8.3) or (8.4), and static competitive proﬁt maximizing or costD minimizing behavior, then

X1 f .x1;1; ; xN;1/ X0 D f .x1;0; ; xN;0/ (8.5) P1 c.p1;1; ; pN;1/ for all t 1; 0 P0 D c.p1;0; ; pN;0/ D where c.p/ denotes a unite cost function (the minimum cost of producing one unite of output y given prices p) (see Diewert 1976 pp. 182–183 for a proof of (8.5) ). Under the above assumptions X1=X0 is the ratio of output and P1=P0 is the ratio of unite cost of output for the two periods t 1; 0. In this sense Laspeyres quantity and price indexes are exact for both a linear productionD function satisfying (8.3) and a linear production function satisfying (8.4). In either case linear production functions (with either zero or perfect substitution between inputs are very unrealistic and highly restrictive. Linear production functions can provide only a ﬁrst order approximation to an arbitrary production function.1 This suggests that Laspeyres indexes may often lead to substantial errors in aggregation over commodities, and it is unlikely that aggregate data col- lected in this manner inherits properties of proﬁts maximization or cost minimization from disaggregate data. In the next two sections we obtain more positive results by demonstrating that particular quantity and price indexes are correct for Translog and Generalized Leontief functional forms.

8.2 Exact indexes for Translog functional forms

Economists have frequently advocated the use of Divisia indexes rather than Laspeyres indexes for aggregating over commodities (e.g. Hulten 1973, pp. 1017–1026). A

1 It can easily be shown that Laspeyres indexes also provide a ﬁrst order approximation to an arbitrary (true) index (see Deaton1980, pp.173–174). 90 8 Index Numbers and Flexible Functional Forms

Divisia quantity index for (e.g.) inputs can be deﬁned in continues time by the line Z Â N Ã Xt X @xi .t/=@t wi .t/xi .t/ integral exp si .t/ where si .t/ PN : X0 D i 1 xi .t/ Á j 1 wj .t/xj .t/ The following Tornqvist¨ indexD is the most commonly used discrete approximationD to the Divisia quantity index for inputs:

Â Ã N Â Ã X1 X xi;1 log si log X0 D xi;0 i 1 D ! (8.6) 1 wi;1xi;1 wi;0xi;0 where si i 1; ;N: Á 2 PN C PN D j 1 wj;1xj;1 j 1 wj;0xj;0 D D The Tornqvist¨ quantity index (8.6) is exact for a Translog production function with constant returns to scale. In other words, assuming static competitive cost minimizing behavior, ÂX Ã Âf .x /Ã log t log t (8.7) Xs D f .xs/ for all periods s, t when f .x/ is Translog constant returns to scale and the quantity index is calculated as in (8.6). Such an index, which is exact for a constant returns (or variable returns) to scale ﬂexible form for f .x/, is termed superlative.

Proof. First we establish an algebraic result for a quadratic function

X 1 X X g.´/ a0 ai ´i aij ´i ´j .aij aj i for all i; j / D C i C 2 i j D or, in matrix notation,

T 1 T g.z/ a0 a z z A z .A symmetric/ (8.8) D C C 2 Then

T 1 T 1 T g.z1/ g.z0/ a .z1 z0/ z A z1 z A z0 D C 2 1 2 0 T 1 T 1 T a .z1 z0/ z A.z1 z0/ z A.z1 z0/ D C 2 1 C 2 0 (8.9) since A is symmetric

1 T .a Az1 a Az0/ .z1 z0/ D 2 C C C which implies

Â ÃT 1 @g.z1/ @g.z0/ g.z1/ g.z0/ .z1 z0/ (8.10) D 2 @z C @z 8.2 Exact indexes for Translog functional forms 91

Moreover (8.10) implies (8.8), i.e. (8.8) is correct if and only if (8.10) is PN correct. Since a Translog production function log f a0 i 1 ai log xi PN PN D C@ log f .xD/ @f.x/=@xCi aij log xi log xj is quadratic in logs, (8.10) and i 1 j 1 @ log xi D f .x/=xi .i D 1; D;N/ imply D N Â Ã Â Ã 1 X @f.x1/ xi;1 @f.x0/ xi;0 xi;1 log f .x1/ log f .x0/ log : D 2 @xi;1 f .x1/ C @xi;0 f .x0/ xi;0 i 1 D (8.11) Static competitive cost minimization implies @f.x/ wi .i 1; ;N/, @xi D @C.w;y/=@y D PN @f.x/ and constant returns to scale implies f .x/ xi (Euler’s theorem). i 1 @xi Substituting these to (8.11), D D

Â Ã N ! Â Ã f .x1/ 1 X wi;1xi;1 wi;0xi;0 xi;1 log N N log (8.12) f .x0/ D 2 P C P xi;0 i 1 j 1 wj;1xj;1 j 1 wj;0xj;0 D D D where right hand side is the Tornqvist¨ input quantity index (8.6).

Moreover, the Tornqvist¨ quantity index (8.6) is exact only for a Translog constant returns to scale production function (this follows from the equivalence between (8.10) and a quadratic function g.z/ (8.8)). The assumption of a constant returns to scale production function appears to be crucial to the interpretation of the particular Tornqvist¨ quantity index (8.6) as exact. This assumption is not crucial only in the case of pair-wise comparisons of aggregate inputs, i.e. if the aggregate index (8.6) is to be calculated only for two time periods t 0; 1 (see Diewert 1976, Op. cit.). DNevertheless, a quantity index closely related to (8.6) can be interpreted as exact for a general Translog production function. Deﬁne the following quantity index:

Â Ã N Â Ã Â Ã X1 1 X wi;1xi;1 wi;0xi;0 xi;1 log log (8.13) X0 D 2 py;1f .x1/ C py;0f .x0/ xi;0 i 1 D where py;t price of the (single) output in period t. This deviates from the Á Tornqvist¨ quantity index (8.6) only in that wi;t xi;t is divided by total revenue PN py;t f .xt / rather than by total cost i 1 wi;t xi;t . In the case of constant returns to PN @f.x/ D PN scale f .x/ xi (Euler’s theorem) and in turn py f .x/ wi xi . i 1 @xi i 1 Thus (8.11)D reducesD to the Tornqvist¨ index (8.6) in the case of constantD returnsD to scale and competitive proﬁt maximization. Unlike the index (8.6), the above quantity index (8.13) is exact for a general (variable returns to scale) Translog production function. This result requires the assumption of proﬁt maximization rather than simple cost minimization, in contrast to (8.6).

Proof. Substituting the ﬁrst order conditions for static competitive proﬁt maximization @f.xt / wi;t .i 1; ;N/ into (8.11), @xi D py;t D 92 8 Index Numbers and Flexible Functional Forms

Â Ã N Â Ã Â Ã f .x1/ 1 X wi;1xi;1 wi;0xi;0 xi;1 log log (8.14) f .x0/ D 2 py;1f .x1/ C py;0f .x0/ xi;0 i 1 D i.e. (8.13) is exact for the general Translog f .x/.

This index (8.13) can be interpreted as an alternative Tornqvist¨ index approximating the continuous Divisia quantity index. Given an input quantity index X1=X0 such as (8.6) or (8.13), we can easily construct a corresponding implicit input price index using the following factor reversal equation: Â ÃÂ Ã PN W1 X1 i 1 wi;1xi;1 D (8.15) W X D PN 0 0 i 1 wi;0xi;0 D i.e. the product of the input price and quantity indexes is equal to the ratio of total expenditure on the N disaggregate inputs for the corresponding time periods t D 0; 1. Assuming X1=X0 f .x1/=f.x0/ (8.7), (8.15) implies DB Â Ã PN W1 i 1 wi;1xi;1=f.x1/ D (8.16) W D PN 0 i 1 wi;0xi;0=f.x0/ D i.e. .W1=W0/ can be interpreted as the index of average costs of production. Obvi- ously if a quantity index is superlative then the corresponding implicit price index is superlative in an analogousB manner. Alternatively an input price index can be calculated directly rather than by using (8.15). Now assume that the production function is constant returns to scale and that PN the cost function C.w; y/ y c.w/ is Translog: log c.w/ a0 i 1 ai log wi PNCPN D D C D C i 1 j 1 aij log wi log wj . Deﬁne the following Tornqvist¨ price index for inputs:D D Â Ã N Â Ã W1 X wi;1 log si log (8.17) W0 D wi;0 i 1 D 2 where s .s1; ; sN / are calculated as in (8.6). Proceeding as in the proof of (8.7), weD obtain the result Â Ã Â Ã W1 c.w1;1; ; wN;1/ log log (8.18) W0 D c.w1;0; ; wN;0/ which is analogous to (8.16). Thus the Tornqvist¨ price index (8.17) is exact for a Translog unit cost function. In the absence of constant returns to scale in production, the assumption of a Translog cost function C.w; y/ implies

2 Since ﬂexible functional forms are not self-dual (e.g. a Translog production function does not imply a Translog cost function, or vice-versa), the price index (8.17) for a Translog unit cost function is not equivalent to the implicit price index (8.16) for a Translog constant returns to scale production function. 8.2 Exact indexes for Translog functional forms 93

Â Ã N Â Ã W1 X wi;1 log si log W0 D wi;0 i 1 (8.19) D Â Ã C.w1;1; ; wN;1; y1/ log D C.w1;0; ; wN;0; y0/ i.e. the Tornqvist¨ index (8.17) is equal to the ratio of total costs in different time periods t 0; 1: Thus, in the absence of constant returns to scale, the direct Tornqvist¨ index (8.17)D cannot strictly be interpreted as a price index for inputs since it depends upon a measure y1; y0 of input quantities x1; x0 as well as upon input prices w1; w0. In contrast, consider the index .W1=W0/ calculated implicitly from (8.15) using the quantity index (8.13), science this index satisﬁes (8.16) for a general Translog production function and proﬁt maximizing behavior, it can be interpreted as the ratio of average costs of production

Â Ã Â Ã W1 C.w1;1; ; wN;1; y1/=y1 log logC (8.20) W0 D C.w1;0; ; wN;0; y0/=y0 even in the absence of constant returns to scale. The derivations of exact index number equations are complicated somewhat in the case of multiple outputs y .y1; ; yM /. Generalizations of the Translog production function doB not appearD to provide an entirely satisfactory basis for index number formulas. For example, assume a Translog transformation function

T T T X X X log y1 a0 ai log í aij log í log ´j (8.21) D C C i 1 i 1 j 1 D D D where z .y2; ; yM ; x1; ; xN / .T M 1 N/ and static competitive profit maximizationÁ D C M N X X max pi yi wi xi y;x (8.22) i 1 i 1 D D s.t. y1 y1.z/ D Then the quadratic identity (8.11) implies

Â Ã N Â Ã M Â Ã yi;1 X x xi;1 X y yi;1 log si log si log yi;0 D xi;0 yi;0 i 1 i 2 DÂ D Ã x 1 wi;1xi;1 wi;0xi;0 (8.23) where si Á 2 pi;1yi;1 C pi;0yi;0 Â Ã y 1 pi;1yi;1 pi;0yi;0 si Á 2 pi;1yi;1 C pi;0yi;0 94 8 Index Numbers and Flexible Functional Forms

(8.23) defines quantity indexes that are exact for the Translog transformation func- PM y tion (8.21). However the output quantity index i 2 si log .yi;1=yi;0/ incorporates D y the level of the first output y0 only indirectly via the definitions of the weights si for output 2; ;M . A more satisfactory approach to the derivation of index numbers in the case of multiple outputs may be in terms of the cost function C.w; y/. Assume a Translog joint cost function

N M X X log C a0 ai log wi bi log yi D C C i 1 i 1 D D N N M M X X X X aij log wi log wj bij log yi log yj (8.24) C C i 1 i 1 i 1 i 1 D D D D T T X X cij log wi log yj C i 1 i 1 D D and static competitive proﬁt maximizing behavior. Then the quadratic identity (8.11), Shephard’s Lemma and @C.w; y/=@yi pi .i 1; ;M/ imply D D Â Ã N Â Ã M Â Ã C1 X x wi;1 X y yi;1 log si log si log C0 D wi;0 C yi;0 i 1 i 1 D D ! 1 w x w x where sx i;1 i;1 i;0 i;0 (8.25) i Á 2 PN C PN i 1 wi;1xi;1 i 1 wi;0xi;0 D D ! 1 p y p y sy i;1 i;1 i;0 i;0 : i Á 2 PN C PN i 1 wi;1xi;1 i 1 wi;0xi;0 D D PN x The ﬁrst sum i 1 si log .wi;1=wi;0/ can be interpreted as a Tornqvist¨ price D PM y index log .W1=W0/ for all inputs, and the second sum i 1 si log .yi;1=yi;0/ can D be interpreted as a Tornqvist¨ quantity index log .Y1=Y0/ for all outputs. In order to see this, rewrite equation (8.25) as Â Ã Â Ã C1 W1 Y1

C0 D W0 Y0 Â Ã Â Ã W1 XN wi;1 where log sx log (8.26) W i 1 i w 0 D D i;0 Â Ã Â Ã Y1 XN y yi;1 log s log Y i 1 i y 0 D D i;0 i.e. the index of total costs is always equal to the product of the input price index and the output quantity index. This conﬁrms that Y1=Y0 can be interpreted as an output 8.2 Exact indexes for Translog functional forms 95 quantity index and W1=W0 can be interpreted as an index of the contributions of inputs to the average cost of aggregate output. The corresponding implicit Tornqvist¨ input quantity and output price indexes can then be calculated from the equations

Â Ã Â Ã PN W1 X1 i 1 wi;1xi;1 D W X D PN 0 0 i 1 wi;0xi;0 D (8.27) Â ÃÂ Ã PM P1 Y1 i 1 pi;1yi;1 D P Y D PM 0 0 i 1 pi;0yi;0 D B 3 These implicit indexes .X1=X0/ and .P1=P0/ are also superlative. Finally, consider the problem of deriving index numbers for aggregation of commodities in the case of consumer behavior.A Assuming that the utility function u.x/ is homothetic (or equivalently constant returns to scale) and that the unit cost function e.p/ min px D x s.t. u.x/ 1 is Translog, then the Tornqvist¨ price index D B B Â Ã N ! Â Ã P1 1 X pi;1xi;1 pi;0xi;0 pi;1 log N N log (8.28) P0 D 2 P C P pi;0 i 1 j 1 pj;1xj;1 j 1 pj;0xj;0 D D D and the corresponding implicit quantity index

Â Ã PN ! Â Ã X1 i 1 pi;1xi;1 P1 log log D log (8.29) X D PN P 0 j 1 pj;0xj;0 0 D are exact (see the discussion of (8.17)–(8.18)). Alternatively suppose that the consumer’s utility function u.x/ is Translog as well as homothetic: B N N N X X X log u a0 ai log xi aij log xi log xj (8.30) D C C i 1 i 1 j 1 D D D u.x/ constant returns to scale implies that the indirect utility function V .p; y/ is constant returns to scale in expenditure y, so that (using Euler’s theorem) V .p; y/ @V .p;y/ D @y y. Applying the quadratic identity (8.10) to (8.30), and using the ﬁrst order @u.x/ @V .p;y/ conditions pi .i 1; ;N/ and in turn @V .p; y/=@y u=y, @xi @y we obtain the resultD D D

3 In the case of constant returns to scale in production, W1=W0 can also be interpreted as a price index of the contributions of inputs to the marginal cost of aggregate output. 96 8 Index Numbers and Flexible Functional Forms

Â Ã N Â Ã u.x1/ X xi;1 log si log u.x0/ D xi;0 i 1 D ! (8.31) 1 pi;1xi;1 pi;0xi;0 where si Á 2 PN C PN j 1 pj;1xj;1 j 1 pj;0xj;0 D D

The right hand side of (8.31) deﬁnes a Tornqvist¨ quantity index log.X1=X0/ which is exact for a homothetic Translog utility function u.x/, and the corresponding im- plicity price index can be calculated using (8.29).

8.3 Exact indexes for Generalized Leontief functional forms

Results in the previous section demonstrated that many index numbers closely related to popular Tornqvist¨ indexes can be rationalized in terms of underlying Translog functional forms. This section brieﬂy illustrates that different index number formulas are implied by Generalized Leontief functional forms. Nevertheless, since both classes of functional forms provide a second order approximation to a true form, the various index number formulas should lead to similar results at least for small changes in quantities and prices. First, assume a constant returns to scale generalized Leontief production function

N N X X f .x/ aij pxi pxj (8.32) D i 1 j 1 D D and static competitive proﬁt maximization. This implies

PN 1=2 f .x1/ i 1 si;0.xi;1=xi;0/ D f .x / D PN 1=2 0 j 1 sj;1.xj;0=xj;1/ D (8.33) wi;t xi;t where si;t Á PN k 1 wk;t xk;t D i.e. the quantity index number X1=X0 corresponding to the right hand side of (8.33) is exact for the production function (8.32).

Proof. Competitive proﬁt maximization and constant returns to scale imply p @f.x/ @xi D PN wi and pf .x/ i 1 wi xi , so that D D wi;t vi;t Á PN j 1 wj;t xj;t D (8.34) @f.x /=@x t i;t .i 1; ;N/ D f .xt / D 8.4 Two-stage aggregation with superlative index numbers 97 for all time periods t. Substituting the derivatives of (8.32) into (8.34),

1=2 PN .xi;t / j 1 aij pxj;t vi;t D (8.35) D f .xt / multiplying vi;0 by pxi;1pxi;0 and summing over i 1; ;N , D N PN PN X i 1 j 1 pxi;1 aij pxi;0 pxi;1 vi;0 pxi;0 D D (8.36) D f .x0/ i 1 D and similarly,

N PN PN X i 1 j 1 pxi;0 aij pxj;1 pxi;0 vi;1 pxi;1 D D (8.37) D f .x1/ i 1 D p Dividing (8.36) by (8.37) (noting ai;j aj;i and pxi;1pxi;0 xi;1=xi;0 xi;0), D D P 1=2 .xi;1=xi;0/ vi;0 xi;0 f .x1/ i : (8.38) P 1=2 D f .x / xj;0=xj;1 vj;1 xj;1 0 j Alternatively , assume constant returns to scale in production and a Generalized Leontief unit cost function c.w/ C.w; y/=y: D N X c.w/ aij pwi pwj (8.39) D i 1 D These assumptions and competitive proﬁt maximization imply

c.w / PN s .w =w /1=2 1 i 1 i;0 i;1 i;0 (8.40) D 1=2 c.w0/ D PN j 1 sj;1 wj;0=wj;1 D where s0, s1 are deﬁned as in (8.33) (the proof is analogous to the above proof of (8.33)). Thus the above price index for inputs is exact for a Generalized Leontief unit cost function (8.39).

8.4 Two-stage aggregation with superlative index numbers

All of the above index number formulas and their relations to Translog and General- ized Leontief functional forms were essentially calculated as a one stage aggregation procedure. For example all inputs 1; ;N were aggregated directly into a single input quantity index and a single input price index rather than into several quantity 98 8 Index Numbers and Flexible Functional Forms and price subindexes. On the other hand, commodities typically are aggregated into various subindexes for use in econometric models. This leads to the following important question: are two stage aggregation produces (using index numbers formulas) exact or approximately exact? For example, suppose that the Tornqvist¨ quantity index number formula (8.6) and the corresponding implicit price index formula are applied separately to two subsets of inputs (NA and NB inputs, respectively, where NA NB N , the total number of inputs), resulting in two quantity indexes QA;QBC D

Â A Ã A X1 Q log A Á X0 NA A A A A ! A ! 1 X wi;1xi;1 wi;0xi;0 xi;1 log D 2 PNA A A C PNA A A xA i 1 j 1 wj;1xj;1 j 1 wj;0xj;0 i;0 D D D (8.41) Â B Ã B X1 Q log B Á X0 NB B B B B ! B ! 1 X wi;1xi;1 wi;0xi;0 xi;1 log D 2 PNB B B C PNB B B xB i 1 j 1 wj;1xj;1 j 1 wj;0xj;0 i;0 D D D and corresponding implicit prices indexes pA, pB . Then the same quantity index number formula (8.6) is applied to the subindexesQ Q QA, QB , pA, pB , resulting in a quantity index Q: Q Q y ÂX Ã Q log 1 y Á X0 1 Â pAQA pAQA Ã ÂQA Ã Q1 1 Q0 0 log 1 (8.42) D 2 pAQA pB QB C pAQA pB QB QA Q1 1 CQ1 1 Q0 0 CQ0 0 0 1 Â pB QB pB QB Ã ÂQB Ã 2 Q1 1 Q0 0 log 1 C 2 pAQA pB QB C pAQA pB QB QB Q1 1 CQ1 1 Q0 0 CQ0 0 0 Are the results of one stage and two stage aggregation identical? For particular, is the quantity index Q calculated in (8.41)–(8.42) exact for a constant returns y to scale Translog production function y f .x1; : : : ; xN /, i.e. does Q equal D y log.f .x1/=f.x0//? In general the two stage aggregation procedure (8.41)–(8.42) is not exact, i.e. Q y ¤ log.f .x1/=f.x0//. This result is not surprising, since results in the previous chapter indicated that two stage budgeting is correct only under strong restrictions on the structure of production or utility functions (e.g. homothetic weak separability). On the other hand the two stage aggregation procedure (8.41)–(8.42) is approximately exact, i.e. Q approximates log .f .x /=f.x //. Moreover, two stage aggre- y 1 0 gation procedures based on known superlative index number formulas are approximately exact. This can be explained very brieﬂy as follows (see Diewert 1978). 8.4 Two-stage aggregation with superlative index numbers 99

The Vartia I price index P V and quantity index QV , where

N Â Ã V X pi;1 log P si log Á pi;0 i 1 D N Â Ã V X xi;1 log Q si log Á xi;0 i 1 D 2 , PN PN 3 pi;1xi;1 pi;0xi;0 j 1 pj;1xj;1 j 1 pj;0xj;0 where si 4 D . D Á5 ; Á log .pi;1xi;1=pi;0xi;0/ PN PN log j 1 pj;1xj;1 j 1 pj;0xj;0 D D (8.43) are know to be consistent in aggregation, i.e. one stage and two stage aggregation procedures based on (8.43) lead to identical index numbers. Unfortunately these Vartia indexes are exact for a constant returns to scale Cobb-Douglas production function y f .x1; : : : ; xN / rather then for a second order ﬂexible functional form.4 D Nevertheless it can be shown that the vartia quantity index QV differentially T approximates the Tornqvist¨ quantity. index Q .X1=X0/ (8.6) to the second order at any point where the prices and quantitiesÁ are the same for the two time periods comparison .p1 p0; x1 x0/, i.e. D D QV .z/ QT .z/ D @QV .z/ @QT .z/

@í D @í (8.44) @2QV .z/ @2QT .z/ for all i; j @í @´j D @í @´j where z .p1; p0; x1; x2/ and p1 p0, x1 x0. Identical results hold for Á V D D T the Vartia price index P and the Tornqvist¨ price index p .W1=W0/ (8.18). These results do not require any assumptions about optimizingÁ behavior log agents. Moreover, these results can be extended to other superlative indexes, e.g. indexes corresponding to Generalized Leontief production functions or cost functions. Therefore two stage aggregation procedures using superlative index number formulas are approximately correct (exact) for relatively small changes in prices and quantities between the comparison time periods t 0; 1. These changes in prices and quantities are usually smaller when indexes areD constructed by chaining observations in successive period rather than using a constant base period. On this basis it is recommended that a Tornqvist¨ quantity macroindex or subindex (8.6), e.g., be constructed as

4 Laspeyres and paasche index numbers are also consistent in aggregation, but the corresponding production are much more restrictive than the Cobb-Douglas (see Section 8.1). 100 8 Index Numbers and Flexible Functional Forms

A ! XNA xi;t QA sA log t 1; ;T (8.45) t i 1 i;t A D D xi;t 1 D A A A A ! A 1 wi;t xi;t wi;t 1xi;t 1 where si;t D 2 PNA A A C PNA A A j 1 wj;t xj;t j 1 wj;t 1xj;t 1 D D (chaining observations in successive periods) rather than as A ! XNA xi;t QA sA log t 1; ;T (8.46) t i 1 i;t A D D xi;0 D A A A A ! A 1 wi;t xi;t wi;0xi;0 where si;t D 2 PNA A A C PNA A A j 1 wj;t xj;t j 1 wj;0xj;0 D D (using a constant base period t 0). D

8.5 Conclusion

The above results indicate that, at least in the case of simple static maximization models, various index number procedures can be used to obtain approximately consistent aggregation of commodities provided that the variation in prices and quantities between comparison periods is small. For time series data this condition can usually be satisfied by chaining observations in successive periods. However in the case of cross-section data there may be substantial variation in quantities or prices between successive periods, and here different superlative index number formulas may lead to significantly different results. In this case it may be useful to compare the variation in the N quantity ratios xi;1=xi;0 to the variation in the N price ratios pi;1=pi;0. Usually there is much less variation in the price ratios than in the quantity A PNA A A A ratios. Then a directly defined price index Pt i 1 si;t log.pi;t =pi;t 1/ is less sensitive to the variation in data than is aD directlyD defined quantity index A PNA A A A A Qt i 1 si;t log.xi;t =xi;t 1/ (since both equations use the same shares st ). ThisD suggestsD that the best strategy in this case is to calculate the price indexes directly and to employ the corresponding implicit quantity indexes (see Allen 1981, pp. 430-435).

References

1. Allen, R. C. and Diewert, W. E. (1981). Direct versus implicit superlative index number for- mulae. The Review of Economics and Statistics, 63(3):430–435. 2. Deaton, A. and Muellbauer, J. (1980). Economics and Consumer Behavior. Cambridge Uni- versity Press. 3. Diewert, W. E. (1976). The Economic Theory of Index Number: A Survey, volume 1. Elsevier Science Publishers. References 101

4. Diewert, W. E. (1978). Superlative index numbers and consistency in aggregation. Economet- rica, 46(4):883–900. 5. Fuss, M., McFadden, D., and Mundlak, Y. (1978). A Survey of Functional Forms in the Eco- nomic Analysis of Production, volume 1. McMaster University Archive for the History of Economic Thought. 6. Hulten, C. R. (1973). Divisia index numbers. Econometrica, 41(6):1017–1025. 102 8 Index Numbers and Flexible Functional Forms 8.6 Laspeyres and Paasche cost of living indexes

min px ) x E.p; u/ s.t. u.x/ u ! D u.x/ homothetic E.p; u/ u E.p; 1/. D „ ƒ‚ … e.p/ True cost of living (COL) index conditional on u: P E.p ; u / 1 1 : P2 D E.p0; u/ Homotheticity COL index is independent of any u : ) P1 u e.p1/ e.p1/ : P0 D u e.p0/ D e.p0/

Laspeyres COL index is > true COL (at u0): Â ÃL P1 x0p1 > E.p1; u / 0 : P0 Á x0p0 E.p0; u0/ „ ƒ‚ … D „ ƒ‚ … Laspeyres COL index true COL index

Paasche COL index is < true COL(at u1): Â ÃP P1 x1p1 E.p1; u/ D 1 : P0 Á x1p0 > E.p0; u1/ „ ƒ‚ … „ ƒ‚ … Paasche COL index true COL index So (assuming homotheticity)

ÂP ÃP ÂP ÃL 1 < true COL < 1 (8.47) P0 P0 Note: (8.47) marks no assumptions about form of preferences (such as Translog), except for homotheticity. If (8.47) does no place close enough bounds on true COL, then we should specify (e.g.) a Tornqvist¨ consumer price index (assuming a Translog CRTS cost function E.p; u/). 8.7 Fisher indexes (for inputs) 103 8.7 Fisher indexes (for inputs)

CRTS C.w; y/ yC.w; 1/ where C.w; 1/ c.w/: unit cost function. Assume! the followingD quadratic c.w/ function:Á

hX X i1=2 c.w/ aij wi wj (8.48) D i j Deﬁne the following price index:

1 " # 2 ÂW ÃF ÂW ÃLÂW ÃP 1 1 1 (8.49) W0 D W0 W0

This is called a Fisher price index (it is a geometric mean of a Laspeyres and Paasche index). We can prove the following result:

Theorem 8.1. Assume CRTS and the quadratic cost function (8.48) (all inputs are at static cost minimizing equilibrium). Then

ÂW ÃF c.w / AC 1 1 1 : W0 D c.w0/ Á AC0 Alternatively deﬁne a Fisher input quantity index as

Â ÃF 1 X1 L P 2 .X1=X0/ .X1=X0/ (8.50) X0 Á „ ƒ‚ … „ ƒ‚ … w0x1=w0x0 w1x1=w1x0 and assume a quadratic CRTS production function

hX X i1=2 y bij xi xj (8.51) D i j Then, assuming all inputs are at static cost minimizing equilibrium, we can show

ÂX ÃF y 1 1 : X0 D y0 Note: A quadratic cost function (8.48) is essentially equivalent to a quadratic production function (8.51). (In contrast, Translog and G.L. forms are not “self-dual”). 104 8 Index Numbers and Flexible Functional Forms

Proof. Assuming CRTS, deﬁne the unit cost function c.w/ C.w; y/=y. Assume D hX X i1=2 c.w/ aij wi wj (8.52) D i j So

1=2 @c.w/ 1hX X i X aij wi wj 2 aij wj .aij aj i / @wi D 2 i j j D P (8.53) aij wj j by (8.52) D c.w/

So P @c.w/=@w aij wj i j (8.54) c.w/ D c.w/2 Cost minimization for all inputs implies Shephard’s Lemma

@c.w/ y xi : (8.55) @wi D P Dividing (8.55) by yc.w/ wi xi , Á i

xi @c.w/=@wi P : (8.56) i wi xi D c.w/ The Fisher input price index is

Â ÃF 1=2 W1 h L P i .W1=W0/ .W1=W0/ W0 D 1=2 Œ.w1x0=w0x0/ .w1x1=w0x1/ D 1=2 Œ.w1x0=w0x0/=.w0x1=w1x1/ D ÄÂ ÃÂ Ã1=2 @c.w0/=@w @c.w1/=@w w1 w0 by (8.56) D c.w0/ c.w1/ Ä 1=2 X X aij wj;0 X X aij wj;1 wi;1 wi;0 by (8.54) i j 2 i j 2 D c.w0/ c.w1/ 1=2 h 2. 2i 1=c.w0/ 1=c.w1/ D c.w1/=c.w0/ D Chapter 9 Measuring Technical Change

In previous chapters we assumed the existence of an unchanged production function y f .x/ as transformation function g.y; x/ 0, where output levels y are deter- minedD by input levels x. In contrast here we allowD for shifts in the technology of the firm or industry that lead to changes in output levels y that cannot be accounted for solely by changes in input levels x. In principle studies of productivity as in this chapter control for changes in the quality of measured inputs. Consequently application of these methods to a particular sector of the economy (e.g. agriculture) will not (in principle) quantify the effect of improved quality of measured inputs on the sector’s output. Improvement in quality of inputs should be analyzed as technical changes in the industries producing those factors. In order to study the effect of improvements in quality of measured inputs as well as changes in levels of unmeasured inputs on agricultural output, it has been suggested that agriculture and manufacturing industries supplying measured inputs to agriculture should be combined into1 a single sector where these inputs are treated as intermediate goods (Kisler and Peterson 1981). This chapter introduces several approaches to the measurement of technical change. Section 9.1 considers econometric production models incorporating a time tread as a proxy for technical change. Section 9.2 summarizes index number procedures for calculating changes in productivity without recourse to econometrics or specification of specific functional forms for the production function, cost function or profit function (in this sense the procedures are non-parametric). Section 9.3 summarizes parametric index number procedures for calculating changes in productivity without recourse to econometrics. These sections 9.2 and 9.3 both assume that there is no variation in utilization rates of capital over time; i.e. the industry or firm is assumed to be in long-run equilibrium. Section 9.4 demonstrates that (under re-

1 In principle it is possible to incorporate technical change into the production function by expand- ing the list of inputs x so as to include all variables that contribute to technical change, e.g. research results of agricultural experiment stations and agricultural extension activities. However these additional inputs are not easily quantiﬁed. This is the primary rationalization for treating technical changes as a residual of output changes that is unexplained by changes in levels of observable inputs.

105 106 9 Measuring Technical Change strictive assumptions) these non-econometric procedures can be modiﬁed to allow for variable utilization rates in capital, and that econometric procedures can (under certain restrictive assumptions) allow for variable utilization rates in capital. Section 9.5 conclude the chapter.

9.1 Dual cost and proﬁt functions with technical change

In both primal and dual econometric models of the industry or firm, technical changes usually is proxied simply by a time trend variable t 1; 2; 3; ;T where T is the number of time periods. By allowing for interactionsD between this time trend and other variables in the model, this specification is designed to proxy more than a regular secular time trend for technical changes over historical time. Of course the time trend may also proxy secular trends in any relevant variables that have been excluded from the model. In addition it is assumed that technical change is exogenous to the industry or firm. A multiple output Translog cost function C C.w; y/ can be generalized to incorporate a time trend t 1; 2; 3; ; as follows:D D N M X X log C ˛0 ˛i log wi ˇi log yi '1t D C C C i 1 i 1 D D N N 1 X X ˛ij log wi log wj C 2 i 1 j 1 D D M M 1 X X ˇij log yi log yj (9.1) C 2 i 1 j 1 D D N M 1 X X ij log wi log yj C 2 i 1 j 1 D D N M X X 2 i t log wi ıi t log yi '2t C C C i 1 i 1 D D Shephard’s lemma implies the following factor share equations:

N M wi xi X X si ˛i ˛ij log wj ij log yj i t i 1; ;N (9.2) Á C D C C C D j 1 j 1 D D

Assuming competitive profit maximization, the first order conditions pj @C.w; y; t/=@yj .j 1; ;M/ and (9.1) imply D D 9.1 Dual cost and profit functions with technical change 107

pj yj @ log C

C D @ log yj N M (9.3) X X ˇj ij log wi ˇj i log yi ıj t j 1; ;M D C C C D i 1 i 1 D D Differentiating (9.1) with respect to t,

@ log C @C.w; y; t/=C.w; y; t/ @t Á @t N M (9.4) X X '1 i log wi ıj log yj 2'2t D C C C i 1 j 1 D D Assuming technical progress (not regress), @ log C=@t < 0. Note that equation (9.4) cannot be estimated directly because the percentage reduction in cost due to technical change (@ log C=@t) is unobserved, and also note that the coefficients '1, '2 of (9.4) cannot be inferred from estimates of factor demand and output supply equation (9.2)–(9.3). In order to obtain estimates of all coefficients of (9.4), it is necessary to estimate directly equation (9.1) defining the Translog cost function. Unfortunately the data will often permit estimation of equation (9.2)–(9.3), but not direct estimation of the cost function equation (9.1). Nevertheless we can easily derive a measure of technical change @C.w; y; t/=@t directly from estimates of the factor share equation (9.2). Since C.w; y; t/ PN Á i 1 xi .w; y; t/wi , D N @C.w; y; t/ X @xi .w; y; t/ wi (9.5) @t Á @t i 1 D Differentiating si .w; y; t/ wi xi .w; y; t/=C.w; y; t/ with respect to t and substi- @s Á @x 2 @x tuting in (9.5) yields @t A @t where matrix A has full rank. Then @t can be @x 1 D @s calculated as @t A using the estimates of @t from the share equations (9.2), @C D and in turn @t can be calculated using (9.5). A single output generalized Leontief cost function with technical change can be specified as

X X 2 X X C y aij pwi pwj y ai wi ty bi wi (9.6) D C C i j i i

2 2 w1 w1 w1 w2 w1 wN 3 C 1 C C C C C w1w2 w2 w2 w2 wN 6 C C C 1 C C C 7 6 7 A 6 : : : : 7 D 4 : : : : 5 w1 wN w2 wN wN .1 wN / C C C C C C which generally has full rank. Deleting the equation for share sN from the estimation of (9.2), PN 1 N 1 i 1 i , Á D 108 9 Measuring Technical Change

PN Here @C.w; y; t/=@t y i 1 bi wi , and all coefﬁcients of this equation can be recovered directly fromD the estimatedD factor demand equations (in contrast to the Translog). The shift in the underlying production function or transformation function due to technical change can easily be calculated from the reduction in cost @C.w; y; t/=@y. Assuming competitive cost minimization and a production function y f .x; t/ for D a single output y, the increase in output @f.x; t/=@t due to technical change can be calculated as follows: @f.x ; t/ @C.w; y; t/ @C.w; y; t/ = (9.7) @t D @t @y or, assuming competitive proﬁt maximization,

@f.x ; t/ @C.w; y; t/ =p (9.8) @t D @t

In the case of constant returns to scale in production C.w; y; t/ Cy .w; y; t/y (Euler’s theorem), and in turn (9.7) implies D

@f.x ; t/ @C.w; y; t/ =y =C (9.9) @t D @t

Thus under constant returns to scale the rate of growth of output (at equilibrium x) is equal to the rate of reduction in cost due to technical change. Similarly in the case of a multiple output transformation function y1 g.y2; ; yM ; x; t/ y1.y; x; t/ and assuming competitive profit maximization,D the shift in transformationÁ functionQ due to technical change @g.y; x; t/=@t can be calculated as Q @g.y; x; t/ @C.w; y; t/ Q =p1 (9.10) @t D @t These relations (9.7)-(9.10) are proved in the next section 9.2. Biases as well as magnitudes of technical change can be calculated from cost functions. For this purpose it is convenient to define Hicks-neutral technical change as technical change that does not alter the firm or industry’s input expansion path. Then the change in cost shares @si .w; y; t/=@t .i 1; ;N/ provides a measure of bias if the production function is homothetic, i.e.D theexpansion in input space is linear from the origin for a given state of technology. However the change in shares @s.w; y; t/=@t does not provide an accurate measure of biases in technical change when the production function is non-homothetic. For example suppose that Hicks-neutral technical change occurs, i.e. technical change does not alter the input expansion path. This technical change simply leads to a re-numbering of the output levels for given isoquants (@f.x; t/=@t > 0 and output y constant implies the firm moves to a lower isoquant). If the expansion path is not linear and the firm’s output level must be held constant as this reindexing occurs, then the firm moves to a different isoquant with a different cost-minimizing 9.1 Dual cost and profit functions with technical change 109 ratio of inputs. Thus the changes in shares @si .w; y; t/=@t .i 1; ;N/ are not equal to zero even though technical change is Hicks-neutral. D In order to correct for non-homotheticity in calculating biases in technical change, note that the change in shares can be decomposed into a scale effect due to movement along the initial expansion path and a bias effect due to the shift in the expansion path. The share equation si si .w; y; t/ can be defined equivalently as D si si .w; i .t/; t/ where i .t/ indexes the isoquant yielding output level y given technologyDO t. Differentiation of this composite function with respect to t yields

@si .w; y; t/ @si .w; i ; t/ @si .w; y; t/ @f.x ; t/ O (9.11) @t D @t C @y @t where @si .w; i ; t/=@t provides a true measure of bias in technical change. Combin- ing (9.7)O and (9.11), a true measure of bias in technical change for a non-homothetic production function can be deﬁned in terms of a cost function as

@s .w; y; t/ @s .w; y; t/ @C.w; y; t/ @C.w; y; t/ i i = i 1; ;N (9.12) @t C @y @t @y D or in elasticity terms as

@ log s .w; y; t/ @ log s .w; y; t/ @ log C.w; y; t/ @ log C.w; y; t/ i i = i 1; ;N @t C @ log y @t @ log y D (9.13) @si .w;y;t/ @ log si .w;y;t/ Homotheticity implies @y @ log y 0 .i 1; ;N/. Similarly in the case of multiple outputs, a trueD measure of biasD in technicalD change can be calculated as

M @ log si .w; y; t/ X @ log s.w; y; t/ @ log C.w; y; t/ @ log C.w; y; t/ = (9.14) @t C @ log yj @t @ log yj j 1 D Technical change can be incorporated into dual proﬁt functions in a manner analogous to the above treatment of cost functions. Here we simply note the following relation between changes in proﬁts and costs due to technical change

@.w; p; t/=@t @C.w; y; t/=@t (9.15) D This relation is proved in the next section 9.2. Finally it is important to note that most of the above discussion applies equally well to short-run and long-run equilibrium models. The above models assumed that all inputs are freely variable and attain static equilibrium levels. In contrast we could postulate, e.g. , a Translog cost function C.w; y; k; t/ conditional on the stocks K of quasi-ﬁxed inputs. In place of (9.7) we have the following relation between the shift @f.x; K; t/=@t in the production function and the shift @C.w; y; k; t/=@t in cost: 110 9 Measuring Technical Change @f.x ; K; t/ @C.w; y; K; t/ @C.w; y; K; t/ = (9.16) @t D @t @y where Cy .w; y; K; t/ p assuming competitive proﬁt maximization. On the other hand the measurementD of biases is complicated somewhat when the cost function is short-run equilibrium in nature.

9.2 Non-Parametric index number calculations of changes in productivity

In this section we consider methods for calculating technical change that require neither econometrics nor the specification of particular functional forms for a production function as cost function. However this index number approach to productivity is defined in terms of continuous time, and errors in approximation result from the use of discrete time data. Moreover these calculations usually assume that all inputs are freely variable and are at static long-run equilibrium levels. Initially assume a single output production function y f .x; t/. Then output at time t is related to inputs at time t and technology as indicatedD by the relation y.t/ f .x.t/; t/. Differentiating with respect to t yields D N X @f.x; t/ @f.x; t/ y xi (9.17) V D @xi V C @t i 1 D @y.t/ @x.t/ where y @t , x @t denotes the change in output and input levels with respectV toÁ time. TheV standardÁ first order conditions for static competitive cost minimization are wi C .w; y; t/ @f.x;t/ 0 .i 1; ;N/, and substituting these y @xi into (9.17) yields D D N X Â wi Ã @f.x; t/ y xi (9.18) V D Cy .w; y; t/ V C @t i 1 D Competitive profit maximization implies Cy .w; y; t/ p, and substituting this into (9.18) yields D N X Âwi Ã @f.x; t/ y xi (9.19) V D p V C @t i 1 D or equivalently N X Âwi xi Ã xi @f.x; t/ y=y V =y (9.20) V D p y xi C @t i 1 D The weightings wi xi =py for input changes xi =xi define a Divisia quantity index for inputs in continuous time. Thus, assuming continuousV time and all inputs are at 9.2 Non-Parametric index number calculations of changes in productivity 111 static long-run equilibrium levels, technical change @f.x; t/=@t could be calculated as the residual in (9.18) or (9.19)-(9.20). However data is measured at discrete time intervals rather than continuously, and equation (9.19) or (9.20) can only be approximated using discrete data. For example integrating (9.19) over an interval t 0; 1 yields D N X Z 1 wi @xi .t/ Z 1 @f.x.t/; t/ y.1/ y.0/ .t/ dt dt (9.21) D p @t C @t i 1 t 0 t 0 D D D However, unless all price ratios wi =p are independent of t .i 1; ;N/, R 1 wi @xi .t/ D closed form solutions for the integrals t 0 p .t/ @t dt generally cannot be defined. (9.20) is most commonly approximatedD using Tornqvist¨ approximations to the Divisia index:

N y.t/ y.t 1/ X 1 Âwi xi wi xi Ã xi .t/ xi .t 1/ TF .t/ .t 1/ y.t/ 2 py C py xi .t/ i 1 D (9.22) R t @f.x.v/;v/ where Tf v t 1 @v =y.v/dv denotes the integral of the residual in (9.19) attributed toÁ technicalD change. There is a further complication in interpreting the calculated shifts in the production function @f.x; t/=@t. The equilibrium point x where the continuous on discrete calculations are made varies over time. Therefore, in order to interpret these results as simple comparable indicators of shifts in the production function over time, it is necessary to assume Hicks-neutral technical change and constant returns to scale. Similar comments apply to the interpretation of other index number calculations of productivity presented in this section and the following section C. In the case of multiple outputs y .y1; ; yM /, we can deﬁne a transformation function normalized on y1: yD1 g.y; x; t/ where y .y2; ; yM /. Differentiating y1.t/ g.y.t/; x.t/; t/ withD respectQ to t yieldsQ Á D Q M N 1 X i X i y gy .y; x; t/y gxi .y; x; t/x @g.y; x; t/=@t (9.23) V D Q Q QV C Q V C Q i 2 i 1 D D

The competitive maximization problem maxy;x; L py wx .g.y; x; t/ 1 i 1 Á iQ i y / has ﬁrst order conditions p p gyi .y; x; t/ 0 .i 2; ;M/, w p gxi .y; x; t/ 0 .i 1; ;N/, and substitutingC these intoQ (9.23)D yieldsD Q D D M N X X .pi =p1/yi .wi =p1/xi @g.y; x; t/=@t (9.24) V D V C Q i 1 i 1 D D However the numerical measure of technical change @g.y; x; t/=@t varies with the choice of output i, i.e. the measure of technical change isQ not symmetric with respect to the normalization of the transformation function. 112 9 Measuring Technical Change

In the case of multiple outputs a move useful measure of technical change may be obtained in terms of the cost function C.w; y; t/. At time t C.w.t/; y.t/; t/ PN i i Á i 1 w .t/x .t/, and differentiating with respect to t yields D N N N X i X i @C.w; y; t/ X i i X i i C i .w; y; t/w Cy .w; y; t/y w x w x w V C i V C @t D V C V i 1 j i 1 i 1 D D D (9.25) i Competitive cost minimization implies Cwi .w; y; t/ x .w; y; t/ .i 2; ;N/ (Shephard’s lemma), and substituting these conditions intoD (9.25) yieldsD

N M @C.w; y; t/ X i i X j w x Cy .w; y; t/y (9.26) @t D V j V i 1 j 1 D D j Competitive proﬁt maximization further implies Cyj .w; y; t/ p .j 1; ;M/, and substituting into (9.26) yields D D N M X X @C.w; y; t/=@t wi xi pi yj (9.27) D V V i 1 j 1 D D or equivalently

N Â i i Ã i M Â j Ã j @C.w; y; t/ X w x x X p yj y =C V i V (9.28) @t D C x C yj i 1 j 1 D D How the simple relations between primal measures and dual cost and proﬁt measures of technical change can easily be established as follows. Comparing (9.18) and (9.26) for the case of a single output and cost minimization establishes

@C.w; y; t/ @f.x ; t/ @C.w; y; t/ (9.29) @t D @t @y or in the case of competitive proﬁt maximization

@C.w; y; t/ @f.x ; t/ p (9.30) @t D @t Similarly comparing (9.24) and (9.27) in the case of multiple outputs and profit maximization establishes @C.w; y; t/ @g.y ; x ; t/ p1 Q (9.31) @t D @t In term of the profit function .w; p; t/ we have the identity .w.t/; p.t/; t/ p.t/y.t/ w.t/x.t/ for profits at time t, and differentiating with respect to t andÁ applying Hotelling’s lemma yields 9.3 Parametric index number calculations of changes in productivity 113

M N @.w; p; t/ X X pi yi wi xi (9.32) @t D V V i 1 i 1 D D Comparing (9.27) and (9.32) establishes

@C.w; y ; t/ @.w; p; t/ (9.33) @t D @t

9.3 Parametric index number calculations of changes in productivity

The Divisia approaches of the previous section required the approximation of continuous time derivatives by discrete differences but did not require the specification of a particular functional form for a production function or cost function. In this section we discuss an alternative approach to calculating changes in productivity: a flexible functional form is assumed for a production function or cost function, and then an index number formula is derived that is consistent with the functional form and with discrete time data. Here there are no errors in approximation due to the use of data for discrete time intervals, but there are errors in approximation to the true functional form for the production on cost function. Here we derive index numbers formulas for technical change corresponding to thus different functional forms: a Translog cost function, a Translog transformation function, and a Translog profit function. First assume a multiple output Translog cost function C.w; y; t/ as in (9.1). The underlying transformation function need not be constant returns to scale. Since this cost function is a quadratic form, the quadratic identity 8.11 of chapter 8 implies 114 9 Measuring Technical Change N Ä 1 X @ log C1 @ log C0 i i log .C1=C0/ log w =w D2 @ log wi C @ log wi 1 0 i 1 D M Ä 1 X @ log C1 @ log C0 i i log y1=y0 C 2 @ log yi C @ log yi i 1 ÄD 1 @ log C1 @ log C0 .t1 t0/ C 2 @t C @t N Ä i i 1 X @C1 w1 @C0 w0 i i i i log w1=w0 D2 @w C1 C @w C0 i 1 D M Ä i i 1 X @C1 y1 @C0 y0 i i (9.34) log y1=y0 C 2 @yi C1 C @yi C0 i 1 ÄD 1 @C1 @C0 =C1 =C0 since t1 t0 1 C 2 @t C @t D C N Ä i i i i 1 X w1x1 w0x0 i i log w1=w0 D2 C1 C C0 i 1 D M Ä i i i i 1 X p1y1 p0y0 i i log y1=y0 C 2 C1 C C0 i 1 ÄD 1 @C1 @C0 =C1 =C0 C 2 @t C @t

i by Shephard’s lemma and @C=@yi p .i 1; ;M/. This index number equation is identical to 8.26 discussed onD pages 8.10-8.11D of chapter 8, except that a time trend variable t 1; 2; 3; is incorporated into the Translog cost function. The ﬁrm is assumedD to be a competitive proﬁt maximizer with all inputs variable and attaining static long-run equilibrium levels. Rearranging (9.34)

( N Ä i i i i Â i Ã) 1 C C Á 1 X w1x1 w0x0 w1 T1 T0 log .C1=C0/ log i 2 C D 2 C1 C C0 w i 1 0 D (9.35) M Ä i i i i Â i Ã 1 X p1y1 p0y0 y1 log i 2 C1 C C0 y i 1 0 D

C @C.wV ;yV ;tV / PN i i V where TV @t =C.wV ; yV ; tV /, CV i 1 wV xV .V 0; 1/ . 1 C CÁ Á D D 2 .T1 T0 / is the average percentage reduction in cost due to technical change at time t C 0; 1. TheD second set of terms within brackets on the right hand side of (9.35) is f g the logarithm of a Tornqvist¨ price index .W1=W0/ for inputs (see equation 8.18), so Á the brackets enclose an implicit Tornqvist¨ quantity index x1=x0 for input f g

A 9.3 Parametric index number calculations of changes in productivity 115 Á x1=x0 .C1=C0/ = .w1=w0/ by the factor reversal equation analogous to 8.16. D The term on the right hand side of (9.35) that is outside the brackets can be interpreted as the logarithm of a quantity index .Y1=Y0/ for outputs (constant returns to scale in production would imply C py and this index would be equivalent to a Tornqvist¨ quantity index for outputs).D Hence the average percentage reduction in costA is the logarithm of the ratio of input and output quantity indexes:

1 C C Á h i T T log .x1=x0/=.Y1=Y0/ (9.36) 2 1 C 0 D This calculated change in cost can easily be related to shifts in the production or transformation function at equilibrium levels of commodities x, y using equations (9.7)-(9.10). Second, assume a constant returns to scaleA Translog production function y f .x; t/ and competitive cost minimization with all inputs variable at long-run equi-D librium levels. Proceeding as in the proof of 8.8, we obtain

Â Ã N Ä i i i i Â i Ã 1 t t y1 1 X w1x1 w0x0 x1 T1 T0 log log i (9.37) 2 C D y0 2 w1x1 C w0x0 x i 1 0 D

t @f.xV ;tV / where TV @t =yV .v 0; 1/. Here the average percentage change in productivity isÁ equal to the logarithmD of the ratio of output and input quantity indexes:

1 t t h T i T T log .y1=y0/=.x1=x0/ (9.38) 2 1 C 0 D

T where .x1=x0/ is the Tornqvist¨ quantity index for inputs. Alternatively if we assume a Translog production function (which is not necessarily constant returns to scale) and competitive proﬁt maximization, then we obtain the closely related index number formula

Â Ã N Ä i i i i Â i Ã 1 t t y1 1 X w1x1 w0x0 x1 T1 T0 log log i (9.39) 2 C D y0 2 p1y1 C p0y0 x i 1 0 D Similarly in the case of multiple outputs assume a Translog transformation function y1 g.y; x; t/ .y .y2; ; yM // and competitive proﬁt maximizing behavior withD allQ inputs variable.Q Á Then

Â Ã M Ä i i i i Â i Ã 1 g g y1 1 X p1y1 p0y0 y1 T1 T0 log log i 2 C D y0 2 p1y1 C p0y0 y i 1 0 D (9.40) N Ä i i i i Â i Ã 1 X w1x1 w0x0 x1 log i 2 p1y1 C p0y0 x i 1 0 D 116 9 Measuring Technical Change

g @g.y ;x ;tV / QV V where TV @t =yV .V 0; 1/. This measure of technical change varies in a simpleÁ manner with the commodityD chosen as numeraire in the transformation function. For example, given the alternative normalization y1 g.y2; ; yM ; x; t/ and y2 h.y1; y3; y4; ; yM ; x; t/, D D 2 M 2 1 1 3 M @g.y ; y ; x; t/=@t .p =p /@h.y ; y ; ; y ; x; t/=@t (9.41) D Third, assume a Translog proﬁt function and competitive proﬁt maximization with all inputs variable at long-run equilibrium levels. The quadratic identity and Hotelling’s lemma establish

Â Ã M Ä i i i i Â i Ã 1 1 1 X p1y1 p0y0 p1 T1 T0 log log i 2 C D 0 2 1 C 0 p i 1 0 D (9.42) N Ä i i i i Â i Ã 1 X w1x1 w0x0 w1 log i C2 1 C 0 w i 1 0 D

@.wV ;pV ;tV / where TV @t =.wV ; pV ; tV /. This measure of the effect of technical change onÁ profits .w; p; t/ is easily related to changes in costs and shifts in the production function or transformation function using equations (9.7)-(9.10). Inter- preting the second and third terms of the right hand side of (9.42) as price indexes for outputs and inputs, respectively, and noting that py wx, it follows that 1 Á 2 T1 T0 is the logarithm of the ratio of implicit quantity indexes for outputs and forC inputs. The behavioral models of the firm employed in sections B and C have assumed static long-run equilibrium and lead to index number formulas such as (9.20) that are defined in terms of the flows xt of all inputs used in production. Since the flow of capital services is not generally observable, it is usually assumed that the flow of capital services is proportional to the stock of capital assets (this assumption may be reasonable at long-run equilibrium). Then the growth rate of the capital service k k flow x =x is equal to the growth rate of the capital stock Kt =Kt . Vt t V The capital stock Kt is often approximated by the perpetual inventory method: Kt It 1 .1 ı/Kt 1 where It 1 denotes gross investment at time t 1 and ı is a D C constant rate of depreciation, and substituting backwards for Kt 1;Kt 2; ;Kt S yields the approximation

S X t S 1 Kt It 1 .1 ı/ C It S (9.43) C S 2 D Thus time series data on capital stock K and in turn the growth rate of the capital service ﬂow is approximated from data on gross investment I and an assumed rate of depreciation ı. Then the rate of technical change is calculated using index number formulas such as (9.20), (9.28), (9.35), (9.37) (e.g. Ball 1985). 9.4 Incorporating variable utilization rates for capital 117 9.4 Incorporating variable utilization rates for capital

There is considerable evidence that utilization rates of capital vary significantly over time. This implies that firms generally are not in long-run equilibrium, and in turn that (a) flows of capital services are not in fixed proportion to the levels of capital stocks and (b) the marginal value product of capital services is not equal to a market wage or rental rate. Nevertheless the index number procedures of sections B-C and many econometric models incorporate these assumptions. One approach to avoiding or reducing these problems in econometric models stems from the assumption that the rate of depreciation for capital varies with the utilization rate of capital. Then the firm’s short-run profit maximization problem can be written as

f f .x;Kt ;Kt / K f k max p wxt p Kt .w; p; p ;Kt / (9.44) f CQ Á Q .xt ;Kt /

f where Kt capital stock predetermined at beginning of period t, Kt position of Á K K Á Kt remaining at end of period t, and p p =.1 r/ is the asset price of capital at the end of period t discounted backQ toÁ the beginningC of period t. The derivatives of the production function are fx./ > 0, fK ./ > 0 and ftf ./ < 0 (an increase f P P P in Kt for a given initial stock Kt implies a lower depreciation rate and utilization rate of the initial stock Kt ). However there is a serious difficulty in implement- ing models such as (9.44) where the depreciation rate as well as utilization rate of capital is treated as endogenous: time series data on capital stocks have invariably been constructed assuming a constant rate of depreciation, and there are substantial econometric problems in the estimation of (9.44) where Kt is unobserved (see Ep- stein and Denny, 1980 for an illustration). Hereafter we shall assume for simplicity that variations in capital utilization do not influence the rate of depreciation. Next consider the effects of short-run equilibrium on non-parametric Divisia index number calculations of technical change, as discussed in section B. Let the production function be y f .x; K; t/ where x denotes flows of variable inputs and K denotes stocks of predeterminedD (quasi-fixed) capital inputs, and assume short-run competitive profit maximization

max pf .x; K; t/ wx .w; p; K; t/ (9.45) x Á Differentiating the identity y.t/ f .x.t/; K.t/; t/ and applying the ﬁrst order i Á conditions pf i .x ; K; t/ w 0 .i 1; ;N/ for a solution to (9.45) yields x D D N @f.x ; K; t/ X Âwi xi Ã xi X @f.x ; K; t/ =y y=y V Kj =y (9.46) @t DV py xi @Kj V i 1 j D j However the marginal products of capital @f.x; K; t/=@K are not observed j directly are equal to price ratios wk =p only in a static long-run equilibrium. 118 9 Measuring Technical Change

Also note that the productivity index (9.46) employs the change in capital stocks K rather than a change in flow of capital services from the stocks. Equation (9.46) V provides an exact measure of change in productivity in continuous time. Thus errors @f.x;K;t/ in calculations of changes in productivity @t under the erroneous assumption of long-run equilibrium are due to mismeasurement of weights for changes in capital stocks kV rather than to mismeasurement of the flow of capital services (the correct weights for K are the unobserved marginal products of capital). In other words, V in productivity studies there is no need to derive capital service flows from capital stocks irrespective of the industry being in long-run or short-run equilibrium. This is an important result since the concept of capital service flows, which has been widely employed in earlier productivity studies, is an artificial construct with little empirical basis. In the special case of constant returns to scale f .x; K; t/ f.x; K; t/ and one capital good K, the marginal product of capital can be calculatedD directly as

" N # @f.x ; K; t/ X Âwi Ã y xi =K (9.47) @K D p i 1 D Using Euler’s theorem and ﬁrst order conditions for (9.45). Then changes in productivity can be calculated directly from a discrete approximation to (9.46). Similarly the variable cost function C C.w; y; K; t/ wx and short-run competitive behavior (9.45) imply D Á

N Â i i Ã i M Â i Ã i @C.w; y; K; t/ X w x x X p yi y X @C.w; y; K; t/ i =C V i V i K =C @t D C x C yi @K V i 1 j 1 i D D (9.48) In the case of constant returns to scale in production and a single capital good K, the unobserved shadow price of capital can be calculated as

0 M 1 @C.w; y; K; t/ X j @C p yj A =K < 0 (9.49) @K D j 1 D (using C.w; y; K; t/ C.w; y; K; t/ and Euler’s theorem). Substituting (9.49) intoD (9.48), the resulting index number formula for technical change can be approximated using data for discrete time intervals. This calculation of technical change under constant returns to scale and a single capital good K requires neither econometrics nor the assumption of long-run equilibrium. Next consider the effects of short-run equilibrium on parametric Divisia index number calculations of technical change, as discussed in section C. Assuming a Translog variable cost function C.w; y; K; t/ and competitive short-run proﬁt maximization ((9.45)), equation (9.35) must be rewritten as 9.4 Incorporating variable utilization rates for capital 119

( N Ä i i i i Â i Ã) 1 C C Á 1 X w1x1 w0x0 w1 T1 T0 log .C1=C0/ log i 2 C D 2 C1 C C0 w i 1 0 D M Ä i i i i Â i Ã 1 X p1y1 p0y0 y1 log i (9.50) 2 C1 C C0 y i 1 0 D Ä j ! 1 X @ log C1 @ log C0 K1 log j 2 @ log Kj C @ log Kj j K0

In general @ log C @C = C cannot be measured without recourse to econo- @ log Kj Á @Kj Kj metric estimation of the factor demand equations. However in the case of a single @ log C quasi-fixed capital good K and constant returns to scale in production, @ log K can be calculated from (9.49). Thus (as in non-parametric Divisia index number formulas) given the assumptions of constant returns to scale and one quasi-fixed input, parametric measures of technical change can be obtained without recourse to econometrics or the assumption of long-run equilibrium. Thus, if there is more than one quasi-fixed input or returns to scale are not constant, econometric methods are required for the measurement of technical change. For example we could postulate a short-run Translog cost function C.w; y; K; t/ analogous to (9.1) and estimate factor share equations for the variable inputs. Then @C.w;y;K;t/ the change in technology @t can be calculated directly from the estimates of the share equations as discussed in section A (page 9.4). Alternatively the shadow prices can be calculated as

N @C.w; y; K; t/ X @xi .w; y; K; t/ wi (9.51) @Kj Á @Kj i 1 D From the estimates of the share equations, and then the change in technology 1 C C 2 T1 T0 can be calculated from the index number equation (9.50). The disad- vantageC of this second approach is that equation (9.50) requires the assumption of short-run competitive profit maximization, whereas estimation of the cost function only requires the weaker assumption of short-run competitive cost minimization. We conclude this section with a brief discussion of empirical measures of capacity utilization based on microeconomic theory. Unexpected changes in output prices or input prices are likely to lead to short-run combinations of variable and quasi- fixed inputs that are inappropriate for the long-run, i.e. under or over-utilization of capacity is likely in the short-run. A fruitful approach to the measurement of capacity utilization is by comparing the observed level of output and “capacity output”. One definition of capacity output is the output level corresponding to the minimum point on the firm’s long-run average cost curve, but this definition is not useful due to difficulties in identifying the long-run average cost curve. A more useful definition of capacity output is the output level yC at which the short-run average total cost curve (with quasi-fixed inputs fixed at their short-run equilibrium levels) is tangent to the long-run average cost curve. If there is constant 120 9 Measuring Technical Change returns to scale in the long-run, then capacity output corresponds to the minimum point on the short-run average total cost curve (see diagram on following page). The rate of capacity utilization CU is then defined as the ratio of actual output y to capacity output yC : CU y=yC (9.52) Á Econometric estimates of a short-run cost function C.w; y; K; t/ can be employed in calculations of a capacity utilization index CU (Berrdt and Herse 1986).

9.5 Conclusion

In this chapter we have discussed several non-econometric and econometric approaches to the measurement of technical change. The non-econometric approaches do not require a long time series of data or the use of highly aggregated data. On the other hand these approaches cannot test hypothesis about technical change, and these approaches can disentangle shifts in productivity and variations in capital utilization rates only in the case of constant returns to scale and a single quasi-ﬁxed input. In contrast econometric approaches require a substantial number of observations and substantial aggregation of commodities, but these approaches can test hypothesis and can calculate changes in productivity given multiple quasi-ﬁxed inputs. In practice non-econometric and econometric approaches to the measurement of technical change can often be viewed as complementary. Most of the effort required to construct non-econometric. Measures of (full) Capacity Output A.

A SRAC AC(w, wK, y, K) LRAC AC(w, wK, y) ≡ ≡

yC C

Fig. 9.1 ﬁgure 1 9.5 Conclusion 121

B. Assuming constant returns to scale in the production function y f .x; K/, f .x; K/ f.x; k/: D D

K K A SRAC0 AC(w, w , y, K0) SRAC1 AC(w, w , y, K1) ≡ ≡

C C y0 y1 C

Fig. 9.2 ﬁgure 2

Short-run average cost function (SRAC)

wx wkK AC.w; wK ; y; K/ min C D x @y s.t. f .x; K/ y D Long-run average cost function (LRAC):

wx wkK AC.w; wK ; y/ min C D x;K y s.t. f .x; K/ y D Index number calculations of technical change will also be required to construct econometric models, and data problems may be indicated more clearly by non- econometric measures of technical change. Thus the following sequence may be appropriate: ﬁrst construct non-econometric index number measures of technical change, and then (data permitting) estimate a short-run equilibrium econometric model. However at least two serious problems remain for the econometric models and Divisia index number formulas outlined here. First technical change is proxied by a simple time trend t 1; 2; 3; (which may interact with other variables) or is calculated as a simpleD residual. Thus there is no theory endogenizing technical 122 9 Measuring Technical Change change to the model. In principle a well constructed theory of demand and supply for changes in technology should improve both econometric and non-econometric measures of technical change. Second, there has been little attempt to incorporate explicitly into the analysis changes in quality of inputs supplied to the industry (sec Berrdt 1983 and Tarr 1982). 4. It should be noted that cross-sectional differences in productivity (e.g. differences in productivity between countries, regions or ﬁrms) can be measured by parametric index number procedures somewhat similar to the procedures discussed in section C (Caves, Christensen and Diewert 1982; Denny and Fuss 1983). Index

C Gorman polar form 65 composite commodity 78 H cost function Hicksian consumer demands 28 short-run Hotelling’s Lemma 12, 14 Generalized Leontief 59 normalized quadratic 59 I Translog 59 cost of living indexes 102 implicitly separable 83 integrability problem 4 D in economics 32 L Divisia index 90 dual cost function 1 Laspeyres index numbers 87 dual expenditure function 28 Laspeyres indexes 102 dual indirect utility function 27 Le Chatelier principle 17 dual proﬁt function 11 dual proﬁt function, restricted 18 M

E marginal frim 21 Marshallian consumer demands 28 equations of motion 24 N Euler’s theorem 4 Normalized Quadratic indirect utility function F 56 normalized quadratic cost function 55 factor reversal equation 92 normalized quadratic proﬁt function 52 Fisher indexes 103 normalized quadratic short-run cost function ﬂexible functional forms 47 59 second order 49 Frobenius theorem 4 P Paasche indexes 102 G primal-dual relations 39

Generalized Leontief functional form 7 Q Generalized Leontief short-run cost function 59 quadratic dual proﬁt function 51

123 124 Index quasi-homothetic 65 T

R Tornqvist¨ index 90 Translog cost function 56 ratio of commodities 85 Translog dual proﬁt function 53 ratio of marginal rates of substitution 85 Translog indirect utility function 57 Roy’s theorem 29 Translog short-run cost function 59 two-stage budgeting 79 S V second order ﬂexible functional forms 49 separable implicitly 83 Vartia I price index 99 weakly 80 Shephard’s Lemma 2 W Slutsky equation 31 sub-utility function 80 Walras Law 63 superlative 90 weakly separable 80