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APEC 5152 - Handout 1 APEC 5152 - Handout 1 February 16, 2017 Micro-Preliminaries Contents 1 Consumer preferences 2 1.1 The indirect utility function . 4 1.2 The expenditure function . 6 1.3 Aggregation . 7 2 Production technologies 8 2.1 The cost function . 9 2.2 The value-added function . 11 2.3 Aggregation . 13 2.3.1 The aggregate cost function and value added function . 13 2.3.2 The aggregate gross national product function . 15 3 Appendix 16 3.1 The Primal-Dual Problem (Envelope Theorem) . 16 3.2 Elasticities and homogenous functions . 18 1 Introduction - microeconomic foundations Throughout these notes, the following notation denotes factor endowments, factor rental rates and output prices. Sectors are indexed by j 2 f1; 2; 3g ; and denote the quantity of sector-j's output by 3 the scalar Yj: Corresponding output prices are denoted p = (p1; p2; p3) 2 R++, with the scalar pj representing the per-unit price of sector-j output. We consider three factor endowments: labor, L; capital, K; and land Z. The corresponding factor rental rates are: w is the wage rate, r is the rate of return to capital, and τ is the unit land rental rate. Represent the vector of factor rental rates by w = (w; r; τ) : 1 Consumer preferences The economy is composed of a large number of atomistic households. Each household faces the η η η η 3 same vector of prices p and the same vector of factor rental rates w. Let υ = (L ;K ;Z ) 2 R++ denote the level of factor endowments held by household-η; with Lη;Kη and Zη representing the η 3 household's endowment of labor, capital and land. The notation υ 2 R++ is a shorthand way of saying each household owns a strictly positive amount of each factor. In most applications that follow we suppress the η superscript of υη and use instead υ = (L; K; Z) : Given factor rental rates w; the household's income is given by w · υ = wL + rK + τZ which is used to purchase qj units of consumption good j at market price pj; j = a; m; s: Given final good prices p; total expenditure on final goods is equal to p · q = p1q1 + p2q2 + p3q3 3 3 where q = (q1; q2; q3) 2 R++: The notation (q1; q2; q3) 2 R++ means the household consumes a strictly positive amount of each consumption good. Then, the household's budget constraint is given by w · υ ≥ p · q M Consumer preferences over goods are represented by the utility function u : R++ ! R+; defined as u (q) : 2 Assumption 1: u (q) satisfies the following properties: 1. u (q) is increasing and strictly concave in q; 2. u (q) is everywhere continuous, and everywhere twice differentiable, 3. u (q) is homothetic1. Assumption 1.1 yields indifference curves that are convex, Assumption 1.2 ensures Marshallian demands are continuous functions, while Assumption 1.3 yields Marshallian demands that are separable in prices and income (see Cornes or Varian). Problem 1 Show that the Cobb-Douglas utility function α1 α2 α3 u = q1 q2 q3 ; where α1 + α2 + α3 = 1 (1) is increasing and twice differentiable in q; and homothetic. The utility function is used to derive two \dual" functions: the indirect utility function and the expenditure function. The first step in deriving the indirect utility function is to choose the level of goods 1, 2 and 3 to maximize utility subject to a budget constraint. The solution to this exercise is a set of utility maximizing Marshallian demand functions. This constrained optimization exercise is standard fare in a typical intermediate microeconomics course. Next, by substituting the optimal Marshallian demands into the utility function one gets the indirect utility function. The expenditure function is also derived in two steps: first, minimize the cost of achieving a given level of utility, yielding cost minimizing Hicksian demand functions { another standard problem solved in a typical intermediate microeconomics course. Then, substitute the optimal Hicksian demand functions back into the original optimization problem (discussed shortly) to get the expenditure function. Both the indirect utility function and expenditure function have very useful derivative properties, the most famous of which are Roy's identity and Shephard's lemma. By Roy's identity, discussed shortly, one can derive Marshallian demand functions using a ratio of partial derivatives of the 1A utility function u (q) is homothetic if u q0 > u q00 if and only if u tq0 > u tq00 for any t > 0: 3 indirect utility function. Shephard's lemma allows us to derive an optimal Hicksian demand function by taking the partial derivative of the expenditure function with respect to the demand function's price. These results obtain for a general class of utility functions { particularly those that satisfy Assumption 1 { and have proven extremely useful in both conceptual and applied research. 1.1 The indirect utility function The indirect utility function gives the household's maximum attainable utility given final good prices, p; and income w · υ, and is defined as V (p; w · υ) ≡ max fu (q): w · υ ≥ p · qg (2) q The indirect utility function inherits the following properties from the direct utility function (see Cornes, pp. 67-70): V1. Homogeneous of degree zero in p and w · υ; V (θp; θw · υ) = V (p; w · υ) ; θ > 0, V2. V (p; w · υ) is convex in p; V3. V (p; w · υ) is continuous and differentiable in p and w · υ; V4. V (p; w · υ) =v(p) w · υ: separable in p and w · υ, By V4, the marginal utility of an additional unit of income is v(p) : V5. Given differentiability, Marshallian demands follow from Roy's identity, vp (p) qj (p)(w · υ) = − j w · υ (3) v (p) where, throughout the text, the subscript on a function indicates a partial derivative, e.g., 2 vpj = @v (p) =@pj and vp1p2 = @ v (p) =@p1@p2: In applied work, the most important properties of the indirect utility function are V1, V2 and V5. By V1, a doubling of input and output prices leaves the level of utility unchanged. V2 implies consumer demand curves are downward sloping (discussed shortly), and V5 says if you have an indirect utility function, you can recover the consumer's demand function using ratios of first derivatives of the indirect utility function. 4 To see these properties in action, introduce the Cobb-Douglas utility function (1) into the opti- mization problem (2). The corresponding Lagrangian is α1 α2 α3 L = q1 q2 q3 + λ [w · υ−p1q1 − p2q2 − p3q3] and, assuming an interior solution, first order conditions are: @L α1 α2 α3 −1 = αiq1 q2 q3 qi + λpi = 0; i = 1; 2; 3 (4) @qi @L = w · υ−p q − p q − p3q = 0 @λ 1 1 2 2 3 Problem 2 Show the solution to (4) yields the Marshallian demand curves ∗ αjw · υ qj (pj)(w · υ) = ; j = 1; 2; 3 (5) pj where q∗ (p ) = αj : j j pj 1. Take the Marshallian demand curves and substitute into the original Cobb-Douglas utility function (1) and simplify. The result is the indirect utility function. 2. Apply Roy's identity and verify that the resulting demands are identical to (5). Problem 3 Use Mathematica to derive the Marshallian demand functions, and indirect demand function you derived in Problem 1. Using Mathematica, verify Roy's identity works. With preferences that are linearly homothetic, the indirect utility function is separable in final good prices and household endowments. Since consumers face the same prices and have identical preferences, we can aggregate across households to get an aggregate { \community" { indirect utility function. Assuming there are N households, the aggregate indirect utility function is given by N X V = v (p) (w · υη) η=1 Accordingly, the total Marshallian demand for good j is N j X η Qj = q (p) (w · υ ) ; 8 j 2 f1; 2; 3g (6) η=1 These functions are the simple aggregation of individual consumer welfare and demands. It also follows from V1 that (6) is homogeneous of degree minus one in prices p and of degree one in income. 5 1.2 The expenditure function The expenditure function gives the minimum cost of achieving utility level q 2 R at given prices p; and is defined as E (p; q) ≡ min fp · q : q ≤ u (q)g (7) q The expenditure function inherits from u (·) ; the following properties: E1. E (p; q) > 0 for any p and q > 0; E2. E (p; q) is non-decreasing in p and q; E3. E (p; q) is concave and continuous in p; E4. E (λp; q) = λE (p; q) ; λ > 0: homogeneous of degree 1 in p, E5. E (p; q) = E (p) q: separable in p and q; E6. Shephard's lemma: If E (p; q) is differentiable in p, then qj = Epj (p; q) = Epj (p) q; j = 1; :::; M E1 says purchasing a strictly positive consumption bundle is costly. E2 says, all else equal, (i) if the price of a consumption good increases, then the cost of achieving the same level of utility increases, or (ii) increasing utility requires an increase in expenditures. By E3, the expenditure function is continuous, with first (partial) derivatives being positive and second partial derivatives being negative. Condition E4 implies demand functions are homogeneous of degree zero in p: E5 results from Assumption 1.3 and implies demand functions are separable in p and q (see Chambers, 1988, chapter 2). Later, we interpret the quantity q to be a composite consumption good, the unit cost of which is E (p).
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