factor prices in general equilibrium 2039 factor prices in general equilibrium The first distinguishing trait of factors is that some- times they do not provide any direct utility and are useful only as inputs in production. Consumers will supply to 1 Introduction the market their entire endowment of such ‘pure’ factors At first glance, the Walrasian general equilibrium model and hence supply will be inelastic with respect to price does not offer a theory of factor prices. Factors are goods changes. As we will see, what matters is local unrespon- supplied by agents to firms who then use them to produce siveness to prices. Perhaps when a factor such as iron ore outputs. In the general equilibrium model, there is no is sufficiently cheap in terms of consumption goods con- such class of goods: one and the same good can simul- sumers will find some direct use for it and hence have an taneously be used as an input by some firms, produced as excess demand that locally varies as a function of prices. an output by other firms, sold by some consumers, and But above some minimum price, consumers will not purchased and consumed by other consumers. Indeed, the consume any iron ore and in this range consumers’ general equilibrium model’s abstraction from the minu- excess demand will be inelastic. Second, technology can tiae of how particular goods are used is one of the theory’s restrict the number of ways in which factors can be pro- great advantages. For many of the classical concerns of the ductively combined. The extreme case occurs with fixed Walrasian tradition – the existence of equilibrium, opti- coefficients – the Leontiev production function – where mality – these details are irrelevant. to produce one unit of a good just one combination of Even if there is a category of factors that consumers sell factors will do. More flexible is the linear activities model and firms buy, it is hard to see any distinctive properties where finitely many constant-return-to-scale techniques of these goods. While factor supply functions can exhibit are available to produce one or more goods. Factors then perverse responses to price changes, so can output may be combined in various configurations but some demand functions. The responses of firms to price factor proportions cannot be used productively (that is, changes are better behaved, and firm factor demands may without disposing of some of the factors). Nonlinear seem to be governed by a distinctive principle: a firm’s activities are qualitatively similar but do not require demand for a factor diminishes in its own price while a constant returns to scale. In all these cases, production firm’s supply of an output increases in its own price. sets have a kinked rather than smooth (differentiable) While correct, these two fundamental rules of producer surface. Consequently factor prices can be adjusted at comparative statics are really reflections of a single law, least slightly from one equilibrium configuration without as Samuelson (1947) showed long ago. Suppose in an changing the quantity of factors that profit-maximizing ‘-good economy that a profit-maximizing firm with producers will demand when producing a given quantity ‘ production set Y R chooses y ¼ðy1; ...; y‘Þ2Y of output (or vector of outputs). In the Leontiev case, when facing prices p ¼ðp1; ...; p‘Þ and y^ 2 Y when fac- picture the multiple price lines that can support the ing p^. Since each decision is profit-maximizing, p Á y model’s L-shaped isoquants. Of course, production sets p Á y^ and p^ Á y^ p^ Á y and hence ðp^ À pÞðy^ À yÞ0. If do not have to exhibit kinks; for example, they will be only one price differs at p compared to p^, say the first, smooth when each output is a differentiable function of then ðp^1 À p1Þðy^1 À y1Þ0. So if p^14p1 then y^1 y1. factor inputs. Any change in relative factor prices will Both of the comparative statics rules now follow from the then lead to a change in factor demand. appropriate sign restrictions on y1 and y^1: when both are Factors of production thus are distinctive in that both positive we conclude that the output of good 1 supplied demand and supply can be unresponsive to certain types by the firm must be weakly increasing in its price, while if of price changes. Factor demand and supply do not have both are negative we conclude that the factor demand for to display this unresponsiveness, but under plausible cir- good 1 must be weakly decreasing in its price (since y^1 cumstances permitted by the general equilibrium model y1 and ðy^1; y1Þ0 imply jy^1jjy1j). It is tempting to they will. Inelastic factor demand and supply in turn can conclude that there is no special general equilibrium lead to an indeterminacy of factor prices. For a simple principle of factor demands, just a specific application example, suppose an economy has one consumption that follows when the sign convention for factors is good, produced by a single linear activity that requires a1 inserted. units of one factor and a2 units of a second factor to yield one unit of output. Set the price of consumption equal to 1, let w1 and w2 be the two factor prices, let the endow- 2 Factor-price indeterminacy ments of the two inelastically supplied factors be e1 0 The demand for and supply of factors can nevertheless and e2 0, and let y be the sole activity usage level. An exhibit distinctive properties, although they are consist- equilibrium ðw1; w2; yÞ0 where the consumption good ent with the generalities pointed out in the previous is produced and has a positive price must satisfy three section. These properties do not matter for the most of conditions: (i) a1w1+a2w2=1 (the activity breaks even), the classical results of general equilibrium theory, but (ii) aiy ei for i=1, 2 (market-clearing for factors), and they can undermine one result, the generic determinacy (iii) aiyoei ) wi ¼ 0 for i=1, 2 (factors in excess supply (local uniqueness) of equilibria. have a 0 price). On the assumption that the demand for 2040 factor prices in general equilibrium output equals factor income, which is a form of Walras’ used when the activity is operated at the unit level. In law, (i)–(iii) imply that the market for output clears. equilibrium the excess demand for each good must be Evidently equilibrium must satisfy y ¼ min½e1=a1; e2=a2. non-positive, each good in excess supply must have a 0 By (iii), the two factors will both have a strictly positive price, each activity must earn non-positive profits, and price only if each activity in use must earn 0 profits. Since deter- minacy and indeterminacy are purely local events, a e e 1 ¼ 2 , (1) search for equilibrium prices and activity near a reference a1 a2 equilibrium can ignore the ‘slack’ equilibrium condi- tions, the market-clearing condition for any good in in which case any w ¼ðw1; w2Þ0 that satisfies (i) will be an equilibrium w: indeterminacy therefore obtains excess supply and the no-positive-profits condition for when (1) holds. We defer for a little while the question of any activity that either makes strictly negative profits whether this knife-edge condition is likely to be satisfied. or utilizes and produces only goods in excess supply: Fixed coefficients and inelastic factor supply do not for small adjustments of prices and activity levels, the always lead to indeterminate factor prices. Prior to the excluded goods will remain in excess supply and the invention of the differentiable production function and excluded activities will continue to make negative profits for awhile thereafter, the standard cure for factor-price or continue to use and produce only goods in excess indeterminacy was to argue that, even if each industry uses supply (and hence continue to break even). Call any good factors in fixed proportions, those proportions will differ not in excess supply and any activity that breaks even and across industries; variations in factor prices will then lead that uses or produces at least one good not in excess to changes in relative output prices, and thus to changes in supply ‘operative’. Given some reference equilibrium with ‘ output demand that feedback to changes in factor demand operative consumption goods, m operative factors, and ‘ (Cassel, 1924; Wieser, 1927). Substitution in consumption n operative activities, let A be the ð þ mÞn activity can thereby play the same equilibrating role as the tech- analysis matrix whose rows and columns correspond to nological substitution of inputs in production. For the the operative goods and activities, let y be the n-vector of ‘ simplest example, suppose we supplement the above operative activity levels, let p be the -vector of prices for single-sector economy with a new sector that uses b units the operative consumption goods, let w be the m-vector 1 of prices for the operative factors, let z(p, w) be the excess of the first factor and b2 units of the second factor to produce one unit of a second consumption good. If we demand function for the operative consumption goods, keep the price of the first consumption good equal to 1, which we assume is homogeneous of degree 0 in (p, w), and finally let e be the m-vector of inelastic supplies of and let pb be the price of the second consumption good, then when both activities break even the equalities the operative factors. Walras’ law then states that p Á zðp; wÞ¼w Á e.