Economics 2010a Fall 2003 Edward L. Glaeser Lecture 9 9. The Producer’s Problem a. Firms and Maximization b. Production Functions c. Supply and Profit Functions d. Cost Functions e. Duality and Producers f. Application: Urban Systems 1. Technology The more tradition approach is to assume: (1) A production correspondence, e.g. f(K,L) or more generally f?Z?, that maps the vector of inputs Z which cost W into a vector of outputs, which are then sold at prices P for total revenues Pf?Z?. In many cases, we think of f?Z? as a function– i.e. only one output– but is doesn’t need to be. We assume first that firms treat prices as given– i.e. they are price takers– i.e. they don’t have market power. (2) We assume that firms maximize profits, and that they have the option (at least in the long run) to exit, i.e. earn zero profits. This is of course a deeply controversial claim. (3) We make some assumption about the number of firms– perhaps free entry of identical firms, perhaps something else. This last assumption gives us a great deal of power– this is the equilibrium assumption in action. Together, profit maximization and free entry of identical firms gives us the following two sets of conditions (assuming that the production function is continuously differentiable and concave). P ?f?Z? ?Z i ? W i for each input marginal revenue equals price. And given these first order conditions: Pf?Z? ? W ? Z ? 0 These two, especially, when combined with demand give you implications about prices. Now back to the MWG set theoretic approach. A production plan is a vector y ? ? L . This includes both inputs and outputs, and an input is a negative element in this vector. An output is a positive element in this vector. Total profits are p ? y The set of all production plans is Y,which is analogous to X, in the consumer chapters. We generally assume that 0 ? Y , so that firms can shut down. Generally, we assume that Y is (1) nonempty (even beyond including 0) (2) closed, i.e. includes its limit points. (3) no free lunch– there is no vector y ? Y where y l ? 0, for all l and y k ? 0 for at least one factor l?k. (4) free disposal– if a vector ?y 1 ,..y k ,..y L ? ? Y where y k ? 0, for then all other vectors ?y 1 ,..x k ,..y L ? ? Y where x k ? y k (you can always get rid of something). (5) irreversibility: if y ? Y then ?y ? Y These properties are more particular: (6) Nonincreasing returns to scale. If y ? Y, then ?y ? Y for all ? ? ?0,1? –you can always scale down. (7) Nondecreasing returns to scale. If y ? Y, then ?y ? Y for all ? ? 1– you can always scale up. (8) Constant returns to scale If y ? Y, then ?y ? Y for all ? ? 0– you can always scale up or down. (9) Additivity (also free entry) if y ? Y and y ? ? Y then y ? y ? ? Y (10) Convexity if y ? Y and y ? ? Y then ?y ? ?1 ? ??y ? ? Y for all ? ? ?0,1? (11) Y is a convex conde. If for any y,y ? ? Y and ? ? 0, ? ? 0, ?y ? ?y ? ? Y MWG Proposition 5.B.1: The production set Y is additive and satisfies the nonincreasing returns condition if and only if it is a convex cone. If it is a convex cone– then the nonincreasing returns condition and additivity immediately follow. Let k ? Max??,?? where k is an integer, then ky,ky ? ? Y by additivity. Also using nonincreasing returns to scale ? k ky ? Y and ? k ky ? ? Y and we’re done. The profit maximization problem is to maximize p ? y subject to y ? Y This yields a supply correspondence y?p? And a profit function ??p? ? p ? y?p? Properties of the profit function MWG Proposition 5.C.1: Suppose that ??.? is the profit function of the production set Y, and that y?.? is the associated supply correspondence. Assume also that Y is closed and satisfies the free disposal property. Then: (1) ??.? is homogeneous of degree one. (2) ??.? is convex (3) y(.) is homogeneous of deree zero (4) If Y is strictly convex, then y(.) is single valued. (5) If y(.) is single valued then ??.? is differentiable and ???p? ? y?p? (6) If y(.) is differentiable then Dy?p? ? D 2 ??p? is a symmetric and positive semidefinite matrix with Dy?p?p ? 0 Homogeneity of degree one in prices occurs because the vector y that maximizes py subject to any constraint is the same vector that maximizes ?py subject to the same constraint for any positive ?. This also gives us homogeneity of degree zero for the supply correspondence. Convexity– ???p ? ?1 ? ??p ? ? ? ???p? ? ?1 ? ????p ? ? for ? ? ?0,1? . Let p ?? ? ?p ? ?1 ? ??p ? and have y,y ? and y ?? denote the optimal production vectors at the three price levels. Then we know that ???p ? ?1 ? ??p ? ? ? ?py ?? ? ?1 ? ??p ? y ?? But py ?? ? py and p ? y ?? ? p ? y ? and we’re done. Cost minimization– now assume a fixed vector of outputs and a production function f(Z). The cost minimization problem is to minimize W ? Z subject to f?Z? ? Q for some fixed level Q. In principle, this quantity constraint could handle a lot of different outputs, but we will certainly focus on the idea of a single output. The correspondence Z?W,Q? is the input demand that satisfies cost minimization. The cost function C?W,Q? ? W ? Z?W,Q? Properties of the Cost Function MWG Proposition 5.C.2: Suppose that C(W,Q) is the cost function of a single output technology Y, with production function F(.) and that Z(W,Q) is the associated conditional factor demand correspondence. Assume also that Y is closed and satisfies the free disposal property. Then: (1) C(.) is homogeneous of degree one in W (2) C(.) is nondecreasing in Q (3) C(.) is a concave function of W (4) C(.) is homogeneous of degree zero in W. (5) If Z?W,Q? consists of a single point at W, then C?W,Q? is differentiable at W and ? W C?W,Q? ? Z?W,Q? (shephard’s lemma) (6) If Z?W,Q? is differentiable at W, then D W Z?W,Q? ? D W 2 C?W,Q? is a symmetric and negative semidefinite matrix with D W Z?W,Q?W ? 0 (7) If F(.) is homogeneous of degree one then Z(.) and C(.) are also homogeneous of degree one. (8) If F(.) is concave, then C(.) is a convex function of Q. The fact that costs satisfty homogeneity of degree one in input prices follows from the fact that if we scale all prices up by the same constant, the solution to the minimization problem is unchanged. This also gives us that input demand is homogeneous of degree zero. Convexity in costs is directly parallel to concavity of the profit function. C??W ? ?1 ? ??W ? ,Q? ? ?C?W,Q? ? ?1 ? ??C?W ? ,Q? ,for? ? ?0,1? . Let W ?? ? ?W ? ?1 ? ??W ? and have Z,Z ? and Z ?? denote the input vectors at the three price levels. Then we know that C??W ? ?1 ? ??W ? ,Q? ? ?WZ ?? ? ?1 ? ??W ? Z ?? But WZ ?? ? WZ and W ? Z ?? ? W ? Z ? and we’re done. The nondecreasing property follows from free disposal. Lastly Shephard’s lemma follows from the fact that: C?W,Q? ? WZ ? ??Q ? F?Z?? which has first order conditions: W j ? ? ?F?Z? ?Z i Differentiate this with respect to any W i and we get ?C ?W i ? Z i ? ?W j ?Z j ?W i ? ?? ?F?Z? ?Z i ?Z j ?W i But all the terms beyond Z i just drop out (this is the envelope theorem in action). A few production functions to know: Leontief: Y ? Min?? 1 Z 1 ,...? L Z L ? Linear Y ? ?? i Z i Cobb-Douglas Y ? ? Z i ? i This is increasing returns if ?? i ? 1, decreasing returns if ?? i ? 1 and constant returns if ?? i ? 1 Remember you can’t just take logs in this case, but you do know that: ? i Y ? W i Z i for all i. This gives you the fact that constant returns technologies are the only ones that yield zero profits. Constant elasticity of substitution: ??? i Z i ? ? 1/? This yields factor demand of Y ? W i ? i 1 1?? Z i , for all i. Systems of cities– supply and demand together: Back to compensating differentials. Define consumer utility over non-housing consumption, housing prices and amenities and assume that homes are homogeneous. This means that utility is U?W ? C,A?, where C is housing prices. Then indifference across space gives us that U?W ? C,A? ? U Or if we assume that that communities fall along a continuum: dW ? dC ? U 1 U 2 dA The increase in wages minus prices must exactly offset the increase in amenities. This is a very famous condition that has been the basis for most of urban economics. But this is only part of the story– there also must be a equilibrium in the product market. Let’s assume that profits equal Pf(L, Z)-WL And assume that free entry pins these down to zero. Then we need to have across space: dPf(L,Z)?P ?f ?Z dZ ? dW ? L ? 0 Change in prices times quantity plus price times change in productivity minus change in wage must equal zero. If prices are constant across space (the traded goods assumption– if not they would have to show up in the consumers side as well–you can do this)– we might as well assume P?1 and then we have dW ? 1 L ?f ?Z Thus changes in prices over space must reflect both sides of the market– changes in productivity and changes in either amenities or housing costs. Now we have 1 L ?f ?Z ? U 1 U 2 dA ? dC Changes in housing costs across space add together change in consumer amenities and producer amenities. Now to close the model, we need to think about equilibrium in the housing supply sector. Perhaps homebuilders face a technology G(Q,D) which is their costs per unit built as a function of local characteristics including density, and let g(Q,D) refer to the derivative of this with respect to quantity. Then we have C?g(Q,D) and dC ? ?g ?Q dQ ? ?g ?Q dD Or 1 L ?f ?Z ? U 1 U 2 dA ? ?g ?Q dD ? ?g ?Q dQ Changes in population across space add together these three different forms of local amenities. If for example, D didn’t change over space, then this tells you that the population response to production amenities depends completely on the supply elasticity of housing.