producer_theory_and_MCS - Lecture Notes on Monotone...

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1 Lecture Notes on Monotone Comparative Statics and Producer Theory Susan Athey updated Fall 2002 These notes summarize the material from the Athey, Milgrom, and Roberts monograph which is most important for basic producer theory. It is not presented in the same order as the monograph, so I have not preserved theorem numbers, etc., but I don’t think it will be too hard to find the relevant sections. Much of the material is drawn from the end of Chapter 2. These notes, as well as the monograph, may have typos and suffer from “copy-and-paste” mistakes as well as notational inconsistencies. We apologize in advance. I. Comparative Statics and Producer Theory Consider a firm with production function F ( k , l ), where k is capital and l is labor. For the moment, let consider the cost-minimization problem. The firm’s problem is as follows: , .. ( ,) min kl stF kl q rk wl = + In the case where output is strictly increasing in each input and the production function F is suitably well-behaved, we can define the isoquant function L ( k , q ) according to the following implicit function: F ( k , L ( k , q )) = q . Thus, we can rewrite the firm’s problem as follows: max k rk wL ( k , q ) (L1) Questions : Does firm’s choice of capital ( k ) increase or decrease monotonically with the input prices ( r , w ) and the level of output required ( q )? Do the answers depend on any properties of the production function? Notice that we in general want conditions that will allow us to answer these questions independent of the particular parameter values. In other words, we want conditions on the production function F that will be sufficient for comparative statics conclusions, and we do not want to rule out ex ante any particular values of q , r , and w .
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2 To answer the comparative statics questions posed above, and others, we will develop a general theory of univariate comparative statics in problems with an additive structure such as the problem (L1). II. Univariate Comparative Statics: The General Setup The general form of the problem is as follows: x * ( θ , S ) arg max x S f ( x , ) + g ( x ) (L2) The choice variable is x , the constraint set is S , the parameter of interest for comparative statics is . Further, all we know about g is that it belongs to a family of functions ( G ), and we thus ask that our results hold for all g G . What family G is relevant depends on the problem. To be concrete, in the producer theory problem, max k rk wL ( k , q (L1) if we wanted to ask: How does capital change with the rental price r ? Then: x is capital, is the rental price, So that we solve max ( , ) x xw L x q −− f ( x , ) = x . t y p i c a l g ( x ) = w L ( x,q ) G ={ w L ( ,q ) | w , q ∈ℜ + , L is “allowable” isoquant} The family of functions G is generated by varying wages, output quantities, and production functions (we might want to specify some allowable class of production functions w/ corresponding isoquants).
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3 Likewise, if we ask: How does capital change with output q ?
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This note was uploaded on 01/07/2012 for the course ECON 6090 at Cornell University (Engineering School).

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producer_theory_and_MCS - Lecture Notes on Monotone...

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