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Unformatted text preview: Definition and Properties of the Cost Function Lecture XIII From Previous Lectures In the preceding lectures we first developed the production function as a technological envelope demonstrating how inputs can be mapped into outputs. Next, we showed how these functions could be used to derive input demand, cost, and profit functions based on these functions and optimizing behavior. In this development, we stated that economist had little to say about the characteristics of the production function. We were only interested in these functions in the constraints that they imposed on optimizing behavior. Thus, the insight added by the dual approach is the fact that we could simply work with the resulting optimizing behavior. In some cases, this optimizing behavior can then be used to infer facts about the technology underlying it. Gorman (1976) Duality is about the choice of the independent variables in terms of which one defines a theory. Chambers (p. 49) The essence of the dual approach is that technology (or in the case of the consumer problem, preferences) constrains the optimizing behavior of individuals. One should therefore be able to use an accurate representation of optimizing behavior to study the technology. The Cost Function Defined The cost function is defined as: Literally, the cost function is the minimum cost of producing a given level of output from a specific set of inputs. This definition depends on the production set V ( y ). In a specific instant such as the Cobb Douglas production function we can define this production set analytically. ( 29 ( 29 , min : x c w y w x x V y = Technology constrains the behavior or economic agents. For example, we will impose the restriction on the technology so that at least some input be used to produce any nonzero level of output. The goal is to place as few of restrictions on the behavior of economic agents as possible to allow for the derivation of a fairly general behavioral response. Not to loose sight of the goal, we are interested in be able to specify the cost function based on input prices and output prices: Is a standard form of the quadratic cost function...
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 Fall '09
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