Lecture notes 4: Theory of production
                                             Hannu Vartiainen
                                                    HECER
                                                   Fall 2015
                                 Hannu Vartiainen HECER     Production
         Producer theory
                Start with a single …rm facing given prices
                Need to describe the technology of the …rm
                Exogenous: prices
                Endogenous: output and input demands
                Aim to understand the optimal production decision of the …rm
                       No attention to organizational nor stratgic aspects
                       Objective to have a model that can be transferred in it its pure
                       form to the general equilibrium framework
                Key di¤erence to the consumer model
                       no income e¤ects
                       everything observable
                                 Hannu Vartiainen HECER     Production
         Primitives:
                Firm with one production good in R+
                                     K
                Input space R
                The primitive of the model: production function
                                                          K
                                                   f : R+ ! R+
                describes the output/input combinations that are
                technologically feasibe
                                 Hannu Vartiainen HECER     Production
         Axiom
         Production function f is continuous, increasing, and quasiconcave
                By monotonicity, if y  y0, then f (y)  f (y0)
                By quasiconcavity, the input requirement set
                                        K
                V(x) = fy 2 R+ : f(y)  xg is convex for all x 2 R+
                Firm’s production function can be represented by the
                production possibility set
                                                             K+1
                                     Y =f(y,x) 2 R+                 : f (y)  xg
                Continuous, increasing, and quasiconcave production function
                corresponds to a production possibility set Y that is
                       convex: if (y,x),(y0,x0) 2 Y, then
                       λ(y,x)+(1+λ)(y0,x0) 2 Y for all λ
                       monotonic: y 2 V(x) and y0  y imply y 2 V(x)
                       closed
                                 Hannu Vartiainen HECER     Production