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Unformatted text preview: he following twin problems: On one hand, in terms of production, a producer needs to consider the primal problem of producing the highest level of output for a fixed cost constraint (output maximization problem). On the other hand, in terms of cost, the same producer needs to consider the dual problem of keeping the lowest level of cost for a fixed production level (cost minimization problem). OUTPUT MAXIMIZATION The primal problem is to get the highest output possible from a given cost level. That is, for a given level of cost, C, the producer must choose K & L to maximize output, Q. Choose K & L to maximize output f(K,L) subject to rK + wL = C
L COST MINIMIZATION The dual problem is to get the lowest factor cost required to produce a given output level. For a given level of output, Q , the producer must choose K & L to minimize cost, C. Choose K & L to minimize cost rK + wL = C subject to f(K,L) = Q
L Highest isoquant for a given isocost Lowest isocost for a given isoquant K K Objective: Highest isoquant Constraint: Fixed isocost Objective: Lowest isocost Constraint: Fixed isoquant 39 As before, we need to satisfy two conditions of producer equilibrium. MRTS = _r_ w rK + wL = C Solving these two equations, we get the producer demands for K & L as functions of factor prices r , w, and cost level, C. K = K(r , w , C) L = L(r , w , C) If we substitute these demands into the objective function, we get the maximum output. Q = f(K(r , w , C) , L(r , w , C)) Q = Q(r , w , C) Q = Q(C) Quantity can now be expressed as a function of the given cost level. PRODUCER'S DUALITY EXAMPLE As before, we need to satisfy two conditions of producer equilibrium. MRTS = _r_ w f(K , L) = Q Solving these two equations, we get producer demands for K & L as functions of r,w, and output level, Q . K = K(r , w , Q ) L = L(r , w , Q ) If we substitute these demands into the objective function, we get the minimum cost. C = r K(r , w , Q ) + w L(r , w , Q ) C = C(r , w , Q ) C = C(Q ) Cost can be expressed as a...
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This note was uploaded on 05/25/2010 for the course ECON 301 taught by Professor Sning during the Spring '10 term at University of Warsaw.
 Spring '10
 sning
 Economics

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