Lecture3 - ECON 301 LECTURE #3 DUALITY OF PRODUCTION AND...

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1 ECON 301 – LECTURE #3 DUALITY OF PRODUCTION AND COST What do we mean by the duality of production and cost? Well, essentially there are many distinct similarities in the logical structure of the theories of production and cost. In reality, production and cost are two sides of the same coin. That is, an efficient producer must consider the 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 COST MINIMIZATION The primal problem is to get the highest The dual problem is to get the lowest output possible from a given cost level. factor cost required to produce a given output level. That is, for a given level of cost, ! , the For a given level of output, Q " , the producer must choose K & L to maximize producer must choose K & L to output, Q. minimize cost, C. Choose K & L to maximize output f (K,L) subject to rK + wL = ! rK + wL = C subject to f (K,L) = Q " Objective: Highest isoquant Objective: Lowest isocost Constraint: Fixed isocost Constraint: Fixed isoquant L K Lowest isocost for a given isoquant L K Highest isoquant for a given isocost
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2 As before, we need to satisfy two conditions As before, we need to satisfy two of producer equilibrium. conditions of producer equilibrium. MRTS = _r_ MRTS = _r_ w w rK + wL = ! f (K , L) = Q " Solving these two equations, we get the Solving these two equations, we get producer demands for K & L as functions producer demands for K & L as of factor prices r , w, and cost level, ! . functions of r,w, and output level, Q " . K = K(r , w , ! ) K = K(r , w , Q " ) L = L(r , w , ! ) L = L(r , w , Q " ) If we substitute these demands into If we substitute these demands into the objective function, we get the the objective function, we get the maximum output. minimum cost. Q = f (K(r , w , ! ) , L(r , w , ! )) C = r K(r , w , Q " ) + w L(r , w , Q " ) Q = Q(r , w , ! ) C = C(r , w , Q " ) Q = Q( ! ) C = C(Q " ) Quantity can now be expressed as a function Cost can be expressed as a function of the given cost level. of the given output level. PRODUCER’S DUALITY EXAMPLE Duality theory provides us with a procedure to construct a cost function from only two pieces of information, namely, a production function and factor prices. For example, suppose we have unit factor prices r = w = 1 and the following Cobb-Douglas production function Q = K 4/5 L 1/5 Q = K 0.8 L 0.2
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3 Using our “short cut” from last class, we get the marginal rate of technical substitution MRTS = 0.8L 0.2K = 4L K We solve the problem using both the primal and dual formulations as follows: Output Maximization (Primal) Using the primal formulation of output maximization, we have the following two
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This note was uploaded on 03/28/2010 for the course ECON 301 taught by Professor Coreyvandewaal during the Winter '09 term at Waterloo.

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Lecture3 - ECON 301 LECTURE #3 DUALITY OF PRODUCTION AND...

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