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Unformatted text preview: function of the given output level. 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 = K4/5L1/5 Q = K0.8L0.2 40 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 equations of producer equilibrium: MRTS = _r_ w rK + wL = C 4L K K+L=C Solving these two equations, we get the factor demands for K & L 4L = _1_ K 1 4L = _1_ K 1 4L = K 4L = K 5L = C L = 0.2C K = 0.8C L = 0.25K 5/4 K = C Substituting these factor demands into the production function, we get the optimal output as a function of the cost term Q = K0.8L0.2 = (0.8C)0.8(0.2C)0.2 = (0.8)0.8(0.2)0.2C0.8 + 0.2 = (0.836511642)(0.724779663)C = 0.606286626C 41 Rearranging the result so that the cost term is on the left hand side, we get the cost term as a function of the output level. C = _____Q_____ 0.606286626 = 1.64938489Q Which is the cost function! Cost Minimization (Dual) Using the dual formulation of cost minimization, we have the following two equations of producer equilibrium: MRTS = _r_ w f(K,L) = Q 4L K K0.8L0.2 = Q Solving these two equations, we get the factor demands for K & L 4L = _1_ 4L = K (4L)0.8L0.2 = Q 40.8 L0.8 L0.2 = Q 3.031433133L = Q K 1 So, L = _____Q _____ 3.031433133 or simply, L = 0.329876977Q 4L = _1_ 4L = K L = 0.25K K0.8(0.25K)0.2 = Q (0.25)0.2 K0.8 K0.2 = Q K 1 0.757858283K = Q So, K = _____Q _____ 0.757858283K
ASIDE: Once we had the demand for labour, L = 0.329876977Q , we could have simply taken the relationship K = 4L and calculated: K = 4(0.329876977Q ) = 1.319507908Q (difference in 9th decimal place is due to machine im...
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- Spring '10