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A firm production function is given by q = f(k,l) = kl. q_0 = 100. w = $20, v = $5. What is the...

Question:

A firm production function is given by q = f(k,l) = kl. q_0 = 100. w = $20, v = $5. What is the value of the Lagrange multiplier ? associated with the cost minimizing input choice? NOTE: write your answer in number format, with 2 decimal places of precision level. Show all work.

Cost:

Cost for the producers refers to the quantitative or the monetary value of the inputs used in the production process. For the consumers cost refers to the monetary value that they have to give up to consumer certain bundle of goods and services. All individuals in the economy desire to reduce the cost need to pay by them.

Answer and Explanation:

The cost is minimized with the given production function as below:

{eq}\begin{array}{c} Q = KL\\ TC = 20L + 5K\\ {\rm{Lagrange}}\,{\rm{Equation,}}\\ l = 20L + 5K + \lambda \left[ {Q - KL} \right]\\ {\rm{Partially}}\,{\rm{derivate,}}\\ \frac{{dl}}{{dK}} = 5 - \lambda L \end{array} {/eq}

{eq}\begin{array}{c} \frac{{dl}}{{dL}} = 20 - \lambda \left( K \right)\\ \frac{{dl}}{{dL}},\frac{{dl}}{{dL}} = 0\\ \left( {\frac{{20}}{5}} \right) = \frac{K}{L}\\ K = 4L \end{array} {/eq}

{eq}\begin{array}{c} Q = L\left( {4L} \right)\\ 100 = 4{L^2}\\ L = 5\\ K = 20 \end{array} {/eq}

{eq}\begin{array}{c} 20 = \lambda K\\ 20 = \lambda \left( {20} \right)\\ \lambda = 1 \end{array} {/eq}

Thus the value of Lagrange multiplier is 1.


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