Suppose the total cost function for a firm is given by \\[ T C=q w^{2 / 3} v^{1 / 3} \\] a. Use Shephard's lemma (footnote 8 ) to compute the constant output demand functions for inputs \(L\) and \(K\) b. Use your results from part (a) to calculate the underlying production function for \(q\)

We are given the total cost function: \\[ T C = q w^{2 / 3} v^{1 / 3} \\] We need to compute the partial derivatives with respect to wages (w) and rental rates (v). Partial derivative with respect to w: \\[ \frac{\partial TC}{\partial w} = q \cdot \frac{2}{3} w^{-1 / 3} v^{1 / 3} \\] Partial derivative with respect to v: \\[ \frac{\partial TC}{\partial v} = q \cdot \frac{1}{3} w^{2 / 3} v^{-2 / 3} \\]

According to Shephard's Lemma, the partial derivatives of the cost function with respect to input prices are equal to the input demand functions. Therefore, we can write the input demand functions for labor (L) and capital (K) as follows: Constant output demand function for labor (L): \\[ L = \frac{\partial TC}{\partial w} \\] \\[ L = q \cdot \frac{2}{3} w^{-1 / 3} v^{1 / 3} \\] Constant output demand function for capital (K): \\[ K = \frac{\partial TC}{\partial v} \\] \\[ K = q \cdot \frac{1}{3} w^{2 / 3} v^{-2 / 3} \\]

Now we have the input demand functions for labor (L) and capital (K). To find the underlying production function for output (q), we need to eliminate the labor and capital terms from the demand functions. First, rearrange the demand functions to solve for w and v: From the labor demand function, \\[ w = \left(\frac{3L}{2qv}\right)^{3/2} \\] From the capital demand function, \\[ v = \left(\frac{3K}{qw}\right)^{3/2} \\] Now, substitute the expression for w from the labor demand function into the capital demand function to eliminate w: \\[ v = \left(\frac{3K}{q\left(\frac{3L}{2qv}\right)^{3/2}}\right)^{3/2} \\] Simplify the expression to find the production function, which represents the relationship between the quantity of output (q) and the inputs of labor (L) and capital (K): \\[ q = \sqrt[3]{\frac{9K^2L^3}{4}} \\] This is the underlying production function for output (q) in terms of labor (L) and capital (K).