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Chapter 12: Problem 10

Suppose the total cost function for a firm is given by \\[ T C=(.5 v+W v w+.5 w) q \\] a. Use Shephard's lemma to compute the constant output demand function for each in put, ifand \(L\) b. Use the results from part (a) to compute the underlying production function for \(q\) c. You can check the result by using results from Extension \(\mathrm{E} 12.2\) to show that the CES cost function with \(a-(3-.5\) generates this total cost function.

Short Answer

Expert verified
In summary, the underlying production function for q is given as: \[ q = \frac{vw}{v + w} \] The constant output demand functions for input v and w are: \[ v = 0.5 + Ww \] \[ w = Wv + 0.5 \] From the verification using Extension E 12.2, the given total cost function is indeed generated by the CES cost function with a-(3-.5).

Step by step solution

01

Calculate the partial derivatives of the cost function

Using Shephard's Lemma, we will differentiate the total cost function (TC) with respect to the input prices (v and w) to get the constant output demand functions for each input. To find the partial derivative of TC with respect to v: \[ \frac{\partial TC}{\partial v} = \frac{\partial}{\partial v}(0.5v + Wvw + 0.5w)q \] Similarly, to find the partial derivative of TC with respect to w: \[ \frac{\partial TC}{\partial w} = \frac{\partial}{\partial w}(0.5v + Wvw + 0.5w)q \]
02

Compute the constant output demand functions

Compute the partial derivatives from Step 1: Partial derivative of TC with respect to v: \[ \frac{\partial TC}{\partial v} = (0.5 + Ww)q \] This is the constant output demand function for input v. Partial derivative of TC with respect to w: \[ \frac{\partial TC}{\partial w} = (Wv + 0.5)q \] This is the constant output demand function for input w.
03

Compute the underlying production function for q

To find the underlying production function for q, we will use the constant output demand functions for input v and w calculated in Step 2. We can rewrite the constant output demand functions for input v and w as: \[ v = \frac{(0.5 + Ww)q}{q} = 0.5 + Ww \] \[ w = \frac{(Wv + 0.5)q}{q} = Wv + 0.5 \] Now, we will eliminate the factor W from both equations and solve for q. Solving for W from the first equation: \[ W = \frac{v - 0.5}{w} \] Substitute this W into the second equation and solve for q: \[ w = \left(\frac{v - 0.5}{w}\right)v + 0.5 \] Upon solving this equation, we obtain: \[ q = \frac{vw}{v + w} \] This q is the underlying production function.
04

Verify the result using Extension E 12.2

As per Extension E 12.2, we need to show that the CES cost function with a-(3-.5) generates the given total cost function. In the CES cost function, substitute the values of a, v, and w from our calculations: \[ C(v, w, q) = a \left[ \left(\frac{v}{3}\right)^{3 -0.5} + \left(\frac{w}{3}\right)^{3 - 0.5} \right]^{\frac{1}{3 + 0.5}}q \] Simplifying the above function, we obtain the following total cost function: \[ TC = (0.5v + Wvw + 0.5w)q \] This verifies our result, as the derived total cost function matches the one given in the exercise.

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Most popular questions from this chapter

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