Question

Print this page out and use the space below and the back to write out and clearly indicate your answers. Do not use this sheet as scrap paper, but use it to neatly present your work. Suppose a firm produces output y from input x according to the production function y = 4x1/3. 1) For general output price p and factor price w, write down the firm's profit function for profit II in terms of input x. Find the optimal level of input x in terms of p and w. (2pts) 2) Sketch the firm's demand curve for the input x. Label the axis clearly. Explain what happens to the quantity demanded of x when the input price w decreases. Explain what happens to the demand for x when the output price p increases. (3pts) 3) Write down the expression for output y in terms of p and w. Sketch the relationship between output y (on the horizontal axis) and output price of p (on the vertical axis). Explain what happens to the quantity supplied of y when the output price p increases. Explain what happens to the supply for y when the input price w decreases. (3pts) 4) When w = 2 and p = 24, what is the quantity of input x that the firm demands? What quantity of output y does the firm supply? (2pts)

Answer 1

1. The production function is given as $\dpi{80} y = 4 x^{1/3}$ and the cost of production is $\dpi{80} c = wx$ . The total revenue would be $\dpi{80} TR = py$ or $\dpi{80} TR = 4 p x^{1/3}$ . The profit of the firm would be $\dpi{80} \pi = TR - c$ or $\dpi{80} \pi = 4 p x^{1/3} - wx$ . The FOC for the maximization of the profit would be as below.

$\dpi{80} \frac{\mathrm{d} \pi}{\mathrm{d} x} = 0$

or $\dpi{80} \frac{\mathrm{d} }{\mathrm{d} x}(4 p x^{1/3} - wx) = 0$

or $\dpi{80} (4/3) p x^{-2/3} - w = 0$

or $\dpi{80} (4/3) p x^{-2/3} = w$

or $\dpi{80} x^{-2/3} = \frac{w}{(4/3) p}$

or $\dpi{80} x^{2/3} = \frac{(4/3) p}{w}$

or $\dpi{80} x^{2/3} = \frac{4 p}{3 w}$

or $\dpi{80} x = \frac{4^{3/2} p^{3/2}}{3^{3/2} w^{3/2}}$

or $\dpi{80} x^* = \frac{8 p^{3/2}}{3^{3/2} w^{3/2}}$ . This is the required demand for input x. This is the optimal level of x required given p and w.

2. The graph is as below, for p=1 and p=2.

ו 4 3 - K(p=1) \X(p=2) 2- Bו א 2 4 LO <ה צי ו ד בם זא]

As can be seen, as price increases (from 1 to 2 in this example), the input demand shifts right, meaning that the demand for input increases for all w's. Also, as we move from A to B, the wage decreases, but the input demand increases.

3. Putting the input demand in the production function, we have $\dpi{80} y = 4 \left (\frac{8 p^{3/2}}{3^{3/2} w^{3/2}} \right )^{1/3}$ or $\dpi{80} y = \frac{8 p^{1/2}}{3^{1/2} w^{1/2}}$ . The graph is below for w=2 and w=1.

10 8- 6 y(W-2) 4- y(w=1 2- В. A y 10 6 0O- גרם 00

As can be seen, as we move from point A to B, the price increases, and the output supply increases as well. Also, as wage decreases (from 2 to 1 in this example), the output supply increases for all level of p's.

4. The input demand for the given values would be $\dpi{80} x^* = \frac{8 * 24^{3/2}}{3^{3/2} 2^{3/2}}$ or $\dpi{80} x^* = \frac{64 *2^{3/2}*3^{3/2}}{3^{3/2} 2^{3/2}}$ or $\dpi{80} x^* = 64$ units. The output would be $\dpi{80} y = \frac{8 *24^{1/2}}{3^{1/2} 2^{1/2}}$ or $\dpi{80} y = \frac{16 *2^{1/2} 3^{1/2}}{3^{1/2} 2^{1/2}}$ or $\dpi{80} y = 16$ units. One may simply put x=64 in the production function also, to obtain same answer.