Question

Short Run Cost Curves: Consider two firms, producing different products, with the following production functions:

q=5KL (1)

q=5(KL).5 (2)

a. For a short-run situation in which K=100, and given wage = 3 and cost of capital = 1, derive expressions short run total cost for each production function. (Start by using the production function to develop an expression for L in terms of q, and then substitute that, and the given parameters, into the generic expression for Total Cost = wL + γK to get cost as a function of q.) Also derive the short run average cost and short run marginal cost curves for both functions (with cost as a function of q in each case).

b. Plot each short-run total cost curve on a separate graph of cost (on the vertical axis) vs. q (on the horizontal axis). Label these curves “TC1.” Now suppose (for each case) that the cost of capitalgoes up to 2. (Continue to assume we’re in the short run and K can not be altered). How would thischange alter your total cost curve? Plot these curves and label them “TC2.”

c. Finally, beginning from your original TC1 curve, suppose that wages fall to 2. How would this alter your TC curve? Plot these new curves and label them “TC3” in each case.

Answer 1

(a) For the short run situation, the production functions would be as $\dpi{80} q= 5 (100)L$ or $\dpi{80} q= 500 L$ and $\dpi{80} q= 5((100)L)^{0.5}$ or $\dpi{80} q= 50 L^{0.5}$ .

For the first one, the labor demand would be as $\dpi{80} q= 500 L$ or $\dpi{80} L^*= \frac{q}{500}$ . The cost of production is $\dpi{80} C = wL + rK$ or $\dpi{80} C= 3L + 100$ , and the cost function would be $\dpi{80} C= 3L^* + 100$ or $\dpi{80} C= 3*\frac{q}{500} + 100$ or $\dpi{80} C= \frac{3q}{500} + 100$ . The SRAC would be $\dpi{80} AC_1= \frac{C}{q} = \frac{3}{500} + \frac{100}{q}$ and the SRMC would be $\dpi{80} MC_1= \frac{\mathrm{d} C}{\mathrm{d} q} = \frac{3}{500}$ .

For the second one, the labor demand would be as $\dpi{80} q= 50 L^{0.5}$ or $\dpi{80} L^*= \frac{q^2}{2500}$ . The cost of production is $\dpi{80} C = wL + rK$ , and the cost function would be $\dpi{80} C= 3L^* + 100$ or $\dpi{80} C= 3*\frac{q^2}{2500} + 100$ or $\dpi{80} C= \frac{3 q^2}{2500} + 100$ . The SRAC would be $\dpi{80} AC_2= \frac{C}{q} = \frac{3 q}{2500} + \frac{100}{q}$ and the SRMC would be $\dpi{80} MC_2= \frac{\mathrm{d} C}{\mathrm{d} q} = \frac{6 q}{2500}$ .

(b) The graph for first production function would be as below.

200 150 TC1 H00 50 2000 4000 6000 8000 10000

The graph for second production function would be as below.

200 150 TC1 400 50 200 100 300 400 500

If the price of capital is 2, we have the cost of production as $\dpi{80} C= 3L + 200$ , and the cost function would be $\dpi{80} C= 3*\frac{q}{500} + 200$ or $\dpi{80} C= \frac{3q}{500} + 200$ . The graph is as below.

400 300 TC2 200 TC1 400 2000 4000 6000 8000 10000 4

For the second production function, we have the cost of production as $\dpi{80} C= 3L + 200$ , and the cost function would be $\dpi{80} C= 3*\frac{q^2}{2500} + 200$ or $\dpi{80} C= \frac{3q^2}{2500} + 200$ . The graph is as below.

400 300 TC2 200 TO1 400 200 100 300 400 500

(c) The cost of production would be $\dpi{80} C= 2L + 100$ , and the cost function would be $\dpi{80} C= 2*\frac{q}{500} + 100$ or $\dpi{80} C= \frac{2q}{500} + 100$ . The graph is as below.

400 300 200 TC1 ТСЗ 400 2000 4000 6000 8000 10000

For the second production function, we have the cost of production as $\dpi{80} C= 2L + 100$ , and the cost function would be $\dpi{80} C= 2*\frac{q^2}{2500} + 100$ or $\dpi{80} C= \frac{2q^2}{2500} + 100$ . The graph is as below.

400 300 TC1 200 TC3 400 200 100 300 400 500