Question

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1. Short Run Cost Curves: Consider two firms, producing different products, with the following production functions: q=5KL (1) q=5(KL)-S (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 Lin terms of q, and then substitute that, and the given parameters, into the generic expression for Total Cost = WL + YK 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). (10) 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 capital goes up to 2. (Continue to assume we're in the short run and K can not be altered). How would this change alter your total cost curve? Plot these curves and label them "TC2."(5) 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. (5) 2. Perfect Competition and Profit Maximization: Continue using the production function and cost curves from equation (2) above. Assume this firm operates in a perfectly competitive market, and that the market clearing price in this market is $10. a. On a new graph, plot the Marginal Revenue curve, with MR as a function of q, and the Marginal Cost curve, with Marginal Cost as a function of q (with marginal cost based on your answer to part (a) above). What is the profit maximizing level of output for this firm? (10) b. Suppose that the cost of capital increases from 1 to 2. How would this affect the profit- maximizing value of q? Explain. (5) C. Suppose the wage increases from 3 to 4 - how would this affect the profit-maximizing level of q? Explain. (5)

Answer 1

a. Firm one and two's production functions in the short run transform to:
$\dpi{80} \fn_phv \small q_1=500L, q_2=50\sqrt{L}$

Using these production functions, amount of labor employed by these firms in the short run is:
$\dpi{80} \fn_phv \small L_1=\frac{q_1}{500}, L_2=(\frac{q_2}{50})^2$

The cost function for firm one is:
$\dpi{80} \fn_phv \small C_1=3L_1+K_1=\frac{3q_1}{500}+100$

The average and marginal cost functions are:
$\dpi{80} \fn_phv \small AC_1=\frac{3}{500}+\frac{100}{q_1}, MC_1=\frac{3}{500}$

The cost function for firm two is:
$\dpi{80} \fn_phv \small C_2=3L_2+K_2=3*(\frac{q_2}{50})^2+100$
The average and marginal cost functions are:
$\dpi{80} \fn_phv \small AC_2=\frac{3q_2}{(50)^2}+\frac{100}{q_2}, MC_2=\frac{6q_2}{50}$

b. Plotting these cost functions:
TC1 for firm 1:

150 100 -50 100 0 150 200 50

TC1 for firm 2:

200 150 100 -50 200 250 0 100 150 50

When the cost of capital increases to 2, total cost functions of the two firms are:
$\dpi{80} \fn_phv \small C_1=\frac{3q_1}{500}+200, C_2=3*(\frac{q}{50})^2+200$

TC2 for firm 1:

200 150 100 50 150 200 250 100 0 50

TC2 for firm 2:

300 250 200 150 100 50 250 200 300 0 100 150 50

c. When wages fall to 2, cost functions for the two firms are:
$\dpi{80} \fn_phv \small C_1=\frac{2q_1}{500}+200, C_2=2*(\frac{q}{50})^2+200$

TC3 for firm 1:

250 200 150 100 50 250 350 400 0 50 100 150 200 300

TC3 for firm 2:

300 250 200- 150 100 50 150 0 100 200 250 300 50