Question

5. A firm produces widgets with production function: q-2vKL. In the short run, the firm's amount of capital is fixed at K = 100. The rental rate is v = 1 and the wage for L is w= 4. (a) Find the firm's short-run total cost curve (SRTC), short-run average cost curve (SRAC), and the short-run marginal cost (SMC) function. (b) Graph the firm's SAC and SMC using the following levels of production: q 25 and q= 100. (c) Find the firm's total cost curve as a function of q, w, v, and K. (d) What level of K should be chosen to minimize total cost (note: K will be a function of q,w, v)? (e) Use the above results to calculate the long-run total cost of production.

Answer 1

a)

Put K=100

Squaring both sides, we get

Short run total cost=SRTC=100*1+4L=100+4L=100+4*q²/400=100+q²/100

Short run average cost=SRAC=SRTC/q=(100+q²/100)/q=(100/q)+(q/100)

Short run marginal cost function=d(SRTC)/dq=2q/100=q/50

b)

We can make the following schedule with the help of above mentioned formulas.

q	SRAC	SMC
0
10	10.10	0.20
20	5.20	0.40
25	4.25	0.50
30	3.63	0.60
40	2.90	0.80
50	2.50	1.00
60	2.27	1.20
70	2.13	1.40
80	2.05	1.60
90	2.01	1.80
100	2.00	2.00
110	2.01	2.20
120	2.03	2.40
130	2.07	2.60

c)

Squaring both sides, we get

Total Cost, TC=vK+wL

Differentiate TC with respect to K, we get

Put

d)

We have derived in the above part

Minimum TC is given by