Question

You are a producer of HW for ECON 301. You use labor time in hours (L) and capital (K) (such as a desk, calculator, or pencil) as inputs to produce HW using q = 2KZ2. Suppose K is fixed at 1 unit in the short-run. The opportunity cost of your time (w) is $8 per hour. The rental rate per unit of capital (() is $50. a) Derive the short-run cost function, i.e. TC as a function of q. b) Derive AC, AFC, AVC and MC functions. c) Sketch the cost function, FC and VC on the same diagram, and AC, AVC and MC functions on a separate diagram

Answer 1

Given

q=2KL^2

Set K=1

q=2L^2

or L^2=0.5q

L=0.7071*q^0.5

a)

TC=r*K+w*L=50*1+8*L=50+8*(0.7071*q^0.5)=50+5.6568q^0.5

b)

In this case,

Fixed Cost=FC=$50

Total Variable Cost=VC=5.6568q^0.5

AFC=FC/q=50/q

AVC=TVC/q=5.6568q^0.5/q=5.6568/q^0.5

AC=AFC+AVC=(50/q)+(5.6568/q^0.5)

MC=dTC/dq=2.8284/q^0.5

c)

q	FC	VC=5.6568q^0.5
0	50	0.00
10	50	17.89
20	50	25.30
30	50	30.98
40	50	35.78
50	50	40.00
60	50	43.82

$ 10 20 30 40 50 60 70

q	AFC	AVC=5.6568/q^0.5	AC=AFC+AVC	MC=2.8284/q^0.5
0
10	5.00	1.79	6.79	0.89
20	2.50	1.26	3.76	0.63
30	1.67	1.03	2.70	0.52
40	1.25	0.89	2.14	0.45
50	1.00	0.80	1.80	0.40
60	0.83	0.73	1.56	0.37