Question

Using the data table:

1. Does the data suggest that the two inputs are perfect complements? Explain.
2. Does the data suggest that the two inputs are Perfect Substitutes? Explain.
3. Does the data suggest that the production function is Cobb-Douglas? Explain.
4. What type of returns to scale do you observe? Explain.
5. Complete Table 2.

Assuming that the price of labor is w = 10 and the price of capital is k=40, use Table 1 to do the following:

1. Complete Table 3.
2. If the firm can sell the good being produced at $73 per unit, determine whether the firm should produce and sell 1, 2, or 4 units.
3. Complete Table 4 by determining the cheapest way to produce the given amounts of output in Table 4 (using the information you have from Table 1).
   TABLE 1: Data TABLE 2: APL and MP TABLE 4: Long Run Costs APL MPL TC ATC 4.00 3.00 8.00 16.00 1.83 2.43 2.76 2.43 2.60 2.76 4.00 TABLE 3: Short Run Costs TC AFC AVC ATC 16 8.00 9.00 16.00 8.00 9.00 16.00 9.00 16.00 8.00 16.00 16.00 18 16 2.43 16 16 2.76 4.00 5.83 9.00 16 32
   

Answer 1

If capital and labor are complements, a decrease in K always leads to a decrease in L, since capital increases the incremental returns to labor. In contrast, when capital and labor are substitutes, a decrease in K leads to a weak increase in L, and so capital and labor will move in opposite directions. From the table, we can see that they are substitutes.

The Cobb-Douglass production function is of the form: Y = K^AL^B Where Y=output, K=amount of capital, L=amount of labor. Again, from the table, we see that increase in K and L by factors of 2 change the output by values to some power of 2. This indicates a Cobb-Douglass production function.

If A+B = 1, the production function exhibits constant returns to scale. If A+B >1, the production function exhibits increasing returns to scale If A+B <1, the production function exhibits decreasing returns to scale.

If we use Y=K^A L^B to understand whether the production exhibits increasing, decreasing or constant returns to scale, we want to look at how 2Y compares to Y(2K,2L) = (2K)^A (2L)^B . Does doubling both inputs exactly double output? If (2K)^A (2L) ^B > 2Y the production function exhibits increasing returns to scale. If (2K)^A (2L)^B < 2Y the production function exhibits decreasing returns to scale. If (2K)^A (2L)^B = 2Y the production function exhibits constant returns to scale. This data gives constant returns to scale since for K =1, L = 4, Y = 1.00. For K=2,L=8, Y=2.00.

APL = Q/L and MPL is the change in APL due to increase in L by 1.

K	L	Q	APL	MPL
4	0	1.00	-	-
4	1	1.56	1.56	1.56
4	2	1.83	0.915	0.27
4	3	2.05	0.683	0.22
4	4	2.25	0.5625	0.20
4	5	2.43	0.456	0.18
4	6	2.60	0.433	0.17
4	7	2.76	0.394	0.16

Next, we compute Table 3. Fixed cost is cost of capital (4*40 = 160). Variable costs are costs of labor.
MC = change in TC/change in Q
AFC = FC/Q

K	L	Q	FC	VC	TC	MC	AFC	AVC	ATC
4	0	1.00	160	-	160	-	160	-
4	1	1.56	160	10	170	17.86	102.56	6.42	108.98
4	2	1.83	160	20	180	37.04	87.43	10.93	98.36
4	3	2.05	160	30	190	45.45	78.05	14.63	92.68
4	4	2.25	160	40	200	50	71.11	17.78	88.89
4	5	2.43	160	50	210	55.55	65.84	20.58	86.42
4	6	2.60	160	60	220	58.82	61.54	23.08	84.62
4	7	2.76	160	70	230	62.5	57.97	25.36	83.33
4	16	4.00	160	160	320	72.58	40	40	80
4	32	5.83	160	320	480	87.43	27.44	54.88	82.32
4	64	9.00	160	640	800	100.95	17.78	71.11	88.89

Now for Table 4, for each value of Q we look at cost minimizing values in Table 1.
Thus, for Q = 1, we have K = 1 and L = 4 as minimum cost option, giving TC = 80.
For Q = 2, K = 2, L = 8. Thus, TC = 160.
For Q = 4, K = 4, L = 16, thus, TC = 320.

If we sell each unit at 73,
Q = 1 ] 73 - 80 = -7
Q = 2 ] 146 - 160 = -14
Q = 4 ] 292 - 320 = -28

Hope this helped!