Question

You are planning to estimate a short- run production function for your firm, and you have collected the following data on labor usage (L) and output (Q): Labor usage (L) Output (Q)

3 1

7 2

9 3

11 5

17 8

17 10

20 15

24 18

26 22

28 21

30 23

a. Please key in the data into MS Excel for regression analysis. Estimate your firm’s short-run production function. Do the parameter estimates have the appropriate algebraic signs? Are they statistically significant at the 5 percent level? (Hint: Run the production function as Q = AL 3 +BL 2 )

b. At what point do you estimate marginal product (MP) begins to fall?

c. Calculate estimates of average products (AP) and marginal products (MP) when the firm employs 20 workers.

d. When the firm employs 20 workers, is short-run marginal cost (MC) rising or falling? How can you tell?

Answer 1

a.

Output (Q)	L^2	L^3
1	9	27
2	49	343
3	81	729
5	121	1331
8	289	4913
10	289	4913
15	400	8000
18	576	13824
22	676	17576
21	784	21952
23	900	27000

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.996664476
R Square	0.993340078
Adjusted R Square	0.881488976
Standard Error	1.277662242
Observations	11
ANOVA
	df	SS	MS	F	Significance F
Regression	2	2191.308213	1095.654106	671.1836214	1.23183E-09
Residual	9	14.69178723	1.632420803
Total	11	2206
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
L3	-0.00074509	0.00019768	-3.769163007	0.004422153	-0.001192274	-0.000297906
L2	0.048730818	0.005274689	9.238614311	6.8893E-06	0.036798642	0.060662993

The production function, Output (Q) = -0.000745L^3+0.048*L^2
Yes with the increase in labor the output increases
Yes, they are statistically significant at 5 percent level

b.

MP =change in Q/change in L

MP begins to fall after labor usage of 20

Labor usage (L)	Output (Q)	MP
3	1
7	2	0.25
9	3	0.50
11	5	1.00
17	8	0.50
17	10
20	15	1.67
24	18	0.75
26	22	2.00
28	21	-0.50
30	23	1.00

c.

Labor usage (L)	Output (Q)	MP	AP
3	1		0.33
7	2	0.25	0.29
9	3	0.50	0.33
11	5	1.00	0.45
17	8	0.50	0.47
17	10		0.59
20	15	1.67	0.75
24	18	0.75	0.75
26	22	2.00	0.85
28	21	-0.50	0.75
30	23	1.00	0.77

At labor = 20, AP=0.75, MP=1.67

d. At, labor = 20, the MP starts to fall, which means the production starts to decline which cause the cost or MC to raise