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

22 15

24 18

26 22

28 21

30 20

33 19

34 17

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 = AL3 +BL2)

b. Please obtain the average product (AP) function and marginal product (MP) function.

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

Answer 1

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?

Ans:- Data of labour uses and Output are already given . As per regression run we have to find out which one is Dependent and which one is Independent variable and Dependent variable is always a function of independent variable in the econometric functions.So In this case Output is dependent and Labour usage is Independent variable. In the hind they mention that we have to calculate the square and cube of labor usage and that is also fall in the independent variable.

The below table is showing the calculation of Square of Labour Usage and Cube of Labour Usage

Q	L	L^2	L^3
1	3	9	27
2	7	49	343
3	9	81	729
5	11	121	1331
8	17	289	4913
10	17	289	4913
15	22	484	10648
18	24	576	13824
22	26	676	17576
21	28	784	21952
20	30	900	27000
19	33	1089	35937
17	34	1156	39304

After run the regression in Excel the summary output is mentioned below;

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.98780403
R Square	0.975756801
Adjusted R Square	0.967675735
Standard Error	1.40523345
Observations	13
ANOVA
	df	SS	MS	F	Significance F
Regression	3	715.3047936	238.4349312	120.7460473	1.37747E-07
Residual	9	17.77212943	1.974681048
Total	12	733.0769231
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	4.9707668	2.471331018	2.011372318	0.075161432	-0.619772354	10.56130595	-0.619772354	10.56130595
L	-1.457879615	0.531976978	-2.740493808	0.022829121	-2.661295145	-0.254464086	-2.661295145	-0.254464086
L^2	0.154731911	0.031571155	4.901053309	0.000846417	0.083312998	0.226150824	0.083312998	0.226150824
L^3	-0.002965512	0.000549104	-5.400635049	0.000432647	-0.004207672	-0.001723352	-0.004207672	-0.001723352

After getting all the variable of L, L^2 and L^3 we have to calculate the Output

Q(Output) = 4.9708 - 1.4579*L + 0.1547*L^2 - 0.0029*L^3

All the three value of Labour usage's P value is less than 0.05 but Labour Usage(L) and Labour Usage Cube (L^3) has negative sign of coefficient which means labour usage increase the output will fall but the Labour usage Square(L^2) is statistically significance which not only P value is less than 0.05 the coefficient also positive sign which tell when the labour usage increase the output also increase.

b. Please obtain the average product (AP) function and marginal product (MP) function.

Ans:- After getting the Output equation we can calculate the Marginal and Average Function

Q(Output) = 4.9708 - 1.4579*L + 0.1547*L^2 - 0.0029*L^3

Marginal product(MP) is nothing but its a addition in output due to increase a variable input bye one unit. so the equation of MP is

dQ/dL = - 1.4579+ 0.3094L - 0.0087 L^2

Average Product(AP) is the total out divided by total labour usage and the equation looks like ;

Average Product(AP) = Q/L

Total Output is 161 and Total Labour Usage is 261

= 161/261 = 0.6168

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

Ans:- In our above problem we already got the Marginal Product function which is explain below

Marginal Product ( MP) dQ/dL = - 1.4579+ 0.3094L - 0.0087 L^2

If the Firm employs 20 workers then equation will be

Marginal Product ( MP) dQ/dL = - 1.4579+ 0.3094(20) - 0.0087 (20)^2

= -1.4579 + 6.188- 3.480

= 1.2501

Q(Output) = 4.9708 - 1.4579*L + 0.1547*L^2 - 0.0029*L^3

= 4.9708 - 1.4579*(20) + 0.1547*(20)^2 - 0.0029*(20)^3

= 4.9708 - 29.158 +61.88- 23.2

=14.4928

Average Product (AP) = Q/L

= 14.4928/20

= 0.7646