Question

I have attached a tough question that I need help with please.

You collect the following production data for your firm:

Q          L

241       2

1189     6

459       3

64         1

1539     9

1102     5

1514   7

755     4

1117     11

1550     10

a.  Which functional form (linear, quadratic, cubic) is most suitable to your data? Construct a scatter diagram but be sure to just do the dots; don't include the lines that connect them. Then, play around with the trendline feature and include what you consider to be the best trendline.

b.  Using OLS, estimate the firm's short-run production function. Comment on the strength of the regression results.

c.  Calculate the Q, AP, and MP for L = 8 workers.

d.  At 8 workers, is MC rising or falling, and how do you know?

Answer 1

a. Below is the scatter diagram drawn using excel -

Production function 1800 1600 1400 1200 1000 800 600 400 200 0 0 2. 6 10 12 Labor Output

Below are different polynomials fitted to the data -

Production function 1800 1600 1400 1200 1000 800 Linear (Q) Poynomial of order 2 600 Polynomial of order 3 400 200 0 2 6 10 12 -200 Labor ......... ndino

Both cubic and quadratic functions fit better to the data set than linear functions. Hence, cubic function fits better than quadratic polynomial.

b. Below are OLS regression results-

Regression Statistics
Multiple R	0.9948
R Square	0.9895
Adjusted R Square	0.9843
Standard Error	68.5755
Observations	10

Dependent variable - Q	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	-18.6135	123.6929	-0.1505	0.8853	-321.28	284.05	-321.28	284.05
L	26.5239	87.3341	0.3037	0.7716	-187.17	240.22	-187.17	240.22
L^2	60.9362	16.9693	3.5910	0.0115	19.41	102.46	19.41	102.46
L^3	-4.8793	0.9455	-5.1607	0.0021	-7.19	-2.57	-7.19	-2.57

The R-Square of the model is 0.9895 which means that 98.95% of the variation in output is explained by this model. However, the coefficient L is not statistically significant at 5% level. Similarly, coefficient L^2 and L^3 is statistically significant at 5% level.

C.

Q = -18.6135 + 26.5239*Q+60.9362*Q^2 - 4.8793*Q^3

When L = 8 workers, Q = -18.6135 + 26.5239*8+60.9362*8^2 - 4.8793*8^3

Q = -18.61+ 212.19 + 3899.92 - 2498.20
Q = 1595 (approx)

AP = Q/L
AP (at L= 8) = 199.375

MP = dQ/dL
= 26.5239+2*60.9362*Q - 3*4.8793*Q^2
= 26.52+121.87*Q - 14.6379*Q^2

MP( at L = 8) = 26.52+121.87*8 - 14.6379*8^2

=26.52+ 974.96 - 936.83

= 64.65

d. At 8 workers, AP is greater than MP and hence diminishing returns to labor has already kicked in. Thus, the MC is rising at 8 units of workers. This happens due to diminishing returns to labor due to which MC begins to rise.