Help me on this one please?

labor usage 3,7,9,11,17,17,20,24,26,28,30
output 1,2,3,5,8,10,15,18,22,21,23

a. does a cubic equation appear to be a suitable specification, given these data? You may construct a scatter
diagram to help with this question.

b.using the computer software for regression analysis, estimate your firm's short-run production function using thr data given here. Do the parameter estimates have the appropriate algebraic sighns? Are they statistically significant at the 5 percent level?

c.At what point do you estimate marginal product begins to fall?

D.calculate estimates of total, average, and marignal products when the firm employs 23 workers.

e.whenthe firm employs 23 workers, is short-run marginal cost (SMC) rising or falling? How can you tell.

I answered thsi very question back in July. Cutting and pasting my response back then:

(On July 23 to TrickyEconomics)
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You still have two values of 17 labor with different outputs. That said, here goes.

A) I wish there was more data, but a skatterplot kinda looks like an S-curve, which can be generated with a Cubic. So, yes.
B) for y= ax + bx^2 + cx^3 + d
I get (parameter and t-statistic):
a=-.81426 (-1.30)
b=0.10442 (2.46)
c=-.00183 (-2.17)
d=2.9235 (1.14)

Such parameters generate the shape shown in the skatter plot. With A cubic, its hard, a-priori to predict what the sign ought to be.
Only b is significant at the 5% level
C) the cubic is the total production function. Take the first dirivitive - this becomes the MP function. It has a maxima about 19. (check my math)
D) Plug 23 into the estimated cubic function and the derived MP function
E) since MP is falling at 23, SMC must be rising.

I hope this helps

Output Labor X = L3 Y = L2

units/m hrs/m
1 3 27 9
2 7 343 49
3 9 729 81
5 11 1331 121
8 17 4913 289
10 17 4913 289
15 20 8000 400
18 24 13824 576
22 26 17576 676
21 28 21952 784
23 30 27000 900






a) Does a cubic euation appear to be a suitable specification, given these data? You may wish to construct a scatter diagram to help you answer
Yes a scatter diagram gives the s shape which is appropriate to indicate a cubic equation would be suitable.

b) Using a computer and software for regression analysis, estimate your firms's short-run production function using the data given here.
Do the parameter estimates have the appropriate algebraic signs? Are they statistically significant at the 5 percent level?

Yes the parameter estimates have the right algebraic signs
รข= -.00075 < 0 and b = .048731 > 0

k-1=2-1=1
n-k=11-2=9
Therefore the critical value of f is 5.12 at the 5% level as compared to the f ratio of 671.1836
The p values are insignificant and the chance of making a type 1 error are null.
The t values of -3.76916 and 9.238614 both exceed the critical t value for 9 degrees of freedom at the 5% level of 2.262 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% Lower 95.0% Upper 95.0%
Intercept 0 #N/A #N/A #N/A #N/A #N/A #N/A #N/A
X Variable 2 -0.00074509 0.00019768 -3.769163007 0.004422153 -0.001192274 -0.000297906 -0.001192274 -0.000297906
X Variable 2 0.048730818 0.005274689 9.238614311 6.8893E-06 0.036798642 0.060662993 0.036798642 0.060662993

a. To determine if a cubic equation is a suitable specification, you can construct a scatter diagram using the given data points. Plot the labor usage (x-axis) against the output (y-axis), and see if there is a clear pattern or relationship between the variables.

b. To estimate the short-run production function using regression analysis, you will need computer software. Input the labor usage as the independent variable and the output as the dependent variable. The software will generate parameter estimates for the equation. Check if the parameter estimates have the appropriate algebraic signs (e.g., positive for inputs that increase output). Additionally, you can perform statistical tests to determine if the parameter estimates are statistically significant at the 5 percent level.

c. To estimate the point at which marginal product begins to fall, you need to analyze the regression results. Look for the coefficient of the labor input variable (in this case, workers) that is statistically significant and negative. This means that as labor increases beyond that point, the marginal product starts to decrease.

d. To calculate estimates of total, average, and marginal products when the firm employs 23 workers, use the estimated short-run production function equation from part (b). Simply plug in 23 for the labor input and calculate the respective values for total, average, and marginal products.

e. To determine if the short-run marginal cost (SMC) is rising or falling when the firm employs 23 workers, you need to understand the relationship between marginal product and marginal cost. If the marginal product is decreasing (as estimated in part c), then the short-run marginal cost is likely increasing. This is because the cost of producing additional units of output is rising as the marginal product of labor declines.