Please help me on this one!
You are planning to estimate a short-run production function for your firm, and you have the following data on labor usage and output:
Labor Output
3 1
7 2
9 3
11 5
17 8
17 10
20 15
24 18
26 22
28 21
30 23
A- Does a cubic equation appear to be a suitable specification, given these data? You may wish to construct a scatter diagram to help you answer this question.
B- Using a computer and software for regression analysis, estimate your firm short term production function using the data given here. Do the parameter estimates have appropriate algebraic signs? Are they statiscally 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 marginal products when the firm employs 23 workers?
E- When the firm employs 23 workers, is short run marginal cost (SMC) rising or falling? How can you tell??
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
my bad, the parameter for c in my response should be -.00183
Hi,economyst. Can you please show me the step by step solution for D)
thanks
A- To determine whether a cubic equation is a suitable specification for the data, you can construct a scatter plot of labor usage (independent variable) against output (dependent variable). This will help visualize the relationship between the two variables and decide if a cubic equation is appropriate.
B- To estimate the short-term production function, you need to use regression analysis with appropriate software (e.g., Excel, R, Python). The parameter estimates will tell you the relationship between labor and output, and their algebraic signs will indicate whether there is a positive or negative relationship.
To check if the parameter estimates are statistically significant at the 5 percent level, you will also need to look at the p-values associated with the estimates. If the p-value is less than 0.05, you can conclude that the estimate is statistically significant at the 5 percent level.
C- To estimate the point at which marginal product starts to fall, you need to examine the trend in the data. Look for the point at which the increase in output starts to slow down as labor usage increases. This will indicate that marginal product is decreasing. You can usually observe this by analyzing the first and second derivatives of the cubic equation, which will show the rate of change of the output and how it changes over different levels of labor.
D- To calculate the estimates of total, average, and marginal products when the firm employs 23 workers, you would need to input the labor usage of 23 into the short-run production function equation that you estimated in part B. This will give you the estimated output (total product) for that labor usage.
To calculate average product, divide the estimated output by 23. And to calculate marginal product, you can either take the derivative of the estimated production function with respect to labor and evaluate it with labor usage equal to 23, or you can calculate the difference in output for labor usage of 24 and 23.
E- To determine whether short-run marginal cost (SMC) is rising or falling when the firm employs 23 workers, you will need to analyze the relationship between marginal cost and output.
If the estimated marginal cost is increasing as output increases, then SMC is rising. Conversely, if the estimated marginal cost is decreasing as the output increases, then SMC is falling. You can determine this by comparing the estimates of marginal cost at different levels of output, such as when the firm employs 23 workers versus when it employs 24 workers. If the estimated marginal cost is higher for 23 workers compared to 24 workers, then SMC is rising. If the estimated marginal cost is lower for 23 workers compared to 24 workers, then SMC is falling.