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.
a. To determine if a cubic equation is a suitable specification, we can construct a scatter diagram.
1. Plot the labor usage values (-3,7,9,11,17,17,20,24,26,28,30) on the horizontal axis.
2. Plot the output values (1,2,3,5,8,10,15,18,22,21,23) on the vertical axis.
3. Connect the plotted points to visualize the relationship between labor usage and output.
By examining the scatter diagram, we can get an idea of the relationship between the variables. If there is a clear pattern or trend, it indicates that a particular equation might be suitable for the data.
b. To estimate the firm's short-run production function using regression analysis:
1. Use computer software capable of conducting regression analysis.
2. Input the labor usage values as the independent variable (X) and the output values as the dependent variable (Y).
3. Run the regression analysis to obtain regression coefficients for the equation.
- Check the signs of the parameter estimates. If they align with economic intuition (i.e., positive for inputs that increase output and negative for inputs that decrease output), then they have appropriate algebraic signs.
- Assess the statistical significance of the coefficients. If the p-values associated with the coefficients are less than 0.05 (assuming a 5% significance level), it indicates statistical significance.
c. To estimate the point where marginal product begins to fall:
1. Calculate the marginal product of labor for each level of labor usage.
- Marginal product is the change in output divided by the change in labor usage.
2. Examine the pattern of marginal product in relation to labor usage.
- Look for the point where the marginal product starts decreasing or shows a diminishing return to labor.
d. To calculate estimates of total, average, and marginal products when the firm employs 23 workers:
1. Use the estimated short-run production function from the regression analysis.
2. Substitute the labor usage value of 23 into the equation to obtain the estimated total product.
3. Divide the estimated total product by 23 to get the average product.
4. Calculate the change in total product when labor usage increases by one unit to get the marginal product.
e. To determine if the short-run marginal cost (SMC) is rising or falling when the firm employs 23 workers:
1. Calculate the marginal cost of production.
- Marginal cost is the change in cost divided by the change in output.
2. Assess the trend of the marginal cost curve at the point where 23 workers are employed.
- If the marginal cost is increasing, SMC is rising.
- If the marginal cost is decreasing, SMC is falling.
- Evaluate the slope of the marginal cost curve or compare the values at adjacent points to determine the direction.