Wilpen Company, a price-setting firm, produces nearly 80 percent of all tennis balls purchased in the United States. Wilpen estimates the U.S. demand for its tennis balls by using the following linear specifications:

Q= a + bP + cM + dPr
Where Q is the number of cans of tennis balls sold quarterly, P is the wholesale price Wilpen charges for a can of tennis balls, M is the consumer’s average household income, and Pr is the average price of tennis rackets. The regression results are as follows:

Dependent Variable: Q
Observations: 20
R-Square: 0.8435
F-Ratio: 28.75
P-Value on F: 0.001

Variable

Intercept-Parameter Estimate 425120, Standard Error 220300, T-Ratio 1.93, P-Value 0.0716

P- Parameter Estimate -37260.6, Standard Error 12587, T-Ratio -22.96, P-Value 0.0093

M- Parameter Estimate 1.49, Standard Error 0.3651, T-Ratio 4.08, P-Value 0.0009

PR- Parameter Estimate -1456, Standard Error 460.75, T-Ratio -3.16, P-Value 0.006

a. Discuss the statistical significance of the parameter estimates a^, b^, c^, and d^ using the p-values. Are the signs of b^, c^, and d^ consistent with the theory of demand?
Wilpen plans to charge a wholesale price of $1.65 per can. The average price of a tennis racket is $110, and consumer’s average household income is $24, 600.
b. What is the estimated number of cans of tennis balls demanded?
c. At the values of P, M, and Pr given, what are the estimated values of the price (E^), income (E^m), and cross-price elasticity’s (E^xr) of demand?
d. What will happen, in percentage terms, to the number of cans of tennis balls demanded if the price of tennis balls decreases 15 percent?
e. What will happen in percentage terms, to the number of cans of tennis balls demanded if average household income increases by 20 percent?
f. What will happen, in percentage terms, to the number of cans of tennis balls demanded if the average price of tennis rackets increases by 20 percent?

So, do you have a question?

Yes, sorry it wasn't added in...

a. Discuss the statistical significance of the parameter estimates a^, b^, c^, and d^ using the p-values. Are the signs of b^, c^, and d^ consistent with the theory of demand?
b. What is the estimated number of cans of tennis balls demanded?
c. At the values of P, M, and Pr given, what are the estimated values of the price (E^), income (E^m), and cross-price elasticity’s (E^xr) of demand?
d. What will happen, in percentage terms, to the number of cans of tennis balls demanded if the price of tennis balls decreases 15 percent?
e. What will happen in percentage terms, to the number of cans of tennis balls demanded if average household income increases by 20 percent?
f. What will happen, in percentage terms, to the number of cans of tennis balls demanded if the average price of tennis rackets increases by 20 percent?

Discuss the statistical significance of the parameter estimates a^, b^, c^, and d^ using the p-values. Are the signs of b^, c^, and d^ consistent with the theory of demand?

What is the estimated number of cans of tennis balls demanded?

At the values of P, M, and Pr given, what are the estimated values of the price (E^), income (E^m), and cross-price elasticity’s (E^xr) of demand?

What will happen, in percentage terms, to the number of cans of tennis balls demanded if the price of tennis balls decreases 15 percent?

What will happen in percentage terms, to the number of cans of tennis balls demanded if average household income increases by 20 percent?

What will happen, in percentage terms, to the number of cans of tennis balls demanded if the average price of tennis rackets increases by 20 percent?

To determine the statistical significance of the parameter estimates (a^, b^, c^, and d^), we need to look at their respective p-values. The p-value represents the probability of observing a coefficient as extreme as the one estimated in the regression if the true coefficient were zero. A p-value less than the significance level (usually 0.05 or 0.01) indicates that the coefficient is statistically significant.

Looking at the p-values provided:
- For the intercept (a^), the p-value is 0.0716 which is greater than the typical significance level of 0.05. Therefore, it is not statistically significant.
- For the price (b^), the p-value is 0.0093 which is less than 0.05, indicating that the coefficient is statistically significant.
- For average household income (c^), the p-value is 0.0009 which is also less than 0.05, indicating statistical significance.
- Lastly, for the average price of tennis rackets (d^), the p-value is 0.006 which is less than 0.05, indicating statistical significance.

The signs of b^, c^, and d^ are consistent with the theory of demand. According to the theory, an increase in price (P) should result in a decrease in the quantity demanded (Q), so a negative sign for b^ is expected. An increase in consumer's income (M) and average price of tennis rackets (Pr) is expected to increase the quantity demanded, so positive signs for c^ and d^ are also expected.

Now, let's move on to answering the specific questions:

b. To estimate the number of cans of tennis balls demanded (Q) at the given values of P ($1.65), M ($24,600), and Pr ($110), we can use the regression equation:
Q = a^ + b^P + c^M + d^Pr
Substituting the values:
Q = 425120 + (-37260.6)*(1.65) + 1.49*(24600) + (-1456)*(110)
= 425120 - 61371.99 + 36594 + (-160160)
= 249182.01
Therefore, the estimated number of cans of tennis balls demanded is approximately 249,182.

c. To calculate the estimated price elasticity (E^), income elasticity (E^m), and cross-price elasticity (E^xr) of demand, we can use the following formulas:
- Price Elasticity (E^) = b^ * (P/Q)
- Income Elasticity (Em^) = c^ * (M/Q)
- Cross-Price Elasticity (Exr^) = d^ * (Pr/Q)

Using the estimated number of cans demanded from part b:
E^ = (-37260.6/249182) * 1.65 = -0.2458 (approximately)
Em^ = (1.49/249182) * 24600 = 0.1473 (approximately)
Exr^ = (-1456/249182) * 110 = -0.6374 (approximately)

d. To calculate the percentage change in the number of cans of tennis balls demanded if the price decreases 15 percent, we use the price elasticity (E^) calculated earlier. The percentage change can be calculated using the following formula:
Percentage Change = (E^ * % change in price)

Given that the price of tennis balls decreases by 15 percent, the percentage change would be -15.
Percentage Change = (-0.2458 * -15) = 3.687 (approximately)
Therefore, there will be an estimated 3.687 percent increase in the number of cans of tennis balls demanded.

e. To calculate the percentage change in the number of cans of tennis balls demanded if average household income increases by 20 percent, we use the income elasticity (Em^) calculated earlier. The percentage change can be calculated using the following formula:
Percentage Change = (Em^ * % change in income)

Given that average household income increases by 20 percent, the percentage change would be 20.
Percentage Change = (0.1473 * 20) = 2.946 (approximately)
Therefore, there will be an estimated 2.946 percent increase in the number of cans of tennis balls demanded.

f. To calculate the percentage change in the number of cans of tennis balls demanded if the average price of tennis rackets increases by 20 percent, we use the cross-price elasticity (Exr^) calculated earlier. The percentage change can be calculated using the following formula:
Percentage Change = (Exr^ * % change in price of tennis rackets)

Given that the average price of tennis rackets increases by 20 percent, the percentage change would be 20.
Percentage Change = (-0.6374 * 20) = -12.748 (approximately)
Therefore, there will be an estimated 12.748 percent decrease in the number of cans of tennis balls demanded.