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

Economics- Managerial - Christopher, Wednesday, July 1, 2009 at 9:45pm

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?

Thanks for your vote of confidence.

1) Look at the T-Ratio and especially the P-Value. You are testing whether the parameter is significantly different from zero. In all parameters except the intercept, the estimate of the parameter is significantly different from zero at the 1% level (P-value < .01). For the intercept, the P-value is .0716. For the intercept, at the 10% confidence level, you could say the estimate is different from zero.
1b) We would expect a negative relationship between sales and price or sales and price of rackets. (As price goes up, Q goes down) We would expect a positive relationship for income. The estimates are properly that.
2) you did not provide levels for P,M, and Pr. But, this step is otherwise easy. Just plug into your equation. That is Q=425120 - 37260.6*P + 1.49*M - 1456*Pr.
2b) Raise price by a small amount (e.g., 1%) what happens to Q. Elasticity is (%change Q)/(%change P).
Ditto for income and Pr.
3) Repeat steps in 2b cept with the percentages given.

To discuss the statistical significance of the parameter estimates and whether the signs of b^, c^, and d^ are consistent with the theory of demand, we need to look at the p-values.

The p-value measures the statistical significance of the parameter estimate. A p-value less than 0.05 indicates that the parameter estimate is statistically significant.

For the parameter estimates:

a^ (Intercept):
- The p-value is 0.0716, which is greater than 0.05.
- Therefore, the intercept estimate is not statistically significant.

b^ (Price):
- The p-value is 0.0093, which is less than 0.05.
- Therefore, the price estimate is statistically significant.
- The negative sign of the price estimate is consistent with the theory of demand, indicating an inverse relationship between price and quantity demanded.

c^ (Income):
- The p-value is 0.0009, which is less than 0.05.
- Therefore, the income estimate is statistically significant.
- The positive sign of the income estimate is consistent with the theory of demand, indicating a positive relationship between income and quantity demanded.

d^ (Price of rackets):
- The p-value is 0.006, which is less than 0.05.
- Therefore, the price of rackets estimate is statistically significant.
- The negative sign of the price of rackets estimate is consistent with the theory of demand, indicating an inverse relationship between the price of rackets and quantity demanded.

Now, let's move on to estimating the number of cans of tennis balls demanded.

The estimated number of cans of tennis balls demanded (Q hat) can be calculated using the regression equation:
Q = a^ + b^P + c^M + d^Pr

Given the values of P, M, and Pr, we can substitute them into the equation and calculate Q hat.

To calculate the estimated values of the price elasticity (E^), income elasticity (E^m), and cross-price elasticity (E^xr) of demand, we need the respective parameter estimates.

Elasticity measures the responsiveness of quantity demanded to changes in price, income, or the price of related goods.

Price elasticity (E^) can be calculated as:
E^ = (b^ * P/Q) * 100

Income elasticity (E^m) can be calculated as:
E^m = (c^ * M/Q) * 100

Cross-price elasticity (E^xr) can be calculated as:
E^xr = (d^ * Pr/Q) * 100

To calculate the percentage change in the number of cans of tennis balls demanded, we use the following formula:
[(Q_new - Q_initial) / Q_initial] * 100

Now let's answer each question step-by-step:

1. The statistical significance of the parameter estimates and the signs' consistency with the theory of demand have been discussed above.

2. To calculate the estimated number of cans of tennis balls demanded (Q hat), substitute the given values of P, M, and Pr into the regression equation Q = a^ + b^P + c^M + d^Pr.

3. To calculate the estimated values of the price (E^), income (E^m), and cross-price elasticity's (E^xr), substitute the respective parameter estimates and the given values of P, M, and Pr into the elasticity formulas.

4. To calculate the percentage change in the number of cans of tennis balls demanded if the price decreases by 15%, substitute the new price into the regression equation and calculate Q_new. Then, use the formula [(Q_new - Q_initial) / Q_initial] * 100 to determine the percentage change.

5. To calculate the percentage change in the number of cans of tennis balls demanded if average household income increases by 20%, substitute the new income into the regression equation and calculate Q_new. Then, use the formula [(Q_new - Q_initial) / Q_initial] * 100 to determine the percentage change.

6. To calculate the percentage change in the number of cans of tennis balls demanded if the average price of tennis rackets increases by 20%, substitute the new price into the regression equation and calculate Q_new. Then, use the formula [(Q_new - Q_initial) / Q_initial] * 100 to determine the percentage change.

To evaluate the significance of the parameter estimates and answer the other questions, we need to analyze the regression results provided.

1. Statistical Significance and Consistency with Theory of Demand:

a^ (Intercept): The p-value for a^ is 0.0716. Since this p-value is greater than 0.05 (assuming a significance level of 0.05), we do not have strong evidence to reject the null hypothesis that the intercept (a^) is statistically significant. The sign of a^ being positive suggests that there may be a base level of demand for tennis balls even at a price of zero.

b^ (Coefficient of P): The p-value for b^ is 0.0093, which is less than 0.05. Therefore, we can conclude that b^ is statistically significant. The negative sign of b^ is consistent with the theory of demand, indicating an inverse relationship between the price of tennis balls (P) and the quantity demanded (Q).

c^ (Coefficient of M): The p-value for c^ is 0.0009, which is less than 0.05, indicating that c^ is statistically significant. The positive sign of c^ suggests that there is a positive relationship between consumer's average household income (M) and the quantity of tennis balls demanded (Q). This is consistent with the theory of demand, as higher income may lead to higher demand for tennis balls.

d^ (Coefficient of Pr): The p-value for d^ is 0.006, which is less than 0.05, indicating that d^ is statistically significant. The negative sign of d^ implies that there is an inverse relationship between the average price of tennis rackets (Pr) and the quantity demanded (Q) of tennis balls. This is consistent with the theory of demand, as higher prices of tennis rackets may decrease the demand for tennis balls.

2. Estimated Number of Cans Demanded:
To estimate the number of cans of tennis balls demanded (Q), you need to know the values of P, M, and Pr. Substitute these values into the regression equation:
Q = a^ + b^P + c^M + d^Pr

3. Estimated Elasticities:
To calculate the price elasticity (E^), income elasticity (E^m), and cross-price elasticity (E^xr) of demand, you need to use the coefficient estimates and the given values of P, M, and Pr.

Price Elasticity (E^): It is calculated as the percentage change in quantity demanded (Q) divided by the percentage change in price (P). Substitute the values into the formula: E^ = (b^ * P/Q) * (P/Q)

Income Elasticity (E^m): It is calculated as the percentage change in quantity demanded (Q) divided by the percentage change in income (M). Substitute the values into the formula: E^m = (c^ * M/Q) * (M/Q)

Cross-Price Elasticity (E^xr): It is calculated as the percentage change in quantity demanded (Q) divided by the percentage change in the price of tennis rackets (Pr). Substitute the values into the formula: E^xr = (d^ * Pr/Q) * (Pr/Q)

4. Percentage Change in Quantity Demanded:
To determine the percentage change in the number of cans of tennis balls demanded, you would need to calculate the price elasticity (E^) using the given information on the price decrease. Then multiply the price decrease by the estimated value of E^ and express it as a percentage.

5. Percentage Change in Quantity Demanded with Income Increase:
Repeat the same process as in 4, but this time, use the income elasticity (E^m) and the given information on the income increase.

6. Percentage Change in Quantity Demanded with Price of Tennis Rackets Increase:
Once again, follow the process described in 4, but this time, use the cross-price elasticity (E^xr) and the given information on the price increase of tennis rackets.

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?

The p-value associated with b is consistent with the theory of demand - if the prices rise, the quantity sold will then drop.
The p-value associated with c is consistent with the theory of demand – when consumers have more (increase in income) they may purchase more tennis balls.
The p-value associated with d is consistent with the theory of demand – if Wilpen increase the price of their tennis rackets then they will sell less of them and consumers will buy less, as well as less tennis balls (no rackets, no balls).

b. What is the estimated number of cans of tennis balls demanded?
Q = a + b*P + c*M + d*Pr
Q = 425120 – 37260.6 P + 1.46 M – 1456 PR
Q = 425120 – 37260.6 * 1.65 + 1.46 * 24600 – 1456 * 110
Q = 239396.01

c. At the values of P, M, and Pr given, what are the estimated values of the price (E), income (Em), and cross-price elasticities (Exr) of demand?
The estimated value for the price elasticities is
E -37260.6(1.65/Q) = -37260.6(1.65/ 239396.01) = -0.257

The estimated values of the income (EM) of demand is
EM = 1.46(24600/Q) = 1.46(24600/239396.01) = 0.150

The estimated values of the cross-price elasticities (EXr) of demand is
EXr = -1456.0(110/Q) = -1456.0(110/ 239396.01) = -0.669

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?
Percentage change in Q = E * percentage change in price = -0.257 * (-15%) = 0.04
The number of cans of tennis balls demanded will rise by 4%

e. What will happen, in percentage terms, to the number of cans of tennis balls demanded if average household income increase by 20 percent?
Percentage change in Q = EM * percentage change in income = 0.150 *20% = 0.03
The number of cans of tennis balls demanded will rise by 3%

f. What will happen, in percentage terms, to the number of cans of tennis balls demanded if the average price of tennis rackets increases 25 percent?
Percentage change in Q = EXR * percentage change in racket price = -0.669 * 25% = -0.167
The number of cans of tennis balls demanded will drop by 16