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 specification: 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 consumers’ average household in-come, 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 ˆ , , , and using the p- values. Are the signs of , and 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 consumers’ 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 ( ), income ( M), and cross- price elasticities ( 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 25 percent?
a. The p-values for the parameter estimates a^ (intercept), P, M, and PR are 0.0716, 0.0093, 0.0009, and 0.006 respectively. Generally, a p-value less than 0.05 is considered statistically significant, indicating that the variable has a significant impact on the dependent variable. In this case, the parameter estimates for P, M, and PR all have p-values less than 0.05, indicating that they are statistically significant.
The signs of the parameter estimates are negative for P and PR, and positive for M. According to the theory of demand, when the price of a product increases, the quantity demanded decreases, hence a negative sign for P is consistent with the theory. Similarly, when the price of a related good (tennis rackets) increases, the quantity demanded decreases, hence a negative sign for PR is also consistent with the theory. On the other hand, when the consumer's income increases, the quantity demanded of a normal good (like tennis balls) increases, hence a positive sign for M is consistent with the theory.
b. To estimate the number of cans of tennis balls demanded, we use the regression equation:
Q = a^ + P * P + M * M + PR * PR
Using the given values:
P = $1.65
M = $24,600
PR = $110
Q = 425120 + (-37260.6 * 1.65) + (1.49 * 24600) + (-1456 * 110)
Q = 425120 + (-61291.59) + (36654) + (-160160)
Q = 258,322.41
Therefore, the estimated number of cans of tennis balls demanded is approximately 258,322.
c. To calculate the elasticities of demand, we use the following formulas:
Price elasticity (EP) = (∂Q/∂P) * (P/Q)
Income elasticity (EM) = (∂Q/∂M) * (M/Q)
Cross-price elasticity (EPR) = (∂Q/∂PR) * (PR/Q)
Using the regression results, the parameter estimates give us the partial derivatives:
∂Q/∂P = -37260.6
∂Q/∂M = 1.49
∂Q/∂PR = -1456
Plugging in the values:
EP = (-37260.6 * 1.65) * (1.65 / 258322.41) = -0.0356
EM = (1.49 * 24600) * (24600 / 258322.41) = 0.1424
EPR = (-1456 * 110) * (110 / 258322.41) = -0.0621
Therefore, the estimated price elasticity of demand (EP) is approximately -0.0356, the estimated income elasticity of demand (EM) is approximately 0.1424, and the estimated cross-price elasticity of demand (EPR) is approximately -0.0621.
d. To calculate the percentage change in the number of cans demanded due to a 15% decrease in price, we use the price elasticity of demand:
Percentage change in quantity demanded = EP * Percentage change in price
Percentage change in quantity demanded = -0.0356 * (-15) = 0.534%
Therefore, the number of cans of tennis balls demanded will decrease by approximately 0.534% if the price of tennis balls decreases 15%.
e. To calculate the percentage change in the number of cans demanded due to a 20% increase in average household income, we use the income elasticity of demand:
Percentage change in quantity demanded = EM * Percentage change in income
Percentage change in quantity demanded = 0.1424 * 20 = 2.848%
Therefore, the number of cans of tennis balls demanded will increase by approximately 2.848% if average household income increases by 20%.
f. To calculate the percentage change in the number of cans demanded due to a 25% increase in the average price of tennis rackets, we use the cross-price elasticity of demand:
Percentage change in quantity demanded = EPR * Percentage change in price of tennis rackets
Percentage change in quantity demanded = -0.0621 * 25 = -1.5525%
Therefore, the number of cans of tennis balls demanded will decrease by approximately 1.5525% if the average price of tennis rackets increases 25%.
a. The p-values associated with the parameter estimates are as follows:
- p-value for Intercept: 0.0716
- p-value for P: 0.0093
- p-value for M: 0.0009
- p-value for PR: 0.006
The p-value is a measure of the statistical significance of a parameter estimate. In this case, all p-values are less than 0.05, which indicates that the parameter estimates are statistically significant. Therefore, we can reject the null hypothesis that the parameters are zero.
The signs of the parameter estimates are as follows:
- The intercept has a positive sign (+425120), indicating that even when all other variables are held constant, there is a base demand for tennis balls.
- The parameter estimate for P is -37260.6, indicating that there is a negative relationship between the price of tennis balls and the quantity demanded. This is consistent with the theory of demand, as higher prices usually lead to lower quantity demanded (law of demand).
- The parameter estimate for M is 1.49, indicating that there is a positive relationship between consumers' average household income and the quantity demanded. This is also consistent with the theory of demand, as higher income generally leads to higher purchasing power and demand for goods.
- The parameter estimate for PR is -1456, indicating that there is a negative relationship between the price of tennis rackets and the quantity demanded of tennis balls. This suggests that tennis balls and tennis rackets are complementary goods, as a higher price for tennis rackets leads to a decrease in the demand for tennis balls.
b. To estimate the number of cans of tennis balls demanded, we can use the regression equation:
Q = a ˆ + b ˆP + c ˆM + d ˆPR
Substituting the given values:
Q = 425120 - 37260.6 * 1.65 + 1.49 * 24600 - 1456 * 110
By calculating this equation, you will obtain the estimated number of cans of tennis balls demanded.
c. To calculate price elasticity of demand (E) and cross-price elasticity of demand (XR), we can use the following formulas:
E = (ΔQ/Q) / (ΔP/P)
XR = (ΔQ/Q) / (ΔPR/PR)
Since the question asks for estimated values, we can use the parameter estimates from the regression results. The estimated values of elasticity can be calculated by multiplying the parameter estimate by the corresponding variable and dividing it by the average values of Q, P, and PR, respectively.
- Estimated price elasticity of demand (E):
E = -37260.6 * (1.65/Q) * (1/Q) * 100
- Estimated income elasticity of demand (M):
M = 1.49 * (24600/Q) * 100
- Estimated cross-price elasticity of demand (XR):
XR = -1456 * (110/Q) * (1/Q) * 100
d. To calculate the percentage change in the number of cans of tennis balls demanded if the price decreases 15 percent, we can use the price elasticity of demand (E) obtained in step c.
Percentage change in quantity demanded = E * (-15)
e. To calculate the percentage change in the number of cans of tennis balls demanded if average household income increases by 20 percent, we can use the income elasticity of demand (M) obtained in step c.
Percentage change in quantity demanded = M * (20)
f. To calculate the percentage change in the number of cans of tennis balls demanded if the average price of tennis rackets increases 25 percent, we can use the cross-price elasticity of demand (XR) obtained in step c.
Percentage change in quantity demanded = XR * (25)