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 R- SQUARE F- RATIO P- VALUE ON F
OBSERVATIONS: 20 0.8435 28.75 0.001
PARAMETER STANDARD
VARIABLE ESTIMATE ERROR T- RATIO P- VALUE INTERCEPT 425120.0 220300.0 1.93 0. 0716 P - 37260.6 12587 22.96 0.0093 M 1.49 0.3651 4.08 0.0009 PR -1456.0 460.75 -3.16 0.0060
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?
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To discuss the statistical significance of the parameter estimates, we need to look at the p-values associated with each estimate. The p-value represents the probability of observing a t-ratio as extreme as the one calculated, assuming the null hypothesis that the parameter estimate is equal to zero.
In this case, the p-values associated with the parameter estimates are as follows:
- p-value for the intercept (P): 0.0716
- p-value for the price (P): 0.0093
- p-value for the average household income (M): 0.0009
- p-value for the average price of tennis rackets (PR): 0.0060
If a p-value is less than the typically used significance level of 0.05, we can reject the null hypothesis that the parameter estimate is equal to zero and conclude that the estimate is statistically significant. Conversely, if the p-value is greater than 0.05, we fail to reject the null hypothesis and conclude that the estimate is not statistically significant.
In this case, the p-values associated with the price (P), average household income (M), and average price of tennis rackets (PR) are all less than 0.05. Therefore, we can conclude that these estimates are statistically significant.
To determine whether the signs of the estimates (a-hat, b-hat, c-hat) are consistent with the theory of demand, we need to consider the expected relationship between the variables and the demand for tennis balls.
- The intercept (a-hat) represents the level of demand for tennis balls when all other variables are zero. A positive intercept indicates that there is some demand for tennis balls even with zero price, income, and average price of tennis rackets. This could be due to factors such as brand loyalty or other non-price factors. However, the p-value for the intercept is greater than 0.05, so we cannot conclude that it is statistically significant.
- The price coefficient (b-hat) represents the expected change in demand for tennis balls when the price of tennis balls changes. In this case, the estimate of -37260.6 suggests an inverse relationship between price and demand. This is consistent with the theory of demand, as consumers are expected to demand less of a good when its price increases. The p-value associated with this estimate is less than 0.05, indicating its statistical significance.
- The income coefficient (c-hat) represents the expected change in demand for tennis balls when consumers' average household income changes. The estimate of 1.49 suggests a positive relationship between income and demand. This is consistent with the theory of demand, as consumers with higher incomes are likely to have higher purchasing power and demand more tennis balls. The p-value associated with this estimate is less than 0.05, indicating its statistical significance.
- The cross-price coefficient (d-hat) represents the expected change in demand for tennis balls when the average price of tennis rackets changes. The estimate of -1456.0 suggests an inverse relationship between the average price of tennis rackets and demand for tennis balls. This is consistent with the theory of demand, as consumers may substitute tennis rackets for tennis balls when the price of tennis rackets increases. The p-value associated with this estimate is less than 0.05, indicating its statistical significance.
b. To calculate the estimated number of cans of tennis balls demanded, we can use the regression results and plug in the given values:
- P (price) = $1.65 per can
- M (average household income) = $24,600
- PR (average price of tennis rackets) = $110
The estimated number of cans of tennis balls demanded (Q) can be calculated using the regression equation:
Q = a-hat + (b-hat * P) + (c-hat * M) + (d-hat * PR)
Plugging in the values, we get:
Q = 425120 + (-37260.6 * 1.65) + (1.49 * 24600) + (-1456.0 * 110)
Simplifying the equation, we get:
Q = 425120 + (-61480.49) + (36654) + (-160160)
Calculating the sum, we get:
Q ≈ 263133
Therefore, the estimated number of cans of tennis balls demanded is approximately 263,133.
c. The price elasticity of demand (E) can be calculated using the following formula:
E = (b-hat * P/Q) * (1/P)
Substituting the given values, we get:
E = (-37260.6 * 1.65/263133) * (1/1.65)
Simplifying the equation, we get:
E ≈ -0.168
Therefore, the estimated price elasticity of demand is approximately -0.168. Note that the negative sign indicates an inverse relationship between price and demand.
Similarly, the income elasticity of demand (EM) can be calculated using the formula:
EM = (c-hat * M/Q) * (1/M)
Substituting the given values, we get:
EM = (1.49 * 24600/263133) * (1/24600)
Simplifying the equation, we get:
EM ≈ 0.021
Therefore, the estimated income elasticity of demand is approximately 0.021.
The cross-price elasticity of demand (EXR) can be calculated using the formula:
EXR = (d-hat * PR/Q) * (1/PR)
Substituting the given values, we get:
EXR = (-1456.0 * 110/263133) * (1/110)
Simplifying the equation, we get:
EXR ≈ -0.056
Therefore, the estimated cross-price elasticity of demand is approximately -0.056. Note that the negative sign indicates an inverse relationship between the price of tennis rackets and the demand for tennis balls.
d. To calculate the percentage change in the number of cans of tennis balls demanded if the price of tennis balls decreases by 15 percent, we can use the own-price elasticity of demand.
The percentage change in quantity demanded (∆Q%) can be calculated using the formula:
∆Q% = (E * ∆P)%,
where E is the own-price elasticity of demand and ∆P is the percentage change in price.
Substituting the values, we get:
∆Q% = (-0.168 * -15)%
Simplifying the equation, we get:
∆Q% ≈ 2.52%
Therefore, the number of cans of tennis balls demanded will decrease by approximately 2.52% if the price of tennis balls decreases by 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.
The percentage change in quantity demanded (∆Q%) can be calculated using the formula:
∆Q% = (EM * ∆M)%,
where EM is the income elasticity of demand and ∆M is the percentage change in income.
Substituting the values, we get:
∆Q% = (0.021 * 20)%
Simplifying the equation, we get:
∆Q% ≈ 0.42%
Therefore, the number of cans of tennis balls demanded will increase by approximately 0.42% if average household income increases by 20%.
f. To calculate the percentage change in the number of cans of tennis balls demanded if the average price of tennis rackets increases by 25 percent, we can use the cross-price elasticity of demand.
The percentage change in quantity demanded (∆Q%) can be calculated using the formula:
∆Q% = (EXR * ∆PR)%,
where EXR is the cross-price elasticity of demand and ∆PR is the percentage change in the average price of tennis rackets.
Substituting the values, we get:
∆Q% = (-0.056 * 25)%
Simplifying the equation, we get:
∆Q% ≈ -1.4%
Therefore, the number of cans of tennis balls demanded will decrease by approximately 1.4% if the average price of tennis rackets increases by 25%.