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?

See my post to John-Q above.

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 that the coefficient estimate is statistically different from zero. A smaller p-value indicates that there is strong evidence to suggest that the coefficient is statistically significant.

For the parameter estimates in this case:

a^ (Intercept): p-value = 0.0716
b^ (Wholesale price, P): p-value = 0.0093
c^ (Average household income, M): p-value = 0.0009
d^ (Average price of tennis rackets, Pr): p-value = 0.006

Based on the p-values, we can conclude that the parameter estimates b^, c^, and d^ are statistically significant at conventional significance levels (e.g., 0.05 or 0.01). However, the intercept term a^ does not appear to be statistically significant, as its p-value is greater than 0.05.

Regarding the signs of b^, c^, and d^, we can compare them to the theory of demand. According to the theory of demand:
- b^ (P) should have a negative sign, indicating an inverse relationship between price and quantity demanded. In this case, b^ is negative (-37260.6), consistent with the theory.
- c^ (M) should have a positive sign, indicating a positive relationship between income and quantity demanded. In this case, c^ is positive (1.49), consistent with the theory.
- d^ (Pr) should have a negative sign, indicating an inverse relationship between the price of tennis rackets and quantity demanded. In this case, d^ is negative (-1456), consistent with the theory.

To determine the estimated number of cans of tennis balls demanded (Q), we can use the regression equation:
Q = a^ + b^P + c^M + d^Pr

Substituting the given values for P, M, and Pr, and using the parameter estimates:
Q = 425120 - 37260.6P + 1.49M - 1456Pr

To calculate the estimated values of the price elasticity of demand (E^), income elasticity of demand (E^m), and cross-price elasticity of demand (E^xr), we need to use the formula for elasticity:

Elasticity = (∂Q/∂X) * (X/Q)

where X represents the variable in question (price, income, or the price of tennis rackets).

For price elasticity (E^):
E^ = (∂Q/∂P) * (P/Q)
Using the parameter estimate b^ for P, and substituting the given values:
E^ = -37260.6 * (P/Q)

For income elasticity (E^m):
E^m = (∂Q/∂M) * (M/Q)
Using the parameter estimate c^ for M, and substituting the given values:
E^m = 1.49 * (M/Q)

For cross-price elasticity (E^xr):
E^xr = (∂Q/∂Pr) * (Pr/Q)
Using the parameter estimate d^ for Pr, and substituting the given values:
E^xr = -1456 * (Pr/Q)

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 price elasticity formula and the given value for the price decrease:

Percentage change = E^ * (-15%)
Substitute the value of E^ from the calculation above.

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 formula and the given value for the income increase:

Percentage change = E^m * (20%)
Substitute the value of E^m from the calculation above.

Similarly, 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 can use the cross-price elasticity formula and the given value for the price increase:

Percentage change = E^xr * (20%)
Substitute the value of E^xr from the calculation above.