# The regression results

The problem that I am having difficulty with is below:

Wilpen Company, a price-setting firm, produces nearly 80 percent of all tennis balls purchased in the United States. Wilpen estimates the US demand for its tennis balls by using the following liner specification: 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 consumers' average household income, 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

INTERCEPRT 425120.0 220300.0 1.93 0.0716

P -37260.6 12587 -22.96 0.0093

M 1.46 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,b,c, and d using the p-values. Are the sign of b,c and d consistent with the theory of demand? (Here's what we can say about the estimate for the intercept, a_hat (reads "a hat", which is the estimate for the parameter a in equation Q = a + bP + cM + dP_R): the p_value corresponding to a_hat is 0.0716, which means that it is significantly different from zero if we are satisfied with a 10% level of significance. Perform the same analysis for b_hat, c_hat, and d_hat. You also need to say something about the signs of b_hat, c_hat, and d_hat and whether they conform to expectations.)

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? (Plug the given values and parameter estimates in equation Q = a + b*P + c*M + d*P_R. (symbol * means multiplication)

c) At the value 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 elasticity is E_hat = -37260.6(1.65/Q), where Q is the quantity as computed in part b. Use the same procedure to find (E_M)_hat and (E_XR)_hat)

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? (Use the estimate for price elasticity, E_hat, obtained in part c, to find the percentage change in quantity. Note E_hat = (% change in Q) / (% change in price), and we are given (% change in price) = -15%.)

e) What will happen, in percentage terms, to the number of cans of tennis balls demanded if average household income increases by 20 percent? (Same procedure as in part d, except that we use (E_M)_hat)

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? (Same procedure as in part d, except that we use (E_XR)_hat)

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#### Solution Summary

The regression results are embedded.

Interpreting the Given Regression Results.

A multiplicative demand function of the form: Qd = a*P^b1*Y^b2*Po^b3 is estimated using cross-sectional data and 224 observations. The regression results were as follows:

Constant (a) Price(P) Income(Y) Price of other good (Po)

Coefficient 0.02248 -0.2243 1.3458 0.1034

Standard Error 0.01885 0.0563 0.5012 0.8145

a. How should the coefficients be interpreted in this equation?

b. What is the quantity demanded if price is $10, income is $9000, and price of the other good is $15?

c. Is demand elastic or inelastic? How can you tell? What impact would a price increase have on total revenue and on total profit?

d. How are these two goods related? Should the firm be concerned about a change in the price of the other good?

e. Is this product a luxury, necessity, or inferior good?