Question

The demand function for JZ bicycles in Super City has been estimated to be:

Q = 4,000 - 11P + 30I

Below is additional information concerning this Regression:

Standard Error (SE) of the Intercept coefficient = 700

Standard Error (SE) of the price coefficient = 0.64

Standard Error (SE) of the Income coefficient = 1.40,

R-Square = 0.64,

Adjusted R-Square = 0.61,

F-Statistic = 31.402,

Q is quantity demanded of JZ bicycles per year, P is the price of JZ bicycles in dollars, and I is annual income in thousands of dollars.

A. Which of the two independent variables (if any) appears to be statistically significant (at the 5% level) in explaining bicycle sales?

B. What proportion of the variation in JZ bicycle sales is explained by the regression equation?

C. Give an economic interpretation of each estimated regression coefficient, i.e., P and I.

D. When P = $300 and I = 30 (use 30 and not 30,000 in the calculations), calculate the following and characterize the demand for JZ bicycles (i.e., elastic, inelastic or unitary elastic) and whether JZ bicycles is an inferior, normal and basic, or normal and luxury:

- The point Price Elasticity of Demand.

- The Income elasticity of demand.

Answer 1

The regression equation is given to be . The number of observations, and hence the degree of freedom is also not given.

(A) The t-statistic of the coefficient would be for the null and alternate . The price coefficient's t-value would be . The critical t-value at 5% significance level for 10 degree of freedom (at any degree of freedom indeed) is , and as out calculated t is less than -2.228 (absolute value of calculated t is greater than absolute value of critical t), we may reject the null in this case. The income coefficient's t-value would be . The calculated t is again greater than the critical t at least 10 degree of freedom (at any degree of freedom indeed), at 5% significance, meaning that we may reject the null.

Both the independent variables are statistically significant in explaining bicycles.

(B) The R-square is , where TSS is total variation in dependent variables, and ESS is total variation explained by independent variable. The R-square is given to be 0.64, meaning that about 64% of the variation in JZ bicycles sales is explained by the independent variable.

(C) The coefficient of price variable is -11, meaning that for a unit increase in price, the bicycles demanded is reduced by 11 units, on average (meaning expectedly).

The coefficient of income variable is 30, meaning that for a unit increase in income (by a unit would mean a $1000), the bicycles demanded is increased by 30 units, on average (bicycle units are not in 1000s).

(D) The regression equation is . The expected sales of bicycles for the given values would be .

• The point price elasticity would be or or or or , and for the given values, we have or . This means that for a unit percent increase in price, the demand is expected to decrease by 2.0625 percent.
• The income elasticity would be or or ro , and for the given values, we have . This means that for a unit percent increase in income, the demand is expected to increase by 0.5625 percent. The elasticity is less than 1, and hence we may say that the demand is inelastic. Also, as the elasticity is more than zero, we may say that the good is a normal good (a good with positive income elasticity), and as the elasticity is not greater than 1, we may say that the good is a necessary or basic good. Hence, the bicycles is a normal and basic good.