ECO 550
2
Demand Estimation
Sample Analytical Problem
The following questions refer to this regression equation. Standard errors are in parentheses.
QD = 15,000 – 10 P + 1500 A + 4 PX + 2 I
(5,234) (2.29) (525) (1.75) (1.5)
R2 = 0.65
N = 120
F = 35.25
Standard error of Y estimate = 565
Q = Quantity demanded
P = Price = 7,000
A = Advertising expense, in thousands = 54
PX = Price of competitor’s product = 8,000
I = Average monthly income = 4,000
1. Calculate the elasticity for each variable and briefly comment on what information this gives you in each case.
Based on values given above,
QD = 15,000 – 10 (7,000) + 1500 (54) + 4 (8,000) + 2 (4,000) =
= 15,000 – 70,000 + 81,000 + 32,000 + 8,000 = 66,000
Price elasticity = -10(7,000/66,000) = -1.06. Demand is elastic (at this point).
Advertising elasticity = 1500(54/66,000) = 1.23. The product is elastic with respect to advertising; a 1 % increase in advertising expense will lead to a greater than 1 % increase in sales.
Cross elasticity = 4(8000/66,000) = 0.48. Cross elasticity is positive, implying that the products are substitutes, but it is less than 1, suggesting that they are not particularly good substitutes and the competitor’s price has little impact on the firm’s sales.
Income elasticity = 2(4000/66,000) = 0.12. The product is income inelastic; thus it is a normal good (necessity), and is not particularly responsive to income fluctuations.
2. Calculate t-statistics for each variable and explain what this tells you.
Price: 10/2.29 = 4.37
Advertising: 1500/525 = 2.86
Competitor’s price: 4/1.75 = 2.29
Income: 2/1.5 = 1.33
All variables are statistically significant with the exception of income. Thus, we can conclude that the other variables do have an impact on the quantity demanded of this product.
3. How is the R2 value calculated and what information does it give you?
R2 = RSS/TSS = 1 – (ESS/TSS), where
TSS = sum of squared deviations of the sample values of Y from their mean, RSS = sum of squared deviations of the estimated values from their mean, and
ESS = sum of the squared deviations of the sample values from their estimated values
The R2 value tells you what percentage of the variations in the dependent variable is explained by variation in the independent variables, or the “goodness of fit” of the equation. In this case, 65 % of the variation in the quantity demanded is explained by variation in the independent variables.
4. How would you evaluate the quality of this equation overall? Do you have any concerns? Explain.
The overall equation is significant, as shown by the F-test. The R2 value is reasonably high. One variable is not significant. It might be desirable to re-estimate the equation without it. The sample is sufficiently large (N = 120). There are no significant concerns.
5. Should this firm be concerned if macroeconomic forecasters predict a recession? Explain.
Based on income elasticity from this equation (0.12), no. The good is income inelastic, so a recession should not cause a significant decrease in sales. Note also that income is not statistically significant in this equation, making it even less of a concern.
6. The firm is considering changing its price to $9,000. Predict the quantity demanded at that price, all other things equal.
At a price of $9,000, the point estimate of quantity demanded would be
QD = 15,000 – 10 (9,000) + 1500 (54) + 4 (8,000) + 2 (4,000) =
= 15,000 – 90,000 + 81,000 + 32,000 + 8,000 = 46,000
7. How could a manager use the information contained in this regression equation?
A manager might note that the demand is elastic, and thus that sales might respond to price decrease. Likewise, sales should respond to increases in advertising. Sales are less likely to be impacted by income changes or by changes in the price of the competitor’s product. The equation could be used to forecast expected sales based on changes in one or more variables. The equation could be used to help in coordinating production plans or with other parts of the firm.