Question

Regression equation for Case 3.0:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.957
R Square	0.915
Adjusted R Square	0.908
Standard Error	5.779
Observations	52
ANOVA
	df	SS	MS	F	Significance F
Regression	4	16947.86487	4236.9662	126.8841	1.45976E-24
Residual	47	1569.442824	33.392401
Total	51	18517.30769
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	39.08190	15.31261	2.55227	0.014012	8.27693	69.88687
X-Price	-7.37039	0.98942	-7.44921	1.71E-09	-9.36084	-5.37994
Y-Price	-5.42813	0.33793	-16.06289	1.03E-20	-6.10796	-4.74831
Z-Price	4.05067	0.33949	11.93173	7.95E-16	3.36771	4.73363
Income	0.00288	0.00038	7.57448	1.11E-09	0.00212	0.00364

Questions and analysis:

1.    Forecast the quantity demanded when own price is $10, the price of Y is $15, the price of Z is $24, and household income is $42,000. Construct an approximately 95% confidence interval around your estimate.

        Sales forecast:__________                                Confidence interval:________ to ________

2.    Is Y a substitute or complement for model X? Is Z a substitute or complement to X? Is X a normal or inferior good?

Y is ___________________

Z is ___________________

X is ___________________

3.    Which independent variables are statistically significant at the 5% level?

	Statistically Significant?
	Yes	No
Price of X	_____	_____
Price of Y	_____	_____
Price of Z	_____	_____
Income	_____	_____

4.    Calculate own price elasticity of demand when own price is $10, the price of Y is $15, the price of Z is $24, and household income is $42,000. Is demand elastic or inelastic at this point?

Elasticity = ____________

Elastic or inelastic? _____________

5.    Suppose the marginal cost of model X is a constant $5 per unit. Find the profit maximizing price and quantity for the producer of model X, once again assuming the price of Y is $15, the price of Z is $24, and household income is $42,000.

Optimal Price: __________

Optimal Quantity: ___________

6.    Calculate cross price elasticity between the model X and the price of Y when own price is $10, the price of Y is $15, the price of Z is $24, and household income is $42,000. Suggest a strategic response: how should the producer of model X respond when the producer of Y raises prices (be as specific as possible)?

Elasticity = _______________

Strategic response:

Answer 1

1. The regression equation is:

Quantity= 39.08190- 7.37039*(XPrice)- 5.42813*(YPrice)+ 4.05067*(ZPrice)+ 0.00288*(Income)

Thus, the forecasted value of the quantity demanded will be:

39.08190- 7.37039*(10)- 5.42813*(15)+ 4.05067*(24)+ 0.00288*(42000)

= 39.08190- 73.7039- 81.42195+ 97.21608+ 120.96

= 102.13213 ~ 102

To calculate the prediction interval of a multiple linear regression model, the following formula can be applied:

prediction interval= Y_forecast ± tValue_?/2 * prediction error

where prediction error= SE_regression * sqrt (1+ distance value)

The prediction error is always slightly greater than the SE_regression. Since we have not been given the distance value in the question, we can multiply the SE_regression with 1.1 to get an approximate value of the prediction error.

Thus, prediction error= 5.779* 1.1= 6.3569

tValue_{0.025, 51}= 2.007584

Hence, prediction interval= 102.13213 ± 2.007584 * 6.3569

12.7620107296

89.370 < Y_forecast < 114.894

2. As the price of X increases, the quantity of X falls down. As the price of Y increases, the quantity of X falls down. Since the quantity sold of X falls as the price of Y increases, X and Y are complementary goods.

Similarly, as the price of Z rises, the quantity sold of good X rises. This means that X and Z are substitutes to each other.

An inferior good is a good whose demand decreases as the income increase. As the income is increasing, the demand for X is increasing. Thus, X is a normal good.

3. The null hypothesis of the independent variables is that they are not significant and the alternate hypothesis is that they are significant. Generally, if the p value is less than the significance level, the null hypthesis is rejected.

Given significance level is 5% or 0.05.

X price p-value= 1.71E-09= very small number (<0.05)= significant

Y price p-value= 1.03E-20= very small number (<0.05)= significant

Z price p-value= 7.95E-16= very small number (<0.05)= significant

income p-value= 1.11E-09= very small number (<0.05)= significant

4. Price elasticity is a measure of the responsiveness of the quantity demanded to a change in price. More specifically, it is the ratio of the percentage change is quantity to a percentage change in price.

The price of X is given as 10, price of Y is 15, price of z is 24 and the income is 42,000. We are required to find the price elasticity for X here. Let us find the percentage change in X and percentage change in Y. At the given values, the quantity demanded is 102.13213 (answer 1). Say, the price changes by 10% to 11. The new value of quantity demanded is 94.76174. The change in quantity demanded is -7.37039, thus the percentage change is -(7.37039/102.13213 ) * 100

= -7.216%

Thus, the elasticity is -7.216/10= -0.7216 ( or 0.7216 more generally)

This value is less than 1. This means that the quantity change is not as responsive as price change. Hence, we can say that the demand is inelastic.

5. Marginal cost is fixed at 5.

Let us assume that the condition for maximizing profits is marginal revenue = marginal cost ( I am not sure if this is the condition. If it is not, you may fit in the function on your own; process remains the same).

Total revenue = Price * quantity

Total revenue= Price * (175.83603- 7.37039 *price )

Therefore total revenue= 175.83603Price - 7.37039Price²

Marginal revenue= differentitate (total revenue with respect to price) = 175.83603-14.74078Price

175.83603-14.74078Price= 5

Therefore, 170.83603= 14.74078Price

Price~ 11.6

Quantity~ 175.83603- 7.37039 *11.6 ~ -90.418015

( NOTE: Please make sure that the profit maximization happens at MR=MC)

6. Cross price elasticity is very similar to price elasticity, only difference being that the change in price is of Y, and we measure the change in quantity of X.

Say, y=15. The quantity demanded is 102.13213. Say, price changes by 10% to 16.5. New quantity demanded is 183.55408 - 89.564145= 93.989935. Percentage change in quantity is 93.989935-102.13213

= (-8.142195/ 102.13213) * 100= -8% Therefore, cross price elasticity is -8/10= -0.8. Inelastic