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	-3.42813	0.21342	-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:

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 exactly should the producer of model X respond when the producer of Y raises its price, say by $1?

Cross Price Elasticity = _______________

Strategic response:

Answer 1

The regression estimate would be .

Qx = 39.0819- 7.37039 PX - 3.42813PY + 4.05067P2 +0.002881

5. The marginal cost of X is MC=5. For the given prices and income, we have
Qx = 39.0819-7.37039 PX-3.42813*15+4.05067* 24+0.00288*42000
or
Qx = 205.83603- 7.37039 PX
or
Px = 27.927427178 -0.1356780310x
as the (inverse) demand for X.

The total revenue would be
TR=0xPx = 27.9274271780 x -0.1356780310
and the marginal revenue would be
_ dặTR) MR= dox
or
d MR= (27.9274271780x -0.1356780310) dox
or
MR = 27.927427178-0.2713560620x
.

The profit maximization would be where the MC is equal to the MR, ie where or
5 = 27.927427178 -0.2713560620x
or $\widehat Q_X = 84.492039754 \approx 84.492$ untis. The corresponding price would be
P = 27.927427178 -0.13567 8031 * 84.492039754
or
Px = 16.463713589 16.46
dollars.

Hence, the optimal price is $16.46 and the optimal quantity is 84.492 units of X.

6. For given price of X and Z, and income, we have $\widehat Q_X = 39.0819 - 7.37039 *10 - 3.42813 P_Y + 4.05067 *24 + 0.00288*42000$ or $\widehat Q_X = 183.55408 - 3.42813 P_Y$ . The cross price elasticity between X and Y is $e_c = \frac{\mathrm{d} \widehat Q_X / \widehat Q_X}{\mathrm{d} P_Y / P_Y}$ or $e_c = \frac{P_Y}{\widehat Q_X} * \frac{\mathrm{d} \widehat Q_X}{\mathrm{d} P_Y}$ or $e_c = \frac{P_Y}{\widehat Q_X} * \frac{\mathrm{d} }{\mathrm{d} P_Y}(183.55408 - 3.42813 P_Y)$ or
HETSTHE-)*
or $e_c = (- 3.42813)\frac{P_Y}{\widehat Q_X}$ .

For price of Y be 15, the quantity would be $\widehat Q_X = 183.55408 - 3.42813 *15 = 132.13213$ . The cross price elasticity would hence be $e_c = (- 3.42813)\frac{15}{132.13213}$ or $e_c = - 0.389170673 \approx - 0.3892$ . This means that for a marginal unit percent increase in price of Y, the quantity of X would reduce by 0.3892%.

The stratergic response would depend on the found cross price elasticity. For a rise in price of Y by $1 would mean that the price of Y increased by $\frac{1}{15}*100 = 6.666666667$ percent. The demand for X would reduce by $0.389170673*6.666666667 = 2.594471153 \approx 2.5945$ percent. As demand reduces, the producer must increase the price of X.

The quantity would hence reduce to $132.1321(1 - \frac{2.5945}{100}) = 128.703932665$ , and since
Qx = 205.83603- 7.37039 PX
, we have the required price as
Px = 10.465131063 10.47
dollars.

Hence, cross price elasticity is -0.3892 and the strategic response would be to to increase the price by $0.47.