Question

0005555000055000003331 5552222555577552223332 112222222222222222233222222222 000555 555888 222999 005005 554552 888338990 222223 3338886 8886667, 7 7 7 7 7 8 8 8 8 8 8 9 99 5005555555055500002227 25522222275777, 5 2 2 2 1 1 10 223333333333333333344333333333 745138121 4693-7710 862289 379205 Q662 3. 4 3 3 3 3 5 6 2 0 0 2 3 2 38691479 43 518 2712509 1 9 3 5 0 3 3 2 3 3 3 2 7 93> 8 2 812125183457 43171 012-35-22 012345678901234567890 1 2 3 4 5 6 7 8 9 e666677, 7788889 9900001111222233 ddh nrts ddh nrts ddh nrts ddh nrts ddh nrts ddh nrts ddh nrts u-2341 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 CASE 2- COsT STRCTURE and PRICING: Sting Ra PoolVac, Inc. manufactures and sells a single product called the "Sting Ray," which is a patent-protected automatic cleaning device for swimming pools. PoolVac's Sting Ray faces its closest competitor, Howard Industries, also selling a competing pool cleaner Using the last 30 quarters of production and cost data, PoolVac wishes to estimate its average variable costs using the following quadratic specification: The quarterly data on average variable cost (AVC), and the quantity of Sting Rays produced and sold each quarter (Q) are presented in the data file. RoolVac also wishes to use its sales data for the last 30 quarters to estimate demand for its Sting Ray. Demand for Sting Rays is specified to be a linear function as the following in which its price (P), average income for households in the U.S. that have swimming pools (M), and the price of the competing pool cleaner sold by Howard Industries (PH) QUESTIONS 1. Run the log-linear regression to estimate the demand function for Sting Rays Evaluate the statistical significance of the three estimated coefficients of parameters by using a significance level of 5 percent. Discuss the elasticities (price elasticity of demand, income elasticity and cross-price elasticity) to define the characters of Sting Ray. (20%) 2. The manager at RooVac, Inc. believes Howard Industries is going to price its automatic pool cleaner at $250, and average household income in the U.S. is expected to be $65,000. Please run a multiple linear regression then explore the inverse demand function (i.e. Price is dependent variable) and marginal revenue (MR) function (Hint: Half-way rule). (15%) 3. Given your MC function in question 2 and MR function in question 5, what is the profit maximizing unit price RooVac should charge for Sting Ray? (Hint Solve the quadratic equation by quadratic formula (10%) 2a

Answer 1

For log-linear regression, the dependent & independent variables are all firstly converted into logarithmic termsusing excel function =LN().

LnP	LnM	LnPH	LnQ
5.616771	10.9682	5.164786	7.406711
5.616771	10.9682	5.164786	7.41698
5.703782	10.9682	5.298317	7.166266
5.703782	10.93845	5.298317	7.193686
5.703782	10.93845	5.298317	7.25347
5.703782	10.93845	5.298317	7.228388
5.703782	10.96561	5.298317	7.223296
5.703782	10.96561	5.298317	7.179308
5.783825	10.96561	5.521461	7.170888
5.857933	10.96128	5.521461	6.749931
5.857933	10.96128	5.4161	6.870053
5.783825	10.96128	5.4161	7.121252
5.783825	10.9725	5.4161	6.981006
5.783825	10.9725	5.4161	6.995766
5.783825	10.9725	5.4161	7.108244
5.783825	10.98504	5.521461	7.176255
5.783825	10.98504	5.521461	7.138073
5.926926	10.98504	5.521461	6.566672
5.857933	10.99541	5.521461	7.019297
6.163315	10.99541	5.926926	4.51086
6.163315	10.99541	5.926926	4.919981
5.926926	11.01535	5.521461	6.753438
5.857933	11.01535	5.521461	6.910751
5.768321	11.01535	5.393628	7.191429
5.768321	11.04052	5.393628	7.226936
5.768321	11.04052	5.393628	7.105786
5.743003	11.04843	5.451038	7.186144
5.743003	11.0501	5.451038	7.210818
5.743003	11.0501	5.451038	7.17549
5.726848	11.05129	5.398163	7.16935

1. On executing log-linear regression in MS excel, the result is:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.92911092
R Square	0.863247101
Adjusted R Square	0.847467921
Standard Error	0.248199193
Observations	30
ANOVA
	df	SS	MS	F	Significance F
Regression	3	10.11050077	3.370166924	54.70798025	2.2873E-11
Residual	26	1.601673826	0.061602839
Total	29	11.7121746
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	14.60338525	16.43778571	0.888403433	0.382469391	-19.18496719	48.39173769	-19.18496719	48.39173769
LnP	-4.20906416	1.228144875	-3.427172352	0.002039952	-6.733552106	-1.684576215	-6.733552106	-1.684576215
LnM	1.745219039	1.439378228	1.212481198	0.236230922	-1.213465283	4.703903361	-1.213465283	4.703903361
LnPH	-0.450773553	0.939449118	-0.479827533	0.635362003	-2.381838872	1.480291766	-2.381838872	1.480291766

The log-linear regression equation is: LnQ = 14.60338525 - 4.20906416 LnP + 1.745219039 LnM -0.450773553LnPH

At 5% level of significance, the p-value of related t-statistic of variables LnP, LnM and LnPH are respectively 0.002039952 , 0.236230922 and 0.635362003, thereby implying that it is only for the variable LnP the actual p-value is less than critical p-value of 0.05 thereby indicating that at 5% level of significance, only LnP variable is statistically significant.

The variable LnM and LnPH are not statistically significant at 5% level of significance.

It shall be noted that coefficient of independent variables in log-linear regression model indicate partial elasticity.

The partial own-price elasticity of demand with respect to own price is -4.20906416

The partial income elasticity of demand with respect to income is 1.745219039

The partial cross-price elasticity of demand with respect to competing good price is -0.450773553

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2.

Executing multiple linear regression such that Price is dependent variable as per inverse demand function, the result is:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.989952373
R Square	0.980005701
Adjusted R Square	0.977698667
Standard Error	6.8546699
Observations	30
ANOVA
	df	SS	MS	F	Significance F
Regression	3	59878.21768	19959.40589	424.7902297	3.3834E-22
Residual	26	1221.648986	46.98649944
Total	29	61099.86667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	325.5320851	38.32229414	8.494587614	5.64009E-09	246.7594814	404.3046889	246.7594814	404.3046889
M	-3.26139E-05	0.000655148	-0.049781011	0.960677361	-0.001379289	0.001314061	-0.001379289	0.001314061
Q	-0.078609648	0.009254111	-8.494565111	5.64037E-09	-0.097631746	-0.059587551	-0.097631746	-0.059587551
PH	0.422440906	0.075350192	5.606368013	6.8365E-06	0.267556368	0.577325443	0.267556368	0.577325443

The inverse-demand regression equation is:

P = 325.5320851 - 0.0000326139M - 0.078609648Q + 0.422440906PH

The total revenue equation is:

TR=PQ = 325.5320851Q - 0.0000326139MQ - 0.078609648Q^2 + 0.422440906(PH)Q

The marginal revenue function is given by first derivative of total revenue equation. The result is:

MR = 325.5320851 - 0.0000326139M - 0.157219296Q + 0.422440906PH

Thus, it is observed that slope coefficient of Q in inverse-demand function is half of that in marginal revenue function.

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3.

The question number 5 is not provided. Hence, the profit-maximizing unit price could not be calculated.

The approach to be used is evaluate MC = MR