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 26 quarters of production and cost data, PoolVac wishes to estimate its average variable costs using the following quadratic specification:

AVC = a + bQ + cQ 2

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. PoolVac also wishes to use its sales data for the last 26 quarters to estimate demand for its Sting Ray. Demand for Sting Rays is specified to be a linear function as the following:

Qd = d + eP + fM + gPH

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 appropriate regression to estimate the average variable cost function (AVC) for Sting Rays. Evaluate the statistical significance of the three estimated parameters using a significance level of 5 percent. Be sure to comment on the algebraic signs of the three parameter estimates.

2. Given your answer in 1, show the estimated total variable cost, average variable cost, and marginal cost functions (TVC,?AVC, and?MC) for PoolVac.

3. Apply dummy variable to construct the time-series quarterly sales estimation of Sting Ray (Hint:?Q = A+Bt+?). Please predict the quantity sold in the first quarter 2013.

4. Run the appropriate regression to estimate the demand function for Sting Rays. Evaluate the statistical significance of the three estimated?slope?parameters using a significance level of 5 percent. Discuss the appropriateness of the algebraic signs of each of the three?slope?parameter estimates.

5. The manager at PoolVac, 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. Using the regression results from Question 4, write the estimated demand function (with only?P?as the independent variable), inverse demand function, and marginal revenue (MR) function.

6. Given your MC function in question 2 and MR function in question 5, what is the profit-maximizing unit price PoolVac should charge for Sting Ray?

Answers I have:

1. Run the appropriate regression to estimate the average variable cost function (AVC) for Sting Rays. Evaluate the statistical significance of the three estimated parameters using a significance level of 5 percent. Be sure to comment on the algebraic signs of the three parameter estimates.AVC = 152.88 - 0.0614Q + 0.00002Q2For the statistical significance the rule is: if p value is less than the significance level then we do not accept the null hypothesis; the coefficients is statistically significant.P value for coefficient of P is 0. This is less than .01 or .05. hence the coefficient is significant.P value for coefficient of MAVG is 0.034 This is more than .01, hence the coefficient is not significant at 1% level. But as .034<.5 it becomes significant at level.p value for coefficient of ph is this more than .01 hence the not level. but as .028 level.the q negative avc increases decreases.the positive increases.>

2. Given your answer in 1, show the estimated total variable cost, average variable cost, and marginal cost functions (TVC, AVC, and MC) for PoolVac.AVC = 152.88 - 0.0614Q + 0.00002Q2TVC = 152.88Q - 0.0614Q2+ 0.00002Q3MC = 152.88-0.1208Q + 0.00006Q2

3. Apply dummy variable to construct the time-series quarterly sales estimation of Sting Ray (Hint: Q = A+Bt+?). Please predict the quantity sold in the first quarter 2013.The equation is Q = 1436.372 - 23.9313T + 47.44963D1 + 9.380952D2 + 36.73535t = 29D1 = 1D2 = 0D3 = 0Q = 1436.372 - 23.9313 * (29) + 47.44963 * (1) + 9.380952 * (0) + 36.73535 * (0)Q = 1436.372 - 694.0077 + 47.44963 + 0 + 0We get estimated Q for 1st quarter of 2013 = 789.81393

4. Run the appropriate regression to estimate the demand function for Sting Rays.Evaluate the statistical significance of the three estimated slope parameters using a significance level of 5 percent. Discuss the appropriateness of the algebraic signs of each of the three slope parameter estimates.Qd = 2729 - 10.76 P + 0.0214 MAVG + 3.17 PH? R2 tells us the predictive power of the regression equation. It equals the ratio of explained variation in Q and the total variation in Q. In our case the value is 96.6%, which is high. So 96.6%of variation in Q is explained by P, MAVG and PH.? For overall significance we use the F value. It is 211 in our case. The critical value of F with degrees of freedom (3,22) is 3.05 at 5% levels. Since 211 > 3.05, we do not accept the null hypothesis that the regression is overall insignificant. Our data supports the claim that the regression equation is statistically significant.? The sign of P is negative, showing that the good obeys the law on demand-it is a normal good. As price rises quantity demanded falls.? The sign of MAVG is positive, showing that it is a non-inferior good. As income rise quantity demanded rises.? The sign of PH is positive showing that both firms? gods are substitutes of each other. As PH rises, quantity of competitor?s good falls and our good?s demand rises. As our good substitutes for the competitor?s good.? P coefficient = -10.8 means that for a 1 unit fall in P demand rises by 10.8 units. Or if price rises by 1 unit then demand falls by 10.8 units.? MAVG coefficient is 0.0214. If income rise by $1 then demand rises by 0.0214 units.? PH coefficient is 3.17. If the price of competitor?s goods rises by $1, then our demand rises by 3.17 units.

5. The manager at PoolVac, 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. Using the regression results from Question 4, write the estimated demand function (with only P as the independent variable), inverse demand function, and marginal revenue (MR) function.

6. Given your MC function in question 2 and MR function in question 5, what is the profit-maximizing unit price PoolVac should charge for Sting Ray?

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1 2 Quarter/Year 1st/2006 4 2nd/2006 5 3rd/2006 4th/2006 1st/2007 2nd/2007 3rd/2007 4th/2007 1st/2008 2nd/2008 3rd/2008 4th/2008 1st/2009 2nd/2009 3rd/2009 4th/2009 1st/2010 2nd/2010 3rd/2010 4th/2010 1st/2011 2nd/2011 3rd/2011 4th/2011 1st/2012 2nd/2012 34567 8 9 10 11 12 13 14 15 A 16 17 18 19 20 21 22 23 24 25 26 27 28 B Period (t) Period (t) 12345 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 D E Sting Ray-PoolVac, Inc. AVC Q 109 1647 118 1664 121 1295 102 1331 121 1413 102 1378 105 1371 101 1312 108 1301 113 854 114 963 105 1238 107 1076 104 1092 104 102 116 126 116 139 152 116 127 123 104 114 1222 1308 1259 711 1118 91 137 857 1003 1328 1376 1219 P 275 275 300 300 300 300 300 300 325 350 350 325 325 325 325 325 325 375 350 475 475 375 350 320 320 320 F M 58000 58000 58000 56300 56300 56300 57850 57850 57850 57600 57600 57600 58250 58250 58250 58985 58985 58985 59600 59600 59600 60800 60800 60800 62350 62350 G PH 175 175 200 200 200 200 200 200 250 250 250 225 225 225 225 250 250 250 250 375 375 250 250 220 220 220