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## PoolVac, Inc. manufactures and sells a single product

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 (TVCAVC, 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?

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 + 0.00002Q2
For 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 5% level.
P value for coefficient of PH is 0.028 This is more than .01, hence the coefficient is not significant at 1% level. But as .028<.5, it becomes significant at 5% level.
The coefficient of Q is negative, as AVC increases the value of Q decreases.
The coefficient of Q² is positive, as AVC increases the value of Q² 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 + 0.00002Q2
TVC = 152.88Q – 0.0614Q2+ 0.00002Q3
MC = 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 + 36.73535
t = 29
D1 = 1
D2 = 0
D3 = 0
Q = 1436.372 – 23.9313 * (29) + 47.44963 * (1) + 9.380952 * (0) + 36.73535 * (0)
Q = 1436.372 – 694.0077 + 47.44963 + 0 + 0
We 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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