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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 accounts for 65 percent of total industry sales of automatic pool cleaners. Its closest competitor, Howard Industries, sells a competing pool cleaner that has captured about 18 percent of the market. Six other vary small firms share the rest of the industry’s sales. Using the last 26 months of production and cost data, PoolVac wishes to estimate its unit variable costs using the following quadratic specification:The monthly data on average variable cost (AVC), and the quantity of Sting Rays produced and sold each month (Q) are presented in the table below.PoolVac also wishes to use its sales data for the 26 months to estimate demand for its Sting Ray. Demand for Sting Ray is specified to be a linear function of its price (P), average income for households in the U.S. that have swimming pools (Mavg), and the price of the competing pool cleaner sold by Howard Industries (PH):The table below presents the last 26 months on the price charged for a Sting Ray (P), average of households with pools (Mavg), and the price Howard Industries charged for its pool cleaner (PH): obsAVCQPMavgPH110916472755800017521181664275580001753121129530058000200410213313005630020051211413300563002006102137830056300200710513713005785020081011312300578502009108130132557850250101138543505760025011114963350576002501210512383255760022513107107632558250225141041092325582502251510412223255825022516102130832558985250171161259325589852501812671137558985250191161118350596002502013991475596003752115213747559600375221168573756080025023127100335060800250241231328320608002202510413763206235022026114121932062350220PoolVac, Inc. incurs total fixed costs of $45,000 per month.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. b.Using the regression results from part 1 a, write the estimated total variable cost, average variable cost, and marginal cost functions (TVC, AVC, and MC) for PoolVac.c.Find the quantity where average variable cost is minimized and compute the minimum average variable cost. 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. b.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 part 2 a, write the estimated demand function, inverse demand function (P as a function of Q), and marginal revenue function. 3. Using your estimated cost and demand function from part 1 and 2, what price would you recommend the manager of PoolVac, Inc. charge for its Sting Ray? Given your recommended price, estimate the number of units PoolVac can expect to sell, as well as its monthly total revenue, total cost, and profit. 4. For the profit maximizing solution in question 3, compute the point elasticity of demand for Sting Rays. In the profit maximizing situation in question 3, a 5 percent price cut would be predicted to _______________ (increase, decrease) quantity demanded of Sting Ray by _________ percent, which would cause total revenue to _____________ (rise, fall, stay the same) and profit to _____________ (rise, fall, stay the same). 5. a. For the profit maximizing solution in question 3, compute the income elasticity of demand for Sting Rays. b.Is the algebraic sign of the income elasticity as you expected? Explain. c.A 10 percent increase in Mavg would be predicted to ___________ (increase, decrease) quantity demanded of Sting Rays by __________ percent. 6.a. For the profit maximizing solution in question 3, compute the cross-price elasticity of demand for Sting Rays. b. Is the algebraic sign of the income elasticity as you expected? Explain. c.A 3 percent decrease in PH would be predicted to __________ (increase, decrease) quantity demanded of Sting Rays by _________ percent. 7. If total fixed cost increase from $45,000 to $55,000, what price would you now recommend in order to maximize profits at PoolVac? Compute the number of units sold at this price, total revenue, total cost and profit. 8. If the manager of PoolVac wanted to maximize total revenue instead of profit (a bad idea), the manager would charge a price of ________. At this price, PoolVac’s profit would be ________, which is __________ (higher than, lower than, the same as) the profit in question 3.
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