Pricing and Production Decisions at PoolVac, Inc.

  PoolVac, Inc. manufactures and sells a sin" rel="nofollow">ingle product called the “Stin" rel="nofollow">ing Ray,” which is a patent-protected automatic cleanin" rel="nofollow">ing device for swimmin" rel="nofollow">ing pools.  PoolVac’s Stin" rel="nofollow">ing Ray accounts for 65 percent of total in" rel="nofollow">industry sales of automatic pool cleaners.  Its closest competitor, Howard Industries, sells a competin" rel="nofollow">ing pool cleaner that has captured about 18 percent of the market.  Six other very small firms share the rest of the in" rel="nofollow">industry’s sales.  Usin" rel="nofollow">ing the last 26 months of production and cost data, PoolVac wishes to estimate its unit variable costs usin" rel="nofollow">ing the followin" rel="nofollow">ing quadratic specification:  AVC = a+ bQ+cQ2  The monthly data on average variable cost (AVC), and the quantity of Stin" rel="nofollow">ing Rays produced and sold each month (Q) are presented in" rel="nofollow">in the table below.  PoolVac also wishes to use its sales data for the last 26 months to estimate demand for its Stin" rel="nofollow">ing Ray.  Demand for Stin" rel="nofollow">ing Rays is specified to be a lin" rel="nofollow">inear function of its price (P), average in" rel="nofollow">income for households in" rel="nofollow">in the U.S. that have swimmin" rel="nofollow">ing pools (Mavg), and the price of the competin" rel="nofollow">ing pool cleaner sold by Howard Industries (PH):  Qd = d+ eP + fMavg + gPH The table below presents the last 26 months of data on the price charged for a Stin" rel="nofollow">ing Ray (P), average in" rel="nofollow">income of households with pools (MAVG), and the price Howard Industries charged for its pool cleaner (PH):   obs AVC Q P MAVG PH1 109 1647 275 58000 1752 118 1664 275 58000 1753 121 1295 300 58000 2004 102 1331 300 56300 2005 121 1413 300 56300 2006 102 1378 300 56300 2007 105 1371 300 57850 2008 101 1312 300 57850 2009 108 1301 325 57850 25010 113 854 350 57600 25011 114 963 350 57600 25012 105 1238 325 57600 22513 107 1076 325 58250 22514 104 1092 325 58250 22515 104 1222 325 58250 22516 102 1308 325 58985 25017 116 1259 325 58985 25018 126 711 375 58985 25019 116 1118 350 59600 25020 139 91 475 59600 37521 152 137 475 59600 37522 116 857 375 60800 25023 127 1003 350 60800 25024 123 1328 320 60800 22025 104 1376 320 62350 22026 114 1219 320 62350 220   PoolVac, Inc. in" rel="nofollow">incurs total fixed costs of $45,000 per month.  1. a.  Run the appropriate regression to estimate the average variable cost function (AVC) for Stin" rel="nofollow">ing Rays.  Evaluate the statistical significance of the three estimated parameters usin" rel="nofollow">ing a significance level of 5 percent. b. Usin" rel="nofollow">ing the regression results from part 1 a, write the estimated total variable cost, average variable cost, and margin" rel="nofollow">inal cost functions (TVC, AVC, and MC) for PoolVac.  TVC = __________________________________________  AVC = __________________________________________  MC = ___________________________________________  c. Compute min" rel="nofollow">inimum average variable cost.    Qmin" rel="nofollow">in = ___________ AVCmin" rel="nofollow">in = ______________   2. a. Run the appropriate regression to estimate the demand function for Stin" rel="nofollow">ing Rays.  Evaluate the statistical significance of the three estimated slope parameters usin" rel="nofollow">ing 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 goin" rel="nofollow">ing to price its automatic pool cleaner at $250, and average household in" rel="nofollow">income in" rel="nofollow">in the U.S. is expected to be $65,000.  Usin" rel="nofollow">ing the regression results from part 2 a, write the estimated demand function, in" rel="nofollow">inverse demand function, and margin" rel="nofollow">inal revenue function.     Demand: ____________________________   Inverse Demand: ____________________________   Margin" rel="nofollow">inal Revenue: ____________________________  3. Usin" rel="nofollow">ing your estimated cost and demand functions from parts 1 and 2, what price would you recommend the manager of PoolVac, Inc. charge for its Stin" rel="nofollow">ing 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.     P: ___________   Q: ___________   TR: ___________   TC: ___________  Profit: ___________  4. For the profit-maximizin" rel="nofollow">ing solution in" rel="nofollow">in question 3, compute the poin" rel="nofollow">int elasticity of demand for Stin" rel="nofollow">ing Rays.    E = ______________  In the profit-maximizin" rel="nofollow">ing situation in" rel="nofollow">in question 3, a 5 percent price cut would be predicted to _______________ (in" rel="nofollow">increase, decrease) quantity demanded of Stin" rel="nofollow">ing Rays by ___________ percent, which would cause total revenue to _____________ (rise, fall, stay the same) and profit to _____________ (rise, fall, stay the same).  5. For the profit-maximizin" rel="nofollow">ing solution in" rel="nofollow">in question 3, compute the in" rel="nofollow">income elasticity of demand for Stin" rel="nofollow">ing Rays.   EM = ______________  a. Is the algebraic sign of the in" rel="nofollow">income elasticity as you expected?  Explain" rel="nofollow">in.        b. A 10 percent in" rel="nofollow">increase in" rel="nofollow">in Mavg would be predicted to _______________ (in" rel="nofollow">increase, decrease) quantity demanded of Stin" rel="nofollow">ing Rays by ___________ percent.    6. For the profit-maximizin" rel="nofollow">ing solution in" rel="nofollow">in question 3, compute the cross-price elasticity of demand for Stin" rel="nofollow">ing Rays.  EXR = ______________  a. Is the algebraic sign of the in" rel="nofollow">income elasticity as you expected?  Explain" rel="nofollow">in.       b. A 3 percent decrease in" rel="nofollow">in PH would be predicted to _______________ (in" rel="nofollow">increase, decrease) quantity demanded of Stin" rel="nofollow">ing Rays by ___________ percent.      7. If total fixed costs in" rel="nofollow">increase from $45,000 to $55,000, what price would you now recommend in" rel="nofollow">in order to maximize profits at PoolVac?  Compute the number of units sold at this price, total revenue, total cost and profit:   P: ___________   Q: ___________   TR: ___________   TC: ___________  Profit: ___________   8. If the manager of PoolVac wanted to maximize total revenue in" rel="nofollow">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" rel="nofollow">in question 3.