Pricing and Production Decisions at PoolVac, Inc.
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 very 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:
AVC = a + bQ + cQ2
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 last 26 months to estimate demand for its Sting Ray. Demand for Sting Rays 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):
Q=d+eP+fM +gP d avgH
The table below presents the last 26 months of data on the price charged for a Sting Ray (P), average income of households with pools (MAVG), and the price Howard Industries charged for its pool cleaner (PH):
PoolVac, Inc. incurs total fixed costs of $45,000 per month.
1.a. 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 = 152.8806 – 0.0614Q + 0.0000218Q2
a > 0 }
b < 0 } U-shaped
c > 0 }
All significant at a level of 5% (P < 0.05)
1.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.
TVC = 152.8806Q – 0.06141Q2+0.0000218Q3
AVC = 152.8806Q – 0.06141Q +0.0000218Q2
MC = 152.8806Q – 0.12282Q+0.0000654Q2
1.c. Compute minimum average variable cost.
Qmin = 1408
AVCmin = $109.63
2.a. 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 = 2727.824 – 10.7582P + 0.02142MAVG + 3.165924PH
B < 0 (Law of Demand)
C > 0 (Normal Good)
D > (Substitutes)
All are significant at a level of 5% (P< 0.05)
2.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, and marginal revenue function.
Demand: Qd = 4912.605 -10.7582P
Inverse Demand: P = 456.638 – 0;09295Qd
Marginal Revenue: MR = 456.638 – 0.859Q
3. Using your estimated cost and demand functions from parts 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. E = -10.7582 * (296.02/1728) = -1.814
In the profit-maximizing situation in question 3, a 5 percent price cut would be predicted to Increase (increase, decrease) quantity demanded of Sting Rays by 9.2 percent, which would cause total revenue to rise (rise, fall, stay the same) and profit to rise...