PoolVac, Inc. manufactures and sells a single product called the “Sting Ray,” wh
ID: 1126686 • Letter: P
Question
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):
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):
obs
AVC
Q
P
Mavg
PH
1
109
1647
275
58000
175
2
118
1664
275
58000
175
3
121
1295
300
58000
200
4
102
1331
300
56300
200
5
121
1413
300
56300
200
6
102
1378
300
56300
200
7
105
1371
300
57850
200
8
101
1312
300
57850
200
9
108
1301
325
57850
250
10
113
854
350
57600
250
11
114
963
350
57600
250
12
105
1238
325
57600
225
13
107
1076
325
58250
225
14
104
1092
325
58250
225
15
104
1222
325
58250
225
16
102
1308
325
58985
250
17
116
1259
325
58985
250
18
126
711
375
58985
250
19
116
1118
350
59600
250
20
139
91
475
59600
375
21
152
137
475
59600
375
22
116
857
375
60800
250
23
127
1003
350
60800
250
24
123
1328
320
60800
220
25
104
1376
320
62350
220
26
114
1219
320
62350
220
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. 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.
2. a. Run the appropriate regression to estimate the demand function for Sting Rays. 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.
Is the algebraic sign of the income elasticity as you expected? Explain.
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.
obs
AVC
Q
P
Mavg
PH
1
109
1647
275
58000
175
2
118
1664
275
58000
175
3
121
1295
300
58000
200
4
102
1331
300
56300
200
5
121
1413
300
56300
200
6
102
1378
300
56300
200
7
105
1371
300
57850
200
8
101
1312
300
57850
200
9
108
1301
325
57850
250
10
113
854
350
57600
250
11
114
963
350
57600
250
12
105
1238
325
57600
225
13
107
1076
325
58250
225
14
104
1092
325
58250
225
15
104
1222
325
58250
225
16
102
1308
325
58985
250
17
116
1259
325
58985
250
18
126
711
375
58985
250
19
116
1118
350
59600
250
20
139
91
475
59600
375
21
152
137
475
59600
375
22
116
857
375
60800
250
23
127
1003
350
60800
250
24
123
1328
320
60800
220
25
104
1376
320
62350
220
26
114
1219
320
62350
220
Explanation / Answer
1.
a. AVC = 141 - 0.02Q
As Q incraeses by 1, AVC decreases by 0.02
b. total variable cost = AVC*Q = 141Q - 0.02Q^2
average variable cost = 141 - 0.02Q
marginal cost functions = 141 - 0.04Q
c. the quantity where average variable cost is minimized = 1312
and the minimum average variable cost = 101
2.
a. demand function for Sting Rays
Q= 2570 - 8*P + 0.02*Mavg
As P incraeses by 1, Q decreases by 8 units
b.P = $250, and average household income (Mavg) = $65,000.
the estimated demand function = 2570 - 8*P + 0.02*Mavg = 2570 - 8*250 + 0.02*65000
inverse demand function (P as a function of Q) = (2570 - Q + 0.02*Mavg)/8
Total revenue function = P*Q = (2570 - Q + 0.02*Mavg)/8 * Q
marginal revenue function = (2570 - 2Q + 0.02*Mavg)/8
Note: max. 4 parts
SUMMARY OUTPUT Regression Statistics Multiple R 0.720724 R Square 0.519443 Adjusted R Square 0.49942 Standard Error 8.679171 Observations 26 ANOVA df SS MS F Significance F Regression 1 1954.166 1954.166 25.94209 3.28E-05 Residual 24 1807.872 75.32801 Total 25 3762.038 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 141.07 5.54 25.44 0.00 129.63 152.52 129.63 152.52 Q -0.02 0.00 -5.09 0.00 -0.03 -0.01 -0.03 -0.01 RESIDUAL OUTPUT Observation Predicted AVC Residuals 1 102.1386 6.861423 2 101.7367 16.26327 3 110.4593 10.54073 4 109.6083 -7.60829 5 107.6699 13.33005 6 108.4973 -6.49729 7 108.6628 -3.66276 8 110.0574 -9.05742 9 110.3174 -2.31744 10 120.8838 -7.88377 11 118.3072 -4.3072 12 111.8067 -6.80665 13 115.6361 -8.63606 14 115.2579 -11.2579 15 112.1849 -8.18487 16 110.152 -8.15197 17 111.3102 4.68975 18 124.2641 1.735945 19 114.6433 1.356746 20 138.9198 0.08018 21 137.8325 14.16754 22 120.8129 -4.81286 23 117.3617 9.638338 24 109.6792 13.3208 25 108.5446 -4.54456 26 112.2558 1.744217