We are going to use the Pool Vac case to conduct a mini-project for class. We will use part of one class and all of next week’s labs to work specifically on the project. If you believe that you need additional time, I can try to make arrangements to get a computer lab next Monday.

Assume that you work for the WKIA (We Know It All) Consulting Firm. Pool Vac has come to you to find out several things about its production process and demand. You will write a report of approximately four pages to answer these questions (this would include your output but assume that any graphs would comprise additional pages. You may work alone or with one other person to complete this assignment. (If working with a partner, I will have greater expectations regarding writing & editing, and interpretation of output).

The pdf file is included in this folder but I did copy it onto this WORD file. You can use the answers to develop your report, where you present to Pool Vac your recommendations for pricing. Along the way, you will present information about costs, elasticity, and predicted sales. Finally, you will see Question 8, which is MY question relating to forecasting. Please include information about the forecast from this method.

I am letting the organization of your report to be your choice (although I do expect an introduction and concluding paragraph). Think about what information will be important to your client. In addition to your final report, attach your answers to the questions from the case.

This assignment is worth 15% of your grade. I have high expectations from you because you have a group a good writers with good technical skills. (I’m actually looking forward to grading your assignment!)

CONSULTING PROJECT

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= a0+a1Q + a2Q2

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):

Qd = b0 + b1P + b2Mave + b3Ph

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). I have created a Minitab Worksheet for you: PoolVac, Inc. incurs total fixed costs of $45,000 per month.

OBS AVC Q P MAVG PH Qsq

1 109 1647 275 58000 175 2712609

2 118 1664 275 58000 175 2768896

3 121 1295 300 58000 200 1677025

4 102 1331 300 56300 200 1771561

5 121 1413 300 56300 200 1996569

6 102 1378 300 56300 200 1898884

7 105 1371 300 57850 200 1879641

8 101 1312 300 57850 200 1721344

9 108 1301 325 57850 250 1692601

10 113 854 350 57600 250 729316

11 114 963 350 57600 250 927369

12 105 1238 325 57600 225 1532644

13 107 1076 325 58250 225 1157776

14 104 1092 325 58250 225 1192464

15 104 1222 325 58250 225 1493284

16 102 1308 325 58985 250 1710864

17 116 1259 325 58985 250 1585081

18 126 711 375 58985 250 505521

19 116 1118 350 59600 250 1249924

20 139 91 475 59600 375 8281

21 152 137 475 59600 375 18769

22 116 857 375 60800 250 734449

23 127 1003 350 60800 250 1006009

24 123 1328 320 60800 220 1763584

25 104 1376 320 62350 220 1893376

26 114 1219 320 62350 220 1485961

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.

ANSWER:

Regression Analysis: avc versus q, q**2

The regression equation is

avc = 153 – 0.0614 q + 0.000022 q**2

Predictor Coef SE Coef T P

Constant 152.881 6.605 23.15 0.000

q -0.06141 0.01462 -4.20 0.000

q**2 0.00002181 0.00000810 2.69 0.013

S = 7.73063 R-Sq = 63.5% R-Sq(adj) = 60.3%

Since the P values of all 3 variables are within the 5% confidence interval, each variable should be considered statistically significant in determining the demand of the pool vacuums. The mathematical operators indicate a trend that conforms with the AVC curve: That is, the trend first goes down and then increases again.

b. Using the regression results from 1a, write the estimated total variable cost, average variable cost, and marginal cost functions (TVC, AVC, and MC) for PoolVac.

TVC = AVC * q => 153 (q) – .0614 q(q) + .000022 q2(q)

= 153q – .0614 q2 + .000022 q3

AVC = 153 – 0.0614 q + 0.000022 q2

MC = = 153 – (2).0614 q + (3).000022 q2

= 153 – .1228q + .000066q2

c. Compute minimum average variable cost.

Qmin = = -.0614 + (2).000022q -.0614 + .000044q

Set q = 0: -.0614 = -.000044q q = Qmin = 1395.

AVCmin = Substitute q with 1395.

153 – 0.0614 (1395. ) + 0.000022 (1395. )2

153 – 85.6806 + 42.8401 = 110.1595

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.

Regression Analysis: q versus pown, mavg, phoward

The regression equation is

q = 2729 – 10.8 pown + 0.0214 mavg + 3.17 phoward

Predictor Coef SE Coef T P

Constant 2728.8 531.7 5.13 0.000

pown -10.758 1.330 -8.09 0.000

mavg 0.021420 0.009452 2.27 0.034

phoward 3.166 1.344 2.36 0.028

S = 73.0546 R-Sq = 96.6% R-Sq(adj) = 96.2%

We should look at the P value for each of the slope parameters and in doing so, we find that price is 100% significant, average income (Mavg) is 96.6% (100-.034) and the price of Howard Industries’ product (Ph) is 97.2% significant (100-.028).

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 2a, write the estimated demand function, inverse demand function, and marginal revenue function.

Demand: QD = 2729 – 10.8p + .0214(65,000) + 3.17(250)

2729 – 10.8p + 1391 + 792.5

2729 + 2183.5 -10.8p

QD = 4912.5 – 10.8p

Inverse Demand: QD = 4912.5 – 10.8p

solve for p, and divide both sides by 10.8

Marginal Revenue: MR = TR and TR = P*Q

( )

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.

P: ___________

Q: ___________

TR: ___________

TC: ___________

Profit: ___________

4. For the profit-maximizing solution in question 3, compute the point elasticity of demand for Sting Rays.

Ep = ______________

In the profit-maximizing situation in question 3, a 5 percent price cut would be predicted to _______________ (increase, decrease) quantity demanded of Sting 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-maximizing solution in question 3, compute the income elasticity of demand for Sting Rays.

EM = ______________

a. Is the algebraic sign of the income elasticity as you expected? Explain.

b. A 10 percent increase in Mavg would be predicted to _______________ (increase, decrease) quantity demanded of Sting Rays by ___________ percent.

6. For the profit-maximizing solution in question 3, compute the cross-price elasticity of demand for Sting Rays.

EXR = ______________

a. Is the algebraic sign of the income elasticity as you expected? Explain.

b. A 3 percent decrease in PH would be predicted to _______________ (increase, decrease) quantity demanded of Sting Rays by ___________ percent.

7. If total fixed costs 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:

P: ___________

Q: ___________

TR: ___________

TC: ___________

Profit: ___________

8. The other way we can estimate sales is with a forecasting technique. Estimate two models (of your choosing) to forecast estimated sales in period 27. Include measures of good fit and a 95% confidence interval around your forecast. Select one of the two models to present to your client.

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