case - cost strcture and pricing sting ray

Question

CASE - COST STRCTURE and PRICING: Sting Ray

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 faces its closest competitor, Howard Industries, also selling a competing pool cleaner. Using the last 26 quarters of production and cost data, PoolVac wishes to estimate its average variable costs using the following quadratic specification:

AVC = a + bQ + cQ².

The quarterly data on average variable cost (AVC), and the quantity of Sting Rays produced and sold each quarter (Q) are presented in the data file. PoolVac also wishes to use its sales data for the last 26 quarters to estimate demand for its Sting Ray. Demand for Sting Rays is specified to be a linear function as the following:

Q_d = d + eP + fM + gP_H,

in which its price (P), average income for households in the U.S. that have swimming pools (M), and the price of the competing pool cleaner sold by Howard Industries (P_H).

QUESTIONS

1. 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.

2. Given your answer in 1, show the estimated total variable cost, average variable cost, and marginal cost functions (TVC, AVC, and MC) for PoolVac.

3. Apply dummy variables to construct the time-series quarterly sales estimation of Sting Ray (Hint: Q = A+Bt+D1t...). Please predict the quantity sold in the first quarter 2014.

4. Run the log-linear regression to estimate the demand function for Sting Rays. Evaluate the statistical significance of the three estimated coefficients of parameters by using a significance level of 5 percent. Discuss the elasticities (price elasticity of demand, income elasticity and cross-price elasticity) to define the characters of Sting Ray.

5. 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. Please run a multiple linear regression then explore the inverse demand function (i.e. Price is dependent variable) and marginal revenue (MR) function (Hint: Half-way rule).

6. Given your MC function in question 2 and MR function in question 5, what is the profit-maximizing unit price PoolVac should charge for Sting Ray? (Hint: Solve the quadratic equation by quadratic formula

Sting Ray-PoolVac, Inc.
Quarter/Year	Period (t)	AVC	Q	P	M	PH
1st/2006	1	109	1647	275	58000	175
2nd/2006	2	118	1664	275	58000	175
3rd/2006	3	121	1295	300	58000	200
4th/2006	4	102	1331	300	56300	200
1st/2007	5	121	1413	300	56300	200
2nd/2007	6	102	1378	300	56300	200
3rd/2007	7	105	1371	300	57850	200
4th/2007	8	101	1312	300	57850	200
1st/2008	9	108	1301	325	57850	250
2nd/2008	10	113	854	350	57600	250
3rd/2008	11	114	963	350	57600	250
4th/2008	12	105	1238	325	57600	225
1st/2009	13	107	1076	325	58250	225
2nd/2009	14	104	1092	325	58250	225
3rd/2009	15	104	1222	325	58250	225
4th/2009	16	102	1308	325	58985	250
1st/2010	17	116	1259	325	58985	250
2nd/2010	18	126	711	375	58985	250
3rd/2010	19	116	1118	350	59600	250
4th/2010	20	139	91	475	59600	375
1st/2011	21	152	137	475	59600	375
2nd/2011	22	116	857	375	60800	250
3rd/2011	23	127	1003	350	60800	250
4th/2011	24	123	1328	320	60800	220
1st/2012	25	104	1376	320	62350	220
2nd/2012	26	114	1219	320	62350	220