Cost Structure and Pricing: Sting Ray - Case Study Example

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), the 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. (30%)

From the regression results, the model can therefore be estimated as
AVC = 152.88 -0.0614Q +2.1812*10-5Q2
At a confidence level of 95% (5% level of significance), all three parameters are significant, as they have a p-value of less than 0.05. Notably, an (intercept) has a p-value of 1.96*10-17, while the coefficients b and have p –values of 0.000342 and 0.01299, respectively

The model implies that average variable costs are a function of both Q and the square of Q. The first parameter, the intercept is 152, implying that the expected average cost at constant quantity is 152. As quantity varies, the average cost will vary directly with Q2 (positive parameter) and inversely with the quantity (negative coefficient). Notably an increase in quantity does not necessarily increase average variable cost, the effect depends on which variable(either Q or Q2 ) has a greater influence.
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. (15%)
TVC = AVC * Q

= 152.88Q -0.0614Q2 +2.1812*10-5Q3
AVC = 152.88 -0.0614Q +2.1812*10-5Q2
MC= ∂TC/∂Q
Where TC (total costs) = FC (fixed costs) + TVC
TC = FC +152.88Q -0.0614Q2 +2.1812*10-5Q3
Thus MC = 152.88 – 0.1228Q + 6.5436*10-5Q2

3. Apply a dummy variable to construct the time-series quarterly sales estimation of Sting Ray (Hint: Q =A+Bt+…). Please predict the quantity sold in the first quarter of 2013. (10%)
By setting dummy variable D1 =1 if the first quarter and 0 otherwise
D2 = 1 if 2nd quarter and 0 otherwise
D3 = 1 if 3rd quarter and zero otherwise
The time series model data is specified as:
Qt = A + Bt + c1 D1 +c2D2 + c3D3 + ε
The model is thus estimated as:
Qt = 1436.372 -23.93t + 47.44D1 + 9.38D2 + 36.73D3
For first quarter of 2013, t = 29, D1 =1, D2=D3=0
Q29 =1436.372 -23.93*29 + 47.44*1

= 789.842
4. 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. (20%)
Regression results
The estimated demand function is thus:
Q = 2728.824 -10.76P +0.021M + 3.166PH

At a 5% significant level, all the parameters are statistically significant, since their p-values are less than 0.05.
d=2728.824 with a p-value of 3.8310-5, implying that if all variables included in the model are held constant, the expected company sales would total 2728.824.
e = -10.758(p-value of 4.83*10-8) the negative sign implies an inverse relationship. That is a reduction in price by a unit increases sales by 10.758 units, ceteris paribus, and vice versa.
f= 0.0214 (p-value = 0.0336). a unit change in M directly changes demand by 0.0214, if P and PH are held constant.
h= 3.166Ph and p-value of 0.027, implies that holding price and income constant, If competitors reduce their prices by a unit, customers shift from Sting ray to its substitutes hence demand reduces by 3.166 units and vice versa.

Notably, changes in income (M) and price of competitors have a direct impact on sales (positive coefficient) while own price changes inversely impact demand (negative coefficient)

Using the regression results from Question 4, write the estimated demand function (with only P as the independent variable), inverse demand function, and marginal revenue (MR) function. (15%) Read More