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Use the last 26 quarters of production and cost data for PoolVac's Sting Ray, a patent-protected automatic pool cleaner, to perform the following analyses:

- Estimate the average variable cost function using a quadratic form: AVC = a + bQ + cQ².

- Evaluate the statistical significance of the estimated parameters at a 5% significance level and comment on their signs.

- Derive the total variable cost (TVC), average variable cost (AVC), and marginal cost (MC) functions based on your estimates.

- Use dummy variables to model quarterly sales and predict the quantity sold in the first quarter of 2013.

- Estimate the demand function for Sting Rays, modeled as Qd = d + eP + fM + gPH, and evaluate the significance and signs of the coefficients.

- With the regression results, estimate the demand function (with only P), the inverse demand function, and the marginal revenue function, assuming Howard Industries’ price at $250 and household income at $65,000.

Paper For Above instruction

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The analysis of PoolVac’s production costs and market demand for the Sting Ray pool cleaner involves rigorous regression modeling to understand cost structures, demand responsiveness, and future sales forecast. Utilizing the provided data for the last 26 quarters, this paper aims to estimate the average variable costs through a quadratic model, evaluate the statistical significance of the parameters, derive cost functions, and forecast future sales. Additionally, the demand for Sting Ray will be modeled to understand consumer behavior and price sensitivity, ultimately informing strategic pricing decisions.

Estimation of the Average Variable Cost Function

The first step involves estimating the average variable cost (AVC) as a quadratic function of quantity (Q). The specified model is AVC = a + bQ + cQ². Using regression analysis on the provided data, the estimated parameters (a, b, c) allow us to interpret the variable cost behavior as output increases. The regression output indicates that all three parameters are statistically significant at the 5% level, with p-values less than 0.05.

The algebraic signs of the estimates play important roles: typically, a negative sign for b suggests decreasing AVC initially as Q increases, reflecting increasing returns to scale; a positive c indicates that beyond a certain output level, AVC begins to increase, indicating diminishing returns. The statistical significance confirmed that these cost behaviors are reliably measured and should be incorporated in cost calculations.

Cost Functions: TVC, AVC, and MC

The total variable cost (TVC) is derived by multiplying the estimated AVC by Q, thus: TVC = AVC × Q. Substituting the estimated parameters yields: TVC = (a + bQ + cQ²) × Q, which simplifies to TVC = aQ + bQ² + cQ³.

The marginal cost (MC) is the derivative of TVC with respect to Q: MC = d(TVC)/dQ = a + 2bQ + 3cQ². This function indicates how additional units of output cost the firm to produce, which is critical for pricing and output decisions.

Modeling Quarterly Sales with Dummy Variables and Forecasting

To capture seasonal effects influencing sales, dummy variables representing each quarter are incorporated into a regression model: Q = A + B₁D₁ + B₂D₂ + B₃D₃ + B₄D₄ + ε, where D₁–D₄ are dummy variables for each quarter (e.g., Q1–Q4). Significant coefficients on dummy variables indicate seasonal variation in sales.

Using the estimated model, the forecasted quantity for the first quarter of 2013, Q1 2013, is obtained by plugging in the relevant dummy variables and estimated parameters. This forecast aids in planning production and inventory management.

Demand Function Estimation and Sign Evaluation

The demand function Qd = d + eP + fM + gPH is estimated using regression analysis on the sales data, with P, M, and PH as explanatory variables. The coefficients provide insights into how price, income, and competitor’s price influence demand.

At a 5% significance level, the estimated parameters e, f, and g are statistically significant, with notable signs. The negative sign for e aligns with the law of demand: higher prices lead to lower quantity demanded. The positive f reflects income effect: higher household income increases demand. The sign of g depends on whether the competitor's price acts as a substitute or complement; a negative g suggests substitution effects, where higher competitor prices increase demand for Sting Ray.

Estimating the Demand Function, Inverse Demand, and Marginal Revenue

Using the regression results, the demand function reduces to: Qd = d + eP, assuming other variables are fixed. Solving for P gives the inverse demand function: P = (Qd - d)/e. For the given values of Howard’s price ($250) and household income ($65,000), the demand function simplifies to: P = (Q - d)/e.

The marginal revenue (MR) function, derived from the inverse demand function, is MR = P + Q(dP/dQ). Since P = (Q - d)/e, then dP/dQ = 1/e, leading to MR = P + Q/e = (Q - d)/e + Q/e = (2Q - d)/e.

This MR function indicates the additional revenue from selling one more unit, guiding optimal pricing strategies. By setting MR equal to marginal cost, the firm can find the profit-maximizing output level.

Conclusion

The comprehensive empirical analysis provides crucial insights into the cost structure and market demand for the Sting Ray. The cost estimates inform production efficiency and scale economies, while the demand estimates guide optimal pricing and sales forecasting. Incorporating seasonal variations enhances forecast accuracy, assisting PoolVac in strategic planning. The demand and MR functions form the basis for making informed decisions aimed at maximizing profitability in a competitive landscape.

References

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