Sherwin Williams Company Is Attempting To Develop A Demand

Sherwin Williams Company Is Attempting To Develop A Demand Model Fo

Sherwin-Williams Company is attempting to develop a demand model for its line of exterior house paints. The company's chief economist believes that the most influential variables affecting paint sales (measured in thousand gallons) are promotional expenditure (measured in thousand dollars), selling price (dollars per gallon), and disposable income per household (thousand dollars). Using data from 30 observations, the Research Department estimated a log-linear regression model with the following variables:

• Dependent variable: log y (paint sales)
• Independent variables: log a (promotional expenditure), log p (selling price), log m (disposable income)

The regression results include coefficients, standard errors, t-statistics, p-values, and confidence intervals for each variable, but these are partially shown in the provided data, with some typographical errors. The primary goal is to perform t-tests on each independent variable to assess their statistical significance at the 95% confidence level. For those variables found significant, interpretations of the economic meaning of each coefficient are required.

Paper For Above instruction

Demand modeling is a fundamental aspect of economic analysis, enabling firms to understand the factors that influence consumers' purchasing behaviors and to optimize their marketing strategies accordingly. Sherwin-Williams, a leading manufacturer of exterior house paints, endeavors to develop a demand model that accurately captures the relationships among various economic and marketing variables affecting sales volume. In constructing such a model, several potential predictors are considered, including promotional expenditure, selling price, and disposable income per household.

The data used for this analysis comprises 30 observations, from which a log-linear regression model was estimated. The model's form allows for the interpretation of elasticities, providing insights into how percentage changes in explanatory variables translate into percentage changes in demand. The estimated coefficients, their statistical significance, and economic interpretations are central to understanding and validating the demand relationships.

Assessing Statistical Significance of Variables

To evaluate whether each independent variable significantly influences exterior paint sales, t-tests are conducted based on the estimated coefficients and their standard errors. The null hypothesis for each test posits that the coefficient equals zero (no effect). The test statistic is calculated as the ratio of the coefficient to its standard error. For example, if the coefficient for promotional expenditure (log a) is denoted as β₁, and its standard error as SE₁, then t = β₁ / SE₁.

The critical value for a two-tailed t-test at the 95% confidence level with 28 degrees of freedom (n-3, assuming three predictors) is approximately ±2.048. If the absolute value of the t-statistic exceeds this critical value, the variable is considered statistically significant at this confidence level.

Results and Economic Interpretation

Suppose, for illustration, the estimated coefficients are as follows:

• log a (promotional expenditure): coefficient = -0.16, standard error = 0.875
• log p (selling price): coefficient = 0.03, standard error = 0.341
• constant term: coefficient = 6.82, standard error = 0.003

Using the t-tests, the t-statistics for promotional expenditure and selling price are:

• T for log a = -0.16 / 0.875 ≈ -0.183
• T for log p = 0.03 / 0.341 ≈ 0.088

Both t-values are less than 2.048 in absolute value, indicating that these variables are not statistically significant at the 95% confidence level. Conversely, the constant term's coefficient, with a t-statistic of 3.482, exceeds the critical value, confirming its significance.

Given that promotional expenditure and selling price are not significant predictors within this model, their economic impacts are minimal or potentially confounded by other factors. The negative sign of the promotional expenditure coefficient suggests an inverse relationship, which may seem counterintuitive; however, given its insignificance, no firm conclusion can be drawn. Conversely, a positive coefficient on selling price would imply demand is price-inelastic, consistent with typical demand theory.

Economic Interpretation of Significant Coefficients

Suppose, instead, that disposable income (log m) was found significant with a coefficient of 0.04 and standard error of 0.341. The t-statistic would then be approximately 0.04 / 0.341 ≈ 0.117, which is also insignificant at the 95% level. If any variable's t-statistic exceeds the critical value, its coefficient would be interpreted as follows:

• For a significant promotional expenditure coefficient, a positive value would indicate that higher promotional spending increases demand, aligning with marketing principles. Conversely, a negative coefficient might suggest that increased advertising does not translate into higher sales, possibly due to saturation or ineffective campaigns.
• For a significant selling price coefficient, a negative value would suggest that higher prices decrease demand, consistent with law of demand. The magnitude of this coefficient could be interpreted as the price elasticity of demand; for example, a coefficient of -0.20 implies a 1% increase in price leads to a 0.20% decrease in sales volume.

These interpretations assist managers in pricing strategies and advertising budgeting to optimize sales and revenue.

Conclusion

In developing a demand model, it is crucial to evaluate the statistical significance of variables to ensure meaningful insights. While the basic regression indicates certain variables may not significantly impact exterior paint sales, these findings guide the company to focus on factors that truly influence demand. Further research might involve augmenting the model with additional variables or employing alternative econometric techniques to refine the demand estimation. Ultimately, robust demand models serve as vital tools in strategic decision-making, resource allocation, and competitive positioning within the home improvement and paint industry.

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