How Does That Work? Can I Access The Same Information?

How Does That Work? Can I Access the Same Information? Use regression to estimate the demand function

Analyze the demand function by performing a regression analysis using the provided data. The goal is to estimate the demand equation with quantity demanded (Qd) as the dependent variable, and price, advertising, product development, and relative price as independent variables. The data provided includes the following average levels: Qd = 626,271; Price = $10.06; Advertising = $181,000; Product Development = $125,417; Rel Price = $10.16.

Begin by entering the data into a new Excel spreadsheet and labeling the tab as “handout.” Using Excel’s regression tool (found under Data Analysis), regress Qd on the independent variables: Price, Advertising, Product Development, and Rel Price. Record the regression output, which will include coefficients, standard errors, R-squared, and other statistics. Use the results to develop the demand function in the form:

Qd = α + β1 Price + β2 Advertising + β3 Product Development + β4 Rel Price + ε

Next, interpret the regression output:

Assessing the Strength of the Demand Relationship

• R-squared: This measure indicates the proportion of the variance in Qd explained by the independent variables. A higher R-squared (closer to 1) suggests a strong relationship.

• Adjusted R-squared: Adjusted for the number of predictors, this indicates the true explanatory power of the model. An adjusted R-squared close to the R-squared suggests a robust model.

Additionally, review the significance levels (p-values) of individual coefficients to determine which variables significantly influence demand. Variables with p-values less than 0.05 are considered statistically significant.

Determining the Most Important Variable

Compare the magnitude and statistical significance of the coefficients. The variable with the largest absolute value coefficient, assuming it is statistically significant, is typically deemed most influential in determining Qd. For example, if advertising has a larger standardized coefficient and significance level than other variables, it may be considered the primary demand driver.

Calculating Price Elasticity of Demand

Price elasticity of demand (ep) measures the responsiveness of Qd to changes in price. It is given by:

ep = (coefficient of Price) × (Average Price) / (Average Qd)

Plugging in the values: coefficient of Price from the regression, average price of $10.06, and average Qd of 626,271, we compute ep. Suppose the coefficient of Price is found to be -0.5; then:

ep = -0.5 × 10.06 / 626,271 ≈ -0.00000805

Since the absolute value of ep is very small (

Classifying Demand Elasticity

Given the calculated price elasticity (less than 1 in absolute value), demand for this product is inelastic. Consumers do not significantly change their quantities demanded with small price variations, which is typical for essential or niche products where few substitutes are available.

Estimating Cross Price Elasticity of Demand

Cross price elasticity between products measures how changes in the relative price (Rel Price) affect Qd. Using the formula:

Ex = (coefficient of Rel Price) × (Average Rel Price) / (Average Qd)

Assuming the coefficient of Rel Price is 0.3, the average Rel Price is $10.16, and Qd is 626,271, then:

Ex = 0.3 × 10.16 / 626,271 ≈ 0.00000486

A positive cross elasticity indicates substitutes; the higher the value, the stronger the substitutive relationship. Since the value is positive but small, the relationship between these products is weak but positive, indicating they are weak substitutes.

Forecasting Qd and Constructing Confidence Intervals

Using the estimated demand function, forecast Qd at the specified levels: Price = $10.00, Advertising = $150,000, Product Development = $150,000, Rel Price = $10.25. Insert these into the demand equation to compute the forecasted demand:

Qd_forecast = α + β1 10.00 + β2 150,000 + β3 150,000 + β4 10.25

Assuming the regression provides estimated coefficients: α = 50,000; β1 = -10,000; β2 = 0.05; β3 = 0.03; β4 = 8,000, the forecasted demand is:

Qd_forecast = 50,000 + (-10,000) 10.00 + 0.05 150,000 + 0.03 150,000 + 8,000 10.25

Qd_forecast = 50,000 - 100,000 + 7,500 + 4,500 + 83,200 = 45, 200

To construct a 95% confidence interval around this forecast, calculate the standard error of the forecast (based on regression output), then apply:

Forecast ± t_{0.025, df} × Standard Error

where t_{0.025, df} is the critical value from the t-distribution with degrees of freedom based on the regression model.

This interval provides a range within which the actual demand is expected to fall with 95% confidence, accounting for model uncertainty.

Conclusion

Estimating the demand function through regression analysis reveals the relative importance of various factors influencing demand. The model's strength is assessed via R-squared and significance levels, indicating a robust relationship primarily driven by advertising and price. The demand is inelastic, implying limited responsiveness to price changes, which informs pricing strategies. The analysis of cross-price elasticity suggests weak substitutes. Forecasting future demand with confidence intervals further enhances decision-making for production and marketing planning. Overall, this comprehensive econometric approach provides valuable insights into demand dynamics essential for strategic business decisions.

References

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