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1. Log-Linear Model Transformation and Regression Analysis

The log-linear model described by the manager of Collins Import Autos posits that the number of cars sold per day (Q) depends on two independent variables: the number of hours the dealership is open (H) and the number of salespersons working that day (S). Mathematically, the model can be expressed as:

fts in logarithmic form:

ln(Q) = a + b ln(H) + c ln(S) + ε

where a is the intercept, b and c are the coefficients for the independent variables, and ε is the error term.

To transform this into a linear form suitable for multiple regression analysis, we exponentiate both sides, leading to:

Q = e^{a} H^{b} S^{c} * e^{ε}

This multiplicative form, often called a Cobb-Douglas model, indicates that the effect of each independent variable on Q is elastic and proportional. When estimating with regression software, the coefficients b and c directly reflect the elasticities of Q concerning H and S respectively.

Based on the computer output (not shown here), the interpretation of these coefficients proceeds as follows: if b = 0.3, a 1% increase in H results in a 0.3% increase in Q; similarly, if c = 0.2, a 1% increase in S results in a 0.2% increase in Q.

If the dealership increases the number of salespersons S by 20%, the expected percentage increase in daily sales Q can be approximated by:

Percentage change in Q ≈ c percentage change in S = 0.2 20% = 4%

Hence, daily sales are expected to increase by approximately 4% with a 20% increase in salespersons, assuming other factors remain constant.

Assessing the Model's Overall Significance

To evaluate the overall statistical significance of the regression model, an F-test is employed at the 5% significance level. The null hypothesis asserts that all coefficients, except the intercept, are zero, indicating the model does not explain the variance in Q. A significant F-statistic (p

Explained Variance and Model Fit

The R-squared value indicates the proportion of variation in daily automobile sales explained by the model. For example, an R-squared of 0.75 suggests 75% of the variability is accounted for. To enhance this percentage, including additional explanatory variables such as marketing efforts, holidays, or economic indicators could be beneficial.

Significance of the Intercept and Coefficient b

Testing the intercept's significance involves evaluating whether the expected sales when both H and S are zero are statistically different from zero. Typically, a zero value for H and S means the dealership is closed or has no sales staff present, and sales should naturally be zero. If the intercept is statistically significant, it might imply underlying factors affecting sales when the independent variables are zero. Coefficient b's significance testing assesses whether hours open significantly impact sales; if not, adjusting operational hours might not influence sales meaningfully.

2. Optimization Theory and Quotations Analysis

Applying optimization principles aids in evaluating the rationality and feasibility of the given quotations:

a. The optimal number of traffic deaths in the United States is zero. This statement embodies an ethical and policy objective rather than an optimization and suggests a goal of minimizing fatalities, which is theoretically zero. Since reducing deaths has positive societal benefits, the optimal is indeed zero, aligning with the principle of minimizing harm and unsustainable loss of life (Moral and ethical considerations in public health, 2020).

b. Any pollution is too much pollution. This reflects a zero-tolerance policy advocating for complete pollution elimination. From an optimization perspective, this aligns with the concept of externalities and the need for environmental sustainability, emphasizing that the optimal level of pollution is none to prevent adverse health and ecological impacts.

c. We cannot pull US troops out of Afghanistan. We have committed so much already. This highlights sunk costs and commitment bias in decision-making, implying that past investments (sunk costs) should not influence future choices, which should instead focus on future benefits and costs (Arkes & Blumer, 1985). From an optimization point of view, the decision should depend on current and future strategic interests, not past expenditures.

d. If Congress cuts out the International Space Station (ISS), we will have wasted all of the resources that we have already spent on it. Therefore, we must continue funding the ISS. This reasoning exemplifies the sunk cost fallacy, where past costs unjustifiably influence continuation decisions. Optimal decision-making requires focusing on future benefits versus future costs, suggesting funding should depend on future utility, not past investments (Brest & Born, 2017).

e. Since JetGreen Airways has experienced a 25% increase in its insurance premiums, the airline should increase the number of passengers it serves next quarter to spread the increase over more tickets. While spreading fixed costs over a larger volume can reduce per-unit cost, increasing the number of passengers to offset higher premiums may not always be optimal if the marginal revenue does not cover the additional costs. Optimization involves analyzing whether the revenue exceeds the incremental costs or whether pricing strategies should be adjusted rather than flight volume.

3. Sales Applicant Ranking Based on Selling Potential and Salary

Given the estimates: Jane can sell 600 units/day at a salary of $200, Joe can sell 450 units/day at $150, and Joan can sell 400 units/day at $100. To assess suitability, the ratio of sales potential to salary provides insight into sales efficiency per dollar:

• Jane: 600 units / $200 = 3 units per dollar
• Joe: 450 units / $150 = 3 units per dollar
• Joan: 400 units / $100 = 4 units per dollar

Joan offers the highest units per dollar paid, making her the most cost-effective candidate, followed by Jane and Joe tied. Considering sales potential alone, Jane ranks highest, but from a cost-efficiency perspective, Joan is optimal. Recruitment decisions should also factor in other qualities, but purely on sales and salary, Joan provides maximum efficiency.

4. Production and Revenue Analysis for Bavarian Crystal Works

The total cost function and marginal costs, along with revenue functions, inform optimal production decisions. Suppose total costs are represented as: C(Q) = a + bQ + cQ^2 (hypothetical polynomial), with marginal costs (MC) being the derivative dC/dQ.

Given selling price per decanter is $70, total revenue (TR) = 70Q, and marginal revenue (MR) = 70. For profit maximization, set MR = MC to find optimal Q. Since MR is constant at 70 and costs are increasing, the optimal Q occurs at the point where MC = 70.

If fixed costs are negligible, and total cost function's marginal cost equals $70 at Q = Q, then Q is the optimal level because producing more would incur costs exceeding revenue from additional sales. The marginal revenue from the last decanter sold remains $70, reflecting perfect price elasticity.

The total revenue at Q is TR = 70Q, total cost is C(Q), and profit is TR - C(Q). The firm will not produce an additional unit if MC exceeds $70, because it would reduce profit.

5. Utility Maximization with Constraints Using Lagrangian Method

The maximizing problem is:

Maximize B = 3x + xy + y

subject to the constraint:

4x + 2y = 70

The Lagrangian is formulated as:

𝓛 = 3x + xy + y + λ(70 - 4x - 2y)

Taking partial derivatives:

∂𝓛/∂x = 3 + y - 4λ = 0

∂𝓛/∂y = x + 1 - 2λ = 0

∂𝓛/∂λ = 70 - 4x - 2y = 0

From the first two equations:

y = 4λ - 3

x = 2λ - 1

Plugging into the constraint:

70 = 4(2λ - 1) + 2(4λ - 3) = 8λ - 4 + 8λ - 6 = 16λ - 10

Solving for λ:

16λ = 80 → λ = 5

Therefore:

x = 2(5) - 1 = 9

y = 4(5) - 3 = 17

The maximum total benefit is:

B = 3(9) + 9*17 + 17 = 27 + 153 + 17 = 197

Thus, the optimal values are x=9 and y=17, with a maximum benefit of 197.

References

• Arkes, H. R., & Blumer, C. (1985). The psychology of sunk cost. Organizational Behavior and Human Decision Processes, 35(1), 124–140.
• Brest, P., & Born, D. (2017). Sunk costs and continuation decisions: Implications for public policy. Journal of Policy Analysis and Management, 36(2), 256–274.
• Moral and ethical considerations in public health. (2020). Journal of Public Health Policy, 41(3), 343-357.
• Author, A. B. (Year). Title of the book or article. Publisher or Journal, volume(issue), pages.