Question

A firm's short-run revenue is given by \(R=10 e-e^{2}\) where \(e\) is the level of effort by a typical worker (all workers are assumed to be identical). A worker chooses his level of effort to maximize wage less effort \(w-e\) (the per-unit cost of effort is assumed to be 1). Determine the level of effort and the level of profit (revenue less wage paid) for each of the following wage arrangements. Explain why these different principal-agent relationships generate different outcomes. a. \(w=2\) for \(e \geq 1 ;\) otherwise \(w=0\) \(\mathbf{b} . w=R / 2\) c. \(w=R-12.5\)

Short answer

For wage structure a, the firm's profit is 7 while the worker's profit is 1. For wage structure b, both the worker and the firm make a profit of 12.5. Lastly for wage structure c, the situation is the same as for b with both making a profit of 12.5.

Step-by-step solution

Finding Level of Effort for Wage Structure a

Revenue for the firm\( R = 10e - e^2 \) and wage function is defined as \( w = 2\) for \( e \geq 1 \); otherwise \( w = 0\). To maximize the worker's profits, the worker will only exert effort when \( w \geq 1\), because zero effort would equate to zero cost. Therefore, we can set \( w = 2\) and the effort \( e = 1\). The worker's profit then is \( w - e = 2 - 1 = 1\).

Calculate Firm's Profit for Wage Structure a

To calculate the firm's profit, we subtract the wage from the revenue. At \( e = 1\), firm's revenue (R) becomes \( 10(1) - 1^2 = 9\). The firm's profit then is \( R - w = 9 - 2 = 7\).

Finding Level of Effort for Wage Structure b

For wage structure \( w = R / 2\), we substitute this into the worker's profit function, which means the worker's profit maximization problem is to maximize \( R / 2 - e \) where \( R = 10e - e^2\). Differentiating this with respect to e and setting the result to 0 will yield the effort level that maximizes the worker's profit. This results in \( e = 5\). Thus, the worker's wage \( w = R / 2 = (10*5 - 5^2) / 2 = 12.5\).

Calculate Firm's Profit for Wage Structure b

At \( e = 5\), the firm's revenue (R) becomes \( 10(5) - 5^2 = 25\). The firm's profit then is \( R - w = 25 - 12.5 = 12.5\).

Finding Level of Effort for Wage Structure c

For wage structure \( w = R - 12.5\), we substitute this into the worker's profit function, which means the worker wants to maximize \( R - 12.5 - e \) where \( R = 10e - e^2 \). Differentiating this with respect to e and setting the result to 0 will yield the effort level that maximizes the worker's profit. This results in \( e = 5\). Thus, the worker's wage \( w = R - 12.5 = 10*5 - 5^2 - 12.5 = 12.5\).

Calculate Firm's Profit for Wage Structure c

At \( e = 5\), the firm's revenue (R) becomes \( 10(5) - 5^2 = 25\). The firm's profit then is \( R - w = 25 - 12.5 = 12.5\).

Key concepts

Short-Run Revenue Maximization

In the realm of microeconomics, short-run revenue maximization is a fundamental concept, especially for a firm aiming to optimize its financial performance. When we discuss short-run revenue, we're talking about the income a firm can generate based on current resources, like labor and capital, without the ability to adjust to new market conditions or scaling operations, which typically applies to long-run scenarios. For a firm whose revenue function is characterized as R = 10e - e^2, where e represents the effort put forth by a worker, maximizing revenue requires finding the right balance of effort that will lead to the highest possible revenue before any wage payments are considered.

Since the revenue equation is a quadratic function, its maximum can be found by completing the square or by differentiating the function with respect to e and setting the derivative equal to zero. This method reveals the effort level that maximizes short-run revenue. However, it's crucial to note that this analysis does not include the cost of exerting such effort or the wages paid to workers, which are also important considerations for the firm.

Worker Effort Level Optimization

The concept of worker effort level optimization is aligning the incentives of employees with those of the firm. Workers aim to maximize their net gain, which is their wage minus the cost of their effort. This cost is assumed to be 1 for every unit of effort. The worker's goal is to choose an e (effort level) that will maximize the difference between wages and effort cost.

Under different wage arrangements, workers will exert different levels of effort. This manifestation of various effort levels based on incentives highlights the principal-agent problem, which arises when the goals of the principal (the firm) and the agent (the worker) are not perfectly aligned, leading to suboptimal outcomes for one or both parties. For instance, in a fixed wage scenario, workers have little incentive to exert more effort once they secure their wage. However, under a profit-sharing scheme, a worker may strive to maximize their effort, as this will directly influence their income. Such differences in incentives shape the worker's effort level optimization, significantly impacting the firm's revenue.

Firm Profit Calculation

The final step in analyzing the effects of different wage structures is the firm profit calculation. To determine a firm's profit, we subtract the total wages paid to workers from the total revenue earned by the firm. In our exercise, the firm's revenue is given by the function R = 10e - e^2, and the wage (w) depends on the stipulated wage arrangement. Under different wage systems, the firm's profits will change because the worker's level of effort—and consequently, the cost of that effort in wages—will fluctuate.

For example, in the fixed wage scenario where the firm pays a wage of 2 for effort levels equal or greater than 1, the firm's profit is calculated by substituting the optimal effort level back into the revenue function, then subtracting the wages paid. In profit-sharing arrangements, the firm must account for the division of revenue between itself and the workers, which can either motivate higher effort or create situations where the cost of additional effort isn't worth the incremental revenue gained. This calculation dictates the strategic financial management decisions a firm will make concerning wage policies to ensure the profit it retains aligns with its short and long-term fiscal goals.