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 (reventue less wage paid) for each of the following wage arrangements. Explain why these different principalagent relationships generate different outcomes. a. \(w=2\) for \(e \geq 1\); otherwise \(w=0\) b. \(w=R / 2\) \(\mathbf{c}, w=R-12.5\)

Short answer

For case a, the level of effort is 1 and the profit is 7; for case b, the effort level is 2.5 with a profit of 3.75; and for case c, although the effort level is 5, the wage scheme leads to a zero profit.

Step-by-step solution

Case A: w=2 for e>=1; otherwise w=0

Start by deriving wage as a function of e. It would be w = 2 for e >= 1 and w = 0 for e < 1. Next, find the level of effort that maximizes the equation w-e. This occurs at e=1, which means w=2. To determine the firm’s profit, substitute e=1 into the given revenue equation to get R=10-1=9. The resulting profit \(π\) is \(π = R - w = 9 - 2 = 7\).

Case B: w=R/2

Work out w as a function of e. w equals half of the revenue, so w = R/2 = \(10e - e^2\) /2 = \(5e - 0.5e^2\). Maximize w less effort, which simplifies to \(2.5e - 0.5e^2\). The derivative of this equation is \(2.5 - e\). Setting the derivative to zero gives e=2.5, which in turn gives w=2.5. Substitute e=2.5 into the revenue equation to compute R and then use the values of w and R to find the profit \(π\), which equals \(π = R - w = 3.75\).

Case C: w=R-12.5

Derive the equation for w based on this wage plan: w = R - 12.5 = \(10e - e^2 - 12.5\). To maximize wage less effort, compute the derivative of the function, and set it zero: 10 - 2e = 0, yielding e=5. Substituting E=5 into the wage equation gives w=0. Calculate R when e=5 and then find the profit under this wage scheme by subtracting w from R, yielding π = R - w = 0.

Key concepts

Short-Run Revenue Maximization

Understanding short-run revenue maximization requires an exploration into how businesses aim to achieve the highest possible revenue within a limited timeframe, usually when one or more factors of production cannot be changed. In the exercise provided, the firm’s revenue is depicted by the formula R=10e-e^2, where e represents the level of effort from a typical worker.

Maximizing revenue in the short run involves finding the level of effort that yields the highest revenue before costs are taken into account. It is important to note that in the context of the exercise, the equation can be maximized by differentiating the revenue formula with respect to effort and finding the value of e that sets the derivative to zero. Here, revenue is a function of the worker’s effort, reflecting a real-life scenario where a worker’s contribution directly affects a firm’s income.

However, since labor cost is a significant factor to consider, the wage arrangements affect the worker's decision on the level of effort, thus influencing the revenue outcome. This relationship showcases a principal-agent problem where the best interests of the firm (principal) may not align with the worker's (agent) actions based on the given wage structure.

Wage-Effort Relationship

The wage-effort relationship highlights how a worker's salary influences their motivation to perform tasks, which encompasses the core of the principal-agent problem. In such scenarios, workers (agents) are presumed to aim for wage maximization while exerting the least effort, whereas firms (principals) desire maximum effort that leads to revenue maximization.

The provided exercise gives insight into this concept through different wage arrangements. For example, in case A, a fixed wage incentivizes the worker to meet a minimum effort threshold (e \text{\(geq\)} 1 to earn w=2), but does not encourage greater effort as the wage does not increase with more effort. Consequently, the worker's maximum wage less effort occurs at the precise threshold, without stimulating additional productivity.

In contrast, in case B and C, where the wage is a function of revenue, workers are encouraged to increase their effort, thereby aligning their interests with those of the firm. The wage-effort curve in these instances better aligns with the firm's revenue maximization goal, but still must be balanced to ensure profit for the firm.

Profit Determination

Profit determination is the process of calculating a firm's earnings after all costs have been subtracted from total revenue. It represents the financial gain realized when the amount of revenue gained exceeds the expenses, costs, and taxes needed to sustain the business.

In our exercise, profit (\(\pi\)) is the firm’s revenue (R) minus the wages paid (w). Different wage arrangements detailed in the scenarios affect both the worker's effort and, consequently, the overall profit. For instance, in case A, a simple fixed wage scenario, the firm preserves a comfortable profit margin of \(\pi = R - w = 7\).

Conversely, in cases B and C, wages are tied directly to revenue, which means as effort and thus revenue increases, wages do too, potentially narrowing the profit margin. In case B, the profit ends up being \(\pi = 3.75\), and interestingly, in case C, where the wage is R-12.5, the profit determination results in zero, underscoring the importance of a properly balanced wage-revenue relationship in order to safeguard profits. This illuminates the complexity of wage structures and their impact on both worker performance and firm profitability within the principal-agent framework.