Question

As chairman of the board of ASP Industries, you estimate that your annual profit is given by the table below. Profit (\Pi) is conditional upon market demand and the effort of your new CEO. The probabilities of each demand condition occurring are also shown in the table. $$\begin{array}{|lccc|} \hline \begin{array}{c} \text { MARKET } \\ \text { DEMAND } \end{array} & \begin{array}{c} \text { LOW } \\ \text { DEMAND } \end{array} & \begin{array}{c} \text { MEDIUM } \\ \text { DEMAND } \end{array} & \begin{array}{c} \text { HIGH } \\ \text { DEMAND } \end{array} \\ \hline \begin{array}{l} \text { Market } \\ \text { Probabilities } \end{array} & .30 & .40 & .30 \\ \hline \text { Low Effort } & \Pi=\$ 5 \text { million } & \Pi=\$ 10 \text { million } \Pi=\$ 15 \text { million } \\ \hline \text { High Effort } & \Pi=\$ 10 \text { million } & \Pi=\$ 15 \text { million } \Pi=\$ 17 \text { million } \\ \hline \end{array}$$ You must design a compensation package for the CEO that will maximize the firm's expected profit. While the firm is risk neutral, the CEO is risk averse. The CEO's utility function is Utility \(=W^{5}\) when making low effort Utility \(=W^{5}-100\) when making high effort where \(W\) is the CEO's income. (The -100 is the "utility cost" to the CEO of making a high effort.) You know the CEO's utility function, and both you and the CEO know all of the information in the preceding table. You do not know the level of the CEO's effort at time of compensation or the exact state of demand. You do see the firm's profit, however. Of the three alternative compensation packages below, which do you as chairman of ASP Industries prefer? Why? Package 1: Pay the CEO a flat salary of \(\$ 575,000\) per yearr Package 2: Pay the CEO a fixed 6 percent of yearly firm profits Package 3 3: Pay the CEO a flat salary of \(\$ 500,000\) per year and then 50 percent of any firm profits above \(\$ 15\) million

Short answer

The most suitable compensation package is the one that maximizes the firm's expected profit and incentivizes the CEO to exert high effort, as deduced from comparing the expected utilities for each package under high and low effort.

Step-by-step solution

Calculate Expected Profit for Each Effort Level

To find the expected profits given the CEO's effort level, we'll weight the given profits by their respective probabilities. Note that regardless of effort level, the market demand's probabilities remain consistent. For each effort level (low and high), calculate the expected profit using the formula: \(Expected Profit = Profit_1 * Prob_1 + Profit_2 * Prob_2 + Profit_3 * Prob_3\).

Construct the CEO's Utility Function

Next, we set up the CEO's utility function based on the outcome. The utility, depending on the level of effort, is given as \(U(W) = W^5\) for low effort and \(U(W) = W^5 - 100\) for high effort, where \(W\) is the income in thousands of dollars. With the three packages, set up three different functions for each, based on whether the CEO chooses to exert high or low effort.

Calculate Expected Utility for Each Package

Determine the CEO's expected utility for each package, for both low and high effort. This is done by first calculating the CEO's income under each scenario, plugging it into the appropriate utility function, and then averaging out the utility under the different market conditions based on their probabilities. This needs to be done for all three packages, for both low and high effort.

Compare the Expected Utilities

To incentivize the CEO to put in high effort, the expected utility must be higher under high effort than under low effort for the chosen compensation package. Looking at the expected utilities calculated in Step 3, determine for which package this is the case.

Select the Suitable Compensation Package

With the information obtained from the previous step, select the most suitable compensation package. This should be the one that maximizes expected profit for the firm, while still providing higher expected utility to the CEO when they exert high effort.

Key concepts

Expected Profit Calculation

The concept of Expected Profit Calculation is integral to the decision-making process in business, particularly when outcomes are uncertain. It involves forecasting the likely financial return of various scenarios by multiplying each potential profit by its probability and then summing up these values.

For example, if a company has three possible profit outcomes based on market demand—low, medium, and high—with respective probabilities (0.3, 0.4, 0.3), and respective profits (in millions): \(5 under low demand, \)10 under medium, and $15 under high, the Expected Profit can be calculated as follows:
\[ Expected\text{ }Profit = (5 \times 0.3) + (10 \times 0.4) + (15 \times 0.3) \text{ million} \ = (1.5 + 4 + 4.5) \text{ million} \ = 10 \text{ million}. \]
When determining CEO compensation, understanding these calculations is crucial to devise a strategy that aligns the CEO's incentives with the company’s profitability goals.

CEO Utility Function

The CEO Utility Function symbolizes the satisfaction or happiness a CEO derives from their compensation. It's particularly important because it quantifies the trade-offs a CEO might make between work effort and income. For a risk-averse CEO, the function takes into account not only the financial gain but also the 'disutility' of effort.

In our scenario, the CEO's utility is defined by the function:
\[ U(W) = W^5 \]
for low effort, and
\[ U(W) = W^5 - 100 \]
when making high effort, with \( W \) representing the income in thousands of dollars. The '-100' reflects the utility cost of the additional effort required. Such functions are critical in determining the kind of compensation package that would be appealing to the CEO, taking into account their attitude towards risk and effort.

Risk Aversion in Economics

The principle of Risk Aversion in Economics refers to the preference for certainty over uncertainty with respect to wealth or income. A risk-averse individual, like our CEO, prefers to receive a certain amount of money with surety rather than gamble for a potentially higher, but uncertain, amount. This behavior is governed by the curvature of the utility function—concave in this case.

A risk-averse CEO's utility function rises at a decreasing rate, meaning each additional dollar of income provides a smaller increase in utility. Therefore, they might prefer a guaranteed salary over a high-risk, variable compensation. Understanding the CEO's risk aversion helps in designing a compensation package that encourages the desired level of effort while compensating for the disutility of extra work.

Incentive Design

The art of Incentive Design in a business context lies in structuring rewards to align an employee's actions with the company's goals. In the case of a CEO, whose efforts can significantly affect a company's profits, the challenge is to create a package that motivates high performance without incurring disproportionate costs.

There are several approaches to this, such as offering a flat salary, profit-sharing, or bonuses tied to financial milestones. Each has its implications for risk-sharing between the firm and the CEO. An effective compensation package for a risk-averse CEO, like the one in our example, might combine a secure base salary with performance-based incentives. This would protect the CEO against the downside while also providing an upside for exceptional performance. The goal is to maximize the CEO's expected utility for high effort, thus incentivizing them to work towards improving the firm’s profitability.