Question

Increasing the size of a team that creates a joint product may dull incentives, as this problem will illustrate. \(^{11}\) Suppose \(n\) partners together produce a revenue of \(R=e_{1}+\cdots+e_{n} ;\) here \(e_{i}\) is partner \(i\) s effort, which costs him \(c\left(e_{i}\right)=e_{i}^{2} / 2\) to exert. a. Compute the equilibrium effort and surplus (revenue minus effort cost) if each partner receives an equal share of the revenue. b. Compute the equilibrium effort and average surplus if only one partner gets a share. Is it better to concentrate the share or to disperse it? c. Return to part (a) and take the derivative of surplus per partner with respect to \(n\). Is surplus per partner increasing or decreasing in \(n ?\) What is the limit as \(n\) increases? d. Some commentators say that ESOPs (employee stock ownership plans, whereby part of the firm's shares are distributed among all its workers) are beneficial because they provide incentives for employees to work hard. What does your answer to part (c) say about the incentive properties of ESOPs for modern corporations, which may have thousands of workers?

Short answer

Question: Analyze the behavior of partners and their efforts to produce revenue in different scenarios and discuss the effects of ESOPs on incentives for employees in large corporations. Short Answer: In situations where partners share revenue equally, individual partner's equilibrium effort is directly proportional to the total revenue. However, when only one partner receives the share, the equilibrium effort remains constant. As the number of partners increases, the surplus per partner decreases, approaching zero. This indicates that in large corporations, incentives provided through ESOPs may not be enough to motivate employees to exert higher effort, as their individual stakes in the company's revenue become smaller.

Step-by-step solution

Set up the maximization problem

For each partner, the individual's objective will be to maximize their own surplus, which is their share of revenue minus their effort cost. Let \(x_i\) be the share received by each partner \(i\). Then we have: $$ x_i = \frac{R}{n} $$ The objective of partner \(i\) will be to maximize: $$ \pi_i = x_ie_i - c(e_i) = \frac{e_1 + \dots + e_n}{n}e_i - \frac{e_i^2}{2} $$

Find the equilibrium effort

To find the equilibrium effort for each partner, we need to differentiate the objective function with respect to \(e_i\) and set it equal to zero. We'll then solve for the equilibrium effort \(e_i^*\). $$ \frac{\partial\pi_i}{\partial e_i} = \frac{e_1 + \dots + e_n}{n}- e_i = 0 $$ Solving for \(e_i^*\), we get: $$ e_i^* = \frac{e_1 + \dots + e_n}{n} $$ Since all partners are symmetric in this case, we can say that \(e_i^* = e^*\). Therefore, \(e_1 = e_2 = \dots = e_n\), and the total revenue is \(R = ne^*\).

Calculate the surplus

Having found the equilibrium effort for each partner, we can now calculate the surplus for the entire partnership. The surplus is the total revenue minus the total cost of effort: $$ Surplus = R - \sum_{i=1}^n c(e_i) = ne^* - \frac{1}{2}\sum_{i=1}^n e_i^2 $$ Using the fact that \(e_i^* = e^*\), we can simplify the surplus: $$ Surplus = ne^* - \frac{1}{2}ne^{*2} = ne^*\left(1 - \frac{1}{2}e^*\right) $$ #b. One partner gets the share#

Set up the maximization problem

In this case, only one partner, say partner 1, gets the share. The objective of each partner \(i \neq 1\) will be to maximize their own effort to increase the total revenue. Their individual surplus will be given by: $$ \pi_i = R - c(e_i) = (e_1 + e_2 + \dots + e_n) - \frac{e_i^2}{2} $$

Find the equilibrium effort

Again, we need to differentiate the objective function with respect to the effort of each partner \(i\) and set it equal to zero. We'll then solve for the equilibrium effort \(e_i^*\). In this case, we have: $$ \frac{\partial\pi_i}{\partial e_i} = 1 - e_i = 0 $$ Solving for \(e_i^*\), we get: $$ e_i^* = 1 $$ The total revenue is now \(R = n\) and the average surplus per partner is: $$ \bar{Surplus} = \frac{R - \sum_{i=1}^n c(e_i)}{n} = \frac{n - \frac{1}{2}\sum_{i=1}^n 1}{n} = \frac{n}{2} $$ #c. Surplus per partner#

Differentiate surplus per partner with respect to n

To find the effect on the surplus per partner with respect to the number of partners, we need to differentiate the surplus per partner with respect to \(n\). Starting with surplus per partner: $$ \frac{Surplus}{n} = e^*\left(1 - \frac{1}{2}e^*\right) $$ And the derivative is: $$ \frac{d}{dn}\left(\frac{Surplus}{n}\right) = - \frac{1}{2}ne^{*2} + e^* $$ Since we found \(e^* = \frac{R}{n}\) in part a, we substitute the value and find the limit of the surplus per partner as \(n\) approaches infinity. Using L'Hopital's rule, we find: $$ \lim_{n\to\infty}\frac{R/ne^{*2}-e^*}{n} = \lim_{n\to\infty}-\frac{ne^{*2}}{2} + e^* = 0 $$ Thus, the surplus per partner decreases as the number of partners increase, and the limit of surplus per partner is zero. #d. ESOPs and incentives#

Evaluate the incentive properties of ESOPs

Based on the answer in part (c), we can conclude that the incentive properties of ESOPs decrease as the number of workers increases. In large corporations with thousands of workers, the individual stake in the company's revenue becomes smaller and may not provide enough incentives for employees to exert higher effort.

Key concepts

Joint Product Creation

Joint product creation is an economic concept where multiple individuals work together to create a single product or service. By pooling their efforts, these individuals can create outputs that may not be possible to achieve individually. This idea is central to the microeconomic team production model, which considers the dynamics of how people collaborate.

Imagine a team of graphic designers working together on a single project. The synergy of their combined skills leads to the creation of a stunning design that none could have produced alone. However, an essential aspect of joint product creation is how the efforts of individuals are coordinated and how the output is shared. If the rewards are evenly distributed regardless of individual contribution, there's a risk that some members will exert less effort—a phenomenon known as 'free-riding.' This leads to the need for balancing incentives to maintain productivity within the team.

Equilibrium Effort

In the realm of team production, equilibrium effort refers to the level of effort each team member contributes such that no one has an incentive to deviate or change their input. It's a state where each individual's costs and benefits of effort are perfectly balanced.

To illustrate, consider a small start-up where each founder contributes to the business's overall success. If one founder starts working fewer hours, it may cause the others to adjust their efforts accordingly, disrupting the team's equilibrium. Achieving this equilibrium is crucial for ensuring that all members are motivated to contribute optimally without feeling overburdened or underappreciated.

Surplus Maximization

Surplus maximization is a key objective in economic theory and refers to the process of maximizing the difference between revenue generated and costs incurred. In a team setting, each member's goal is to contribute to the team's revenue while also taking into consideration their own costs, such as the physical or mental effort required to perform tasks.

For instance, in a factory setting with workers on an assembly line, the surplus is maximized when each worker contributes the best they can without incurring excessive fatigue or dissatisfaction. Ultimately, surplus maximization not only benefits the individuals involved but also enhances the overall productivity and profitability of the enterprise.

Employee Stock Ownership Plans (ESOPs)

Employee Stock Ownership Plans (ESOPs) are programs that provide a company's workforce with ownership interest in the company. They are intended to align the interests of the employees with those of the company's shareholders, theoretically encouraging workers to improve the company's performance because they directly benefit from its success.

Through ESOPs, employees get the chance to be shareholders and often enjoy a sense of empowerment and increased job satisfaction. However, the effectiveness of ESOPs can be diluted in very large companies where the personal impact of an individual employee's effort on the company's stock price is minuscule, potentially diminishing the intended incentive effects.

Incentive Effects

Incentive effects relate to how particular policies or practices influence the motivation of individuals within an organization. These effects are critical for maintaining productivity, as they help to ensure that team members are motivated to put forth their best effort.

Effective incentive structures can range from monetary rewards, like performance bonuses, to non-monetary perks, such as recognition and career advancement opportunities. However, as seen in ESOPs for large corporations, if the incentive is too small or if the link between effort and reward is weak, it can fail to motivate employees, resulting in less effort and lower productivity. This highlights the need for carefully designed incentive schemes tailored to team size and individuals' capacity to influence outcomes.