6.3 The Principal-Agent Model
The principal-agent model is a classic of modern economics. The idea is simple: someone (the “Principal”) wants to hire someone (the “Agent”) to do something for them, but can’t perfectly monitor them to make sure they do a good job. How can the Principal incentivize the Agent to put forward effort?
For example, suppose Paul has recorded an adorable video of him teaching economics to his cat, and wants the video to go viral. He hires a social media manager, Anna (with initial “A” for “Agent”) to boost his video online. In this case Paul (with initial “P”) is the Principal, and Anna (with initial “A”) is the Agent.
Paul’s choice of contract
Paul’s choice is how to write a contract that specifies the wage $w$ he pays Anna.
- He could pay her $w$ no matter what happens.
- He could pay her $w_E$ if she puts in a good effort at marketing his video, and $w_N$ if she doesn’t.
- He could pay her $w_H$ if the video goes viral, and $w_L$ if it doesn’t.
Anna’s choice of effort
Having seen Paul’s contract, Anna faces a choice: exert effort ($E$) in marketing Paul’s video or not ($N$). She can also reject the contract ($R$) and get utility of $\underline u = 20$ from some outside option.
- If Anna doesn’t exert effort (chooses $N$), the probability his video goes viral is ${1 \over 2}$ and she incurs a (utility) cost of effort of $c = 10$.
- If she does exert effort (chooses $E$), the probability his video goes viral is ${1 \over 4}$ but she has no cost.
- If she rejects the contract, she gets a utility of 20, and the probability his video goes viral is zero.
Note that the fundamental nature of Anna’s choice is that exerting effort is costly; so Paul’s challenge is to design a contract that provides enough incentives for Anna to incur that cost; but at the same time, is good enough overall that she doesn’t reject it.
Payoffs
Going viral is worth some amount $V$ to Paul; his utility is reduced by the amount he pays Anna. He is risk neutral, so his utility is just \(u_P(V,w) = \begin{cases}V - w & \text{ if the video goes viral}\\-w & \text{ if the video doesn't go viral}\end{cases}\) Anna is risk averse; her utility from money is $\sqrt{w}$. So her payoff is \(u_A(w,\text{effort}) = \mathbb{E}[\sqrt{w} | \text{ effort chosen}] - \text{cost of effort}\) or $\underline u$ if she rejects the contract.
As usual, we can write the payoffs as a function of the choices of the individuals:
- If Paul pays her $w$ regardless, his expected payoff is \(u_P = \begin{cases}{1 \over 2}V - w & \text{ if she exerts effort}\\ {1 \over 4}V - w & \text{ if she doesn't exert effort}\\ 0 & \text{ if she rejects the contract}\end{cases}\) and her payoffs are \(u_A = \begin{cases}\sqrt{w} - 10 & \text{ if she exerts effort}\\ \sqrt{w} & \text{ if she doesn't exert effort}\\ 20 & \text{ if she rejects the contract}\end{cases}\)
- If Paul can observe her effort, and pays her $w_E$ if she exerts effort and $w_N$ if she doesn’t, his expected payoff is \(u_P = \begin{cases}{1 \over 2}V - w_E & \text{ if she exerts effort}\\ {1 \over 4}V - w_N & \text{ if she doesn't exert effort}\\ 0 & \text{ if she rejects the contract}\end{cases}\) and her payoffs are \(u_A = \begin{cases}\sqrt{w_E} - 10 & \text{ if she exerts effort}\\ \sqrt{w_N} & \text{ if she doesn't exert effort}\\ 20 & \text{ if she rejects the contract}\end{cases}\)
- Finally, if Paul cannot observe her effort, but pays her $w_H$ if the video goes viral and $w_L$ if it doesn’t, then his expected payoff is \(u_P = \begin{cases}{1 \over 2}(V - w_H) + {1 \over 2}(-w_L) & \text{ if she exerts effort}\\ {1 \over 4}(V - w_H) + {3 \over 4}(-w_L) & \text{ if she doesn't exert effort}\\ 0 & \text{ if she rejects the contract}\end{cases}\) and her payoffs are \(u_A = \begin{cases}{1 \over 2}\sqrt{w_H} + {1 \over 2}w_L - 10 & \text{ if she exerts effort}\\ {1 \over 4}\sqrt{w_H} + {3 \over 4}w_L & \text{ if she doesn't exert effort}\\ 20 & \text{ if she rejects the contract}\end{cases}\)
Whew! That was a lot. But we now have enough information to figure out which kind of contract Paul should write, depending on his value of $V$.
Contract 1: Fixed Wage
If Paul pays Anna a fixed wage, then there is no reason for her to exert costly effort, since she’ll be paid the same amount no matter what. This is called the moral hazard problem: if a Principal pays the same amount regardless of effort, the Agent has no incentive to put forward good effort.
Anna will accept this contract as long as $\sqrt{w} \ge 20$, or $w \ge 400$. As usual, we’re going to assume that if she’s indifferent between two options, she takes the one Paul wants. (You could also imagine that he would pay her $w = 400.01$ or some such, just to tip the balance.)
Is it worth it for Paul to offer this contract? Given that she won’t exert effort, his expected payoff is ${1 \over 4}V - 400$, since he goes viral with probability ${1 \over 4}$ and he has to pay $w = 400$ no matter what. So this is worth it to him as long as $V \ge 1600$.
Contract 2: Observable Effort
Suppose Paul can observe Anna’s effort level. In this case, he can write a contract specifying a wage and effort level. (Equivalently, he can specify that if the effort level is not chosen, the wage will be zero.) When would he want to set a contract requiring high effort, and when will they want to set a contract requiring low effort?
We’ve already seen that if $V \ge 1600$, it’s worth it to him to hire her even knowing she would put no effort in. In that case Paul would specify a wage of $w_N = 400$ and he would get a payoff of ${1 \over 4}V - 400$.
If he sets a contract requiring high effort, he will have to ensure that the payoff to the contract is at least as high as Anna’s outside option: \(\sqrt{w_E} - 10 \ge 200 \Rightarrow w_E \ge 900\) In this case, Paul’s expected payoff would be ${1 \over 2}V - 900$. This is better than his payoff with low effort if \(\begin{aligned} {1 \over 2}V - 900 &\ge {1 \over 4}V - 400\\ {1 \over 4}V &\ge 500\\ V &\ge 2000 \end{aligned}\) Therefore, if Paul can observe Anna’s effort, his optimal contract would be:
- If $V < 1600$, don’t bother hiring her
- If $1600 \le V < 2000$, hire her for $w_N = 400$ but don’t require effort
- If $V \ge 2000$, hire her on a contract that specifies she exerts effort, and pays her $w_E = 900$.
Contract 3: Unobservable effort
If the agent’s effort is unobservable, the problem becomes more interesting. In this case the principal (Paul) cannot base the wage on how much effort is put forward, but can make the wage contingent on whether the project is successful or not: that is, a wage $w_H$ if the video goes viral, and a wage $w_L$ if it doesn’t. Recall from above that Anna’s utility in this case was \(u_A = \begin{cases}{1 \over 2}\sqrt{w_H} + {1 \over 2}w_L - 10 & \text{ if she exerts effort}\\ {1 \over 4}\sqrt{w_H} + {3 \over 4}w_L & \text{ if she doesn't exert effort}\\ 20 & \text{ if she rejects the contract}\end{cases}\)
In this case, assuming the principal wants the agent to exert effort, the wage structure needs to do two things:
- the difference between $w_L$ and $w_H$ must be high enough to incentivize the agent to exert effort. This is the incentive compatibility constraint: \(\begin{aligned}u_A(\text{effort}) &\ge u_A(\text{no effort})\\ \tfrac{1}{2} \sqrt{w_H} + \tfrac{1}{2}\sqrt{w_L} - 10 &\ge \tfrac{1}{4}\sqrt{w_H} + \tfrac{3}{4}\sqrt{w_L}\\ \tfrac{1}{4}\sqrt{w_H} - \tfrac{1}{4}\sqrt{w_L} &\ge 10\\ \sqrt{w_H} - \sqrt{w_L} &\ge 40 \end{aligned}\)
- the overall value of the contract must be high enough that the agent doesn’t reject the contract: that is, that the value of accepting the contract (and putting forward effort) is greater than the outside option. This is the participation constraint: \(\begin{aligned}u_A(\text{effort}) &\ge u_A(\text{reject contract})\\ \tfrac{1}{2} \sqrt{w_H} + \tfrac{1}{2}\sqrt{w_L} - 10 &\ge 20\\ \tfrac{1}{2}\sqrt{w_H} + \tfrac{1}{2}\sqrt{w_L} &\ge 30\\ \sqrt{w_H} + \sqrt{w_L} &\ge 60 \end{aligned}\)
We now have two equations in two unknowns: \(\begin{aligned}\sqrt{w_H} - \sqrt{w_L} &\ge 40\\ \sqrt{w_H} + \sqrt{w_L} &\ge 60 \end{aligned}\) Solving yields \(\begin{aligned}w_H^\star &= 2500\\ w_L^\star &= 100\end{aligned}\)
If Paul offers this contract, and Anna accepts it an exerts effort, what is his payoff? Well, his video goes viral with probability ${1 \over 2}$, in which case he gets a payoff of $V - 2500$; on the other hand, with probability ${1 \over 2}$ his video doesn’t go viral, but he just has to pay 100, so he gets a payoff of $-100$. Therefore his expected payoff is \(u_P = \tfrac{1}{2}(V - 2500) + \tfrac{1}{2}(-100) = {1 \over 2}V - 1300\) This is better than just paying a flat rate of 400 for low effort if \(\begin{aligned} \tfrac{1}{2}V - 1300 \ge \tfrac{1}{4}V - 400\\ \tfrac{1}{4}V \ge 900\\ V \ge 3600 \end{aligned}\)
Remember that if Paul could have observed Anna’s effort, he would have hired her and encouraged effort if $V \ge 2000$; now he needs $V \ge 3600$ in order to make it worth while to try to encourage effort. Essentially, because Anna is risk averse, Paul needs to offer her a lot of money to accept the risk that there’s a 50% chance he won’t pay her; and he needs to pay her something (at least in this case) even if she’s unsuccessful. He’s basically writing a costly insurance policy; so it needs to be really, really valuable to him to