Principal Agent Model

Asymmetric information can lead to inefficiencies.

We devote our analysis to the principal-agent model—in which there is only one party on each side of the market. The party who proposes the contract is called the principal. The party who decides whether or not to accept the contract and then performs under the terms of the contract (if accepted) is called the agent.

Two models of asymmetric information are studied most often.

In a first model, the agent’s actions taken during the term of the contract affect the principal, but the principal does not observe these actions directly. The principal may observe outcomes that are correlated with the agent’s actions but not the actions themselves. This first model is called a hidden-action model, which is also called a moral hazard model.

In a second model, the agent has private information about the state of the world before signing the contract with the principal. The agent’s private information is called his type. The second model is thus called a hidden-type model, and also an adverse selection model.

In a full-information environment, the principal could propose a contract to the agent that maximizes their joint surplus and captures all of this surplus for herself, leaving the agent with just enough surplus to make him indifferent between signing the contract or not. This outcome is called the first best. The first best is a theoretical benchmark that is unlikely to be achieved in practice because the principal is rarely fully informed.

The outcome that maximizes the principal’s surplus subject to the constraint that the principal is less well informed than the agent is called the second best.

Adding further constraints to the principal’s problem besides the informational constraint—for example, restricting contracts to some simple form such as constant per-unit prices—leads to the third best, the fourth best, and so on, depending on how many constraints are added.

Hidden Actions

Suppose a corporation delegates its operation to one representative manager. The owner (she), who plays the role of the principal in the model, offers a contract to the manager (he), who plays the role of the agent.

The manager decides whether to accept the employment contract and, if so, how much effort $e\geqslant 0$ to exert.

Assume the firm’s gross profit $\pi_g$​ takes the following simple form: $$ \pi_g = e + \epsilon $$ The gross profit of the firm is increasing in the manager’s effort $e$ and also depends on a random variable $\epsilon \sim N(0, \sigma^2)$. The cost of the manager $c(e)$ is increasing $c^\prime(e) > 0$ and convex $c^{\prime\prime}(e)>0$ .

Let s be the salary—which may depend on effort and/or gross profit, depending on what the owner can observe—offered as part of the contract between the owner and manager.

Suppose the principal is risk neutral and wants to maximize the expected value of her profit $$ E(\pi_n) = E(e+\epsilon-s) = e-E(s) $$ Assume the manager is risk averse $U(z)=-e^{A(s-c(e))}$, and his expected utility is $$ E(U) = E(s) - \frac{A}{2} \text{Var}(s) - c(e) $$

First best (full-information case)

With full information, the owner pay the manager a fixed salary $s^\ast$ if he exerts the first-best level of effort $e^\ast$. Hence $E(s^\ast)=s^\ast$ and $\text{Var}(s^\ast)=0$ .

The owner faces an optimization problem $$ \begin{aligned} \max\; & e^\ast-s^\ast \\ \text{ s.t. } & s^\ast - c(e^\ast) \geqslant 0 & (\text{IR}) \end{aligned} $$ where the constraint $\text{IR}$ makes sure the manager accept this job contract.

Therefore, the owner optimally pays the lowest salary satisying the equation $s^\ast=c(e^\ast)$, and earns her net profit $$ e^\ast - c(e^\ast) $$ which is maximize for $e^\ast$ satisfying the first-order condition $$ c^\prime(e^\ast)=1 $$

If $c(e)=e^2/2$, then $e^\ast=1, s^\ast=1/2$.

Second best (hidden-action case)

If the owner can observe the manager’s effort, then she can implement the first best by simply ordering the manager to exert the first-best effort level. If she cannot observe effort, the contract cannot be conditioned on $e$.

However, the owner can observe the firm’s profit and hence offers a salary that is linear in gross profit $$ s(\pi_g) = a+b\pi_g $$ The fixed component $a$ can be thought of as the manager’s base salary and $b$ as the incentive pay in the form of stocks, stock options, and performance bonuses.

The owner-manager relationship can be viewed as a three-stage game.

• First, the owner sets the salary, which amounts to choosing $a$ and $b$.

• Second, the manager decides whether or not to accept the contract.

• Third, the manager decides how much effort to exert conditional on accepting the contract.

The subgame-perfect equilibrium of this game can be solved by backward induction.

The manager’s expected utility from the linear salary is $$ \begin{aligned} & E\left(a+b \pi_g\right)-\frac{A}{2} \operatorname{Var}\left(a+b \pi_g\right)-c(e) \\ & = a+b e-\frac{A b^2 \sigma^2}{2}-c(e) \end{aligned} $$ Note that $$ \begin{gathered} E\left(a+b \pi_g\right)=E(a+b e+b \epsilon)=a+b e+b E(\epsilon)=a+b e\\ \operatorname{Var}\left(a+b \pi_g\right)=\operatorname{Var}(a+b e+b \epsilon)=b^2 \operatorname{Var}(\epsilon)=b^2 \sigma^2 ; \end{gathered} $$ The first-order condition for the e maximizing the manager’s expected utility yields the $\text{(IC)}$ constraint $$ c^\prime(e)=b $$ The $\text{IR}$ constraint for the manager is $$ a \geqslant c(e)+\frac{A b^2 \sigma^2}{2}-b e $$ The owner choosing parameters $a$ and $b$ to maximize her expected surplus $$ e-(a+be) = e(1-b)-a $$ subject to two constraints $\text{(IR)}$ and $\text{(IC)}$. It turns out that $\text{(IR)}$ is an equality.

So the owner’s objective can be rewritten as $$ e-c(e)-\frac{A \sigma^2\left[c^{\prime}(e)\right]^2}{2} $$ The second-best effort $e^{\ast \ast}$ satisfies the first-order condition $$ c^{\prime}\left(e^{\ast\ast}\right)=\frac{1}{1+A \sigma^2 c^{\prime \prime}\left(e^{\ast\ast}\right)} < 1 \Longrightarrow e^{\ast\ast} < e^\ast $$ indicating the second-best effort is less than first-best effort.

Finally, the parameters $a$ and $b$ can be resolved by $\text{(IR)}$ and $\text{(IC)}$.

Comparison

By comparing the two results, we have the following two key implications:

First, the presence of hidden information raises a possibility of shirking and inefficiency that is completely absent in the standard model. The manager does not exert as much effort as he would if effort were observable.

Second, although the manager can be regarded as an input like any other (capital, labor, materials, and so forth) in the standard model, he becomes a unique sort of input when his actions are hidden information. It is not enough to pay a fixed unit price for this input.

Hidden Types

Consider a monopolist (the principal) who sells to a consumer (the agent) with private information about his own valuation for the good.

Let the utility of a consumer of type $\theta$ from consuming a bundle of $q$ units of a good with a total tariff $T$ be $$ U=\theta v(q) - T $$ Assuming $v^\prime(q)>0$ and $v^{\prime\prime}(q) < 0$. The consumer’s type is given by $\theta$, which can be high $\theta_H$ with probability $\beta$ and low $\theta_L$ with probability $1-\beta$. The high type enjoys consuming the good more than the low type $(0 < \theta_L < \theta_H)$ .

For simplicity, we are assuming that there is a single consumer in the market. The analysis would likewise apply to markets with many consumers, a proportion $\beta$ of which are high types and $1-\beta$ of which are low types.

Suppose the monopolist has a constant marginal and average cost $c$ of producing a unit of the good. Then the monopolist’s profit from selling a bundle of $q$ units for a total tariff of $T$ is $$ \Pi = T - cq $$

First-best nonlinear pricing

In the first-best case, the monopolist can observe the consumer’s type $\theta$ before offering him a contract.

For each type, the participation constraint may be written as $$ \theta v(q) - T \geqslant 0 $$ The monopolist will choose $T=\theta v(q)$. Substituting this value of $T$ into the monopolist’s profit function yields $$ \theta v(q) - cq $$ Taking the first-order condition and rearranging provides a condition for the first-best quantity: $$ \theta v^\prime(q)=c $$ Hence the first-best quantity offered to the high type ($q_H^\ast$) is the quantity satisfying $\theta_H v^\prime(q_H^\ast)=c$, and its tariff is set to be $T_H=\theta_H v(q^\ast_H)$. So is the low type.

Second-best nonlinear pricing

Now suppose that the monopolist does not observe the consumer’s type when offering him a contract but knows only the distribution $P(\theta=\theta_H)=\beta=1-P(\theta=\theta_L)$ .

The first-best contract would no longer ‘‘work’’ because the high type obtains more utility by choosing the bundle targeted to the low type (B) rather than the bundle targeted to him (A). In other words, choosing A is no longer incentive compatible for the high type.

The second-best contract is a menu that targets bundle $\left(q_H, T_H\right)$ at the high type and $\left(q_L, T_L\right)$ at the low type. The contract maximizes the monopolist's expected profit, $$ \beta\left(T_H-c q_H\right)+(1-\beta)\left(T_L-c q_L\right) $$ subject to four constraints: $$ \begin{gather} \theta_L v\left(q_L\right)-T_L \geq 0, \tag{IR-L}\\ \theta_H v\left(q_H\right)-T_H \geq 0, \tag{IR-H}\\ \theta_L v\left(q_L\right)-T_L \geq \theta_L v\left(q_H\right)-T_H, \tag{IC-L}\\ \theta_H v\left(q_H\right)-T_H \geq \theta_H v\left(q_L\right)-T_L, \tag{IC-H} \end{gather} $$ The first two are participation constraints for the low and high type of consumer, ensuring that they accept the contract rather than forgoing the monopolist’s good. The last two are incentive compatibility constraints, ensuring that each type chooses the bundle targeted to him rather than the other type’s bundle.

The constraints $\text{(IR-L)}$ and $\text{(IC-H)}$ hold with equality in the second best.

Note that $$ \text{(IR-L)} + \text{(IC-H)} \Rightarrow \theta_H v\left(q_H\right)-T_H \geq \theta_Lv\left(q_L\right)-T_L \geq 0 $$ So $\text{(IR-H)}$​ can be safely ignored.

Meanwhile, $$ \begin{aligned} \text{(IR-L)} + \text{(IC-H)} \text{ holds with equality} & \Rightarrow \theta_H v(q_H) - T_H = \theta_H v(q_L) - \theta_Lv(q_L) \\ & \!\!\!\!\!\!\!\!\! \Rightarrow \theta_L v(q_H) - T_H = (\theta_L - \theta_H)[v(q_H)-v(q_L)] \leq 0 \end{aligned} $$

Hence $\text{(IC-L)}$ can be ignored.

With $\text{(IR-L)}$ and $\text{(IC-H)}$​ $$ \begin{gathered} T_L = \theta_Lv(q_L) \\ T_H = \theta_H[v(q_H)- v(q_L)] + T_L = \theta_H[v(q_H) - v(q_L)] + \theta_Lv(q_L) \end{gathered} $$ By substituting these expressions for $T_L$ and $T_H$ into the monopolist’s objective function, we convert a complicated maximization problem with four inequality constraints into the simpler unconstrained problem of choosing $q_L$ and $q_H$ to maximize $$ \beta\left\{\theta_H\left[v\left(q_H\right)-v\left(q_L\right)\right]+\theta_L v\left(q_L\right)-c q_H\right\}+(1-\beta)\left[\theta_L v\left(q_L\right)-c q_L\right] $$ Taking the first order conditions yields $$ \begin{gathered} \theta_L v^{\prime}\left(q_L^{\ast \ast}\right)=c+\frac{\beta\left(\theta_H-\theta_L\right) v^{\prime}\left(q_L^{\ast \ast}\right)}{1-\beta} \\ \theta_H v^{\prime} (q^{\ast\ast}_H) = c \end{gathered} $$ The condition for the high type is identical to the first best, implying that there is no distortion of the high type’s quantity in the second best.

The equation implies $\theta_L v^\prime(q_L^{\ast\ast}) > c$. Since $v(q)$ is concave, we see that the second-best quantity is lower than the first best, verifying the insight that the low type’s quantity is distorted downward in the second best to extract surplus from the high type.