Private Information and Incentives

1 Communication of Private information

In this section we discuss models where the agent has private information that is not observable to the principal, which introduces challenges in aligning incentives. The agent’s private information can relate to various factors, such as their skill, expertise, or the profitability of investment opportunities. This informational asymmetry means that the agent holds an information rent which allows him to potentially extract more value than he could if the principal were fully informed. The key challenge for the principal is to design contracts that minimize these rents while still providing incentives for the agent to act in the firm’s best interest.

Let x and y be the outcome and additional metric, both observable at the end of the game. The agent’s private information signal m has a priori probability density function g(m). As before, the agent’s effort is e. Once the signal is received, the density function of the outcome and the other metric is updated to be f(x,y|e,m). The models differs in three aspects: First, when does the agent receive the signal m, second, whether the agent can leave the contract after observing m, and third, whether the agent can communicate m to the principal in a truthtfull way.

1.1 Principal’s Problem

We start with the scenario where the agent acquires private information after signing the contract but before choosing his action. He is unable to leave after observing the signal and cannot communicate the signal to the principal. After receiving the private information, the agent can adjust his effort level on the signal, meaning e(m). The optimal compensation w(x,y) cannot depend on the signal m because the agent cannot communicate it to the principal.

The principal’s problem is

\begin{aligned} \max_{w(x,y),e(m)} \quad & \mathbb{E}_{x, y, m} \left[ G[x - w(x,y)] | e(m) \right] \\ \text{subject to} \quad & \mathbb{E}_{x, y, m} \left[ U[w(x, y)| e(m)] - V[e(m)] \right] \geq \underline{U} \text{ for all } m, \quad & \text{(PCs)}\\ & e(m) \in \arg\max \mathbb{E}_{x, y | m} \left[ U[w(x, y) | e] - V(e) \right] \text{ for each } m \quad & \text{(ICCs)} \end{aligned}

Notice that now the principal’s problem include a set of PCs and ICCs, each of them for a different signal m. To avoid that the agent to leave after observing bad realizations of the signal, the principal should offer a contract that is better than the outside options for every realization, not just in expectations. This implies that the agent obtains an excess level of utility, called information rent, derived from the fact that the agent earns the minimal acceptable level of utility just in the realization of the worst signal, and earns in excess to this minimal level in the other realizations. The principal can reduce the information rent by forcing the agent to truthfully report the private information. In those cases, the principal can use this information as an additional metric to adjust the contract to reduce the information rent.

However, given the private nature of the information and the incentive scheme, the principal knows that the agent has incentive to misreport the signal. Let \color{blue}\hat{m}(m) be the report that the agent sends after observing m. The principal can design a different contract that depends on the reported signal, w(x,y,\textcolor{blue}{\hat{m}}). The principal’s problem is then

\begin{aligned} \max_{w(x,y,\textcolor{blue}{\hat{m}}),e(m), \textcolor{blue}{\hat{m}(m)}} \quad & \mathbb{E}_{x, y, m} \left[ G[x - w(x,y,\textcolor{blue}{\hat{m}})] | e(m) \right] \\ \text{subject to} \quad & \mathbb{E}_{x, y| m} \left[ U[w(x, y,\textcolor{blue}{\hat{m}})| e(m)] - V[e(m)] \right] \geq \underline{U} \text{ for all } m, \\ & e(m) \in \arg\max \mathbb{E}_{x, y | m} \left[ U[w(x, y,\textcolor{blue}{\hat{m}}) | e] - V(e) \right] \text{ for each } m, \quad \\ & \hat{m}(m) \in \arg\max \mathbb{E}_{x, y | m} \left[ U[w(x, y,\textcolor{blue}{\hat{m}}) | e] - V(e) \right] \text{ for all } m \\ \end{aligned}

The first set of constraints consists of the minimal acceptable utility constraints, the second set comprises the incentive compatibility constraints on the agent’s effort, and the third set includes the incentive compatibility constraints on the agent’s communication strategy.

1.2 Relevation Principle

As the set of possible strategies is very large, the problem for the principal is difficult to solve. A shorcut developed in the literature is the Relevation Principle. The revelation principle states that any proposed mechanism that involves nontruthful communication by the agent can be duplicated or beaten in terms of expected utilities by an equilibrium mechanism in which truthful reporting is induced. The cost for forcing the agent to tell the trut is that the principal must commit to not use the information as fully as he would if the truthful message did not have to be motivated. The revelation principle is a very helpful tool for researchers because it reduces the number of alternative reporting strategies needed to be considered. This allows researchers to focus on reporting strategies that encourage truthful reporting. To apply the relevation principal to our problem, we should substitute the report by the true signal, \textcolor{blue}{\hat{m}}=\textcolor{green}{m}, and set the contract such that it ensures that the best reporting strategy for the agent is to tell the truth.

\begin{aligned} \max_{w(x,y,\textcolor{green}{m}),e(\textcolor{green}{m}), \textcolor{green}{m}} \quad & \mathbb{E}_{x, y, m} \left[ G[x - w(x,y,\textcolor{green}{m})] | e(m) \right] \\ \text{subject to} \quad & \mathbb{E}_{x, y| m} \left[ U[w(x, y,\textcolor{green}{m})| e(m)] - V[e(m)] \right] \geq \underline{U} \text{ for all } m, \\ & e(m) \in \arg\max \mathbb{E}_{x, y | m} \left[ U[w(x, y,\textcolor{green}{m}) | e] - V(e) \right] \text{ for each } m \quad \\ & m(m) \text{ is the } \hat{m}(m) \in \arg\max \mathbb{E}_{x, y | m} \left[ U[w(x, y,\textcolor{green}{m}) | e] - V(e) \right] \text{ for all } m \\ \end{aligned}

In these models, the principal’s welfare loss is due to the information rent of the agent, which exists even in contexts with risk neutral agents. For this reason, mos of the model examine private information issues using risk neutral agents, so they can avoid the complexities that result when risk aversion and risk sharing issues are also present.

2 Budgeting

The literature has studied the optimal budgeting of projects when the agent has private information about the true cost of the project. The main takeaway from these studies is that the optimal budgeting involves the principal providing more resources than the actual cost of the project to incentivize the agent to reveal the true cost. In other words, in these settings, a budget that consider more funds for organizational slack is efficient. This is because the agent has an incentive to overstate the cost of the project to extract more resources than needed. The principal can reduce the information rent by forcing the agent to tell the truth. The cost for forcing the agent to tell the truth is that the principal must commit to not use the information as fully as he would if the truthful message did not have to be motivated. The optimal solution involves the principal rejecting certain profitable projects, a phenomenon known as capital rationing. While both effects, the organizational slack and the capital rationing, are seen as ex post inefficiencies, they can also manifest as ex ante efficient organizational designs in information assymetry settings. For early contributions see Antle and Eppen (1985) and Antle and Fellingham (1997).

The model assumes that the agent privately learns the minimal investment amount, $ I_0=mx$. Below I_0, the project is not feasible. If the agent receives at least I_0=mx, then the project generates a cash flow x. Notice that the project consists of a linear production technology. The factor m is interpreted as the cost of the project. In these models, the agent does not have the resources to invest in the project, and this capital is supplied by a risk-neutral principal who does not know m. The agent can communicate m to the principal. The principal transfers to the agent the capital z to cover the initial investment amount and the agent’s compensation. However, the agent is incentivized to overstate the true cost to divert a fraction of z for its own benefit, referred to as organizational slack in the literature. There is no incentive to understate the cost below its true value because in that case the capital supplied will not be enough to start the investment, and thus, there is no cash flow.

Notice we can recast the investment costs m to the rate of return on investment r=1/m-1. Therefore, we can interpret this problem as private information on the investment costs or the rate of return. Of course, the rate of return is negative if m_i>1, meaning that the investment is not profitable for a principal because the cost of capital here is normalized to 1.

We assume a discrete number of potential information signals that the agent can privately observe about the true cost m_1 < m_2 < \ldots < m_n. The least favorable signal is m_n because it represents the higher cost. The ex-ante probability of observing m_i is g_i.

2.1 Principal and Agent’s Preferences

The principal is risk-neutral with utility function G(x,z)=x - z. The agent is also risk neutral with utility function U(w)=z-m_ix, meaning the money transfered by the principal minus the cost project observed privately. As in the initial setting, we assume that the agent can leave after observing a bad signal. Notice that the principal makes the decision on the transference z needed to obtain x based on the communication of the reported costs \hat{m} provided at the beginning.

2.2 First Best Solution

Let’s ignore for now the information assymetry in the true cost of the project m. The principal’s problem is to maximizes the expected value of net profits. Normalizing the unit value of output to be 1, the principal’s problem is

\begin{aligned} \max_{x_i, z_i} \quad & \sum_{i=1}^{N} (x_i - z_i) g_i \\ \text{subject to} \quad & z_i - m_i x_i \geq \underline{U} \quad \text{for all } i = 1, \dots, N, \\ & 0 \leq x_i \leq x_{\max}, \\ & 0 \leq z_i. \end{aligned} \tag{1}

Thus the problem for the principal is to find the optimal combination of (cash flow, transference) (x_i,z_i) that maximizes the expected value of the principal’s profits subject to the agent’s minimal acceptable utility constraint in any scenario of the cost of the project.

In Equation 1, the first restriction is the minimal acceptable utility constraint for the agent regardless of the signal observed. The second and third restrictions set bounds on the cash flows and transference for the solution to exist (notice the linearity in the production technology and lack of agent’s wealth).

The first restrictions implies that z_i^*=\underline{U}+m_i x_i. Inserting this into the objective function, we obtain
\begin{aligned} \max_{x_i} \quad & \sum_{i=1}^{N} (1-m_i)(x_i g_i)-\underline{U} \\ \end{aligned}

which is maximized when x_i=x_{\max} if m_i\leq 1 and x_i=0 if m_i> 1. The intuition is that the expected value of the principal’s profits is at its maximum when the project generates the highest possible cash flows if the signal indicates that the project is profitable. Consistently, the optimal transfer is

z_i = \begin{cases} m_i x_{\max} + \underline{U} & \text{for } m_i \leq 1, \\ \underline{U} & \text{for } m_i > 1. \end{cases}

When the project cost is high compared to the cost of capital, the principal should transfer the minimum acceptable utility. On the other hand, when the cost is in a profitable range, the principal should increase the transfer amount to fund the profitable investment, leading to largest cash flows.

2.3 Communication Strategies

Suppose the agent privately sees signal m_i but reports a higher cost m_j to obtain more the resources than needed ((m_j-m_i)x_j). However, there is a limit to how much he will overstate. If the agent claims the cost is too high (over 1), the principal will not provide any resources. Thus, the agent will report a cost slightly under 1 for all m_i<1. If the agent observes a cost realization above 1, there is no incentive to lie because the project will not be funded anyway. As previously mentioned, the principal does not need to worry about the agent understating the costs. If he does, the project will not receive the minimal funding to produce the required output, so any lie in this direction will always be uncovered.

2.3.1 Principal’s Problem

The principal’s problem is to define a set of contract (x_i,z_i) that depends on the reported signal, z_i(\hat{m}) such that the agent’s best reporting strategy is to tell the truth. The principal’s problem is then

\begin{aligned} \max_{x_i, z_i} \quad & \sum_{i=1}^{N} (x_i - z_i) g_i \\ \text{subject to} \quad & z_i - m_i x_i \geq \underline{U} \quad \text{for all } i = 1, \dots, N, \\ & z_i - m_i x_i \geq z_j - m_i x_j \quad \text{for all } i, j, \\ & 0 \leq x_i \leq x_{\max} \quad \text{for all } i, \\ & 0 \leq z_i. \end{aligned} \tag{2}

Notice that this version adds the second set of N^2 restrictions, which ensures that the agent does not have an incentive to lie about the cost of the project (the ICCs, also called truth-telling constraints). Obviously, the agent’s reported costs m_j does not affect the underlying true cost of the project m_i. This is why the true cost m_i is at each side of the ICCs. In other words, the ICCs implies that when the agent observes the cost m_i, he is better off with the contract (x_i,z_i) than with any other combitation (x_j,z_j). This set of constraints can be reduced to the set of 2(N - 1) “adjacent” constraints, meaning that the manager must prefers the communicate the true cost to the cost immediately above and the cost immediately below dominates.

The upper bound restrictions in the cash flow can also be simplified to a single restriction x_N\leq X_{\max}.

The agent’s slack z_i-m_i x_i, decreases weakly in i because as m_i​ increases (i.e., higher costs), both x_i​ and z_i decreases in i. In the worst realization of i, the minimum acceptable utility is is z_N-m_N x_N=\theta.

Let k^* be the value of i that maximizes the principal’s problem. This implies that x_i=x_{\max} for i=1,\ldots,k^* and x_i=0 for all i=k^*+1,\ldots,N. Since z_1=m_1 x_1+\theta, and then z_{i}-z_{i-1}=m_i(x_i-x_{i-1}) for i=2, \ldots, N, the transfer policy is

z_i = \begin{cases} m_{k^*} x_{\max} & \text{for } i=1,...,k^*, \\ 0 & \text{for } i=k^*+1,...,N \end{cases}

Compare with the first best, we see that here the agent receives slack for all cost realization below the threshold (m_{k^*} x_{\max}>m_{i} x_{\max} for all i<k^*). The principal provides additional more funds than the actual investment cost to incentive the agent to reveal the truth. But, how much is the new cutoff? the cutoff is less than 1.0. This implies that the optimal solution involves the principal rejecting certain profitable projects (capital rationing). When the principal adjusts the cutoff, it impacts the flexibility the agent has for all lower levels below that point. Therefore, the principal’s trade-off is to balance the cost of setting the cutoff too high (paying for slack) and too low (foregoing profitable projects).

We can see this by solving numerically the linear program in Equation 2 considering the real cost m being a uniform discrete variable defined between 0.1 and 1.5, and \theta=0.

from IPython.display import Markdown
from tabulate import tabulate
import numpy as np
from scipy.optimize import linprog
import pandas as pd

# Parameters
m = np.arange(0.1, 1.6, 0.1)
N = len(m)
theta = 0
x_max = 10
g = np.array([1/N] * N) # Uniform distribution of signals

# Objective function: maximize sum of (x_i - z_i) * g_i
# Notice this is equivalent to minimiza  sum of z_i * g_i -x_i * g_i 
# c is a one-dimensional vector of size 2N: [x_1, z_1, x_2, z_2, ..., x_N, z_N]
c = np.zeros(2 * N)
for i in range(N):
    c[2 * i] = -g[i]  # Coefficient for x_i
    c[2 * i + 1] = g[i]  # Coefficient for z_i

# Inequality constraints for participation and incentive compatibility
A = []
b = []

# Participation constraints: z_i - m_i * x_i >= theta (agent's minimal utility constraint)
# The constraint is a one-dimensional vector of size 2N to multiply with [x_1, z_1, x_2, z_2, ..., x_N, z_N]
for i in range(N):
    constraint = [0] * (2 * N)  # Create a 2N-dimensional row, for matching dimension with ICCs
    constraint[2 * i] = -m[i]   # Coefficient for x_i
    constraint[2 * i + 1] = 1   # Coefficient for z_i
    A.append(constraint)
    b.append(theta)

# Incentive compatibility constraints I: [z_i - m_i * x_i] -[z_{i-1} - m_{i} * x_{i-1}] >= 0 
# The constraint is a one-dimensional vector of size 2N to multiply with [x_1, z_1, x_2, z_2, ..., x_N, z_N]
for i in range(1, N):
    constraint = [0] * (2 * N)  # Create a 2N-dimensional row
    constraint[2 * i] = -m[i]  # Coefficient for x_i
    constraint[2 * i + 1] = 1  # Coefficient for z_i
    constraint[2 * (i - 1)] = m[i]  # Coefficient for x_{i-1}
    constraint[2 * (i - 1) + 1] = -1  # Coefficient for z_{i-1}
    A.append(constraint)
    b.append(0)

# Incentive compatibility constraints II: [z_i - m_i * x_i]-[z_{i+1} - m_{i} * x_{i+1}] >= 0 
for i in range(N - 1):
    constraint = [0] * (2 * N)  # Create a 2N-dimensional row
    constraint[2 * i] = -m[i]  # Coefficient for x_i
    constraint[2 * i + 1] = 1  # Coefficient for z_i
    constraint[2 * (i + 1)] = m[i]  # Coefficient for x_{i+1}
    constraint[2 * (i + 1) + 1] = -1  # Coefficient for z_{i+1}
    A.append(constraint)
    b.append(0)

# Convert A and b to numpy arrays for linprog function: A*constraints <= b
A = np.array(A) * -1  # Multiply A by -1 to convert the inequality to the form Ax <= b
b = np.array(b) * -1  

# %%
# Bounds: 0 <= x_i <= x_max, 0 <= z_i
x_bounds = [(0, x_max)] * N
z_bounds = [(0, None)] * N
bounds = []
for i in range(N):
    bounds.append(x_bounds[i])
    bounds.append(z_bounds[i])

# Solve the linear programming problem
res = linprog(c, A_ub=A, b_ub=b, bounds=bounds, method='highs')

# %%
# Check if the optimization was successful
if res.success:
    x = res.x[::2]  # Extract the budget values from the solution, they are in even indices
    z= res.x[1::2]
    # the slack is transfers{i} -m{i}*cashflow{i}
    slack =  z- m *   x
    investment= m * x
    # Create a dataframe for the results
    df = pd.DataFrame({'Cost m': m,  'Budget Z': z, "Investment": investment , 'Slack': slack, 'Cash flow X': x,})
    # Print the results as markdown using the tabulate package
    display(Markdown(tabulate(df, headers='keys', tablefmt='pipe', showindex=False)))
else:
    print("Optimization failed. Check the constraints or problem formulation.")

Table 1: Numerical Example

Cost m	Budget Z	Investment	Slack	Cash flow X
0.1	5	1	4	10
0.2	5	2	3	10
0.3	5	3	2	10
0.4	5	4	1	10
0.5	5	5	0	10
0.6	-0	-0	0	-0
0.7	-0	-0	0	-0
0.8	-0	-0	0	-0
0.9	-0	-0	0	-0
1	0	-0	0	-0
1.1	0	-0	0	-0
1.2	0	-0	0	-0
1.3	0	-0	0	-0
1.4	0	-0	0	-0
1.5	0	-0	0	-0

In Table 1 we can see the two key insights from this type of models.

1. The optimal cutoff is m_{k^*}=0.5, which implies that the principal will not fund any project with a minimal cost above 0.5. Notice that for 0.6\leq m < 1, the profits of the investment outweight the costs, so, the principal is rationing the resources.
2. For m\leq 0.4, the principal still transfers m_{k^*}x_{max} to the agent, which is the more the resources needed for the project. The agent is rewarded for communicating the the true minimal cost of the project.

3 Suggestions for Further Reading

Several studies have looked at budgeting capital investment across time in contexts where the manager has better informaton. See for example, Rogerson (1997). A benefit of looking at multi-period settings is that we can differentiate cash flows from accounting earnings. In that sense,Reichelstein (2000) compares performance measures based only on realized cash flows with the residual accounting income; while Durra and Reichelstein (2002) apply the inter-temporal setting to analyze the optimal capital charge for investment decision for divisional managers with superior information.

References

Antle, Rick, and Gary D. Eppen. 1985. “Capital Rationing and Organizational Slack in Capital Budgeting.” Management Science 31 (2): 163–74. http://www.jstor.org/stable/2631513.

Antle, Rick, and John Fellingham. 1997. “Models of Capital Investments with Private Information and Incentives: A Selective Review.” Journal of Business Finance & Accounting 24 (7-8): 887–908. https://doi.org/10.1111/1468-5957.00140.