Preface
From Tirole (2006), we have seen the power of diversification in mitigating credit rationing, through cross-pledging. By reducing the total agency rent, the investors will be willing to invest in the entrepreneur with two uncorrelated projects with the same outcome and probability distributions. The purpose of this section is to simulate the data and examining whether the entrepreneur’s contract \(\{R_2, 0, 0\}\) is optimal.
1. Model Setup: Two Independent, Identical Projects
1.1 Financing Projects Individually
The risk-neutral entrepreneur investor has two independent, identical projects. For each project, it generates a cash flow \(r = R\) under the project success at time t, with probability \(p\) conditional the effort level \(E\), and generates nothing if the project fails. Meanwhile, if the entrepreneur exerts high effects on the project, i.e. \(E = 1\), then the probability of success of this project is \(p = p_H \in (0,1]\). Otherwise, \(p = p_L \in [0,1)\) if the entrepreneur shirks (exerts low effect) on the project. We also assume \(p_H > p_L\).
In addition, the entrepreneur has the same amount of assets \(A\) for each project and the total investment for each project is \(I\). And there are infinitely many risk-neutral investors in the market. This means that the entrepreneur has to obtain outside financing to start the project and also has the bargaining power when constructing the contract. However, we consider the situation that the entrepreneur will face financial constraints, not able to obtaining outside financing because of credit rationing, when the two projects are managed by two independent entrepreneurs. The settings go as follows:
We denote the private benefit for the entrepreneur when shirking on the project is \(B\). And the project gives positive NPV when the entrepreneur exerts effect, but gives negative NPV when the entrepreneur shirks. i.e. \[ p_H R > I > p_L R \]
We focus on the case that the entrepreneur with only one project faces financial constraints. This means that a contract satisfying the incentive constraint will fail the participation constraint, and so the project cannot be financed. Therefore, we can write the question as \[ \max_{R_b} R_b p_H \quad \text{ s.t. } \] \[ \text{(PC)} \quad p_H (R - R_b) \geq I - A \iff A \geq p_H R_b - (p_H R - I) \quad \tag{1} \] \[ \text{(IC)} \quad R_b p_H \geq R_b p_L + B \iff R_b \geq \frac{B}{\Delta p} \text{, where } \Delta p = p_H - p_L. \tag{2} \]
Combining the (PC) and (IC), we know that the contract fails to exist if and only if \[ A < \bar{A} = p_H \frac{B}{\Delta p} - (p_H R - I) = I - p_H[R - \frac{B}{\Delta p}]. \] In other words, the agency rent is too big to satisfy the participation constraint and this lead to the financial constraint.
1.2 Financing Projects Collectively by Diversification
Now, let see what will happen if the entrepreneur, with \(2A\) assets, seeks financing for this new combined project - the combination of previous two projects. First, we need to add additional assumptions and clarifications.
Now the entrepreneur can take three actions, namely exerting efforts on both projects \((E = 2)\), exerting efforts on only one project \((E = 1)\) and shirking on both projects \((E = 0)\). And the private benefits towards the entrepreneur from this combined project is \((2 - E)B\), depending on the effort level \(E \in \{0,1,2\}\).
There will be three different outcomes (states) for this combined project. The distribution of this combined project’ cash flows \(r \in \Omega = \{2R, R, 0\}\) conditional on the effort level \(E\) is \[ \begin{equation} \mathbf{P}(r | E) = \begin{cases} p^1 p^2 \quad & \text{, if } r = 2R \\ p^1 (1-p^2) + (1-p^1) p^2 \quad & \text{, if } r = R \\ (1-p^1) (1-p^2) \quad & \text{, if } r = 0, \\ \end{cases} \end{equation} \] where \(p^1\) and \(p^2\) are the probability of success conditional on the level of efforts exerted on the first and second project respectively. In the current setting, \(p^1\) and \(p^2\) should be the same conditional on the same effort level towards the corresponding project. This is just to make further analyses less complex at this stage.
Then we can build up the new setting as \[ \max_{R_b(r)} \sum_{r \in \Omega} R_b(r) \mathbf{P}(r | E = 2) \quad \text{ s.t. } \] \[ \text{(PC)} \quad \sum_{r \in \Omega} (r - R_b(r)) \mathbf{P}(r | E = 2) \geq 2 (I - A) \tag{3} \] \[ \text{(} \text{IC}_1 \text{)} \quad \sum_{r \in \Omega} R_b(r) \mathbf{P}(r | E = 2) \geq \sum_{r \in \Omega} R_b(r) \mathbf{P}(r | E = 1) + B \tag{4} \] \[ \text{(} \text{IC}_2 \text{)} \quad \sum_{r \in \Omega} R_b(r) \mathbf{P}(r | E = 2) \geq \sum_{r \in \Omega} R_b(r) \mathbf{P}(r | E = 0) + 2B \tag{5} \]
And all conditional distributions are presented below:
| \(r\) | \(\mathbf{P}(r|E = 2)\) | \(\mathbf{P}(r|E = 1)\) | \(\mathbf{P}(r|E = 0)\) |
|---|---|---|---|
| \(2R\) | \(p_H p_H\) | \(p_H p_L\) | \(p_L p_L\) |
| \(R\) | \(2 p_H (1-p_H)\) | \(p_H (1-p_L) + (1-p_H) p_L\) | \(2 p_L (1-p_L)\) |
| \(0\) | \((1-p_H) (1-p_H)\) | \((1-p_H) (1-p_L)\) | \((1-p_L) (1-p_L)\) |
2. Verify Optimal Contract(s)
From Tirole (2006), we know the optimal contract in this case is in the form of \(R_b = \{R_2, R_1, R_0\}\) such that \(R_2 > 0 = R_1 = R_0\). i.e. (one of) the optimal contract is \(R_b = \{R_2, 0, 0\}\). From Table 1 we have that,
\[ R_2 p_H^2 \geq R_2 p_H p_L + B \iff R_2 \geq \frac{B}{p_H \Delta p} \quad \text{(} \text{IC}_1 \text{)} \] \[ R_2 p_H^2 \geq R_2 p_L^2 + 2B \iff R_2 \geq \frac{2B}{(p_H + p_L) \Delta p} = \frac{B}{\frac{p_H + p_L}{2} \Delta p} \quad \text{(} \text{IC}_2 \text{)} \] Since \(p_H > p_L\), we have \(p_H > \frac{p_H + p_L}{2}\). Therefore, if \(R_2\) in contract \(R_b\) satisfies (IC\(_2\)), (IC\(_1\)) must also be satisfied.
However, please note the contract does not provide incentive for exerting effort on one project \((E = 1)\) over shirking on both \((E = 0)\). The corresponding incentive constraint is \[ R_2 p_H p_L \geq R_2 p_L^2 + B \iff R_2 \geq \frac{B}{p_L \Delta p}. \quad \text{(} \text{IC}_3 \text{)} \]
And since \(p_H > \frac{p_H + p_L}{2} > p_L\), the contract \(R_b\) satisfies (IC\(_1\)) and (IC\(_2\)) may not satisfy (IC\(_3\)). And this is the direction we are trying to explore under the case of moral hazard, with unobservable limited ability \(E \leq \bar{E}\), where \(\bar{E}\) is the maximum effort level.
From the incentive constraints above (including (IC\(_1\)) and (IC\(_2\))) and Eq. (3), (4) and (5), we can see that the contract with the minimum agency rent is \[ R_2 = \frac{2B}{(p_H + p_L) \Delta p}. \] And we plug this into the participation constraint and get that the maximum pledgeable income is \[ 2R p_H - R_2 p_H^2 = 2R p_H - 2 \frac{p_H B}{(p_H + p_L)\Delta p} p_H = 2 p_H [R - (1-d_2)\frac{B}{\Delta p}] \geq 2(I - A) \] \[ \implies A \geq \bar{\bar{A}} = I - p_H [R - (1-d_2)\frac{B}{\Delta p}] \text{, where } d_2 = \frac{p_L}{p_H + p_L} \in (0, \frac{1}{2}) \] And we can compare this \(\hat{A}\) with \(\bar{A}\) from section 1.1 with single project, where \[ \bar{A} = I - p_H[R - \frac{B}{\Delta p}]. \] We can see that the diversification reduces the minimum agency rent and bring down the minimum asset required from \(\bar{A}\) to \(\bar{\bar{A}}\). Therefore, if the asset holding \(A \in (\bar{\bar{A}}, \bar{A})\), the entrepreneur will obtain the external financing with the new combined project.
3. Find Optimal Contract(s)
Following Tirole (2006) and the logic in section 2, we know that the contract \(R_b = \{R_2, 0, 0\}\) satisfies the (PC), (IC\(_1\)) and (IC\(_2\)) and is definitely an optimal contract when the borrower (entrepreneur) has the bargaining power. However, the more interesting question is whether this optimal contract is unique?
Now let’s start with an example:
The list of parameters values:
- \(p_H =\) 0.6: conditional prob of success under efforts
- \(p_L =\) 0.5: conditional prob of success under shirking
- \(R =\) 100: project’s cash flow under success
- \(B =\) 5: private benefits under shirking on each project
- \(I =\) 55: total investment required for each project
- \(A =\) 20: assets contributed by the entrepreneur in each project
Please note that the asset \(A\) is predetermined to satisfy the requirement presented in section 1 that \(A \in (\bar{\bar{A}}, \bar{A})\). But in reality, we know that the entrepreneur is always optimal to provide enought assets just passing the threshold.
Then we can plug the values into the question for the combined project, under the assumption of limited liabilities for both parties. This indicates that \(R_0 = 0\).
\[ \max_{R_b(r)} 0.36 R_2 + 0.48 R_1 \quad \text{ s.t. } \] \[ \text{(PC)} \quad 0.36 (200 - R_2) + 0.48 (100 - R_1) \geq 70 \tag{6} \] \[ \text{(} \text{IC}_1 \text{)} \quad 0.36 R_2 + 0.48 R_1 \geq 0.3 R_2 + 0.5 R_1 + 5 \tag{7} \] \[ \text{(} \text{IC}_2 \text{)} \quad 0.36 R_2 + 0.48 R_1 \geq 0.25 R_2 + 0.5 R_1 + 10 \tag{8} \]
We will first visualize the constraints in a 2D graph.
If we denote the objective function as \(g(R_b) = \sum_{r \in \Omega} R_b(r) \mathbf{P}(r | E = 2)\), then in this example, we can write \(g(R_b) = 0.36 R_2 + 0.48 R_1 \implies R_2 = k R_1 + a\). And this function is parallel to the (PC) boundary. So, to maximize the objective function, all contracts in this set \(\{R_b = (R_2, R_1, 0): 0.36 R_2 + 0.48 R_1 = 50, R_1 \in (0, 33.3333333)\}\) are optimal, indifferent contracts to a risk-neutral entrepreneur. And the red area in the graph represents the candidate contracts satisfying PC and two IC constraints.
4. Conclusion
Therefore, we can conclude from this exercise that the contract \(\{R_2, 0, 0\}\), \(R_2 \in (0, 200)\) is a solution for the above optimization problem; however, this optimal contract is not unique if the two IC constraints are slack. Meanwhile, we can say that if \(\{R_2, 0, 0\}\) contract cannot be realized, then there exists no equilibrium solution to the above problem.
5. Extension and Discussion
From the above analysis, we can see that the objective function plays an important role in determining the optimal contract. When the objective function changes to a different format, for instance adding an upper bound on the maximum efforts level for the entrepreneur, this contract \(\{R_2, 0, 0\}\) may not be optimal anymore. And we will go to explore this dimension in our paper.