1. Framework Setup
Like in the previous file “TCP_Project_Premodel”, we consider the following setting. The risk-neutral entrepreneur now still has two independent, identical projects. Each project 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 \(p_H > p_L\). And same as before, we still consider the case that the entrepreneur can contribute asset amount \(A\) in each project and needs external financing to start the project, which requires total investment \(I\). We also assume that there are infinitely many risk-neutral investors and the entrepreneur has the bargaining power. The private benefit for the entrepreneur shirking on each project is \(B\). Meanwhile, we still hold the assumption that the NPV of each project is positive if the entrepreneur exerts effort on the project, and that is negative if the entrepreneur shirks. i.e. \(p_H R > I > p_L R\).
1.1 Additional Assumptions
Different from the previous case, we have several new assumptions:
The each project will no longer face financial constraints individually. i.e. \[ A \geq \bar{A} = I - p_H[R - \frac{B}{\Delta p}] \] Ceteris paribus, this assumption guarantees that the combined project can obtain external financing based on our analyses in the previous document.
We bring adverse selection into the model. We assume that there are two types of entrepreneurs - Good and Bad. The first type (Good) is the one without effort levels as in the previous example, who can exerts efforts on all projects. The second type (Bad), who is also the interest of this project, can only exerts limited efforts and there is an uppper bound for the maximum level of efforts \(\bar{E}\) can be exerted by the entrepreneur to the combined project. In this two projects world, the effort level for the first type is \(E \in \{0,1,2\}\), with \(E = 2\) being exerting efforts on both projects. And the second type can exerts effort no higher than \(\bar{E} = 1\), which means the entrepreneur can only exerts effort on maximum 1 project. i.e. \(E \in \{0,1\}\). And the type of entrepreneurs is only known by the entrepreneurs themselves, not by any other agents in the economy in any way.
This setting is equivalent to assigning a new private benefit function \(B(E)\) with increasing marginial benefits of efforts (i.e. \(\frac{\partial B}{\partial E} > 0\)), replacing the old one with zero marginial benefits of efforts. And we can set the incremental private benefit on the second project going to infinity to mimic our setting. The main reason for this setup is to relax the previous implicit assumption of infinite efforts in the baseline model for the diversification. And in reality, the manager (interchangeable with the entrepreneur hereafter) may choose to diversify their existing businesses, e.g. entering a different industry, even though they do not have the knowledge, experience or expertise to develop a successful business in that industry.
We may extend this further to a continuous effort level model.
1.2 Background Information
Now we consider a two-period setting. The entrepreneur currently successfully runs one project and exerts efforts on this project. Now at the start of the first period, the manager can choose to diversify by starting a new project, which is independent of the existing one and has the same potential outcomes as described in 1. Framework Setup. At the start of the each period, conditional on the project(s) receiving financing, the entrepreneur will choose its effort level \(E_t\), where \(t \in \{1,2\}\). Although \(E_t\) and \(\bar{E}\) are not observable to the investors, the investors can receive a signal at the end of each period \(X_t\). We assume that this signal \(X_t\) contains all available and accessible information to the investors, associated with the project operation. For simplicity, we assume that the only accessible signal to the investors is the success or failure outcome of two projects at the end of the first period.
Then at the beginning of the second period, the entrepreneur will decide whether to spin off the firm into two separate entities, each managing only one project. If the entrepreneur chooses to spin off, then there will be a lump-sum cost to the corporation, which reduces the total NPV, and an additional cost to the entrepreneur who may lose some private benefits. But such a spin-off may increase the total NPV, even after incurring the costs, and create benefits to both parties. Meanwhile, if choosing to continuously operate the combined project, the entrepreneur may face financial constraints conditional on the signal \(X_t\). As the investors will update their beliefs about entrepreneur’s ability (i.e. efforts) to managing both projects. In other words, based on the signal and the prior belief, the investor may classify the entrepreneur into the bad type and refuse to continue to finance the company.
PS: We may also extend this further to the firm with long-term and short-term debts, where the short-term debt needs to be refinanced at the beginning of each period.
Therefore, throught this model, we want to show the under what situation a company will choose to spin off and what are the benefits created by such action.
2. Model Construction (A Sandbox)
2.1 Financing in Period 1
The case with a Good entrepreneur will be the same as in the previous document. Now we are interested in the case with a Bad entrepreneur. And we also assume that the company receives finance only if the investors identify the entrepreneur as a Good type and such decision can be represented by an indicator variable, which equals to 1 if the investors think the entrepreneur is more likely to be a Good type. In other words, the investors can only identify the entrepreneur into a Good or Bad type, and cannot be in between such as being a Good type with 80% chance and a Bad type with 20% (This assumption will be relaxed in the following sections). So, at the beginning of the first period, the investors believe that the entrepreneur is a Good type and agree to finance project. Then we can formulate this optimization problem as \[ \max_{R_b(r)} \sum_{r \in \Omega} R_b(r) \color{red}{\mathbf{P}(r | E = 1)} + \color{red}{B} \quad \text{ s.t. } \] \[ \text{(PC)} \quad \sum_{r \in \Omega} (r - R_b(r)) \mathbf{P}(r | E = 2) \geq 2 (I - A) \tag{1} \] \[ \text{(} \text{IC}_1 \text{)} \quad \sum_{r \in \Omega} R_b(r) \mathbf{P}(r | E = 2) \geq \sum_{r \in \Omega} R_b(r) \mathbf{P}(r | E = 1) + B \tag{2} \] \[ \text{(} \text{IC}_2 \text{)} \quad \sum_{r \in \Omega} R_b(r) \mathbf{P}(r | E = 2) \geq \sum_{r \in \Omega} R_b(r) \mathbf{P}(r | E = 0) + 2B \tag{3} \]From the above setup, we can see that the objective function for the entrepreneur is adjusted based upon the limited efforts of a Bad type. While, the participation constraints and the incentive constraints remain to be the same, as the probability distributions of project’s cash flow conditional on the effort level do not alter in the investors’ view.
Although the more rigorous way to construct the objective function is to choose the maximum between the agency rent conditional on \(E = 1\) and that conditional on \(E = 0\), we will show in the following section 2.3 Analyses in Period 1 that the two objective function will return the same optimal contract in this two-project world.
| > Section for Lagrangian |
In a more intuitive way, we know the set of potential contracts satisfying the PC and two IC constraints from previous study and we just need to find contract(s) providing the highest value for the objective function. Also, because the investors will participate if and only if the PC is satisfied and the entrepreneur will take the whole NPV of the project by the design of our model, the (PC) constraint must bind. Under the limited liability assumptions for both agents, we can rewrite the objective function and (PC) constraints as \[ R_2 = - \frac{p_H (1-p_L) + (1-p_H) p_L}{p_H p_L} R_1 + a \text{, where } a \text{ is a constant, and} \tag{4} \] \[ R_2 = - \frac{2 p_H (1-p_H)}{p_H^2} R_1 + b = - \frac{2 (1-p_H)}{p_H} R_1 + b \text{, where } b \text{ is a constant.} \tag{5} \] Since \(- \frac{p_H (1-p_L) + (1-p_H) p_L}{p_H p_L} < - \frac{2 (1-p_H)}{p_H}\), we know that the contract \(R_b = \{R_2, 0, 0\}\) is no longer an optimal contract. But rather a new contract \(R_b' = \{R_2', R_1', 0\}\) will be the unique optimal contract.
2.2 Visualize Optimal Contract in Period 1
The list of parameters values:
- \(p_H =\) 0.6: conditional prob of success under efforts
- \(p_L =\) 0.4: conditional prob of success under shirking
- \(R =\) 100: project’s cash flow under success
- \(B =\) 5: private benefits under shirking on each project
- \(I =\) 50: total investment required for each project
- \(A =\) 7: assets contributed by the entrepreneur in each project (This value is different from that in the previous document.)
We plug these values into the optimization problem and generate the following graph.
The blue point on the dashed blue line gives the highest objective value for the entrepreneur and the optimal contract is \(R_b' = \{470/9, 95/3, 0\}\) in period 1. Please also note that this contract can be proposed by the Good type entrepreneur as well, given that this \(R_b'\) contract and the original \(R_b = \{R_2, 0, 0\}\) contract are indifferent to the Good type entreprenur.
2.3 Analyses in Period 1
Given the optimal contract identified in 2.2 Visualize Optimal Contract in Period 1, we now need to uncover the true constraints and analyse accordingly.
* Entrepreneur’s Objective Function
Because of adverse selection, there is a mismatch between the constraints considered by the investors, when they decide the contract to finance the project, and the true constraints. Therefore, it is unknown whether the entrepreneur will choose to shirk on both projects or just one. If we assume that exerting the maxmium effort is optimal for the entrepreneur, we have shown in 2.2 Visualize Optimal Contract in Period 1 that the optimal contract is \(R_b' = \{R_2', R_1', 0\} = \{470/9, 95/3, 0\}\). Then if we assume that the optimal action for the entrepreneur is to shirk on both projects (i.e. \(E = 0\)), the objective function in the optimization problem will be \[ \max_{R_b(r)} \sum_{r \in \Omega} R_b(r) \color{red}{\mathbf{P}(r | E = 0)} + \color{red}{2B}. \quad \tag{6} \] And if we translate such an objective function into a line in the two dimensional graph - \((R_1, R_2) \in [0, R] \times [0, 2R]\), the line can be rewritten as \[ R_2 = - \frac{2 p_L (1-p_L)}{p_L^2} R_1 + c = - \frac{2 (1-p_L) p_H}{p_L p_H} R_1 + c \text{, where } c \text{ is a constant.} \tag{7} \] And comparing with Eq (4), we can see that \(- \frac{2 (1-p_L) p_H}{p_L p_H} < - \frac{p_H (1-p_L) + (1-p_H) p_L}{p_H p_L} < - \frac{2 (1-p_H)}{p_H}\). This intuitively indicates that the contract \(R_b' = \{R_2', R_1', 0\}\) will still be the unique optimal contract for the entrepreneur. Therefore, we can conclude that this contract \(R_b' = \{R_2', R_1', 0\}\) will be optimal independent to entrepreneur’s action. But the action choice of the entrepreneur will be collectively determined by the agency rent and the private benefits. The following graph will support the previous arguments.
And it is clear that we can find out the action \(E^*\) taken by the entrepreneur with this contract \(R_b'\) is \[ E^* = \arg \max_{E \in \{0, 1, 2\}} \Big \{ \max \{\sum_{r \in \Omega} R_b'(r) \mathbf{P}(r | E = 1) + B \text{ , } \sum_{r \in \Omega} R_b'(r) \mathbf{P}(r | E = 0) + 2B\} \Big \} \] Since then agency rent also links to the private benefit \(B\), we know the first and second components in the maximization function can also be rewritten as a function of \(B\). In the above example, we know that the maximum utilities conditional on \(E = 0\), \(E = 1\) and \(E = 2\) are 33.56, 34 and 34 respectively, and the manager will choose \(\color{red}{E = 1}\) in this case. i.e. exerts effort on only one project. While the agency rent for the entrepreneur when managing only one project is 17. This is smaller than the expected utlity realised under the diversification and supports our previous assumption that the entrepreneur will only diversify if can obtain higher personal utility.
(The following argument needs more rigorous prove!)
The purpose of this section will be (1) find the close-form expression of the optimal contract (2) searching whether the following \(\bar{B}\) exists.
Date: 2022-06-12
And we know (assume with confident at this stage) there exists a threshold \(\bar{B}\) such that \[ E^* = \begin{cases} 1 \quad &\text{, if } B \leq \bar{B} \\ 0 \quad &\text{, if } B > \bar{B} \end{cases} \] Intuitively speaking, this is not very clear to see, but we can continue our work, conditional on entrepreneur’s optimal action. But no matter the choice of action, the NPV of this combined project, managed by the Bad manager, will be smaller than that managed by a Good manager. And since the manager will only consider this diversification strategy if the utility is higher in this case.
* Investors’ Participation Constraints
Please note that we will focus on the case that \(E = E^* = 1\) in all following analyses.
Since the optimal contract binds the misspecified participation constraint Eq. (1), we have \[ \sum_{r \in \Omega} (r - R_b'(r)) \mathbf{P}(r | E = 2) = 2 (I - A). \tag{8} \] Now we consider the true NPV of investors’ investments and we can write it as \[ \sum_{r \in \Omega} (r - R_b'(r)) \mathbf{P}(r | E = E^*) - 2 (I - A) = \sum_{r \in \Omega} (r - R_b'(r)) \Big [\mathbf{P}(r | E = E^*) - \mathbf{P}(r | E = 2) \Big ], \tag{9} \] where the second equality comes from Eq (9). And it is possible that the true NPV of investors’ investment be weakly smaller than 1.
Still using the previous example, we plot the true binding PC constraints in the above graph. The red dashed line is the true binding PC constraints when the entrepreneur exerts effort on only one project, and the green dashed line is the true binding PC constraints under the entrepreneur shirking on both projects. And the NPV of investors’ investment is non-negative if and only if the optimal contract \(R_b'\) lies in the area below the corresponding dashed line. In the above case, we know that NPVs of investors’ investments under \(E = 1\) and \(E = 0\) are -15 and -29.56 respectively. This is consistent with the red dashed line (PC under \(E = 1\)) and the green dashed line (PC under \(E = 0\), not presented in the graph but is at the left bottom). However, this result is sensitive to the initial asset \(A\) contributed by the entrepreneur and also \(p_H\) and \(p_L\). The purpose of this part is to demonstrate that such case exists.
And with the assumption that the entrepreneur will take action \(E = 1\) as a Bad type, the investors will be more likely to identify his/her skill type. This information can be incorporated into the signal \(X_t\).
2.4 Investors’ Belief Update
In this section, we analyse how the investors’ belief on the type of
the entrepreneur changes at the beginning of period 2, given the signal
\(X_0\) at the end of period 1. The
signal \(X_0\) here assumes to be the
only accessible information of the project towards investors and we
define it to be the outcome of two projects, namely \(2R\), \(R\) or 0. i.e. \(X_0 \in \{2R, R, 0\}\). We define the
variable \(T \in \{G, B\}\) as the type
of the entrepreneur, where \(G\)
denotes the Good type and \(B\) denotes
the Bad type. And investors’ belief at the beginning of period 1 is
\[
\mathbb{P}(T) = \begin{cases}
q \quad &\text{, if } T = G \\
1-q \quad &\text{, if } T = B,
\end{cases}
\] where \(q > 0.5\).
And
\[
\mathbb{P}(X_0 | T = G) = \begin{cases}
p_H^2 &\text{, if } X_0 = 2R \\
2 p_H (1-pH) &\text{, if } X_0 = R \\
(1 - p_H)^2 &\text{, if } X_0 = 0.
\end{cases}
\] \[
\text{ and }
\mathbb{P}(X_0 | T = B) = \begin{cases}
p_H p_L &\text{, if } X_0 = 2R \\
p_H (1-pL) + (1-p_H) p_L &\text{, if } X_0 = R \\
(1 - p_H)(1 - p_L) &\text{, if } X_0 = 0.
\end{cases}
\] The intuition here is that the probability distribution of the
signal \(X_0\), conditional on the
manager being a Good type, is the same as the distribution of the
project cash flow when the manager exerts efforts on both project. This
relationship is determined by both the PC and two IC constraints when
forming the contract. And align with our previous setting, the
conditional probability distribution of \(X_0\) for a Bad type manager is the
distribution of the project cash flow when the manager exerts efforts on
only one project.
Therefore, investors can use the Bayes rule to update their beliefs
towards the type of the entrepreneur at the beginning of the period 2,
before deciding whether to continue finance the project. And we
have
\[
\mathbb{P}(T = G | X_0) = \frac{\mathbb{P}(X_0 | T = G) \mathbb{P}(T =
G)}{\mathbb{P}(X_0)}
= \frac{\mathbb{P}(X_0 | T = G) q}{\mathbb{P}(X_0 | T = G) q +
\mathbb{P}(X_0 | T = B) (1-q)}. \tag{10}
\] Specifically, we have \[
\mathbb{P}(T = G | X_0 = 2R) = \frac{p_H^2 q}{p_H^2 q + p_H p_L (1-q)} =
\frac{1}{1 + \frac{p_L}{p_H} \frac{1-q}{q}} > \frac{1}{2} \tag{11}
\] \[
\mathbb{P}(T = G | X_0 = R) = \frac{2 p_H (1-p_H) q}{2 p_H (1-p_H) q +
[p_H (1-p_L) + (1-p_H) p_L] (1-q)}
\] \[
= \frac{1}{1 + \frac{p_H (1-p_L) + (1-p_H) p_L}{2 p_H (1-p_H)}
\frac{1-q}{q}} \tag{12}
\] \[
\mathbb{P}(T = G | X_0 = 0) = \frac{(1-p_H)^2 q}{(1-p_H)^2 q + (1-p_H)
(1-p_L) (1-q)} = \frac{1}{1 + \frac{1-p_L}{1-p_H} \frac{1-q}{q}} >
\frac{1}{2} \tag{13}
\]
In Eq (11), as \(p_L < p_H\) and \(q < 0.5\), \(\frac{p_L}{p_H} \frac{1-q}{q} < 1\) and \(\mathbb{P}(T = G | X_0 = 2R) > \frac{1}{2}\). In Eq (12), it is possible that \(\mathbb{P}(T = G | X_0 = R) < \frac{1}{2}\) with \(\frac{p_H (1-p_L) + (1-p_H) p_L}{2 p_H (1-p_H)} \frac{1-q}{q} > 1\). And the same logic goes in Eq (13) that \(\mathbb{P}(T = G | X_0 = 0) < \frac{1}{2}\) if \(\frac{1-p_L}{1-p_H} \frac{1-q}{q} > 1\). From the above graph, we can see how will the posterior probability changes. Please note that the exact threshold for determining whether a manager is more of a Good than Bad type is extraneously decided by the investors and it may not necessarily be 0.5.
2.5 Financing in Period 2
As shown in 2.4 Investors’ Belief Update, investors’ belief towards whether the manager is a Good type is updated when receiving the signal \(X_0\). If the investors still identify the manager as a Good type, this company will continue to be financed and operate in Period 2. However, if the entrepreneur is identified as a Bad type, i.e. realizing the entrepreneur can exert effort on maximum one project, then the company may face financial constraints. And we will analyse this scenario.
First, we rewrite the optimization problem in the case that investors know the entrepreneur is a Bad type. \[ \max_{R_b(r)} \sum_{r \in \Omega} R_b(r) \mathbf{P}(r | E = 1) + {B} \quad \text{ s.t. } \] \[ \text{(PC)} \quad \sum_{r \in \Omega} (r - R_b(r)) \mathbf{P}(r | E = 1) \geq 2 (I - A) \tag{14} \] \[ \text{(IC)} \quad \sum_{r \in \Omega} R_b(r) \mathbf{P}(r | E = 1) + B \geq \sum_{r \in \Omega} R_b(r) \mathbf{P}(r | E = 0) + 2B \] \[ \Rightarrow \sum_{r \in \Omega} R_b(r) \Big(\mathbf{P}(r | E = 1) - \mathbf{P}(r | E = 0)\Big ) \geq B \tag{15} \] In this case, the investors will invest in the project if and only if the entrepreneur can be contracted and incentivized to prefer exerting efforts on one project than shirking on both. So, the objective function to the entrepreneur is conditional on \(E = 1\) and the new (PC) and (IC) are formulated accordingly. Then from the (PC) and (IC) we have \[ \sum_{r \in \Omega} r \mathbf{P}(r | E = 1) - 2 (I - A) \geq \sum_{r \in \Omega} R_b(r) \mathbf{P}(r | E = 1) \] \[ \text{and } \sum_{r \in \Omega} R_b(r) \mathbf{P}(r | E = 1) \Big(\frac{\mathbf{P}(r | E = 1) - \mathbf{P}(r | E = 0)}{\mathbf{P}(r | E = 1) }\Big ) = \sum_{r \in \Omega} R_b(r) \mathbf{P}(r | E = 1) L(r|E = 1) \geq B, \] where \(L(r|E=1) = \frac{\mathbf{P}(r | E = 1) - \mathbf{P}(r | E = 0)}{\mathbf{P}(r | E = 1)}\) is the likelihood ratio. And let’s first use the previous example to see whether a contract can be formed in this case.
We can see that in this case, the entrepreneur faces financial constraints as no contract can satisfy both (PC) and (IC) and the company can no longer receive external financing.
Therefore, the entrepreneur faces the choice between spin-offs and raising funds throught other channels. The second option here may be to issue more equities, which was not incorporated in our initial setting. Here, just for simplicity, we consider only the first option. Once being identified as a Bad type, the entrepreneur can either keep the combined firm and face bankruptcy because of credit rationing, or spin off the firm and operate separately. It is clear to see that the entrepreneur will always choose the plan maximising the individual utility. And in this case, the first option, which leads to bankruptcy, gives a non-positive payoff towards the risk-neutral entrepreneur. While in the second option, the entrepreneur can choose to spin off the combined project into two independent and identical projects, as we assume at the very beginning of the project. In this case, the type of the entrepreneur will no longer influence the optimal contract and the entrepreneur can now manage only one of the project. The other project will be managed by a newly hire manager, but we assume the existing manager still receive the remainder of the project cash flow after deducting the debt repayment to investors and the minimum agenecy rent to the new manager. (We also assume that the entrepreneur is protected by the limited liability in each project.) After the spin-off, we denote the project maanged by the entrepreneur as project \(a\) and the project managed by the new manager as proejct \(b\). Based on our assumption at the very beginning 2.2 Visualize Optimal Contract in Period 1, where we purposefully set the amount of asset contributed by the entrepreneur in each project \(A > \bar{A}\), both project \(a\) and \(b\) will receive external financing from the investors. And in project \(a\), the agenecy rent to the entrepreneur is equivlent to the NPV of proejct \(a\). And in project \(b\), the entrepreneur, acting as an equity holder, expects to gain the NPV of project \(b\) net the expected agency rent to the new manager. Therefore, the total amount of expected cash flow to the entrepreneur will be strictly positive in this setting, and agreeing a corproate spinoff is the optimal choice. Of course we can also consider the cost of spinoff \(C\), and evaluate the trade-off between this cost to the entrepreneur (e.g. cost of hiring and public relation) and the benefit shown above. But this won’t necessary alter the logic of spin-offs we presented above.
The analysis of the entrepreneur’s benefit in the project \(b\) is only valid if the entrepreneur still owns the company. i.e. The contracts to investors and the new manager are determined by the entrepreneur. Then from the previous document with single project, we know that the expected cash flow to the entrepreneur is \[ \max \Big \{ p_H R - I - p_H \frac{B}{\Delta p}, 0 \Big \} \] Please note that this value is sensitive to value of \(A\), as it determines whether the NPV of the project is higher than the agency rent required. For instance, if \(A = \bar{A}\), then this expected cash flow to the entrepreneur from project \(b\) will be zero as well.
Also, the new manager may not be the same as the entrepreneur in this case. If we denote the same of parameters for the new mamager as \(\theta = \{p_H', p_L', \Delta p' = p_H' - p_L'\}\), the expected cash flow to the entrepreneur will be \[ \max \Big \{ p_H' R - I - p_H' \frac{B}{\Delta p'}, 0 \Big \}. \] And the entrepreneur should always choose the manager who can generate the highest expected cash flow to the entrepreneur.
2.6 Conclusion of the General Framework
The basic story of the previous analysis goes as this. The entrepreneur has two independent and identical projects. Both projects do not suffer from financial constraints, given the assets contributed by the entrepreneur. And now the entrepreneur wants to diversify and combines two projects into a single company. The entrepreneur can either be a Good or Bad type, but this information is unknown to investors. We assume that, at the first period, investors identify the entrepreneur as a Good type and finance the corresponding company. The entrepreneur can obtain a higher utility given this diversification, which consists the agency rent and the private benefit. At the end of the first period, investors receive a signal \(X_0\) and they use it to update their beliefs about the type of entrepreneur. If they identify the entrepreneur as a Bad type, the entrepreneur may face financial constraints as they don’t have enough pledgeable income. Then at the beginning of the second period, the entrepreneur will choose to spin off the company into two, based on the projects, and continue to manage the old project \(a\) and hire a new manager for the other project \(b\). This corporate spin-off leads to a utility gain comparing to the case of bankruptcy, in the absence of the cost of spin-offs.