State Variable
The cumulative output of the project evolves according to:
dX_t = A_t dt + \sigma dB_t
The continuation value W_t is the total utility that the principal expects the agent to derive from the future after a given moment of time t:
W_t = \mathbb{E}^A \left[ \int_t^\infty e^{-r(s-t)} \left( u(C_s) - h(A_s) \right) ds \mid \mathcal{F}_t \right],
where: - r is the discount rate, - u(C_s) is the utility from consumption C_s, - h(A_s) is the disutility from effort A_s, - \mathbb{E}^A denotes the expectation under the probability measure induced by the agent’s effort strategy A, - \mathcal{F}_t represents the information available up to time t.
Define the agent’s total expected utility up to time t as:
V_t = r \int_0^t e^{-rs} \left( u(C_s) - h(A_s) \right) ds + e^{-rt} W_t.
This formulation accounts for both the accumulated utility up to time t and the discounted continuation value W_t.
Under the agent’s optimal effort strategy, V_t should be a martingale with respect to the filtration \mathcal{F}_t generated by the observable output process X_t. This implies:
\begin{aligned}
dV_t &= r e^{-rt} \left( u(C_t) - h(A_t) \right) dt - r e^{-rt} W_t dt + e^{-rt} dW_t \\
&= 0.
\end{aligned}
The Martingale Representation Theorem states that any martingale can be expressed as an integral with respect to a Brownian motion. Thus, there exists a progressively measurable process Y_t such that:
dV_t = r e^{-rt} Y_t \sigma dZ_t,
where: - \sigma is the volatility of the output process, - Z_t is a standard Brownian motion under the probability measure corresponding to the agent’s effort.
Derive the Dynamics of W_t:
Equating the two expressions for dV_t and solving for dW_t, we obtain:
\begin{align}
dW_t = r \left( W_t - u(C_t(W_t)) + h(A_t(W_t)) \right) dt + r Y_t(W_t) (dX_t-A(W_t)dt).
\end{align}
Here, Y_t represents the sensitivity of the agent’s continuation value to changes in the output process X_t. Notice that dX_t-A(W_t)dt=\sigma dZ_t.
If the agent takes the contracted level of effort, then \mathbb{E}[dX_t-A(W_t)]=0, so in that case, the drift of the continuation value is r \left( W_t - u(C_t(W_t)) + h(A_t(W_t)) \right).
Interpretation:
- Drift: The continuation value grows at the interest rate r, minus the agent’s utility from consumption and the disutility from effort at time t.
- Incentives: If the agent deviate from the contracted effort level, the actual effort impacts the drift of its continuation value through the second term in dW_t.
- The agent’s optimal effort is chosen to maximize the expected change in the continuation value \mathbb{E}[dW_t] minus the cost of effort h(A_t): rY(W_t)A_t-r*h(A_t). As effort is compensated, the principal will set the sensitivity Y(W_t) that the minimum level that induces the agent to choose the contracted effort level.
- when the cost of effort is differentiable, the optimal effort sensitivity in the contract is \gamma(a)=h'(a).
Principal’s Hamilton-Jacobi-Bellman Equation
The principal’s optimal control problem is:
F(W_0)=\max_{a_t>0, c_t} \mathbb{E} \left[ \int_0^\infty e^{-r(u-t)} ( r(a_u - c_u) du \right],
such that:
\begin{align}
dW_t &= r(W_t - u(c_t) + h(a_t)) dt + r \gamma(a_t)( dX_t-a_t dt) \\
C_t &\geq 0 \\
a_t &\in [0,\hat{A}] \\
\gamma(a_t) &\text{ is incentive compatible}
\end{align}
We can convert this optimal control problem into a dynamic problem by defining the principal’s profit function F(W) as:
F(W_t) = \max_{a > 0, c} \mathbb{E}_t \left[ \int_t^{t+s} e^{-r(u-t)} (a_u - c_u)du+ e^{-rs}F(W_{t+s}) \right]
By applying the Ito’s Lemma to e^{-rs}F(W_{t+s}), dividing by s and letting s goes to zero, we obtain the Hamilton-Jacobi-Bellman (HJB) equation for the principal’s profit function F(W):
\begin{align}
r F(W) = \max_{a > 0, c} \left\{ r(a - c) + F'(W) r(W - u(c) + h(a)) + \frac{1}{2} F''(W) r^2 \gamma(a)^2 \sigma^2 \right\}
\end{align}
- r(a - c) is the instantaneous profit flow of the principal,
- F'(W) r(W - u(c) + h(a)) accounts for how the agent’s continuation value changes due to effort and consumption,
- \frac{1}{2} F''(W) r^2 \gamma(a)^2 \sigma^2 captures the impact of volatility and the sensitivity of effort in the contract.
Profits from retiring the agent
To retire the agent with value u(c), the principal can offers him constant c and allows him to choose zero effort. The profits from this for retiring are:
F_0(u(c))=-c
As there is no effort, the continuation value is the same as the agent’s utility function W_t=U_t, meaning that:
F_0(W)=-W^2
Boundary Conditions & Smooth Pasting
The function F(W) must satisfy the following conditions:
Lower boundary condition (zero profit at zero continuation value):
F(0) = 0
Retirement boundary condition (profits from retiring the agent):
F(W_{gp})=-W_{gp}^2
Smooth pasting condition (ensuring differentiability at retirement):
F'(W_{gp}) = -2 W_{gp}
Solution Method
As solving method usually require transforming second-order ODEs into a first-orde ODE, we rewrite the HJB equation as:
F''(W) = \min_{a > 0, c} \frac{F(W) - a + c - F'(W)(W - u(c) + h(a))}{r \gamma (a)^2 \sigma^2 / 2}
In the code below we use shooting methods for solving the model.
