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Consider the following problem \begin{align} V(x_0) := &\max_{(u(t))_{t \geq 0}}\int^\infty_0{\left[e^{-r_1 t}F_1(x(t),u(t)) + e^{-r_2 t}F_2(x(t),u(t))\right]dt}\\ \text{s.t. } &\dot x(t) = f(x(t),u(t))\\ & x(0) = x_0 \end{align} where $x$ is the state, $u$ the control, $F_i(\cdot)$ the payoff function, $f(\cdot)$ the state equation and $x_0$ some initial condition. Suppose that the constant time preference rates differ, i.e. $r_1 \neq r_2$.

  • Is the problem solvable? Why, why not?

I'd agrue that we cannot solve the problem since the Maximum Principle as well as the Hamilton-Jacobi-Bellman approach are not applicable.

Edit: Suppose the setup was symmetric ($r_1 = r_2 = r$). Define the Hamiltonian \begin{align} H(x,u,\lambda) := F_1(x,u) + F_2(x,u) + \lambda f(x,u). \end{align}

Maximum Principle: consider the FOC for the evolution of the costate \begin{align} \dot \lambda = \lambda r - \frac{\partial H(x,u,\lambda)}{\partial x} \end{align} How would this condition be adjusted to asymmetric disount rates?

The Hamilton Jacobi-Bellman equation reads \begin{align} rV(x) = \max_u H(x,u,V'(x)) \end{align}

Again, this works because $r_1 = r_2 = r$. Otherwise I wouln't know what to do.

Edit II Since the discount rates are constant, I was hinted to define $r_1 = r_2 + \delta$. Such that \begin{align} V(t,x(t)) = &\max_{(u(s))_{s \geq t}}\int^\infty_t{e^{-r_1 s}\left[F_1(x(s),u(s)) + e^{-\delta s}F_2(x(s),u(s))\right]ds} \end{align}.

With the usual argument we can derive the following HJB \begin{align} &V(t,x(t)) = \max_{(u(s))_{s \in[t,t+\epsilon]}}\left\{\int^{t+\epsilon}_t{e^{-r_1 s}\left[F_1(x(s),u(s)) + e^{-\delta s}F_2(x(s),u(s))\right]ds} + V(t+\epsilon, x(t+\epsilon))\right\}\\ &0 =\frac{1}{\epsilon} \max_{(u(s))_{s \in[t,t+\epsilon]}}\int^{t+\epsilon}_t\left\{{e^{-r_1 s}\left[F_1(x(s),u(s)) + e^{-\delta s}F_2(x(s),u(s))\right]ds} + \frac{V(t+\epsilon, x(t+\epsilon))-V(t,x(t))}{\epsilon}\right\}\\ &0 \stackrel{\epsilon\to 0}{=}\max_{u(t)}\left\{e^{-r_1 t}\left[F_1(x(t),u(t)) + e^{-\delta t}F_2(x(t),u(t))\right] + V_t(t,x(t)) + V_x(t,x(t))\dot x\right\}\\ &-V_t = \max_{u}\left\{e^{-r_1 t}\left[F_1(x,u) + e^{-\delta t}F_2(x,u)\right] + V_xf(x,u)\right\} \end{align}

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    Why do you think they are inapplicable? – 2017-01-19
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    I edited my post. I hope the issue is somewhat clearer now. – 2017-01-19
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    You should pick up a textbook (or some lecture notes) and go through the derivation of the HJB equation. You should be able to follow it almost exactly, with minor modifications to suit your objective function. – 2017-01-20
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    Well, yes, I know how to derive the HJB equation. There is a step where I express the present value value function $V^{pv}(t,x)$ in terms of the current value value function $V^{cv}(x)$ through multiplying by the discount factor $e^{-rt}$, i.e., $V^{pv}(t,x) = e^{-rt}V^{cv}(x)$. Don't know how to proceed here with asymmetric $r$s. – 2017-01-23
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    There are alternative derivations of the HJB equation that do not make use of that step. – 2017-01-23
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    Would you be so kind to hint me to some notes? I know of a different technique where you perturb the optimal path $u^*$ around some $\epsilon$, such that $u = u^* + \epsilon$ and then derive focs and evaluate them at $\epsilon = 0$? http://economics.stackexchange.com/questions/6435/solving-the-hamilton-jacobi-bellman-equation-necessary-and-sufficient-for-optim – 2017-01-23

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