I am trying to build a model and am having trouble getting a closed form solution for this value function: $$V(h_{t})=\max\limits_{{k_{t+1},\tau_{t},l_{2},I}} ln[w(h_{t})[1-l_{2}-\alpha I]-k_{t+1}-\tau_{t}]+\sigma ln(l_{2})+\beta ln(Rk_{t+1}+\tau_{t+1})+\lambda ln(Rk_{t}+\tau_{t})$$

  • $\lambda,\beta,R,\sigma,\alpha$ are constants
  • $k_{t+1},\tau_{t},l_{2}$ are my controls
  • $h_{t}$ is my state.
  • I is an indicator variable
  • $\tau_{t+1}$ is a function of I

What might be some approaches I could take to get a closed form solution?

I know about the Bellman Equation and the technique to solve one but here my right hand side has no value function so I am at a loss. Any advice would be much appreciated.

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    $\begingroup$ Why not take first order conditions and plug in the optimal solution? $\endgroup$ – Walrasian Auctioneer Mar 11 at 21:52
  • $\begingroup$ I is an indicator variable that takes on a value of 1 if which condition is met? That would make this a possibly stochastic problem. $\endgroup$ – ChinG Mar 12 at 23:12
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    $\begingroup$ @WalrasianAuctioneer Do you mean taking FOC with respect to my controls and then solving for $k^*_{t+1}$,$τ_t^*$ and $l_2^*$ to plug into the right hand side of my value function? $\endgroup$ – Bobby Mar 13 at 2:45
  • $\begingroup$ @ChinG Allow me to give more context. I am trying to model a binary choice made by the agent: I=1 if he makes Choice A and I=0 if makes Choice B. Choosing I=1 will lower the agent's supply of labor as seen from $[1−l_2−αI]$ where $l_2$ represents leisure and time endowment is normalized to 1. At the same time, I=1 brings about higher $\tau_{t+1}$ in the future. At the end of the day I aim to characterize the conditions (i.e. the range of values the model parameters take) in equilibrium under which the agent will prefer one choice over the other. $\endgroup$ – Bobby Mar 13 at 2:57
  • $\begingroup$ Just saw @WalrasianAuctioneer 's comment, and yours. This is a simple optimization problem. You are solving for the indirect utility function- indeed just take the FOC with respect to controls, or set up a Lagrangean. Either works. $\endgroup$ – ChinG Mar 13 at 13:58

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