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Hot answers tagged dynamic-programming

3 votes
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Bellman Equation & Envelope Theorem

The key thing to note here is that in the optimum, $c_t$ will depend on $k_t$. Thus, the value function is \begin{align} V(k_t, t) &= \max\{u(c_t) + \beta V(f(k_t) - c_t, t + 1)\}\\ &= u(c_t(...
2 votes

Showing that reward function is bounded (dynamic programing)

The function $u(F(x)-y)$ is not necessarily bounded on $A$. For example, if $u(x) = F(x) = \sqrt{x}$ then: $$u(F(x)-y) = \sqrt{\sqrt{x}-y},$$ Taking $y = 0$, this gives $u(F(x)) = \sqrt{\sqrt{x}}$, ...
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2 votes
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• 11.7k
1 vote

How can I show that the policy function is non-decreasing?

I'm going to assume that everything is smooth and that the optimal solution $g(x)$ is interior (in $]0, f(x)[$) First you can show that the function $V$ is concave as the Bellman operator maps concave ...
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1 vote
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Understanding Duality between Individual and Collective Maximization in Macroeconomic Models

The proof of the first welfare theorem is almost the same as the one you are familiar with from MWG. The main difference is that if you have recursive budget constraints, you have to show that you can ...
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1 vote

How can I show that the optimal savings are 0 for all time periods?

As you noticed, positive savings are only beneficial if the next period's consumption is lower. But this can only happen with positive savings if there are positive savings next period. By the same ...
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1 vote

Regarding the arbitrariness of states and controls

The change from $c$ as a decision variable towards $k'$ as a decision variable is by a simple change of variables. In the original setup, $k$ is the state and $c$ is the control (decision) variable. ...
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1 vote

Technical question about grid setting in dynamic programming models

In that case, I think you can try to approximate the value of the steady-state $k^*$. If your capital accumulation function is implicitly defined, say $k_{t+1}=\phi(k_{t+1},k_t)$: You can still prove ...
• 21

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