3
votes
Accepted
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 $u(F(x)) = \sqrt{\sqrt{x}}$, ...
- 10.3k
2
votes
Accepted
Applying dynamic programming to constrained utility
You try to solve
$$
\max_{x,y} \int_0^\infty e^{-\rho t} A U(x,y) \textrm{d} t
$$
where $U(x,y) = x^\beta y^{1-\beta}$ and $x = z(1-y)^\alpha - c$. (I think there should be a minus sign in front of $...
- 10.3k
2
votes
Accepted
How can I show that the policy function is non-decreasing?
Suppose $x'>x$. We'll show that $g(x')\geq g(x)$. Suppose that is not the case and we have $x'>x$ but $g(x')<g(x)$. Since $0\leq g(x') < g(x) \leq f(x)< f(x')$, $g(x')$ is feasible in ...
- 8,206
1
vote
Does this contraction mapping map strictly concave functions into strictly concave functions?
No. The maximum of two concave functions is usually not concave, so this is pretty hopeless.
Here is an explicit counterexample: Let $\bar{k}=1$, $W(k)=\sqrt{k}$, $f(k)=k$, $\beta=0.999$. Let $V(k)=k+...
- 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 ...
- 10.3k
1
vote
Accepted
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 ...
- 11.7k
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 ...
- 11.7k
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. ...
- 10.3k
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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