Questions tagged [dynamic-programming]
94 questions
6
votes
1 answer
Optimal stopping (reference request)
I am interested in the following optimal stopping problem:
On each day, a number $a_i$ is drawn from a (possibly fixed) distribution.
I can either stop now, getting a payoff of $a_i$, or wait for a later
draw.
In principle, this could go on…
afreelunch
- 856
- 3
- 9
3
votes
1 answer
How can I show that the optimal savings are 0 for all time periods?
Consider an infinitely-lived agent’s consumption-saving problem. The agent receives $e > 0$ units of endowment every period, can save via an asset with constant return $R$. The agent is endowed with $s_0$ units of the asset initially. At period $t$,…
Ludwig Gershwin
- 373
- 1
- 8
3
votes
2 answers
In Blackwell's condition for T to be a contraction mapping, we require that satisfies discounting. What is the intuition of discounting?
The discounting condition is as follow:
There exists some $\beta \in (0, 1)$ such that $[T(f + a)](x) ≤ (T f)(x) + βa$, for all $f ∈ B(X), a ≥ 0, x ∈ X$.
While the monotonicity condition makes sense, I can't give a nice meaning to this property.
Tecon
- 129
- 8
2
votes
0 answers
Taking limit of a sequence
I am given a following dynamic programming problem;
$$
\sup_{k_{t+1}}\sum_{t=0}^{t=\infty}\delta^t(ak_t-\frac{b}{2}k_t^2-\frac{c}{2}(k_{t+1}-k_t)^2)
$$
where $f(k_t) = ak_t-\frac{b}{2}k_t^2$ is the production function of a firm and…
Sher Afghan
- 616
- 4
- 15
1
vote
1 answer
Does this contraction mapping map strictly concave functions into strictly concave functions?
Consider the following functional equation:
$$TV(k)=\max[W(k),\beta V(f(k))]$$
where $\beta\in (0,1)$, $W(k)$ is continuous, increasing, bounded, and strictly concave function defined on $[0,\bar{k}]$, and $f(k)$ is a continuous, increasing, and…
user45416
1
vote
0 answers
Advantages of using Bellman equations
Suppose we wish to solve a simple infinite horizon cake eating problem,
such that:
$$
\max_{\left\{ c_{t}\right\} _{t=0}^{\infty}}\sum_{t}^{\infty}\beta^{t}u\left(c_{t}\right)
$$
subject to:
$$
c_{t}+a_{t+1}=a_{t}
$$
where $a_{t}$ is the total…
Kwame Brown
- 311
- 1
- 7
1
vote
1 answer
Dynamic programming with housing consumption and labor
I try to solve the following maximization problem of a representative household with dynamic programming. However, my last result is not similar to the solution. Could any one help me?
$$\max\limits_{C_{t},H_t,N_t} E_0 \sum_{t=0}^{\infty}…
DB225681
- 13
- 3
0
votes
2 answers
How can I show that the policy function is non-decreasing?
Consider the following functional equation:
$$V(x)=\max_{y\in [0,f(x)]}[u(f(x)-y)+\beta V(y)]$$
where $u$ is continuous, strictly increasing, and strictly concave; the function $f$ is continuous and strictly increasing, and $\beta\in (0,1)$. Let…
Ludwig Gershwin
- 373
- 1
- 8
0
votes
1 answer
Technical question about grid setting in dynamic programming models
I already know that expanding the grids of state variables around the steady states works, however in my toy model it's hard to get steady states analytically, so I cannot determine the boundary for variables. Is there any other method to set the…
Barry
- 23
- 3