Questions tagged [dynamic-programming]

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Optimization in discrete time

I have made optimizations in continuous time that belong to the control theory, for example one case: $\max(\min)V[u(t)]=\int_0^Tf(t,x(t),u(t))dt$ constraint to: $\dot x=g(t,x(t),u(t))$ Where: $x(t)$: ...
4
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0answers
37 views

Value function iteration with habit

I would like to know how I could write a value function when there are habits in preferences. I have the following equations: $$ u\left(C, t, H_{t}, L_{t}\right)=\frac{\left(C_{t} / H_{t}^{\kappa}\...
4
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194 views

Solution to Dynamic Programming (Bellman Equation) Problem

Could someone please provide pointers on how to solve the below? If any theoretical approximations are possible, that would be very helpful. If numerical solutions are the right approach, could you ...
3
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0answers
24 views

Value function iteration with stochastic productivity's standard deviation

Hello I would like to know how would you discretize the AR(1) process of technology in a standart RBC model when there is stochastic productivity's standard deviation. Namely I have: Technology $Z_t$ ...
3
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0answers
124 views

What are the boundary value conditions for generic HJBs in economics?

Consider a routine continuous time optimization problem: $ V(t,a_{t}) := \max \int_{\tau=t}^{\tau = T} e^{-\rho (\tau -t)} u(c_{\tau})d\tau $ $\text{ s.t. }$ $\dot{a}_{t} = y + ra_{t} - c_{t}$, $a_{...
3
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0answers
670 views

Market clearing condition with Walras law

I have a diamond overlapping model The question is as follows Let us consider an infinitely lived production economy populated at time t by $N_t$ identical and perfectly competitive adult ...
2
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0answers
54 views

Cake eating problem

Consider an infinitely lived agent born in time zero, endowed with a cake of size $x_0$. The cake is storable (without depreciation) and infinitely divisible. The agent derives contemporary utility ...
2
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0answers
79 views

Linearization of the dynamic system (I did it, but I have a mistake that I cannot catch. Help me please)

I have the following dynamic system in discrete time For p is price, d is demand and s is supply. $$p_{t+1}-p_t= a(d_t-s_t)$$ $$s_{t+1}-s_t=bp_ts_t-ws_t$$ $$d_t= k-gp_t$$ I have to linearize this ...
2
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0answers
47 views

Why is there a Lagrangian Multiplier in the Dynamic Programming Problem of the RBC model?

Suppose that the household faces the following problem: $\underset{ c_t , k_{t+1}, n_t } \max \mathbb{E}_0 \sum_{t=0}^{\infty} \beta^t \ln c_t + \ln (1 - n_t)$ subjected to $ k_{t+1} = A_t k_t ^{\...
2
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0answers
98 views

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 ...
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38 views

Coase conjecture and Stokey model

The durable goods monopolist can charge different prices by every periods. Critical type consumers are indifferent between buying today and buying tomorrow. So I construct a Bellman equation like ...
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218 views

Natural borrowing/debt limit and other borrowing constraints

When confronted with the simple household consumption maximization problem under uncertainty (and with Arrow security sequential trading) $$\max_{\{c_t(s^t),a_{t+1}(s^t,s_{t+1})\}_{t=0}^{\infty}}\...
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282 views

Explanation of Dynamic Programming “Guess and Verify” Technique

According to my textbook, the analytical technique for solving a Bellman's Equation is as follows: Guess a form for $V_0(x)$ Solve the maximization problem with respect to the control and obtain a ...
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0answers
99 views

Stochastic Dynamic Programming: Deriving the Steady-State for a Lottery

I am working through the basic examples of the stochastic RBC models in the book by McCandless (2008): The ABCs of RBCs, pp. 71 - 75 A Standard Stochastic Dynamic Programming Problem Here is a ...
1
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0answers
146 views

How to solve a variation of Merton's optimal portfolio problem?

Does anyone know how to solve the following problem? I have tried to solve this but I'm lost since I have never dealt with a Stochastic Dynamic Programming problem with many variables. $max_{c_{t},\...
1
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0answers
74 views

Hamilton-Jacobi-Bellman with heterogeneous discount rates

Let $i=1,2$ denote the players, $x$ the state and $u_i$ the control of player $i$. The state equation reads $\dot x = f(x,u_1,u_2)$ and the objective function is given by $F_i(x,u_1,u_2)$. Now I'd ...
1
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103 views

Dynamic programming: verification principle

Consider the Gale cake problem with $u(c) = log(c)$, so that the problem becomes:$$\max_{x}\sum_{t=0}^{\infty} \beta^{t}log(x_t-x_{t+1})\: sub\: 0<x_{t+1}\leq x_t,\: x_0>0\ given$$ It is ...
0
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53 views

Cost-optimal p2p-trade in a community of households

I’m trying to solve the following problem and I’ve been working on it for a long time already: I want to optimize electricity-costs in a smart grid. There’s producer and consumer households in the ...
0
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34 views

Dynamic programming and Difference equations applications

I'm asked by my teacher to prepare a presentation with economic applications of Dynamic Programing (Bellman Equation) and Difference equations. I'm not sure what this things are used for in economics ...