Questions tagged [dynamic-programming]

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3
votes
2answers
140 views

Resources to derive economic forecasts

What publicly available resources are there, with sample code, that can be used to build my own macroeconomic model? A search on Github shows that some people have posted code, but I think we can do ...
3
votes
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 $Z_t$ ...
4
votes
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}\...
2
votes
1answer
251 views

The Principle of Optimality and the Bellman Equation

Given (1) and (2), is it possible to show the existence of a Bellman equation (3), using Bellman's Principle of Optimality? $$\ max \Sigma\beta^s U(C_t)$$ Subject to the following resource ...
2
votes
1answer
71 views

Exercise 4.7 in SLP (dynamic programming)

Exercise 4.7 (b) : Show that under Assumptions 4.10 and 4.11, $T:H(X) \to H(X)$. $H(X)$ is the set of continuous and homogeneous of degree one functions and $Tf(x) = \sup_{y \in \Gamma(x)} \{F(x,y) + ...
2
votes
1answer
79 views

Proving that the utility is concave

Consider a household which solves the following problem: $$v(k,r,w)=\underset{c,l\in B{(k,r,ω)}}{\ {max}} \{u(c,l)\}$$ where $u : R_+^2 \rightarrow R$ is a strictly concave, twice continuously ...
0
votes
0answers
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 ...
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 ...
21
votes
3answers
692 views

When Optimal Control fails (?)

In order to "ask my question", I have to solve a model first. I will omit some steps but still, this will unavoidably make this post very long -so this is also a test to see whether this community ...
0
votes
0answers
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 ...
3
votes
1answer
152 views

A Cake Eating Problem in Continuous Time: Hamiltonian or HJB?

Your standard continuous time cake eating problem is defined as follows: $$\max_{c(t)}\int_0^\infty e^{-rt} \ln (c(t)) dt$$ subject to $$f(k(t))=k(t)$$ $$\dot{k}(t)=-c(t)$$ Approaching this problem by ...
3
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0answers
123 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_{...
1
vote
2answers
156 views

What is state space representation for DSGE modeling

I'm beginning with DSGE modeling, and a mathematical representation (perhaps trivial for most of the people that are more with this topic) is the space-state state representation of a dynamical model, ...
5
votes
0answers
63 views

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: $x(t)$: ...
2
votes
1answer
52 views

Can anyone help me understand the Motrtensen-Pissarides model?

I am seeking help with the Mortensen-Pissarides model in discrete time? Basically, I was given the following in class example (which we didn’t complete do to time constraints associated with the end ...
4
votes
2answers
183 views

Solving a HJB with a probability to transit to a new state

I am trying to solve the problem of a firm facing the possibility of a future tax, in continuous time. The firm maximizes $V(k)=\int_{t=0}^{\infty}e^{-rt} \pi_t dt$ with $\pi_t=f(k_t)-i_t$ and $\dot{k}...
3
votes
1answer
145 views

More than one Bellman Equation

I'm attending to my first dynamic optimization course, and what I don't fully graps yet is that sometimes we have to use more than one bellman equation. How do you realize that? I mean how do you know ...
2
votes
2answers
253 views

Dynamic programming, optimal consumption-savings (finite horizon) problem

Let $w_t$ denote a consumer's wealth at time $t$ and $c_t$, the amount she chooses to consume, so her savings exiting this time period are $w_t-c_t$. Given this savings decision, her savings $w_{t+1}$ ...
1
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0answers
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 ...
1
vote
1answer
80 views

What is the result of the Bellman Equation

I'm just starting with dynamic optimization and although I understant the proof's of the theorem I'm not able to fully understand whether the bellman equation is a function , a function valuated at ...
2
votes
2answers
70 views

Why do game theorists use a discounted payoff of this form?

Excuse the click-baity title. I notice the discounted payoff in the game theory literature usually takes the form $$\sum_{t=1}^\infty\lambda(1-\lambda)^{t-1}R_t$$ This differs from the discounted ...
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
votes
1answer
344 views

The Cake Eating Problem with Depreciation (Modelling difficulties)

How does one go about modelling the cake eating problem with depreciation? (i.e The cake goes bad over time) The problem I have is the following. Lets define a cake eating problem sequentially as: ...
1
vote
2answers
299 views

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,...
1
vote
1answer
157 views

A profit maximization problem (whole problem has been solved, I just have question about interpretation)

I would like to discuss with you about the following production function. $$y=f(t_m, t_l)=\rho t_m^m(n+t_l)$$ where $0<m<1 $ and $n>0$ are fixed parameters. $t_m$ is manager time. $t_l$ ...
0
votes
1answer
91 views

Sequential Price Competition for Perfect Complements

There are two goods, $1$ and $2$ produced by two firms at zero marginal costs. The goods are perfect complements. The demand for each goods is: $Q_1=Q_2=a-(p_1+p_2)$. The prices are set sequentially, ...
0
votes
3answers
159 views

What is unknown in Bellman Equation?

\begin{align} V(W)=\max\limits_{W'\in[0,W]}\qquad& u(W-W')+\beta V(W')\qquad\forall W \end{align} $\textbf{My Question}$: Why is the unknown in the Bellman equation $V(W)$ itself? Isn't the ...
2
votes
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 ^{\...
6
votes
1answer
195 views

How do I begin to approach this dynamic discrete choice model?

I'm working through an old problem set (that sadly I don't have solutions for) and I got stuck. It is a dynamic model of entrepreneurship and invention. I'm looking for guidance on this model as well ...
1
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0answers
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}}\...
1
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0answers
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 ...
0
votes
1answer
80 views

Capital accumulation

I have following dynamic optimization problem $$V(k_0)=\max \sum_{t=0}^{\infty} b^t((1-a)ln c_t+a ln(l_t))$$ Subject to $l_t+e_t=h$ and $y=Ak_t^pe_t^{1-p}$ and $k_{t+1}=i_t+(1-x)k_t$ Where c is ...
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 ...
4
votes
0answers
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 ...
1
vote
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
vote
1answer
103 views

A reference for most used utility functions in macroeconomic problems of intertemporal optimization

I'm looking for a reference with the most used utility functions in macroeconomic problems of intertemporal optimization. The reference should preferably include a list of properties of those ...
14
votes
1answer
2k views

Solving the Hamilton-Jacobi-Bellman equation; necessary and sufficient for optimality?

Consider the following differential equation \begin{align} \dot x(t)=f(x(t),u(t)) \end{align} where $x$ is the state and $u$ the control variable. The solution is given by \begin{align} x(t)=x_0 + \...
4
votes
1answer
34 views

Question regarding Carlstrom and Feurst (1997)

I am reading through Carlstrom and Feurst's 1997 paper Agency Costs, Net Worth, and Business Fluctuations: A Computable General Equilibrium Analysis and had a question, although it could apply to ...
4
votes
2answers
164 views

Optimisation using value function

I have the following optimisation problem: max $E_{0}\sum_{t=0}^{\infty}[log(c_{t}) + log(m_{t})]$ subject to $y + \frac{M_{t-1}}{p_{t}} + R_{t-1}\frac{B_{t-1}}{p_{t}} = c_{t} + m_{t}+b_{t}+\tau_{t}$ ...
1
vote
1answer
160 views

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\...
3
votes
1answer
718 views

Bellman equation for this dynamic programming problem

For the following problem \begin{equation}\max_{(\tilde{c}_t,\tilde{a}_{t+1+s})}\sum_{s=0}^{\infty}\beta ^su(\tilde{c}_{t+s})\end{equation} s.t. the following restrictions $\begin{equation}...
4
votes
1answer
2k views

Solution to the Bellman equation is a fixed point

I have recently started studying dynamic optimization. I cannot quite wrap my head around the fact that the value function of the Bellman equation is a fixed point of a contraction mapping. As far my ...
2
votes
1answer
133 views

Dynamic programming problem with dimension over 1000

I am working on a project which need to solve a dynamic programming problem with dimension over 1000. In past literature, there exist several methods like Smolyak algorithm and Sparse grid method that ...
1
vote
1answer
141 views

How to deal with Prescott's formulation of time to build in his original RBC model?

So I was replicating the results obtained in section 4 of Prescott's original paper, which derives optimality conditions in steady state without shock. I hope to solve the social planner's ...
1
vote
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
vote
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 ...
7
votes
3answers
752 views

Solution Method for Infinite-Horizon Maximization Problem

Full disclosure: this problem was part of a final exam that none of our class could really solve definitively. Below the general form is a specific utility function we worked with that I'll try to ...
2
votes
1answer
544 views

Update of value function in continuous time - HJB

When solving (numerically, by value function iteration) a dynamic programming problem in discrete time, such as $$V_1(a) = \max_{c} \ u(c) + \dfrac{1}{1+\rho}V_0(a')$$ we maximize with respect to ...
3
votes
1answer
68 views

Joint Dynamic Programming: Group Activity

Here we have two agents who can spend their time doing some group activity ($h$) or staying at home ($l$). Each agent $i$ is trying to maximize their respective dynamic programming problem: \begin{...
3
votes
1answer
290 views

Prove the uniqueness of steady state

I have a difference equation $$ p_t^{1-\alpha}=\alpha\sigma \left(y-p_t-\frac{(\sigma p_{t-1}^\alpha+b)p_t^{1-\alpha}}{\alpha\sigma} \right) $$ where $\alpha \in [0,1]$ and everything else is $&...