Questions tagged [dynamic-programming]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
0 votes
0 answers
13 views

One shot-deviation property for games of imperfect information

The equivalence between subgame perfection and one-deviation property is typically stated for games of perfect information (where information sets are singletons). Does the Blackwell-style argument ...
2 votes
0 answers
43 views

Mathematical Prerequisites for Recursive Macroeconomic Theory (Thomas J Sargent, Lars Ljungqvist)

I'm a math grad who is interested in learning more about economics for fun. Reading through RMT, I saw some interesting math (in chapter 2) around using "invariant functions" to determine ...
  • 23
1 vote
0 answers
34 views

Are there any ``sophisticated'' mathematical modelling where they solve for the utility function?

Are there any references in literature of any ``sophisticated'' mathematical modelling where they solve for the utility function under specific conditions using differential equations theory? In such ...
0 votes
0 answers
19 views

Difference between complete market and incomplete market in terms of welfare loss

What do you think about the relative welfare of an incomplete market and complete market in the recursive competitive economy with both physical and human capital accumulation — I think that in the ...
2 votes
0 answers
44 views

When do Economists use the Linear Quadratic Regulator in simulating DSGE models

I was watching some excellent videos on DSGE models by Klaus Pretner. The author was able to solve some simple model such as the Ramsey-Cass-Koopman's model, and a New Keynesian model with frictions, ...
  • 385
1 vote
0 answers
37 views

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}+...
0 votes
0 answers
40 views

catching up with Johnes utility in the recursive equilibrium model

Consider an infinite horizon representative agent economy in which the agents enjoy consumption as usual but also cares about its relative level of consumption compared to last period’s average level ...
3 votes
0 answers
71 views

Current best methods for solving dynamic optimization problems in high dimensional state spaces

I was wondering what the current best methods are for solving dynamic optimization problems in high dimensional state spaces. Let me lay out the common cases where I would do something like this in ...
  • 385
5 votes
1 answer
107 views

When does it make sense to use variational methods, versus dynamic programming, versus nonlinear control methods so solve DSGE models

I come from a statistics and applied math background, but have been looking at some problems related to macroeconomic DSGE models lately. So I am still trying to understand the ideas and economics ...
  • 385
0 votes
0 answers
22 views

Policy function iteration method in continuous time (with shocks)

Is there any reference available on the algorithm of policy function iteration method in continuous time, when we have uncertainty in the model? Currently, my conclusion is that the combination of ...
  • 1
4 votes
0 answers
41 views

What assumptions can be made to ensure convexity in this optimization problem?

This question is a continuation of the question I asked at: How can I show convexity of this value function? Where I came to the conclusion that more assumptions are required to show that the ...
4 votes
1 answer
238 views

How can I show convexity of this value function?

I have set up an optmization problem as follows: $$V(A)=\max_{l, C} \quad u(C,l)$$ Where the only constraint is as follows: $$C=f(l,A)$$ Here $u$ is the utility function which captures social welfare. ...
6 votes
1 answer
200 views

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 ...
3 votes
2 answers
183 views

How can I formulate the following optimization problem?

I want to set up an optmiization problem for global warming in which a planner determines how much carbon dioxide gas is emitted. Let's say we reduce this problem down to two periods, then I ...
user avatar
1 vote
1 answer
177 views

simplification of FOC

This is the first-order condition of a dynamic programming problem where I am trying to get the Euler equation from a sequential problem. (1) $$\frac{\partial V(d_2)}{\partial d_3} = \frac{-1}{d_2-d_3}...
2 votes
0 answers
40 views

More equations than endogenous variables in RBC model

I recently came across my late father's bachelor's thesis, in which a RBC model is described. It appears to be a variant of Hansen's 1985 model with indivisible labor. I wrote the model into Dynare to ...
  • 21
1 vote
0 answers
34 views

How to define a dynamic programming problem in an incomplete information game?

How can I define a problem of dynamic programming, to use the Hamilton-Jacobi-Bellman equation in order to solve the utility maximization problem of the generic agent of a dynamic game with incomplete ...
  • 448
2 votes
0 answers
59 views

Closed form solution to consumption-saving model in discrete time

Consider the simplest consumption-savings model of the following form: $$ \max_{\{c_t,a_{t+1}\}_t}\mathbb{E_0}\sum_{t\geq 0} \beta^tu(c_t) \\ a_{t+1} + c_t = (1+r)a_t + y_t \\ y_t \mid y_{t-1} \sim F $...
  • 59
4 votes
1 answer
183 views

Dynamic programming in infinite horizon model

Using an infinite horizon model, a dynamic programming approach uses a fixed point to solve the model: $V = \Gamma(V)$. How do I interpret the meaning of $V$? For example, when we decide a investment ...
  • 195
2 votes
0 answers
65 views

Easy introduction to Markov processes and dynamic programming [closed]

I am taking advanced macro course in Fall. Could you please advise me a simple introduction to Markov processes and dynamic programming? I mean easy. Thanks!
  • 113
3 votes
2 answers
172 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 ...
  • 3,761
4 votes
0 answers
41 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 $Z_t$ ...
  • 437
4 votes
0 answers
53 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}\...
  • 437
2 votes
1 answer
100 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) + ...
  • 393
0 votes
0 answers
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 ...
  • 1
2 votes
1 answer
150 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 ...
2 votes
0 answers
135 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 ...
0 votes
0 answers
58 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
1 answer
461 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 ...
  • 7,910
2 votes
1 answer
352 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 ...
  • 564
4 votes
0 answers
159 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 votes
2 answers
535 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, ...
  • 727
5 votes
0 answers
128 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: $x(t)$: ...
  • 727
2 votes
1 answer
65 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 ...
  • 87
3 votes
2 answers
202 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
1 answer
294 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 ...
1 vote
0 answers
52 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 ...
  • 23
2 votes
1 answer
134 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
2 answers
93 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 ...
  • 265
2 votes
0 answers
84 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
1 answer
530 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: ...
  • 7,910
3 votes
2 answers
433 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 $w_{t+1}$ ...
  • 448
1 vote
1 answer
165 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
1 answer
169 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, ...
  • 401
0 votes
3 answers
230 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
0 answers
130 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 ^{\...
3 votes
2 answers
581 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,...
  • 119
1 vote
0 answers
296 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}}\...
  • 111
1 vote
0 answers
417 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
2 answers
121 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 ...
  • 63