Questions tagged [dynamic-programming]
The dynamic-programming tag has no usage guidance.
91
questions
2
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1
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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 $[0,\bar{k}]$,...
1
vote
2
answers
130
views
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 ...
- 365
2
votes
1
answer
67
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Understanding Duality between Individual and Collective Maximization in Macroeconomic Models
I'm currently studying macroeconomic models, specifically from the book "Recursive Macroeconomic Theory." In Chapter Seven, it is mentioned that some economic models involving firms and ...
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3
votes
1
answer
225
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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 ...
- 365
1
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1
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32
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Showing that reward function is bounded (dynamic programing)
I have the following dynamic programming problem:
$$\max_{\{x_t,y_{t+1}\}_{t=0}^\infty}\sum_{t=0}^\infty \beta^tu(F(x_t)-y_{t+1})\;\;\;\;\;\text{s.t}\;\;\;\;y_{t+1}\in\Gamma(x_t)$$
where $\Gamma(x)=\{...
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3
votes
1
answer
90
views
Applying dynamic programming to constrained utility
I am trying to solve problem that looks like this; there is utility function that takes $x$ and $y$ as inputs, $x$ is produced by production function that depends on labor $l+y=1$. $x, y$ depend on $t$...
- 1,751
0
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0
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23
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Estimation of learning model with finite horizon forward-looking individuals
I am stuck with the estimation of learning model where individuals are forward-looking in a finite horizon.
Specially, a user is watching a TV program containing many episodes, and he doesn't know the ...
- 11
1
vote
1
answer
65
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Regarding the arbitrariness of states and controls
I am trying to better understand the process of deriving Euler Equations using
the first order condition of the problem on the right hand side of a Bellman equation
and the Benveniste-Scheinkman ...
2
votes
0
answers
120
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Stochastic control of jumps of random size
Consider the problem of maximizing expected lifetime utility
$$
V(a_t) \equiv \max_c\mathrm{E}_t \int_t^\infty e^{\rho (s - t)}u(c_t)\mathrm{d}t
$$
subject to a state process $\mathrm{d}a_t$ which is ...
0
votes
1
answer
36
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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 ...
- 23
1
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1
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136
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Habit forming Model & State Variables
Taking the habit formation model of consumption, as a standard dynamic programming problem.
Bellman Value Function for Habit Model
Max$\sum_{t=1}^Tβ^tu(c_t - γc_{t-1})$ $\qquad \qquad (1)$
s.t.
$w_{t+...
- 897
4
votes
2
answers
704
views
Bellman Equation & Envelope Theorem
I'm unsure where the envelope theorem comes into play when i differentiate the Bellman Equation with respect to $k_t$.
To me it looks like the regular chain rule and in fact the exact opposite of the ...
- 897
3
votes
1
answer
66
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Simulation of a dynamic search and matching model
I want to simulate a search and matching model very close to the canonical model developed by Pissarides. I am interested in representing the trajectory of the unemployment rate when the unemployment ...
- 31
0
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0
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39
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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 ...
3
votes
0
answers
104
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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 ...
- 155
1
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0
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42
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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 ...
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0
answers
67
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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
66
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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:
$$
c_{t}+...
- 311
3
votes
0
answers
84
views
Current best methods for solving dynamic optimization problems in high dimensional state spaces
I was wondering what the current best methods are for solving dynamic optimization problems in high dimensional state spaces. Let me lay out the common cases where I would do something like this in ...
- 385
5
votes
1
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157
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When does it make sense to use variational methods, versus dynamic programming, versus nonlinear control methods so solve DSGE models
I come from a statistics and applied math background, but have been looking at some problems related to macroeconomic DSGE models lately. So I am still trying to understand the ideas and economics ...
- 385
4
votes
0
answers
43
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What assumptions can be made to ensure convexity in this optimization problem?
This question is a continuation of the question I asked at:
How can I show convexity of this value function?
Where I came to the conclusion that more assumptions are required to show that the ...
- 183
4
votes
1
answer
292
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How can I show convexity of this value function?
I have set up an optmization problem as follows:
$$V(A)=\max_{l, C} \quad u(C,l)$$
Where the only constraint is as follows:
$$C=f(l,A)$$
Here $u$ is the utility function which captures social welfare. ...
- 183
6
votes
1
answer
217
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Optimal stopping (reference request)
I am interested in the following optimal stopping problem:
On each day, a number $a_i$ is drawn from a (possibly fixed) distribution.
I can either stop now, getting a payoff of $a_i$, or wait for a ...
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3
votes
2
answers
194
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How can I formulate the following optimization problem?
I want to set up an optmiization problem for global warming in which a planner determines how much carbon dioxide gas is emitted. Let's say we reduce this problem down to two periods, then I ...
user40867
1
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1
answer
203
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simplification of FOC
This is the first-order condition of a dynamic programming problem where I am trying to get the Euler equation from a sequential problem.
(1) $$\frac{\partial V(d_2)}{\partial d_3} = \frac{-1}{d_2-d_3}...
2
votes
0
answers
52
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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
39
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 ...
- 468
2
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0
answers
81
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Closed form solution to consumption-saving model in discrete time
Consider the simplest consumption-savings model of the following form:
$$
\max_{\{c_t,a_{t+1}\}_t}\mathbb{E_0}\sum_{t\geq 0} \beta^tu(c_t) \\
a_{t+1} + c_t = (1+r)a_t + y_t \\
y_t \mid y_{t-1} \sim F
$...
- 59
4
votes
1
answer
301
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Dynamic programming in infinite horizon model
Using an infinite horizon model, a dynamic programming approach uses a fixed point to solve the model: $V = \Gamma(V)$.
How do I interpret the meaning of $V$? For example, when we decide a investment ...
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2
votes
0
answers
67
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Easy introduction to Markov processes and dynamic programming [closed]
I am taking advanced macro course in Fall. Could you please advise me a simple introduction to Markov processes and dynamic programming? I mean easy. Thanks!
- 113
3
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2
answers
191
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Resources to derive economic forecasts
What publicly available resources are there, with sample code, that can be used to build my own macroeconomic model? A search on Github shows that some people have posted code, but I think we can do ...
- 3,761
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0
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55
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Value function iteration with stochastic productivity's standard deviation
Hello I would like to know how would you discretize the AR(1) process of technology in a standart RBC model when there is stochastic productivity's standard deviation. Namely I have:
Technology $Z_t$ ...
- 457
4
votes
0
answers
73
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Value function iteration with habit
I would like to know how I could write a value function when there are habits in preferences. I have the following equations:
$$
u\left(C, t, H_{t}, L_{t}\right)=\frac{\left(C_{t} / H_{t}^{\kappa}\...
- 457
2
votes
1
answer
112
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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) + ...
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0
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54
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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
232
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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 ...
- 1,048
2
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0
answers
205
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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 ...
- 1,048
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0
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72
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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
625
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 ...
- 8,295
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1
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389
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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 ...
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0
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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_{...
- 340
3
votes
2
answers
735
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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, ...
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0
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146
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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:
$x(t)$: ...
- 821
2
votes
1
answer
81
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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 ...
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2
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219
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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}...
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3
votes
1
answer
480
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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 ...
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1
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0
answers
59
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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
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1
answer
146
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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 ...
- 93
2
votes
2
answers
106
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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
85
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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 ...
- 120