Questions tagged [dynamic-optimization]

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Utility maximization across yield curves?

I'm attempting to solve a utility maximization problem for return-on-investment (ROI) across two different products, where each product experiences a different linear ROI curve. For product one, the ...
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3 votes
0 answers
40 views

Dynamic Information Provision model setup - It generalizes Dirk Bergemann and Stephen Morris

The following model setup is from the paper Dynamic Information Provision: Rewarding the Past and Guiding the Future by Ian Ball. It generalizes both the ideas of strategic information transmission of ...
3 votes
0 answers
56 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 ...
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0 answers
31 views

dynamic macroeconomics: Does the system "state" affect the dynamic optimization problem?

I am new to economics, but come from a statistics and math background, so the control theory and dynamic optimization ideas are familiar to me. What is a little confusing is that in most control ...
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4 votes
0 answers
40 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
210 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. ...
3 votes
2 answers
169 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 ...
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0 votes
1 answer
85 views

What is the 'fixed point problem'?

I understand what the fixed point is, but don't understand what the fixed point 'problem'. Is it resolved by 'fixed point iteration'? I am reading a paper, and the paper mentions that the default ...
2 votes
0 answers
48 views

Necessary conditions in overlapping generations model (OLG)

The consumer at each period maximizes \begin{equation} \displaystyle\sum_{i=1}^{I}\beta^{i-1}U(c^i_{t+i},l^i_{t+i}) , t=0,1,2,3,... \end{equation} subject to \begin{equation} (1+\eta_t)c^{i}_t+...
2 votes
0 answers
36 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 ...
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2 votes
0 answers
47 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 $...
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14 views

Information acquisition problem and desing of information

I am searching some details about the information acquisition problem. I want to understand what this is about and where can I find the beginning of this literature. I think this problem is some kind ...
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0 votes
0 answers
31 views

In the RBC model, why is household's problem dynamic?

Why is capital assumed to be a good accumulated in the current period t but being used in t+1? I think my interpretation is wrong here. I understand that households problem is dynamic because of this ...
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5 votes
0 answers
64 views

Optimal level of consumption in discrete time model with quadratic preferences and infinite horizon

I am trying to derive an expression for the optimal level of consumption in the basic problem: $ max \hspace{1cm} U_t = E_t \left[\sum_{s=t}^{\infty} \beta^{s-t} \left( C_s - a\frac{Cs}{2}^2 \right) ...
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4 votes
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Out-of-Equilibrium Macroeconomic Dynamics using Differential Equations

My OP below asks about the RBC model, but I am actually interested in any out-of-equilibrium macroeconomic model; CGE, DSGE or whatever the correct nomenclature is. The Real Business Cycle model is a ...
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4 votes
0 answers
83 views

Solve the Ben-Porath Model (Optimal Control Problem)

Suppose we have a Ben-Porath style human capital investment model, in which the representative agent maximize her lifetime earnings: $$V(h, a)=\max \int_{a}^{R} e^{-r(t-a)}\left[ w h(t)(1-n(t))-px(t)\...
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4 votes
0 answers
38 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 $Z_t$ ...
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4 votes
0 answers
48 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}\...
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3 votes
0 answers
30 views

Derive optimal wage in New Keynesian-Calvo wage stickiness

Following Costa, 2016 in page 96, developing the labor variety optimal wage decided by the household, the FOC is: $$0=E_t\sum_{i=0}^\infty(\beta\theta_w)^{t+i}\left\{\psi_W\left[L_{t+i}\left(\frac{W_{...
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4 votes
0 answers
65 views

Neoclassical Two-Sector Model of Endogenous Growth: Getting the consumption growth rate

I'm struggling to derive the growth of consumption from a two-sector model with the traditional Cobb-Douglas function. The model I am speaking about incorporates the fractions used by physical and ...
1 vote
1 answer
90 views

reference request - dynamic discrete time optimization methods

Next semester I am taking business cycle theory course. I emailed my teacher and he replied that I need firm understanding of dynamic discrete time optimization methods. I am gonna study mainly from ...
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2 votes
1 answer
35 views

Resolution - Ramsey growth model with per capita variables

Let's say that we have the sum of the utility of a social planner $$\int_{0}^{\infty}U\left(C\right)e^{-\rho t}dt$$ where $C$ is the total consumption. If we want to write this by a per capita ...
1 vote
1 answer
112 views

Introduction of an asset tax in the AK model

Let's start by analyzing the family's problem with the imposition of a tax. Assuming a CRRA utility function: \begin{equation} U = \int_0^\infty e^{-(\rho-n)t} \cdot \left[ \frac{c^{1-\theta}-1}{1-...
5 votes
0 answers
107 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: $x(t)$: ...
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2 votes
1 answer
58 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 ...
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1 vote
0 answers
138 views

Solving a HJB with additional constraints on control and state variables

I am trying to solve a Hamiltonian-Jacobi-Bellman equation with additional constraints on the state and control variables, but I am a bit confused on how to do that. In Intrilligator 2002, it is ...
0 votes
0 answers
60 views

How to optimize this dynamical system? Needing guidelines

I'm trying to solve a growth model, where the author indicates is a dynamical system. I want to ask if someone would help me with some guidelines of how to optimize this, I've been trying to solve it ...
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3 votes
2 answers
200 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}...
1 vote
0 answers
8 views

What would it mean for a parameter region to be E-stable but undeterminate?

What would it mean for a parameter region for a dynamic system to be E-stable, but indeterminate? E-stable: it's stable under learning... so this means, once in equilibrium, you stay there? Or does it ...
2 votes
0 answers
43 views

determinacy vs. indeterminacy of equilibria in dynamic systems

As explained in Hommes (2018), equilibria in dynamic systems, like DSGE models, can either be determinate or indeterminate. A REE (rational expectations equilibrium) is determinate when there exists a ...
2 votes
1 answer
85 views

What is meant by the abbreviation 'MSV solution', used in the context of DSGE modeling?

What is meant by the abbreviation MSV solution, used in the context of adaptive learning in DSGE modeling? E.g. see Bullard and Mitra (2002) minimum state variable (MSV) solutions it is in full, but ...
1 vote
0 answers
48 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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2 votes
1 answer
124 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 ...
1 vote
0 answers
37 views

In a rational expectation framework, do all agents know the true law of motion of the economy?

Expectations are model consistent, on average, in a RE framework, so errors are made, but on average they are zero. thus In a rational expectation framework, do all agents know the true law of ...
0 votes
1 answer
254 views

Are overlapping generation (OLG) models extensions of a DSGE model?

Are overlapping generation models (OLG) extensions of a dynamic stochastic general equilibrium (DSGE) model? Or aren't these DSGE per se?
4 votes
3 answers
2k views

What is the difference between identification, calibration and estimation?

In fitting theoretical models to data, what is the difference between identification, calibration and estimation?
6 votes
1 answer
321 views

What is the difference between a perfect foresight equilibrium and a rational expections equilibrium?

What is the difference between a perfect foresight equilibrium and a rational expections equilibrium? Why is it the same in case of a non-stochastic model? Can there be a perfect foresight ...
1 vote
1 answer
55 views

Expectational stability: adaptive learning of RE equilibria in dynamic systems

There are two steps in the explanation of the expectational stability concept by Evans and Honkapohja (2001) (see below) that I don't understand. Step 1. What does this formula below mean, ...
2 votes
0 answers
83 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 ...
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1 vote
0 answers
66 views

Writing constraint

A firm accumulates useful knowledge $k$ by investing in R&D activities. Specifically, if the firm invests $r > 0$ dollars into R&D, the stock of useful knowledge grows by about $2\sqrt{r}$ ...
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3 votes
2 answers
360 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 $w_{t+1}$ ...
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0 votes
1 answer
868 views

Maximization problem FOC and Euler equation

Can someone please help me with the Lagragian and the derivation of the following objective function ? Beneath I provide the objective function, the constraint and the Euler equation that results from ...
0 votes
1 answer
89 views

Reference Request - Dynamic Optimization with More Than One State Variable

I would like to understand how to solve dynamic optimization problems involving more than one state variable and state equation (to apply to long-term economic models with more than one capital good). ...
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1 vote
1 answer
31 views

What is the "bequest condition" in a finite-horizon discrete optimization problem?

For a finite-horizon discrete time optimization problem, my textbook provides a condition called the "bequest condition", which I'm not familiar with. Specifically, where the state at time $t$ is ...
3 votes
0 answers
97 views

Firm Dynamic Optimization Problem

A firm has received an order at time $0$ for $M$ units of product to be delivered by time $T$. It seeks a production schedule for filling this order at minimum cost. Let $x(t)$ denote inventory ...
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1 vote
0 answers
113 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 ...
2 votes
2 answers
949 views

Textbook on the mathematics of RBC/DSGE models?

I'm reading David Romer's Macroeconomics. However, what I don't like is that he doesn't go at all into detail about the mathematical underpinnings of RBC/DSGE models. When it comes to the central ...
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0 votes
0 answers
100 views

Good book/article that goes into depth about transversality conditions?

I know how to derive the transversality condition in simple models like the Ramsey model. However, I am looking to develop a deeper understanding of transversality conditions in more complex models. ...
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2 votes
0 answers
157 views

Is there an argument from first principles for the form of the no-ponzi condition?

in the ramsey model, we use the no ponzi condition $$\lim_{t\to\infty}e^{-R_t}a_t\geq 0$$ for assets $a_t$ that a household holds at time $t$. I understand intuitively what the reasoning behind ...
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7 votes
1 answer
213 views

An Optimal Control Model: A Ridiculous Result for a Steady State

I was experimenting with a seemingly simple optimal control problem that generates a system of differential equations. When I calculate the values of the steady state of the system I get some very ...