Questions tagged [dynamic-optimization]

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619 views

Price optimization with demand forecast

I have one year sales data of a retail company and lets say I am forecasting the next month sales for the product. I have got the sales using time series in R. Now I want to forecast the price as well....
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1answer
59 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-...
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2answers
206 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?
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45 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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1answer
45 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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53 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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2answers
166 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}...
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23 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 ...
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20 views

Numerical Backward Induction Optimal portfolio choice

I am currently considering a simple life-cycle problem. We consider a market with equity risk only, which follows a geometric Brownian motion. We seek to maximize the terminal wealth of a CRRA utility ...
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2answers
199 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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0answers
6 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 ...
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1answer
21 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 ...
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0answers
21 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 ...
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33 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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1answer
59 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 ...
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0answers
34 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 ...
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1answer
49 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?
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1answer
131 views

Analytically tractable Ramsey model: how to solve ODE for optimal trajectories

In Brunner and Strulik (2002) the authors claim, that the solution of \begin{align} \dot c &= \frac{c}{\sigma}(\alpha k^{\alpha-1} - \delta - \rho)\\ \dot k &= k^\alpha - \delta k - c \end{...
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1answer
64 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 ...
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11 views

What is the name for the techniques that are used to determine the determinacy of DSGE models with rational expectations?

What is the name for the techniques that are used to determine the determinacy of DSGE models with rational expectations? For some context, see e.g. the work of THE SOLUTION OF LINEAR DIFFERENCE ...
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1answer
38 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, ...
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76 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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0answers
62 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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1answer
478 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 ...
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1answer
77 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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1answer
29 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 ...
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0answers
73 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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0answers
97 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 ...
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1answer
609 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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76 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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0answers
142 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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1answer
158 views

An Optimal Control Model: A Rediculous 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 ...
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0answers
197 views

Optimal Fight Purse and Boxing Strategies

The following is all public information available to all the players in this scenario. The General Setup In the aftermath of the infamous race between the tortoise and the hare, the salty hare went ...
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2answers
10k views

Transversality Condition in neoclassical growth model

In the neo-classical growth model there is the following transversality condition: $$\lim_{t\rightarrow\infty}\beta^{t}u'(c_{t})k_{t+1}= 0,$$ where $k_{t+1}$ is the capital at period $t$. My ...
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1answer
85 views

Money in utility function - Value function

I am reading Walsh's (2003) book on monetary economics. Specifically the chapter on money in utility function. I understand the basics of a value functions but I can't seem to get the same results as ...
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2answers
263 views

Phase Diagram - growth model

The dynamics of Ramsey-Cass-Koopman growth model is usually summarized in phase diagrams with the 2 equations (conventional symbols apply): \begin{align*} \frac{\dot c}{c}=&\frac{r-\rho}{\theta}\\ ...
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0answers
192 views

Dynamic demand model in many good competitive markets and price optimization

This is a question about demand models, price optimization, dynamic pricing, big data, online learning, so I will cross-post in other communities. $\mathbf{Background}$ I am interested in dynamic ...
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2answers
611 views

Preference for consumption smoothing and actual smoothing

The typical dynamic consumption-saving under certainty model can be written as: $$ \max V(c)=\sum_{t=1}^{T} \beta^{t-1}\; u(c_t) $$ Subject to the intertemporal budget constraint $$ \sum_{t=1}^{T}\...
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1answer
316 views

Are there stable improvements of the Ramsey-Cass-Koopmans model?

The Ramsey-Cass-Koopmans neoclassical model of growth is saddle path stable. In other words, it is stable upon perturbation from the steady state on a one-dimensional line, but is unstable towards a ...
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1answer
2k views

Question about Euler condition [closed]

In dynamic macroeconomic model(without production sector), Euler equation is $$U'(C_t)=b(1+r)U'(C_{t+1})$$ I found another equation related with Euler which is called Euler condition. In New ...
3
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1answer
153 views

Dynamic Optimization : Resource Stock as a Sink

I think about a dynamic problem where social planner maximizes the following utility ; $$\underset{c\left(t\right)}{max}\int_{0}^{\infty}u\left(c\left(t\right)\right)e^{-\rho t}$$ subject to two ...
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2answers
771 views

Saddle path equilibrium on financial market with rational expectations

In his 1978 paper introducing the Tobin tax Tobin states that : As a technical matter, we know that a rational expectations equilibrium on markets of this kind is a saddle point. That is, there ...
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1answer
118 views

Monetary policy optimization

I was wondering if anyone could give me some advice / lectures / introduction to stochastic optimization that could be applied to monetary policy. I have heard of the Dynamic stochastic general ...
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2answers
218 views

Dynamic optimisation

Consider a simple dynamic consumption-saving problem. A solution can be characterised using a Lagrangian approach that generates a set of first order conditions and some boundary conditions. An ...
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0answers
95 views

CES utility in dynamic setting

Suppose I have a multiperiod consumption-saving problem with two or multiple goods able to be consumed. If the utility within a period is Constant Elasticity of Substitution, ie. $C = (c_1^\frac{\...
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0answers
70 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 ...
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0answers
79 views

Dynamic dividing a dollar game

The question is a reformulation of an incomplete version. Consider the following dynamic dividing a dollar game where agent 1 claims $x(t)$ of the dollar and agent 2 $y(t)$ (paper). \begin{align} &...
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1answer
475 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 ...
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1answer
61 views

Paper where an integral in the constraint of an optimization problem is treated as infinite sum

I am looking for a paper (or textbook, or even lecture notes example) where there is a problem such as $$ \max f(x) \\ \text{ s.t. } \int g(x) \leq \int c $$ and there is at least some exposition/...
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1answer
128 views

Dynamic optimization with assets as state variable: interpreting capital gains and losses

Given a hamiltonian of the form: \begin{equation} H_{t} = ln(c_{t}) \dot{} e ^{-\rho t} + \lambda_{t}(w+ra_{t}-c_{t}), \end{equation} with $c_{t}$ consumption at time t (the control variable), $\rho &...