Questions tagged [dynamic-optimization]
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60
questions
2
votes
0answers
31 views
Closed form solution to consumption-saving model in discrete time
Consider the simplest consumption-savings model of the following form:
$$
\max_{\{c_t,a_{t+1}\}_t}\mathbb{E_0}\sum_{t\geq 0} \beta^tu(c_t) \\
a_{t+1} + c_t = (1+r)a_t + y_t \\
y_t \mid y_{t-1} \sim F
$...
0
votes
0answers
10 views
Information acquisition problem and desing of information
I am searching some details about the information acquisition problem. I want to understand what this is about and where can I find the beginning of this literature. I think this problem is some kind ...
0
votes
0answers
26 views
In the RBC model, why is household's problem dynamic?
Why is capital assumed to be a good accumulated in the current period t but being used in t+1? I think my interpretation is wrong here. I understand that households problem is dynamic because of this ...
5
votes
0answers
55 views
Optimal level of consumption in discrete time model with quadratic preferences and infinite horizon
I am trying to derive an expression for the optimal level of consumption in the basic problem:
$ max \hspace{1cm} U_t = E_t \left[\sum_{s=t}^{\infty} \beta^{s-t} \left( C_s - a\frac{Cs}{2}^2 \right) ...
4
votes
0answers
61 views
Out-of-Equilibrium Macroeconomic Dynamics using Differential Equations
My OP below asks about the RBC model, but I am actually interested in any out-of-equilibrium macroeconomic model; CGE, DSGE or whatever the correct nomenclature is.
The Real Business Cycle model is a ...
4
votes
0answers
63 views
Solve the Ben-Porath Model (Optimal Control Problem)
Suppose we have a Ben-Porath style human capital investment model, in which the representative agent maximize her lifetime earnings: $$V(h, a)=\max \int_{a}^{R} e^{-r(t-a)}\left[ w h(t)(1-n(t))-px(t)\...
3
votes
0answers
27 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
39 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:
$$
u\left(C, t, H_{t}, L_{t}\right)=\frac{\left(C_{t} / H_{t}^{\kappa}\...
3
votes
0answers
26 views
Derive optimal wage in New Keynesian-Calvo wage stickiness
Following Costa, 2016 in page 96, developing the labor variety optimal wage decided by the household, the FOC is:
$$0=E_t\sum_{i=0}^\infty(\beta\theta_w)^{t+i}\left\{\psi_W\left[L_{t+i}\left(\frac{W_{...
4
votes
0answers
63 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
1answer
78 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 ...
2
votes
1answer
28 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
1answer
79 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
0answers
69 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
54 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 ...
1
vote
0answers
36 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
0answers
59 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 ...
3
votes
2answers
186 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
0answers
7 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
0answers
26 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
1answer
55 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
0answers
41 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 ...
2
votes
1answer
95 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
0answers
36 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
1answer
160 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
2answers
916 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
1answer
214 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
1answer
42 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
0answers
82 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 ...
1
vote
0answers
64 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 $2\sqrt{r}$ ...
2
votes
2answers
276 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 $w_{t+1}$ ...
0
votes
1answer
734 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
1answer
86 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). ...
1
vote
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 ...
3
votes
0answers
85 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 ...
1
vote
0answers
105 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
2answers
786 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 ...
0
votes
0answers
81 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. ...
2
votes
0answers
150 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 ...
7
votes
1answer
180 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 ...
4
votes
0answers
199 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 ...
0
votes
1answer
95 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 ...
0
votes
2answers
742 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....
2
votes
2answers
295 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}\\
...
2
votes
2answers
751 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
$$
\sum_{t=1}^{T}\...
3
votes
1answer
162 views
Dynamic Optimization : Resource Stock as a Sink
I think about a dynamic problem where social planner maximizes the following utility ;
$$\underset{c\left(t\right)}{max}\int_{0}^{\infty}u\left(c\left(t\right)\right)e^{-\rho t}$$
subject to two ...
9
votes
2answers
12k views
Transversality Condition in neoclassical growth model
In the neo-classical growth model there is the following transversality condition:
$$\lim_{t\rightarrow\infty}\beta^{t}u'(c_{t})k_{t+1}= 0,$$
where $k_{t+1}$ is the capital at period $t$.
My ...
3
votes
2answers
916 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
...
3
votes
2answers
246 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 ...
1
vote
0answers
101 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{\...