Questions tagged [dynamic-optimization]
The dynamic-optimization tag has no usage guidance.
66
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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 ...
4
votes
1
answer
185
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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. ...
3
votes
2
answers
155
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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 ...
0
votes
1
answer
57
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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
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0
answers
39
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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
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0
answers
33
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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 ...
2
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0
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39
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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
$...
0
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0
answers
13
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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
0
answers
29
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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
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0
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59
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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
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0
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62
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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 ...
4
votes
0
answers
74
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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)\...
4
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0
answers
29
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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 $Z_t$ ...
4
votes
0
answers
45
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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:
$$
u\left(C, t, H_{t}, L_{t}\right)=\frac{\left(C_{t} / H_{t}^{\kappa}\...
3
votes
0
answers
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Derive optimal wage in New Keynesian-Calvo wage stickiness
Following Costa, 2016 in page 96, developing the labor variety optimal wage decided by the household, the FOC is:
$$0=E_t\sum_{i=0}^\infty(\beta\theta_w)^{t+i}\left\{\psi_W\left[L_{t+i}\left(\frac{W_{...
4
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0
answers
63
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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
85
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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
1
answer
31
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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
108
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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
90
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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:
$x(t)$: ...
2
votes
1
answer
56
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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 ...
1
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0
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82
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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
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0
answers
60
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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
2
answers
190
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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}...
1
vote
0
answers
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
0
answers
29
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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
75
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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0
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41
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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 ...
2
votes
1
answer
118
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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 ...
1
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0
answers
36
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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
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1
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206
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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
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3
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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
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1
answer
267
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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
53
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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
82
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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 ...
1
vote
0
answers
66
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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 $2\sqrt{r}$ ...
3
votes
2
answers
316
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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
1
answer
818
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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
87
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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
1
answer
29
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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
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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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0
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108
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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
881
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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
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0
answers
90
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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
0
answers
154
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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
1
answer
205
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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 ...
4
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207
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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
1
answer
98
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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
2
answers
756
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....
3
votes
2
answers
314
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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}\\
...