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Questions tagged [dynamic-optimization]

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Intertemporal profit maximization

Assume a producer wishes to maximize the net present value, choosing optimal quantities of K and L. variables are time dependent. y is the production function, p is the price of y. K is capital, r is ...
Meg's user avatar
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1 vote
1 answer
107 views

How to solve dynamic problem with 2 production functions?

Suppose we have the following problem: $\max \int_0^\infty \exp(-\rho t) u(c(t))dt$ where $c(t)$ is consumption at time $t$. Subject to: $\dot{k}(t)= f(k(t))- c(t) - \delta k(t)$. where $k$ is capital,...
WilliamT's user avatar
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1 vote
2 answers
120 views

Nonhomogeneous linear dynamic system exercise

I have the following system of equations: \begin{align*} x_{t+1} & = 3x_t + y_t \\ \\ y_{t+1} & = 2 + 5y_t. \end{align*} I know how to solve it when I do not have a constant, but I couldn't ...
Hypatia's user avatar
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3 votes
1 answer
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Has mathematical economics contributed to the mathematics of space exploration?

We see the work of Bellman showing up in both orbital trajectory planning/optimization and, obviously DSGE modeling. Also, in a recent example, the JWST uses ideas surrounding Pareto optimization and ...
whydaho99's user avatar
1 vote
0 answers
55 views

Durable goods in a (two sector) necolassical growth model

i want to add a firm to a neoclassical growth model that produces a durable good which it rents out in each period to the consumers. Right now i'm using the following approach: The firm maximizes: $\...
mfba's user avatar
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0 answers
33 views

Deriving euler equation and transversality condition

$U(c_t)=\sum_{t=0}^{\infty}\beta^t(\{u_0c_t+\frac{u_1}{2}c_t^2\})$ subject to $c_t+k_{t+1}\leq f_0 k_t$ I need to find the euler equations and the transversality conditions. I have currently tried ...
GraceLynn87's user avatar
1 vote
0 answers
61 views

Expected value in budget constraint

I have been reading this paper by Yang Liu, Lukas Schmid and Amir Yaron, which contains a very elegant mechanism that generates an endogenous liquidity premium for US government debt. However, I got ...
Wittgenstein's Poker's user avatar
2 votes
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34 views

What pricing strategies does Amazon use and how do they affect consumers' purchasing decisions?

As a frequent Amazon customer, I have noticed that the prices of products I am interested in buying often fluctuate over time. These changes could either be an increase or decrease in price, and I ...
paulmuaddib's user avatar
1 vote
0 answers
17 views

Dynamic investing problem - Private Equity

I've been thinking about the following problem. Consider an agent who starts out with \$1 and on any given day $t$, is given the opportunity to invest in an asset with expected return $\mu_t$ and ...
user357269's user avatar
1 vote
0 answers
35 views

Optimization Model for Market Clearing using Uniform Pricing

Dear all, can someone please share a simple example for market clearing via Uniform Pricing using an Optimization Model? I am trying to simulate a market using bid values with quantity and price, from ...
Marmik Pancholi's user avatar
2 votes
0 answers
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2 dimensional optimization using fminsearch

I have a complicated capital - debt capital structure optimization problem but to start off simple I just added an extra parameter to the stochastic neoclassical growth model to see how the ...
user41131's user avatar
1 vote
1 answer
33 views

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 ...
EBS's user avatar
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3 votes
0 answers
64 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 ...
Oliver Queen's user avatar
3 votes
0 answers
94 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 ...
krishnab's user avatar
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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 ...
krishnab's user avatar
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4 votes
1 answer
59 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 ...
L. Johnson's user avatar
4 votes
1 answer
326 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. ...
L. Johnson's user avatar
3 votes
2 answers
204 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 ...
user avatar
2 votes
2 answers
396 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 ...
user14261785's user avatar
2 votes
0 answers
72 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+...
Franciscolli's user avatar
2 votes
0 answers
64 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 ...
DDV's user avatar
  • 21
2 votes
0 answers
83 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 $...
Jsck's user avatar
  • 59
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0 answers
50 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 ...
Damien's user avatar
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5 votes
0 answers
110 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) ...
Giorgetto's user avatar
  • 223
4 votes
0 answers
98 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 ...
LBogaardt's user avatar
  • 281
4 votes
0 answers
127 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)\...
Alalalalaki's user avatar
  • 2,419
4 votes
0 answers
56 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 $Z_t$ ...
BAL's user avatar
  • 457
4 votes
0 answers
79 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}\...
BAL's user avatar
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3 votes
0 answers
36 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_{...
manifold's user avatar
  • 843
4 votes
0 answers
77 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 ...
John M. Riveros's user avatar
1 vote
1 answer
113 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 ...
Mr. T's user avatar
  • 113
2 votes
1 answer
44 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 ...
optimal control's user avatar
1 vote
1 answer
145 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-...
Pedro Cunha's user avatar
5 votes
0 answers
164 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: $x(t)$: ...
manifold's user avatar
  • 843
2 votes
1 answer
98 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 ...
Tony456's user avatar
  • 87
1 vote
0 answers
413 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 ...
Mr. Fafa's user avatar
0 votes
0 answers
61 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 ...
RLF's user avatar
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3 votes
2 answers
227 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}...
Mr. Fafa's user avatar
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 ...
Beck Batucada's user avatar
2 votes
0 answers
113 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 ...
Beck Batucada's user avatar
2 votes
1 answer
206 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 ...
Beck Batucada's user avatar
1 vote
0 answers
62 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 ...
modern's user avatar
  • 23
2 votes
1 answer
155 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 ...
mmendina's user avatar
  • 103
1 vote
0 answers
41 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 ...
Beck Batucada's user avatar
1 vote
1 answer
455 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?
Beck Batucada's user avatar
5 votes
3 answers
4k 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?
Beck Batucada's user avatar
6 votes
1 answer
632 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 ...
Beck Batucada's user avatar
1 vote
1 answer
61 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, ...
Beck Batucada's user avatar
2 votes
0 answers
86 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 ...
studentp's user avatar
  • 170
1 vote
0 answers
68 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 $2\sqrt{r}$ ...
studentp's user avatar
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