Questions tagged [dynamic-optimization]
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75
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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 ...
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57
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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 ...
2
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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 ...
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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 ...
- 173
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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 ...
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35
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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 ...
- 31
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1
answer
32
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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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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 ...
- 184
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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 ...
- 385
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38
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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 ...
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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 ...
- 183
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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. ...
- 183
3
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2
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194
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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 ...
user40867
2
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2
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281
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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 ...
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68
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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+...
- 75
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52
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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 ...
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Closed form solution to consumption-saving model in discrete time
Consider the simplest consumption-savings model of the following form:
$$
\max_{\{c_t,a_{t+1}\}_t}\mathbb{E_0}\sum_{t\geq 0} \beta^tu(c_t) \\
a_{t+1} + c_t = (1+r)a_t + y_t \\
y_t \mid y_{t-1} \sim F
$...
- 59
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49
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In the RBC model, why is household's problem dynamic?
Why is capital assumed to be a good accumulated in the current period t but being used in t+1? I think my interpretation is wrong here. I understand that households problem is dynamic because of this ...
- 21
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95
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Optimal level of consumption in discrete time model with quadratic preferences and infinite horizon
I am trying to derive an expression for the optimal level of consumption in the basic problem:
$ max \hspace{1cm} U_t = E_t \left[\sum_{s=t}^{\infty} \beta^{s-t} \left( C_s - a\frac{Cs}{2}^2 \right) ...
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Out-of-Equilibrium Macroeconomic Dynamics using Differential Equations
My OP below asks about the RBC model, but I am actually interested in any out-of-equilibrium macroeconomic model; CGE, DSGE or whatever the correct nomenclature is.
The Real Business Cycle model is a ...
- 281
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108
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Solve the Ben-Porath Model (Optimal Control Problem)
Suppose we have a Ben-Porath style human capital investment model, in which the representative agent maximize her lifetime earnings: $$V(h, a)=\max \int_{a}^{R} e^{-r(t-a)}\left[ w h(t)(1-n(t))-px(t)\...
- 2,339
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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 $Z_t$ ...
- 457
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73
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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:
$$
u\left(C, t, H_{t}, L_{t}\right)=\frac{\left(C_{t} / H_{t}^{\kappa}\...
- 457
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answers
33
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Derive optimal wage in New Keynesian-Calvo wage stickiness
Following Costa, 2016 in page 96, developing the labor variety optimal wage decided by the household, the FOC is:
$$0=E_t\sum_{i=0}^\infty(\beta\theta_w)^{t+i}\left\{\psi_W\left[L_{t+i}\left(\frac{W_{...
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74
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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 ...
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1
answer
104
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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 ...
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1
answer
39
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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 ...
- 2,115
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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-...
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146
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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:
$x(t)$: ...
- 821
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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 ...
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327
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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 ...
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61
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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 ...
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219
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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}...
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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 ...
- 539
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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 ...
- 539
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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 ...
- 539
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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 ...
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1
answer
146
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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 ...
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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 ...
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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?
- 539
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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?
- 539
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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 ...
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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, ...
- 539
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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 ...
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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 $2\sqrt{r}$ ...
- 120
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2
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556
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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 $w_{t+1}$ ...
- 468
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1
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1k
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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 ...
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92
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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). ...
- 8,311
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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 ...
- 201
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107
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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 ...
user16020