Questions tagged [optimal-control]

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3
votes
1answer
41 views

Current valued VS Present Valued Hamiltonian Differing Euler equations

I have been having some difficulties with recovering the same euler equation from the following optimal control problem when comparing the present valued hamiltonian to the current valued hamiltonian. ...
4
votes
0answers
46 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)\...
5
votes
2answers
428 views

Euler equation in Continuous time VS Discrete time

I have seen the euler equation in discrete time for the baseline neoclassical growth model written as: $$\frac{U'(c_{t+1})}{U'(c_{t})}=\frac{1}{\beta(1+r)}$$ however I have also seen the euler ...
7
votes
1answer
130 views

Intuition behind Euler Lagrange equation in economics

When being exposed to the concept of the Euler Lagrange equation as a mathematical concept, many ideas from the physical sciences are used to explain its relevance in terms of choice of shortest path, ...
5
votes
1answer
80 views

Deriving the Euler equation from a Continuous Time Dynamic Programming Problem (HJB)

Solving for the Euler equation in discrete time is farily straight forward with the use of the Benveniste Scheinkman theorem. However for the following standard Ramsey model: $$\max \int_{0}^{\infty}e^...
3
votes
0answers
112 views

What are the boundary value conditions for generic HJBs in economics?

Consider a routine continuous time optimization problem: $ V(t,a_{t}) := \max \int_{\tau=t}^{\tau = T} e^{-\rho (\tau -t)} u(c_{\tau})d\tau $ $\text{ s.t. }$ $\dot{a}_{t} = y + ra_{t} - c_{t}$, $a_{...
1
vote
2answers
143 views

What is state space representation for DSGE modeling

I'm beginning with DSGE modeling, and a mathematical representation (perhaps trivial for most of the people that are more with this topic) is the space-state state representation of a dynamical model, ...
5
votes
0answers
60 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: $x(t)$: ...
8
votes
1answer
127 views

Continuous time optimization with two laws of motion (the Hamiltonian with two laws of motion)

How would we deal with a continuous time optimal control problem with two laws of motion? Suppose we have the following RCK like environment with human capital investment. $$\max_{c(t),k(t),h(t)}\int_{...
5
votes
1answer
214 views

Current Value Hamiltonian VS Present Value Hamiltonian in Economics

I've been looking at a number of optimal control problems and have been wondering under what conditions one should use the current value Hamiltonian over the present value Hamiltonian. Does it depend ...
1
vote
0answers
39 views

Constancy of current value Hamiltonian and numeric computations

I am playing with a simple, continuous time optimal growth problem to learn optimal control. The social planner chooses $c_t$ to maximize: $$\int_0^\infty e^{-\rho t}u(c_t)dt$$ where $$\dot k_t=f(k_t)-...
2
votes
2answers
70 views

Why do game theorists use a discounted payoff of this form?

Excuse the click-baity title. I notice the discounted payoff in the game theory literature usually takes the form $$\sum_{t=1}^\infty\lambda(1-\lambda)^{t-1}R_t$$ This differs from the discounted ...
2
votes
0answers
60 views

Finding optimal path in continuous time

I have the following optimal control problem $$\max_{c_t,l_t} \int_0^{\infty} [ln(c_t)+\theta ln(1-l_t)]e^{-pt}dt$$ st. $$\dot{k_t}=k_t^{1/2}l_t^{1/2}-c_t-\beta l_t$$ $$k_0>0$$ I do big part of ...
2
votes
2answers
248 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 $w_{t+1}$ ...
6
votes
1answer
157 views

Optimal price function: application of calculus of variations

The problem, I am trying to solve is based on the paper by Rochet and Vila 1994 (see literature below). In fact, it is a variant of the seminal paper of Kyle 1985 in the finance/economics literature. ...
3
votes
1answer
134 views

Solution method in Smith (2006) A Closed Form Solution to the Ramsey Model

I am trying to understand the way Smith demonstrates that the general solution to his equation (12) is (13) (see page 6). (12) \begin{eqnarray} \dot{z} &=& (1- \alpha)\left[1-\left(\delta +...
2
votes
1answer
471 views

Optimal control theory: How to maximize Hamiltonian in this case?

The problem is to maximize $\int_0^1 y(t) + u(t)^2 dt$ where $y$ is state and $u$ is control. Further we have $y' = u, y(0) = 5$. I set up the Maximum Principle equations, but, in particular, I need ...
7
votes
1answer
173 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 ...
7
votes
1answer
235 views

Optimization: Dynamic Programming vs Kuhn-Tucker

Considering the standard utility maximization of representative household which lives forever, one may use dynamic programming and Kuhn-Tucker in case of discrete time. For instance, one would like to ...
6
votes
4answers
2k views

Does control system engineering have a place in economics?

Do central banks use some form of engineering-style PID control systems/feedback loops to implement monetary policy? I'm an electrical engineering student taking microeconomics/macroeconomics and a ...
2
votes
1answer
134 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{...
4
votes
1answer
120 views

Stability analysis and dimension of a dynamic control system

I have an optimal control problem where I have two control and one state variable. $$max\int_{0}^{\infty}\left(u\left(c\right)-P_{M}M\right)e^{-\rho t}dt\tag{1}$$ where $P_{M}$ is the unit price of ...
7
votes
1answer
156 views

Simple Derivation of Maximum Principle

Consider the simplest problem of optimal control \begin{align} &\max_u\int^T_0{F(y,u)dt}\\ \text{s.t.} \quad&\dot y = f(y,u)\\ & y(0) = y_0\\ & y(T)~~\text{free} \end{align} ...
2
votes
1answer
77 views

Optimal Stopping

When we solve Bellman equations, I normally would think of the Blanchard Kahn technique. But in the case that I have an optimal stopping problem, or where the decision that the agent has to take is to ...
8
votes
4answers
445 views

Autonomous or non-autonomous optimal control system?

I have a following system with endogeneous discounting. $c,k$ and $h(k)$ are consumption, capital and endogeneous discount function based on physical capital. (the properties of this function are not ...
13
votes
1answer
476 views

Stochastic growth in continuous time

Literature: See Chang (1988) for theoretical part and Achdou et al. (2015) for numerical part respectively. Model Consider the following stochastic optimal growth problem in per capita notation. \...
1
vote
0answers
54 views

discount factor on rival and non-rival goods

I am asking to myself if we must use a different discount rate on rival and non rival goods. In standard Solow-Swann model, we use generally the discount factor $\rho$. More formally, let's give the ...
3
votes
0answers
67 views

Converging Trajectories and Sufficiency for Optimality

(The question is loosely relatet to this thread.) In the paper "Feedback Equilibria for a class of non-linear Differential Games" by Mäler et al. it is stated (p. 14) In fact sufficiency is ...
19
votes
0answers
533 views

How do I use the Malliavin calculus to solve for the optimal trading strategy in the classic Merton problem?

How do I use the Malliavin calculus to solve for the optimal trading strategy in the classic Merton problem? In Duffie's book "Dynamic Asset Pricing," he outlines the "Martingale method" of solving ...
20
votes
3answers
687 views

When Optimal Control fails (?)

In order to "ask my question", I have to solve a model first. I will omit some steps but still, this will unavoidably make this post very long -so this is also a test to see whether this community ...