10
votes
Accepted
Current Value Hamiltonian VS Present Value Hamiltonian in Economics
There is no clear right and wrong about this, it's just a matter of convenience. The current-value Hamiltonian is likely to be more convenient when the objective function includes a discount factor. ...
- 8,311
9
votes
Accepted
Does control system engineering have a place in economics?
Economists have been exploring control theory applications to macro economics for decades. For example, here is a 40 year-old research paper written in 1976 on the topic.
top of page 2 (also numbered ...
- 1,101
7
votes
Accepted
Euler equation in Continuous time VS Discrete time
One way to see (intuitively) the connection between the left hand sides is to write the discrete case as:
$$
\frac{u'(c(t + \tau))}{u'(c(t))},
$$
for $\tau = 1$. Now if we generalise this to a setting ...
- 10.3k
6
votes
Continuous time optimization with two laws of motion (the Hamiltonian with two laws of motion)
In this case you have two state variables, so as you say, you have to check the FOC ($J$ is the Hamiltonian) for $\frac{\partial J}{\partial K}=-\dot\lambda_1$ and $\frac{\partial J}{\partial H}=-\dot\...
- 821
6
votes
Accepted
Current valued VS Present Valued Hamiltonian Differing Euler equations
First, the Present value hamiltonian equals:
$$
{\cal P} = e^{-rt} \frac{c(t)^{1-\theta}}{1 - \theta} + \lambda(t)(Ak(t)^\alpha - \delta k(t) - c(t))
$$
This gives the first order conditions:
$$
\...
- 10.3k
6
votes
Euler equation in Continuous time VS Discrete time
You cannot completely ignore the RHS. Starting with
$$\frac{U'(c_{t+1})}{U'(c_{t})}=RHS,$$
replace $t+1$ by $t+\Delta t$ to get
$$\frac{U'(c(t+\Delta t))}{U'(c(t))}=RHS_{\Delta t},$$
where $RHS_{\...
- 6,550
5
votes
Accepted
Deriving the Euler equation from a Continuous Time Dynamic Programming Problem (HJB)
As already commented, the equation you probably meant is
$$
\rho V(k)= \sup_c \{\, u(c) + V'(k) ( f(k) -\delta k -c ) \,\}.
$$
I have never seen this equation called the HJB equation (probably missing ...
- 2,619
5
votes
Autonomous or non-autonomous optimal control system?
Following Caputo, an Optimal Control problem is autonomous when none of the functions appearing in the description of the problem depends explicitly on the time-variable. But this means that the ...
- 33.4k
5
votes
Autonomous or non-autonomous optimal control system?
What about rewriting the problem in the following way?
$$\underset{\left\{ c_{t}\right\} }{max}\int_{t=0}^{\infty}\left[u\left(c_{t}\right) e^{y_t}\right]e^{-\rho t}dt$$
with the new state variables ...
- 967
5
votes
Accepted
What are the assumptions made about fixed points in the dynamics equations of Recursive macroeconomics?
On pages 53-55 of the Stokey, Lucas, with Prescott (1989) book they discuss the Contraction Mapping Theorem. This theorem guarantees existence and uniqueness of the solution (one fixed point).
The ...
- 487
4
votes
Accepted
Autonomous or non-autonomous optimal control system?
The OP's answer is correct in its conclusion, but he applies a strange argument at the end to arrive there.
Applying brute-force differentiation, the present value Hamiltonian is
$$\mathcal{H}=e^{-...
- 33.4k
4
votes
An Optimal Control Model: A Ridiculous Result for a Steady State
One general issue I see is that you try to include uncertainty in a framework developed for a deterministic setup.
What you do is to use expected income in the equation of motion for human capital. ...
- 33.4k
4
votes
Accepted
Dynamic programming, optimal consumption-savings (finite horizon) problem
Your value function is as follows:
$$
V_t[w] = \max_{c_t \in[0,w]} \left\{u(c_t) + \frac{1}{2}V_{t+1}[\alpha(w_t - c_t)] + \frac{1}{2}V_{t+1}[\beta(w_t-c_t)] \right\}
$$
with the terminal condition
$$
...
- 1,869
4
votes
Optimal price function: application of calculus of variations
Not really an answer, but too long for comment.
The $P$ in your
$$y(u,v)^*= \frac{v-P(y^*)}{P'(y^*)} +u$$
expression from the insider's problem and the $P$ in the expression
$$
\min_{P(\cdot)} \...
- 2,619
3
votes
What is state space representation for DSGE modeling
This question is too broad as it stands. There is a longer answer already, but I think it is possible to deal with a core part of the question easily.
The concrete question is what is a state-space ...
- 9,852
3
votes
Does control system engineering have a place in economics?
For a PID system to work, you need to be (at least approximately) correct about the relationship between the variables you are trying to manage. Unfortunately the relationships between macroeconomic ...
- 1,051
3
votes
Does control system engineering have a place in economics?
A small addendum. Lars Peter Hansen and Thomas J. Sargent wrote the book "Robustness", which is an attempt to apply robust control to economics. They treated robust control from a game theoretical ...
- 9,852
3
votes
Autonomous or non-autonomous optimal control system?
I think I proved in a rigorous way that the system is autonomous for the model that I have written 3-4 days and I think it is useful for the community, especially for those who are working on ...
- 2,115
3
votes
Accepted
Analytically tractable Ramsey model: how to solve ODE for optimal trajectories
The differential equation
$$\dot k = \frac{1}{\sigma} k^\alpha - \delta k$$
has the structure of a Bernoulli equation. We solve it by the following transformation steps:
1) Multiply throughout by $k^{...
- 33.4k
3
votes
Accepted
Stability analysis and dimension of a dynamic control system
Differentiating $(3)$ with respect to time we get
$$u_{cc}\dot c = \dot \lambda \implies \frac {u_{cc}}{u_c} \dot c = \frac {\dot \lambda}{\lambda}$$
Inserting into $(6)$ we obtain
$$\dot M = -\...
- 33.4k
3
votes
Accepted
stochastic optimal control/FOC/Reis(2021)
I think the "standard" approach in solving HJB equations like this is to guess a form for $V$ and then verify. Your first order conditions for $c/a$ and $k/a$ are right.
Now, guess that $V(a,...
3
votes
stochastic optimal control/derive the consumption process
Okay, I know how the authors got their solution. I am just not sure it is justified. I'll present it anyway.
Judging by the appendix in the paper, the authors solve a deterministic problem and then ...
2
votes
Simple Derivation of Maximum Principle
After a comment exchange, let's provide the answer under disrete-time formulation. The problem is now written
\begin{align}
&\max_{\{u\}_1^T, \{y\}_1^T}\sum^T_{t=1}{F(y_t,u_t)}\\
\text{s.t.}...
- 33.4k
2
votes
Accepted
Optimal Stopping
I recommend the manuscript here by Lawrence Evans. The related example is described as 'Rocket Railroad Car'. The first instance is on pages 9-12 where a geometric solution is provided. The choice set ...
- 967
2
votes
Does control system engineering have a place in economics?
Economists use optimal control both in microeconomics and in macroeconomics.
Your question is about economic policy in particular, but policy decisions can be guided both by micro and macro models.
...
- 195
2
votes
Optimal control theory: How to maximize Hamiltonian in this case?
Hamiltonian
\begin{align}
H(y(t),u(t),\lambda(t)) = y(t)+u(t)^2 + \lambda(t) u(t)
\end{align}
First order conditions read
\begin{align}
&H_u = 0 \quad \Longleftrightarrow \quad u(t) = -\frac{\...
- 1,559
2
votes
What is state space representation for DSGE modeling
State Space Representations of Linear Systems:
https://lpsa.swarthmore.edu/Representations/SysRepSS.html
As systems become more complex, representing them with differential equations or transfer ...
- 760
2
votes
Accepted
Why do game theorists use a discounted payoff of this form?
In my experience, it's mainly just for cleanliness for results.
Consider an infinite horizon repeated game, with discounted payoff representation (where I use $\delta = (1-\lambda)$ in your notation)...
- 1,869
2
votes
Multiple solutions to an HJB, how to pin down the optimal "viscosity" solution?
Answer to Q1:
If we re-write FONC as a function of $a$: $u'(c(a))-V_{a} =0$
Differentiate wrt $a$ (as in Walde 2010): $u''(c(a)) c'(a) -V_{aa} =0$
We know from SOSC that $u''(c(a))<0$.
If we ...
- 340
2
votes
Accepted
$H$ is a constant? Maximizing: $\int _0^Te^{-t}f(x,u)dt$ st $x_t=g(t,x,u)$ and $g$ is independent of $t$
Consider the problem:
$$
\begin{align*}
\max_{u} \int_0^T f(x(t), u(t)) dt& \\
\text{ s.t. } &\dot x = g(x(t), u(t)),\\
&\text{ + boundary conditions}
\end{align*}
$$
Assume that $g(x,u)$ ...
- 10.3k
Only top scored, non community-wiki answers of a minimum length are eligible