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10 votes
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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. ...
Adam Bailey's user avatar
  • 8,529
9 votes
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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 ...
FreeMarketUnicorn's user avatar
7 votes
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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 ...
tdm's user avatar
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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\...
manifold's user avatar
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6 votes
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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: $$ \...
tdm's user avatar
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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_{\...
VARulle's user avatar
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5 votes
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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 ...
Fića's user avatar
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5 votes
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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 ...
Michael's user avatar
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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. ...
Alecos Papadopoulos's user avatar
4 votes
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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 $$ ...
Walrasian Auctioneer's user avatar
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)} \...
Michael's user avatar
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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 ...
Wittgenstein's Poker's user avatar
3 votes
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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,...
Wittgenstein's Poker's user avatar
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 ...
Brian Romanchuk's user avatar
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 ...
Mick's user avatar
  • 1,046
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 ...
Brian Romanchuk's user avatar
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. ...
kitsune's user avatar
  • 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{\...
clueless's user avatar
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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 ...
SystemTheory's user avatar
2 votes
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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)...
Walrasian Auctioneer's user avatar
2 votes
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How can I apply the Hamiltonian function and Pontryagin's maximum principle in the context of Optimal Control Theory?

I give below just an outline to orient oneself in the context of Pontryagin’s Principle, answering briefly your questions. First of all, it must be noticed that what is called the Pontryagin's Maximum ...
BakerStreet's user avatar
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2 votes
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$H$ 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 $g(x,u)$ ...
tdm's user avatar
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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 ...
Albert Zevelev's user avatar
1 vote
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When does it make sense to use variational methods, versus dynamic programming, versus nonlinear control methods so solve DSGE models

The main two tools for economists solving infinite-horizon constrained optimization problems in discrete-time, as in your example problem, are: Karush–Kuhn–Tucker (KKT) conditions - which is a ...
Fića's user avatar
  • 487
1 vote

Which industries are widely agreed upon to require central regulation on a national basis, regardless of economic or political system?

I would say first-best answer is public utilities (power, water, sanitation). They have extreme economies of scale and are generally "natural monopolies". Telecommunication has similar ...
RegressForward's user avatar
1 vote

What is the relationship between the HJB and "Hamiltonian"? Why is the Hamiltonian H(p) inside the HJB?

Since you mention Walton, here is something from his notes page 111 Definition 4: The Hamiltonian and is defined $H(t,x,\rho) := min _v \{c(t,x,v) - \rho v\}$ Notice the analogue to the Hamiltonian ...
erik's user avatar
  • 721
1 vote

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

Let's solve the differential equation (12). As a first step, we look for a "simpler" differential equation, namely (A.1). (A.1) can be written $$\frac{\dot{z}}{z}=-(1-\alpha)\left(\delta+\frac{\bar{x}...
GuiWil's user avatar
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1 vote
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Optimization: Dynamic Programming vs Kuhn-Tucker

I would say that the main difference stems from the solution method, which results in your statement about all paths versus only the path at time t being true. Dynamic programming (at least when done ...
Maarten Punt's user avatar
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1 vote

Stochastic growth in continuous time

More of a comment: There should be an expectation operator in the statement of the problem, otherwise problem doesn't make sense. That "...the deterministic and stochastic value function must be the ...
Michael's user avatar
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