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8 votes
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In Blackwell's condition for T to be a contraction mapping, we require that satisfies discounting. What is the intuition of discounting?

Without discounting, you cannot show either $$ T(g + || f - g||) \leq Tg + \beta || f - g|| $$ or $$ T(f + || g - f||) \leq Tf + \beta || g - f|| $$ and thus you cannot demonstrate that $T$ is ...
Kenneth Rios's user avatar
  • 1,259
7 votes
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A Cake Eating Problem in Continuous Time: Hamiltonian or HJB?

The comment by user @MaartenPunt is accurate. I don't think that in general one can identify situations where one should have a clear preference over one formulation over the other. It is more of a ...
Alecos Papadopoulos's user avatar
6 votes
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Optimal stopping (reference request)

This is known as the McCall search model in economics. The original paper shows that the optimal stopping strategy rule is given by a "reservation wage", there is a threshold such that it is ...
Michael Greinecker's user avatar
5 votes
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Dynamic programming in infinite horizon model

There are two interrelated maximisation problems. The first is the infinite horizon maximisation problem: $$ \begin{align*} v(k) = &\max_{a_1, a_2, \ldots} \sum_{t = 0}^\infty \delta^t F(k_t, c_t),...
tdm's user avatar
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5 votes
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How can I show convexity of this value function?

Suppose that $u(C,l)=\sqrt{C}-l^2$ and $f(l,A)=\big(l+g(A)\big)^2$, where $g$ is any function of $A$ that is not convex. Then $$u\big(f(l,A),l\big)=l+g(A)-l^2.$$ The optimal labor supply is given by $...
Michael Greinecker's user avatar
4 votes
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Solution Method for Infinite-Horizon Maximization Problem

Your first question (regarding constraints on the parameters) can be answered through first and second derivative analysis. In order to satisfy strictly increasing, we need $u'>0$ and to satisfy ...
DornerA's user avatar
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4 votes
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What is unknown in Bellman Equation?

The original problem was probably of the form $$\max_{\{W_t\}_{t=1}^\infty}\sum_{t=0}^\infty \beta^t u(W_t-W_{t+1}),$$ $$\mbox{s.t. } W_{t+1}\in[0,W_t] \ \forall \ t, ~~ W_0 \mbox{ given}$$ When ...
Regio's user avatar
  • 4,188
4 votes

A reference for most used utility functions in macroeconomic problems of intertemporal optimization

I think that this article might be helpful: “Exotic Preferences for Macroeconomists” http://www.nber.org/chapters/c6672.pdf They give a thorough explanation of many preference functionals and show ...
thekiciminister'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

The Principle of Optimality and the Bellman Equation

Recall that the Principle of Optimality states that the solution to Our Bellman Functional Equation is the same as the solution to the sequential problem if: Assumption 1: $\Gamma(x)$ (our set of ...
EconJohn's user avatar
  • 8,407
4 votes

Has mathematical economics contributed to the mathematics of space exploration?

In the history of dynamic programming some economic works are considered pioneering contributions to the theory of dynamic programming (and in this sense it can be said that they contributed to ...
BakerStreet's user avatar
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3 votes
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The Cake Eating Problem with Depreciation (Modelling difficulties)

It would seem that the way you've formulated your production function/law of motion has introduced double counting into the problem. Note that substituting 1 and 2 into 3 gives: $$k_{t+1}=(1-\delta)(...
H Rogers's user avatar
  • 638
3 votes
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What is the result of the Bellman Equation

...the result of applying sup operator is a NUMBER... Read it carefully. The equation is $$ v(x_0) = \sup_{ \{x_t \}_{t \geq 1}} \cdots \quad (1) $$ This defines a function $v$, called the value ...
Michael's user avatar
  • 2,619
3 votes
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Optimisation using value function

Starting with your original equation: $max_{c_t, m_t, b_t} E_0\sum_{t=0}^\infty U(c_t, m_t)$ s.t. (1) $y+\frac{m_{t-1}}{1+\pi_t}+\frac{1+i_{t-1}}{1+\pi_t}b_{t-1}=c_t+m_t+b_t+\tau_t$ Here: $R_{t-1} ...
Boaten's user avatar
  • 98
3 votes

Bellman equation for this dynamic programming problem

The "second" constraint appears redundant and it confuses matters. Re-arrange the first one to obtain $$\tilde{a}_{t+1} = \big[\tilde a_t+(1-\delta )Y_t-\tilde{c}_t\big]R_t$$ This tells us that ...
Alecos Papadopoulos's user avatar
3 votes

Solution Method for Infinite-Horizon Maximization Problem

Your first question, if it's literally correct, is easy: The only way for $u'$ to be positive for c=0 is for p=1. if p =1 then sign($\phi$)=sign($\theta$) so that the product is positive. But, since $...
Fix.B.'s user avatar
  • 2,668
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

Exercise 4.7 in SLP (dynamic programming)

It follows from Assumption 4.10. Let $\lambda y \in \Gamma(\lambda x)$ be the solution. Let $\delta = \frac{1}{\lambda}$. By Assumption 4.10, $$ \lambda y \in\Gamma(\lambda x) $$ implies $$ \delta \...
Walrasian Auctioneer's user avatar
3 votes

Resources to derive economic forecasts

You can find excellent examples of codes for DSGE models as well as VAR on QuantEcon. For example, here is an example of VAR model in Python, and here is an example of some simple DSGE model. The ...
1muflon1's user avatar
  • 57.1k
3 votes

Resources to derive economic forecasts

For forecasting with VAR in R there are some good tutorials on econometrics with R. This tutorial from Justin Eloriaga also helped me when we had to make VAR for our quantitative macro assignment. ...
csilvia's user avatar
  • 2,734
3 votes

simplification of FOC

Rearrange $$\frac{-1}{d_2-d_3} + \frac{\beta}{d_3} = 0$$ to $$\frac{\beta}{d_3} = \frac{1}{d_2-d_3}$$ Flip it: $$\frac{d_3}{\beta} = d_2-d_3$$ The equation is now linear in $d_3$.
Giskard's user avatar
  • 29.4k
3 votes
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How can I formulate the following optimization problem?

If you want to determine how much carbon dioxide should be omitted by solving an optimization problem, then a constraint on the quantity of $CO_2$ isn't quite what you need. The normal constraint on ...
Adam Bailey's user avatar
  • 8,529
3 votes
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Bellman Equation & Envelope Theorem

The key thing to note here is that in the optimum, $c_t$ will depend on $k_t$. Thus, the value function is \begin{align} V(k_t, t) &= \max\{u(c_t) + \beta V(f(k_t) - c_t, t + 1)\}\\ &= u(c_t(...
Wittgenstein's Poker's user avatar
2 votes
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Update of value function in continuous time - HJB

You iterate towards a fixed point, so you want to reach a situation where plugging in your current iterated value produces itself. Now using your notation, we are told that we should calculate $$V_{n+...
Alecos Papadopoulos's user avatar
2 votes

Solving the Hamilton-Jacobi-Bellman equation; necessary and sufficient for optimality?

(This perhaps should be considered a comment.) If you have solved the HJB equation, it is sufficient to get the optimal solution. So you do not "have to be concerned with any other optimality ...
Brian Romanchuk's user avatar
2 votes

Question regarding Carlstrom and Feurst (1997)

What you address (and what is done in your referred paper) is called Model Calibration Many economic models use this method to determine one or more parameters: Calibration is a strategy for finding ...
skoestlmeier's user avatar
2 votes

Optimisation using value function

Thanks to @Boaten I was able to find the solution. For those interested here are the steps for deriving the Fisher relation: Combining the FOC $[b_{t}]$ and the envelope for $[b_{t-1}]$ we get $U_{...
user11767's user avatar
  • 661
2 votes

Capital accumulation

$a=0$ means implies the household does not enjoy leisure. In other words, the trade-off between consumption and leisure is switched off. It's not generally true that with $a=0$, utility is higher for ...
E. Sommer's user avatar
  • 1,325
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

Solving a HJB with a probability to transit to a new state

I would leave this as a comment but I cant. You are on the right track. Once you know $V_2(k)$ then you can plug that into to the first hjb and solve. To solve for $V_2$ you need to find the optimal ...
user28714's user avatar
  • 151

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