8
votes
Accepted
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 ...
- 1,229
7
votes
Accepted
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 ...
- 33.2k
6
votes
Accepted
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 ...
- 11.4k
5
votes
Accepted
Prove the uniqueness of steady state
Rearranging the steady state equation
$$
\overline{p}^{\alpha}=\alpha y\overline{p}^{\alpha-1}-
\alpha\overline{p}^{\alpha}-\frac{a+1}{\sigma}
$$
we get
$$
(1 + \alpha)\overline{p}^{\alpha}=\alpha y\...
- 28.4k
5
votes
Accepted
Guess and Verify
Another somewhat canonical form is the value function for risk-sensitive preferences when consumption follows a random walk with drift (there are also versions including capital -- see Backus Ferriere ...
- 1,603
5
votes
Accepted
Dynamic programming in infinite horizon model
There are two interrelated maximisation problems. The first is the infinite horizon maximisation problem:
$$
\begin{align*}
v(k) = &\max_{a_1, a_2, \ldots} \sum_{t = 0}^\infty \delta^t F(k_t, c_t),...
- 9,757
5
votes
Accepted
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 $...
- 11.4k
4
votes
Accepted
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 ...
- 1,558
4
votes
Accepted
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 ...
- 4,188
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
$$
...
- 1,849
4
votes
A reference for most used utility functions in macroeconomic problems of intertemporal optimization
I think that this article might be helpful:
“Exotic Preferences for Macroeconomists”
http://www.nber.org/chapters/c6672.pdf
They give a thorough explanation of many preference functionals and show ...
- 404
4
votes
The Principle of Optimality and the Bellman Equation
Recall that the Principle of Optimality states that the solution to Our Bellman Functional Equation is the same as the solution to the sequential problem if:
Assumption 1: $\Gamma(x)$ (our set of ...
- 8,133
3
votes
Accepted
The Cake Eating Problem with Depreciation (Modelling difficulties)
It would seem that the way you've formulated your production function/law of motion has introduced double counting into the problem. Note that substituting 1 and 2 into 3 gives:
$$k_{t+1}=(1-\delta)(...
- 605
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 ...
- 33.2k
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 $...
- 2,648
3
votes
Accepted
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} ...
- 98
3
votes
Multiple equilibria: which one to select?
I'm not sure I follow the logic on that equation having infinitely many solutions and steady states. In any case, in what follows are some guidelines for equilibrium selection.
It depends a lot on ...
- 10.7k
3
votes
Accepted
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 ...
- 2,609
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 ...
- 9,834
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
$$
\delta \...
- 1,849
3
votes
Resources to derive economic forecasts
You can find excellent examples of codes for DSGE models as well as VAR on QuantEcon.
For example, here is an example of VAR model in Python, and here is an example of some simple DSGE model.
The ...
- 52.8k
3
votes
Resources to derive economic forecasts
For forecasting with VAR in R there are some good tutorials on econometrics with R. This tutorial from Justin Eloriaga also helped me when we had to make VAR for our quantitative macro assignment. ...
- 2,576
3
votes
simplification of FOC
Rearrange
$$\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$$
The equation is now linear in $d_3$.
- 28.4k
3
votes
Accepted
How can I formulate the following optimization problem?
If you want to determine how much carbon dioxide should be omitted by solving an optimization problem, then a constraint on the quantity of $CO_2$ isn't quite what you need. The normal constraint on ...
- 8,251
3
votes
Accepted
Bellman Equation & Envelope Theorem
The key thing to note here is that in the optimum, $c_t$ will depend on $k_t$. Thus, the value function is
\begin{align}
V(k_t, t) &= \max\{u(c_t) + \beta V(f(k_t) - c_t, t + 1)\}\\
&= u(c_t(...
2
votes
Solving the Hamilton-Jacobi-Bellman equation; necessary and sufficient for optimality?
(This perhaps should be considered a comment.)
If you have solved the HJB equation, it is sufficient to get the optimal solution. So you do not "have to be concerned with any other optimality ...
- 9,834
2
votes
References to learn continuous-time dynamic programming
Applied Intertemporal Optimization by Klaus Wälde is a very very nice book, even for those who are not really familiar with mathematics.
The book treats deterministic and stochastic models, both in ...
- 2,115
2
votes
Accepted
Update of value function in continuous time - HJB
You iterate towards a fixed point, so you want to reach a situation where plugging in your current iterated value produces itself. Now using your notation, we are told that we should calculate
$$V_{n+...
- 33.2k
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 ...
- 319
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_{...
- 651
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