14

I believe the problem is that the steady state may not exist, and the system instead exhibits steady growth (depending on parameters). The reason is because the model is equivalent to the standard consumption-saving problem with exogenous and constant interest rate. To see that, first consider the first order condition for labor choice $f_2(k,\ell) = w$ (...


9

For continuous-time stochastic dynamic programming, the small, nontechnical Art of Smooth Pasting by Dixit is a wonderful option. It does a very effective job of conveying the basic intuition. Stokey's more recent The Economics of Inaction is also decent, but for a practical-minded person it probably underperforms Dixit - its much greater length and ...


9

I am posting this as an answer, because it continues on user @ivansml answer... which is the one that identified the catch here, a catch I naively have overlooked (although it is a narrow case, while the interesting part comes after. Nevertheless, it should have been dealt with). Indeed, with exogenous wage rate, and perfectly competitive optimization on ...


8

Consider the version of the paradox from Wikipedia: A casino offers a game of chance for a single player in which a fair coin is tossed at each stage. The pot starts at 2 dollars and is doubled every time a head appears. The first time a tail appears, the game ends and the player wins whatever is in the pot. Thus the player wins 2 dollars if a tail ...


6

Dynamic Programming & Optimal Control by Bertsekas Introduction to Modern Economic Growth by Acemoglu The Acemoglu book, even though it specializes in growth theory, does a very good job presenting continuous time dynamic programming.


6

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 case-specific issue (and maybe for some twisted problems where one of the two may fail for usually technical reasons). See this post for some related discussion, ...


5

Rearranging the steady state equation $$ \overline{p}^{\alpha}=\alpha y\overline{p}^{\alpha-1}- \alpha\overline{p}^{\alpha}-\frac{a+1}{\sigma} $$ we get $$ (1 + \alpha)\overline{p}^{\alpha}=\alpha y\overline{p}^{\alpha-1}- \frac{a+1}{\sigma}. $$ As $\alpha \in [0,1]$, the left hand side of the equation is increasing in $\overline{p}$ and the right hand side ...


5

In an intertemporal maximization problem, we seek to find the optimal sequence of the control and the state variables. It is the recursive nature of the problem that permits us to consider a "typical" point in time and just one condition per variable. For each such problem, we need to find out (carefully) in how many distinct periods a specific ...


5

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 Zin 2014). $$c_t = \mu + c_{t-1} + \sigma_c \varepsilon_{t}$$ Begin with preferences given as Epstein-Zin with a certainty equivalence function of the form $\...


5

I think that the key question is whether this firm is the only firm in the economy. If it is then it is no longer correct for it to take $w$ as given as $w$ will be affected by its own capital accumulation decision. In this case you should make the substitutions that you made before your equation (2) while setting up the Hamiltonian. On the other hand if ...


5

I think Kamien and Schwartz's Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management is pretty well known.


5

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 a contraction mapping in the traditional proof. For intuition, note that Blackwell's sufficiency conditions for a contraction mapping is a fixed-point theorem ...


4

Controlled Markov Processes and Viscosity Solutions by Fleming and Soner includes a number of applications to Finance and Differential Games.


4

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 strictly concave, we need $u''<0$. What does this actually mean? $$u'(\frac{}{})=\phi p\theta c_t^{p-1}e^{\theta c_t^p}>0$$ Since we know that $c_t^{p-1}e^{...


4

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 their usefulness with macroeconomic applications. Im not a specialist in macroeconomics, so I dont know the literature that well, but I think that the most ...


4

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 formulated that way there is no need to solve for any function $V$, but the infinite elements of the sequence ${\{W_t\}}_{t=1}^\infty$ are all choice variables. ...


4

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 feasible values) is non-empty for all $x\in X$. Assumption 2:$\lim_{t\rightarrow\infty}\sum_{t=0}^\infty \beta^t F(x_t,x_{t+1}) $ exists for all $\tilde{x}\in \Pi(...


3

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 $exp()>0$ and $p=1$, the first term of $u''$ cancels out and the only way for u'' to be negative is for $\phi$ to be negative. Therefore $\theta<0$ too. ...


3

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 wealth in the beginning of next period is fully determined by current-period decisions and known states, without any uncertainty whatsoever: we start with the ...


3

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: $R_{t-1} =1+i_{t-1}$ and $1+\pi_t=\frac{P_t}{P_{t-1}}$ Note that in this problem, you have have two state variables, $m_{t-1}$ and $b_{t-1}$, and your main issue have ...


3

A really nice methodology for approximating the HJB is the upwind scheme, which I learnt quite quickly using Ben Moll et al's notes and codes The examples are continuous time versions of familiar heterogenous agents economies models such as Hugget and Aiyagari.


3

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 the context. Here are some criteria, in descending order of relevance Clean Ways Is any of the steady states unstable? If so, it's less likely to be the one ...


3

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: $$k_{t+1}=(1-\delta)(c_t+x_t)+x_t$$ Where investment in period t is counted twice. The correct law of motion is simply: $$ 3. \: k_{t+1}=(1-\delta)x_t $$ And the general form of ...


3

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 $$ \delta \lambda y \in \Gamma(\delta\lambda x) $$ in other words $$ y \in \Gamma(x) $$


3

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 above are just examples, you can find different models on the site that might suit you better. In addition, you should note that in every case you will have to do ...


3

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. PenState also has good sources for econometrics, here are their sources for VAR.


2

There is an old result in dynamic programming due to David Blackwell, according to which stationary problems allow for stationary best responses. So if you would gain by changing your behavior after a certain history, you would gain by changing it at every history corresponding to the same state. For the original reference, see the corollary to Theorem 1 ...


2

A deviation (one-shot or not) can certainly generate a sequence that differs from the optimal one for an arbitrary number of periods. You could treat a dynamic programming problem as a repeated game between one player and chance. The one-shot deviation principle should then carry over from repeated games to dynamic programming.


2

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 discrete and continuous time. I would really say for this book "Dynamic Optimization for dummies". I was not familiar at all with dynamic optimization but this ...


2

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 $U_{c_{t}} = \beta E_{t}\frac{U_{c_{t+1}}(1+i_{t})}{1+\pi _{t+1}}$ Assuming utility is log in both arguments this can be simplified as $\frac{c_{t+1}}{c_{t}} = \...


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