13

The transversality condition may be more easily understood if we start from a problem with finite horizon. In the standard version, our objective is to $$ \max_{\{c_t,k_{t+1}\}_{t=0}^T} \sum_{t=0}^T\beta^t u(c_t) $$ subject to $$ \begin{aligned} f(k_t)-c_t-k_{t+1}&\ge0,\quad t=0,\dots,T &&\text{(resource/budget constraint)}\\ c_t,k_{t+1}&\...


9

The Dynamic Programming ("Bellman' Equation") formulation incorporates the terminal boundary condition ("transversality conditions") needed in case we use the Lagrangian/Euler equation formulation. The initial conditions are still needed in both approaches. To see why, consider the problem $$\text{max}_{\{x_t\}} \sum_{t=0}^{\infty} \beta^t U(c_t)$$ $$s.t....


6

In my opinion, the best derivation is by logic. Think about it this way: If the only thing we are telling the household is to maximize its utility, optimal behaviour would then be just making infinite debt and consume infinitely. This is no sensible solution. We therefore need another optimality condition. This should answer question 2. In a finite horizon ...


6

The mathematical theory behind DSGE models can be found in any textbook on stochastic dynamic optimisation. One common reference that economists use for this is Stokey, Lucas and Prescott. Of course, they focus exclusively on recursive methods, but (perhaps) the lion’s share of dynamic problems in economics are solved in this way. There is also a treatment ...


6

As suggested by 1muflon1: I don't really use dynamic discrete optimization (yet) in my work, but I did explore several books once. Here are some suggestions: Economic Dynamics: Theory and Computation by John Stachurski: It's very easy to fall in love with this guy. He writes casually yet authoritatively. The book covers required real analysis and includes ...


5

You are unfortunately mistaken. DSGE models are at the heart of monetary policy and the most widely used class of models in this field. To work in monetary there is no real way around learning DSGE. A very good book to get started. is Walsh (2010) "Monetary Theory and Policy". I can also recommend Gali's book ("Monetary Policy, Inflation, and the Business ...


4

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. Let $I_{a,t}$ denote the indicator function for attack, taking the value $1$ when there is an attack, and the value $0$ when there isn't. Then, properly, $$\dot{...


4

I'm unsure exactly where your struggle is, in general. To try and address your issues, what kind of boundary conditions might we desire in dynamic problems? Consider the two period consumption-savings problem where we have $$c_1 + s_1 = y_1$$ $$c_2 = y_2 + (1+R)s_1$$ $y_1, y_2$ being endowments in each period, $s_1$ as savings, $R$ for interest, and $c_1, ...


4

I agree with 1muflon1, but allow me to add some more nuance. Identification and calibration can be meant to express a subset of estimation. Any identified coefficient is also an estimate, but not vice-versa. An identified estimate is any estimate that fulfills certain conditions that make it the true number we want. For example, any coefficients from (...


4

Identification and estimation are often used interchangeably (at least that’s my observation from attending conferences and reading papers) but according to econometric literature there is a subtle difference. For example, in the John Stachurski "A Primer in Econometric Theory" the identification is a process of finding out if the parameters are ...


3

The argument relayed in the question as regards consumption smoothing is flawed. Consumption smoothing does not mean consumption equality over periods, but rather, tendency to avoid corner solutions, or near-corner solutions. So it has nothing to do with whether, in the context of this family of models, $\beta (1+r) =1$ or not (after all, in a more ...


3

The Euler Equation typically refers to the interior optimal choice between consumption today and tomorrow (or some similar intertemporal choice). That is, it equalizes the marginal utility of consuming a unit today vs saving that unit in order to consume tomorrow. This yields the first equation you listed (where $\beta=\frac{1}{1+\rho}$): $$ u'(C_{t}) = \...


2

Some papers that come to my mind are ; -Calvo and Obstfeld (1988, Econometrica) -Endres et al (2014, Resource and Energy Economics) -Traeger et al. (2012, European Economic Review) In these papers, the objective function has two integrals but as these models are treating an overlapping generations model, there is a such aggregation that one of the ...


2

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) Mulitply throughout by $k^{-\alpha}$: $$k^{-\alpha}\dot k = \frac{1}{\sigma} - \delta k^{1-\alpha} \tag{1}$$ 2) Define the variable $$z \equiv k^{1-\alpha} \implies \dot z = (1-\...


2

Suppose you have a dynamic system $$ x_{t+1} = Ax_{t} $$ with a stationary point (or steady state as used in growth or RBC literature), say, $x^*$, i.e. $x^{*} = Ax^{*}$. Now, consider the following question. Starting from an initial value $x_0$, how many paths are there leading to the stationary point $x^*$? If there is an unique path going from $x_{0}$ ...


2

This is not a formal definition, but a useful piece of intuition. I think that the best way to think about it is that when there is uncertainty in a model it arises mainly in two forms either there is information that some agents have, but not every agent has it (private information), or there are truly random events that no one knows (in game theory jargon,...


2

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 $i$ as a function of $k$. Then plug $i(k)$ into the 2nd HJB. That will give you a second order ode. Solving that will give you $V_2(k)$ and you go to 1.


2

Minimum state variable (MSV) solution is a special technique used to find an unique equilibrium with desirable properties in DSGE models. Often DSGE models can have multiple paths that will satisfy the conditions given by the system you are modelling. Hence to provide some meaningful results you have to somehow choose between the all possible paths/...


2

...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 function, which is a type of indirect utility function. For given $x = x_0$, the value of $v(x)$ is defined to be the sup on the RHS, taken over feasible ...


2

You can find OLG models that do not classify as DSGE (in particular, the model might not be stochastic) as well as DSGE with overlapping generations (contrary to those with infinitely lived agents). You can find more detail on this on this working paper by Assous and Duarte (2017), as they note In the early 1980s, when the real business cycle ...


2

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 $$ V_{T}[w_T] = \max_{c_T \in [0,w_T]} u(c_T) $$ So, we can solve this via backward induction. Clearly, at the final period $T$, since $u$ is monotonic, we ...


2

I think your math is mostly correct but I have to admit that I am not used to AK models. A short answer for your main question: is it ok to introduce taxation in the model without including the government? Answer: Yes, of course. There are overwhelming literature on dynamic or static theoretical modeling of the effect of tax policy without introducing ...


2

The reason why the $e^{nt}$ term is there is because you want to multiply the whole utility by the number of people. You are actually not substituting consumption into the utility but multiplying the utility of an individual by the number of individuals. So the problem would look like: $$\int_{0}^{\infty}\frac{c^{1-\sigma}}{1-\sigma}e^{nt} e^{-\rho t}dt$$ At ...


1

Taking a variable as given comes from the assumptions of the model and cannot result from any optimisation problem (think simply that when a variable is taken as given it is treated as a constant in your problem so one cannot lead to the other). Here, taking the public good as given means the household is not able to choose the level of public good to ...


1

These are many questions. O.k., so let's go step by step: (Q1) What is a mapping actually? A map is just another term for a function. Here, every "law of motion", the actual one (ALM) and the perceived one (PLM), is characterized by its parameters $a$ and $b$. The ALM depends on the PLM, and the function mapping the PLM-parameters to the ALM-parameters is ...


1

One I can introduce is " ABC of RBC " It was so helpful to me.


1

Although it is true that the First order Conditions for a strictly convex utility function will yield the same solution as that of a strictly concave one, the second order conditions are entirely different. Instead of finding a maximum, you are indeed finding a minimum. In other words, FOCs are necessary but not sufficient. Intuitively Intuitively, a ...


1

"Then, is it possible to say that $\mu$ could take a negative value as natural capital represents a cost for capital accumulation ?" No. $\mu$ can be thought of as the shadow price of natural resources. Being a "price", it has to be non-negative. Note that if we set $\mu=0$ the problem reverts back to the standard model, which has an intuitive explanation: ...


1

Your calculation has two typos a) you type two times the minus sign related to $f'$ and b) you write $(1+\delta)$ instead of $(1-\delta)$. If we correct for these we have $$u_{c}(c_{t},m_{t}) + \beta V_{\omega}(\omega_{t+1})\left[\frac{f'(\omega_{t}-c_{t}-m_{t}-b_{t})}{1+n}(-1)+\frac{1-\delta}{1+n}(-1)\right]$$ Taking out the minus sign and $1/(1+n)$, and ...


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