11

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}&\...


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 ...


5

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....


5

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 ...


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

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 ...


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

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, ...


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

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 ...


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 ...


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

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 ...


1

In control systems engineering, there was the "Aizerman Conjecture" (the transliteration of Aizerman varies) that argued that a linear time-varying system was stable if the parameters (state matrices) at all times corresponded to stable systems. However, counter-examples were found to this conjecture. About the only way to show that a time-varying system is ...


1

In fact, saddle path equilibrium is one of the most common equilibrium in canonical growth models, mainly in dynamic optimization problems. It is possible that he makes a reference to a decentralized equilibrium without cycles around steady state. Otherwise, if there are cycles around a steady-state equilibrium. It means that there are oscillations (see ...


1

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 $$V_{n+1}(a) = V_n(a) + \Delta$$ where $$\Delta = \ u(c(a^*)) + \dfrac{\partial V_n(a)}{\partial a}da_t(a^*) - \rho V_n(a)$$ Insert the second into the first to ...


1

The economic essence of saddle-path stability, is that it requires conscious behavior and decision making from the part of economic agents, in order for the system to remain on the saddle path. Simply put, we cannot do "whatever we like" and still expect the system to settle down. But that is what makes it interesting (and true to reality): on their own, ...


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