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, ...


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

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 ...


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

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) $$ (1-\delta)\sum_{t=0}^{\infty}\delta^t R_t $$ where $0 < \delta < 1$. Suppose I play a strategy that gives me the same payoff, say $a$, for each ...


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.


1

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 representation, if possible with some of the intuition, its uses in economics (control theory) My background is in control systems theory, where the notion of a ...


1

To get things started: The reservation wage is the wage that makes the worker indifferent between taking a job and getting V_e and not taking a job and getting V_u. To find it, solve for the value of w that makes V_e = V_u. Solving the entire model is not possible given the info you provided. You need the block that describes the firms. My sense is that it ...


1

(The second equation for the value function of the unemployed should be $$ v(w,U)= \max \{v(w,E); \,u[c,1]+\beta\int v(w', U) dF(w')\}. \quad (*) $$ ) ...how do you know when your problem solution require more than one Bellman equation? Whenever the state space of the problem contains discrete coordinates, there would be "multiple" value ...


1

Are you referring to repeated games, in which player $i$ obtains a payoff of the form $$(1-\delta)\sum_{t=0}^{\infty}\delta^{t}u_{i}(x_{i}^{t})$$ from the sequence of payoffs $\left\{x_{i}^{t}\right\}$? The $(1-\delta)$ is appended to the front to give the average per period payoff.


Only top voted, non community-wiki answers of a minimum length are eligible