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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.