Questions tagged [bellman-equations]

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When could value functions in Bellman equations be calculated explicitly?

Given the simplest form of a Lucas model, i.e., a Bellman equation given by \begin{align} J(x_t) & = \max_{c_t, x_{t+1}} \{ u(c_t) + \beta E_{\pi} [ J(x_{t+1})] \} \\ & \textrm{ s.t. } ...
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Bellman Equation & Envelope Theorem

I'm unsure where the envelope theorem comes into play when i differentiate the Bellman Equation with respect to $k_t$. To me it looks like the regular chain rule and in fact the exact opposite of the ...
• 897
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Differentiating Bellman equation

Assume that we have a Bellman equation that is $$V(k)=\max_{0\leq k'\leq f\left(k\right)}u\left(f\left(k\right)-k'\right)+\beta V\left(k'\right)$$ The textbook says that if we differentiate with ...
• 311
267 views

Write down budget constraint

Assume an infinite horizon representative agent economy with the following consumer preferences $u(c_t)$ The production technology of this economy uses capital and land, which is fixed amount in ...
• 120
1 vote
115 views

Wealth in the utility function

Suppose there is a representative household of unit mass who lives forever. Preferences are given as: $$\sum\beta^tu(c_t,k_{t-1})$$ Technology is given as: $$k_{t+1}=AF(K_t,L_t)+(1-\delta)K_t-c_t$$ ...
• 1,048
301 views

Dynamic programming in infinite horizon model

Using an infinite horizon model, a dynamic programming approach uses a fixed point to solve the model: $V = \Gamma(V)$. How do I interpret the meaning of $V$? For example, when we decide a investment ...
• 195
159 views

Consider the growth model with inelastic labor supply, full depreciation, log utility and CRS technology with the Bellman equation be defined as follows: $$V(k)=\max(log(k^\alpha-k')+\beta V(k'))$$ st ...
• 1,048
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• 1,781
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Dynamic programming and Difference equations applications

I'm asked by my teacher to prepare a presentation with economic applications of Dynamic Programing (Bellman Equation) and Difference equations. I'm not sure what this things are used for in economics ...
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• 61
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Solution to the Bellman equation is a fixed point

I have recently started studying dynamic optimization. I cannot quite wrap my head around the fact that the value function of the Bellman equation is a fixed point of a contraction mapping. As far my ...
• 43
156 views

What is the difference between comma and plus in the bellman equation?

Bellman equation: $V(x) = max \{F(x,y)+ \beta V(y)\}$ $V(x) = max \{F(x,y), \beta V(y)\}$ When to use the plus and when to use the comma? Would you mind give me an example to explain such ...
• 273
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Update of value function in continuous time - HJB

When solving (numerically, by value function iteration) a dynamic programming problem in discrete time, such as $$V_1(a) = \max_{c} \ u(c) + \dfrac{1}{1+\rho}V_0(a')$$ we maximize with respect to ...
• 43
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Benveniste-Scheinkman condition gives derivative that still depends on the value function

What I mean by the title is often, if we have a value function like $$V(K,I) = \max_{K',I'} F(K') +\beta V(K',I')$$ the First order conditions will give us something that depends on the derivative of ...
• 260
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Optimal Stopping

When we solve Bellman equations, I normally would think of the Blanchard Kahn technique. But in the case that I have an optimal stopping problem, or where the decision that the agent has to take is to ...
• 1,631
1 vote
52 views

Forward-Looking HJB: Rewrite as PDV

We live in continuous time. Let there be some discount rate $D(t)$, which consists of a discount rate, and some death probability. $V(t)$ contains the flow value of, say, being alive. If you are alive,...
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I have the following system of equations $$\rho V(u, \epsilon^i) = F(u, \epsilon^i) + V_u(u, \epsilon^i)g(u, V(u, \epsilon^i) + \lambda^i \left(V(u, \epsilon^{-i}) - V(u, \epsilon^i)\right)$$ with \$...