# Questions tagged [dynamic-programming]

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### Does this contraction mapping map strictly concave functions into strictly concave functions?

Consider the following functional equation: $$TV(k)=\max[W(k),\beta V(f(k))]$$ where $\beta\in (0,1)$, $W(k)$ is continuous, increasing, bounded, and strictly concave function defined on $[0,\bar{k}]$,...
1 vote
130 views

### How can I show that the policy function is non-decreasing?

Consider the following functional equation: $$V(x)=\max_{y\in [0,f(x)]}[u(f(x)-y)+\beta V(y)]$$ where $u$ is continuous, strictly increasing, and strictly concave; the function $f$ is continuous and ...
67 views

### Understanding Duality between Individual and Collective Maximization in Macroeconomic Models

I'm currently studying macroeconomic models, specifically from the book "Recursive Macroeconomic Theory." In Chapter Seven, it is mentioned that some economic models involving firms and ...
225 views

### How can I show that the optimal savings are 0 for all time periods?

Consider an infinitely-lived agent’s consumption-saving problem. The agent receives $e > 0$ units of endowment every period, can save via an asset with constant return $R$. The agent is endowed ...
1 vote
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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 ...
66 views

### Simulation of a dynamic search and matching model

I want to simulate a search and matching model very close to the canonical model developed by Pissarides. I am interested in representing the trajectory of the unemployment rate when the unemployment ...
39 views

### One shot-deviation property for games of imperfect information

The equivalence between subgame perfection and one-deviation property is typically stated for games of perfect information (where information sets are singletons). Does the Blackwell-style argument ...
104 views

### Mathematical Prerequisites for Recursive Macroeconomic Theory (Thomas J Sargent, Lars Ljungqvist)

I'm a math grad who is interested in learning more about economics for fun. Reading through RMT, I saw some interesting math (in chapter 2) around using "invariant functions" to determine ...
1 vote
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### Are there any sophisticated'' mathematical modelling where they solve for the utility function?

Are there any references in literature of any sophisticated'' mathematical modelling where they solve for the utility function under specific conditions using differential equations theory? In such ...
67 views

### When do Economists use the Linear Quadratic Regulator in simulating DSGE models

I was watching some excellent videos on DSGE models by Klaus Pretner. The author was able to solve some simple model such as the Ramsey-Cass-Koopman's model, and a New Keynesian model with frictions, ...
1 vote
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### More equations than endogenous variables in RBC model

I recently came across my late father's bachelor's thesis, in which a RBC model is described. It appears to be a variant of Hansen's 1985 model with indivisible labor. I wrote the model into Dynare to ...
1 vote
39 views

### How to define a dynamic programming problem in an incomplete information game?

How can I define a problem of dynamic programming, to use the Hamilton-Jacobi-Bellman equation in order to solve the utility maximization problem of the generic agent of a dynamic game with incomplete ...
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### What is state space representation for DSGE modeling

I'm beginning with DSGE modeling, and a mathematical representation (perhaps trivial for most of the people that are more with this topic) is the space-state state representation of a dynamical model, ...
146 views

### Optimization in discrete time

I have made optimizations in continuous time that belong to the control theory, for example one case: $\max(\min)V[u(t)]=\int_0^Tf(t,x(t),u(t))dt$ constraint to: $\dot x=g(t,x(t),u(t))$ Where: $x(t)$: ...
81 views

### Can anyone help me understand the Motrtensen-Pissarides model?

I am seeking help with the Mortensen-Pissarides model in discrete time? Basically, I was given the following in class example (which we didn’t complete do to time constraints associated with the end ...
219 views

### Solving a HJB with a probability to transit to a new state

I am trying to solve the problem of a firm facing the possibility of a future tax, in continuous time. The firm maximizes $V(k)=\int_{t=0}^{\infty}e^{-rt} \pi_t dt$ with $\pi_t=f(k_t)-i_t$ and \$\dot{k}...
480 views

### More than one Bellman Equation

I'm attending to my first dynamic optimization course, and what I don't fully graps yet is that sometimes we have to use more than one bellman equation. How do you realize that? I mean how do you know ...
1 vote
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### Coase conjecture and Stokey model

The durable goods monopolist can charge different prices by every periods. Critical type consumers are indifferent between buying today and buying tomorrow. So I construct a Bellman equation like ...
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### What is the result of the Bellman Equation

I'm just starting with dynamic optimization and although I understant the proof's of the theorem I'm not able to fully understand whether the bellman equation is a function , a function valuated at ...
Excuse the click-baity title. I notice the discounted payoff in the game theory literature usually takes the form $$\sum_{t=1}^\infty\lambda(1-\lambda)^{t-1}R_t$$ This differs from the discounted ...
I have the following dynamic system in discrete time For p is price, d is demand and s is supply. $$p_{t+1}-p_t= a(d_t-s_t)$$ $$s_{t+1}-s_t=bp_ts_t-ws_t$$ $$d_t= k-gp_t$$ I have to linearize this ...