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Questions tagged [dynamic-optimization]

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Bellman and Lagrange equation used at the same time

I just encounter a strange maximization problem in Sargent's Recursive Macroeconomic Theory book, when they have Bellman equation and Lagrange equation at the same time. Specifically: $P(v) = \max_{...
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1answer
59 views

Reference Request - Dynamic Optimization with More Than One State Variable

I would like to understand how to solve dynamic optimization problems involving more than one state variable and state equation (to apply to long-term economic models with more than one capital good). ...
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1answer
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What is the “bequest condition” in a finite-horizon discrete optimization problem?

For a finite-horizon discrete time optimization problem, my textbook provides a condition called the "bequest condition", which I'm not familiar with. Specifically, where the state at time $t$ is ...
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Firm Dynamic Optimization Problem

A firm has received an order at time $0$ for $M$ units of product to be delivered by time $T$. It seeks a production schedule for filling this order at minimum cost. Let $x(t)$ denote inventory ...
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Stochastic Dynamic Programming: Deriving the Steady-State for a Lottery

I am working through the basic examples of the stochastic RBC models in the book by McCandless (2008): The ABCs of RBCs, pp. 71 - 75 A Standard Stochastic Dynamic Programming Problem Here is a ...
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1answer
328 views

Textbook on the mathematics of RBC/DSGE models?

I'm reading David Romer's Macroeconomics. However, what I don't like is that he doesn't go at all into detail about the mathematical underpinnings of RBC/DSGE models. When it comes to the central ...
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Good book/article that goes into depth about transversality conditions?

I know how to derive the transversality condition in simple models like the Ramsey model. However, I am looking to develop a deeper understanding of transversality conditions in more complex models. ...
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Is there an argument from first principles for the form of the no-ponzi condition?

in the ramsey model, we use the no ponzi condition $$\lim_{t\to\infty}e^{-R_t}a_t\geq 0$$ for assets $a_t$ that a household holds at time $t$. I understand intuitively what the reasoning behind ...
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An Optimal Control Model: A Rediculous Result for a Steady State

I was experimenting with a seemingly simple optimal control problem that generates a system of differential equations. When I calculate the values of the steady state of the system I get some very ...
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Optimal Fight Purse and Boxing Strategies

The following is all public information available to all the players in this scenario. The General Setup In the aftermath of the infamous race between the tortoise and the hare, the salty hare went ...
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1answer
71 views

Money in utility function - Value function

I am reading Walsh's (2003) book on monetary economics. Specifically the chapter on money in utility function. I understand the basics of a value functions but I can't seem to get the same results as ...
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2answers
350 views

Price optimization with demand forecast

I have one year sales data of a retail company and lets say I am forecasting the next month sales for the product. I have got the sales using time series in R. Now I want to forecast the price as well....
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2answers
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Phase Diagram - growth model

The dynamics of Ramsey-Cass-Koopman growth model is usually summarized in phase diagrams with the 2 equations (conventional symbols apply): \begin{align*} \frac{\dot c}{c}=&\frac{r-\rho}{\theta}\\ ...
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2answers
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Preference for consumption smoothing and actual smoothing

The typical dynamic consumption-saving under certainty model can be written as: $$ \max V(c)=\sum_{t=1}^{T} \beta^{t-1}\; u(c_t) $$ Subject to the intertemporal budget constraint $$ \sum_{t=1}^{T}\...
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1answer
125 views

Dynamic Optimization : Resource Stock as a Sink

I think about a dynamic problem where social planner maximizes the following utility ; $$\underset{c\left(t\right)}{max}\int_{0}^{\infty}u\left(c\left(t\right)\right)e^{-\rho t}$$ subject to two ...
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Transversality Condition in neoclassical growth model

In the neo-classical growth model there is the following transversality condition: $$\lim_{t\rightarrow\infty}\beta^{t}u'(c_{t})k_{t+1}= 0,$$ where $k_{t+1}$ is the capital at period $t$. My ...
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2answers
362 views

Saddle path equilibrium on financial market with rational expectations

In his 1978 paper introducing the Tobin tax Tobin states that : As a technical matter, we know that a rational expectations equilibrium on markets of this kind is a saddle point. That is, there ...
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130 views

Dynamic optimisation

Consider a simple dynamic consumption-saving problem. A solution can be characterised using a Lagrangian approach that generates a set of first order conditions and some boundary conditions. An ...
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0answers
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CES utility in dynamic setting

Suppose I have a multiperiod consumption-saving problem with two or multiple goods able to be consumed. If the utility within a period is Constant Elasticity of Substitution, ie. $C = (c_1^\frac{\...
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Dynamic demand model in many good competitive markets and price optimization

This is a question about demand models, price optimization, dynamic pricing, big data, online learning, so I will cross-post in other communities. $\mathbf{Background}$ I am interested in dynamic ...
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0answers
47 views

Hamilton-Jacobi-Bellman with heterogeneous discount rates

Let $i=1,2$ denote the players, $x$ the state and $u_i$ the control of player $i$. The state equation reads $\dot x = f(x,u_1,u_2)$ and the objective function is given by $F_i(x,u_1,u_2)$. Now I'd ...
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0answers
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Dynamic dividing a dollar game

The question is a reformulation of an incomplete version. Consider the following dynamic dividing a dollar game where agent 1 claims $x(t)$ of the dollar and agent 2 $y(t)$ (paper). \begin{align} &...
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1answer
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Monetary policy optimization

I was wondering if anyone could give me some advice / lectures / introduction to stochastic optimization that could be applied to monetary policy. I have heard of the Dynamic stochastic general ...
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1answer
300 views

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 ...
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1answer
997 views

Question about Euler condition [closed]

In dynamic macroeconomic model(without production sector), Euler equation is $$U'(C_t)=b(1+r)U'(C_{t+1})$$ I found another equation related with Euler which is called Euler condition. In New ...
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1answer
279 views

Are there stable improvements of the Ramsey-Cass-Koopmans model?

The Ramsey-Cass-Koopmans neoclassical model of growth is saddle path stable. In other words, it is stable upon perturbation from the steady state on a one-dimensional line, but is unstable towards a ...
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1answer
50 views

Paper where an integral in the constraint of an optimization problem is treated as infinite sum

I am looking for a paper (or textbook, or even lecture notes example) where there is a problem such as $$ \max f(x) \\ \text{ s.t. } \int g(x) \leq \int c $$ and there is at least some exposition/...
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1answer
88 views

Dynamic optimization with assets as state variable: interpreting capital gains and losses

Given a hamiltonian of the form: \begin{equation} H_{t} = ln(c_{t}) \dot{} e ^{-\rho t} + \lambda_{t}(w+ra_{t}-c_{t}), \end{equation} with $c_{t}$ consumption at time t (the control variable), $\rho &...
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1answer
117 views

Analytically tractable Ramsey model: how to solve ODE for optimal trajectories

In Brunner and Strulik (2002) the authors claim, that the solution of \begin{align} \dot c &= \frac{c}{\sigma}(\alpha k^{\alpha-1} - \delta - \rho)\\ \dot k &= k^\alpha - \delta k - c \end{...