16
votes
Accepted
Transversality Condition in neoclassical growth model
The transversality condition may be more easily understood if we start from a problem with finite horizon.
In the standard version, our objective is to
$$
\max_{\{c_t,k_{t+1}\}_{t=0}^T} \sum_{t=0}^...
- 15.3k
9
votes
Dynamic optimisation
The Dynamic Programming ("Bellman' Equation") formulation incorporates the terminal boundary condition ("transversality conditions") needed in case we use the Lagrangian/Euler equation formulation. ...
- 33.4k
7
votes
Textbook on the mathematics of RBC/DSGE models?
The mathematical theory behind DSGE models can be found in any textbook on stochastic dynamic optimisation.
One common reference that economists use for this is Stokey, Lucas and Prescott. Of course, ...
- 2,156
6
votes
Transversality Condition in neoclassical growth model
In my opinion, the best derivation is by logic. Think about it this way: If the only thing we are telling the household is to maximize its utility, optimal behaviour would then be just making infinite ...
- 870
6
votes
Accepted
reference request - dynamic discrete time optimization methods
As suggested by 1muflon1:
I don't really use dynamic discrete optimization (yet) in my work, but I did explore several books once. Here are some suggestions:
Economic Dynamics: Theory and Computation ...
- 1,705
5
votes
Accepted
What is the difference between identification, calibration and estimation?
I agree with 1muflon1, but allow me to add some more nuance.
Identification and calibration can be meant to express a subset of estimation. Any identified coefficient is also an estimate, but not ...
- 6,128
5
votes
What is the difference between identification, calibration and estimation?
Identification and estimation are often used interchangeably (at least that’s my observation from attending conferences and reading papers) but according to econometric literature there is a subtle ...
- 54.4k
5
votes
Monetary policy optimization
You are unfortunately mistaken. DSGE models are at the heart of monetary policy and the most widely used class of models in this field. To work in monetary there is no real way around learning DSGE.
...
- 6,128
5
votes
What is the 'fixed point problem'?
In mathematics a fixed point is, in general, in a mapping of a space in itself, a point that corresponds to itself.
In particular, fixed points for functions $f(x)$ are value of the variables of the ...
- 3,312
5
votes
Accepted
How can I show convexity of this value function?
Suppose that $u(C,l)=\sqrt{C}-l^2$ and $f(l,A)=\big(l+g(A)\big)^2$, where $g$ is any function of $A$ that is not convex.
Then $$u\big(f(l,A),l\big)=l+g(A)-l^2.$$
The optimal labor supply is given by $...
- 11.7k
4
votes
Dynamic optimisation
I'm unsure exactly where your struggle is, in general. To try and address your issues, what kind of boundary conditions might we desire in dynamic problems?
Consider the two period consumption-...
- 6,578
4
votes
An Optimal Control Model: A Ridiculous Result for a Steady State
One general issue I see is that you try to include uncertainty in a framework developed for a deterministic setup.
What you do is to use expected income in the equation of motion for human capital. ...
- 33.4k
4
votes
Accepted
Dynamic programming, optimal consumption-savings (finite horizon) problem
Your value function is as follows:
$$
V_t[w] = \max_{c_t \in[0,w]} \left\{u(c_t) + \frac{1}{2}V_{t+1}[\alpha(w_t - c_t)] + \frac{1}{2}V_{t+1}[\beta(w_t-c_t)] \right\}
$$
with the terminal condition
$$
...
- 1,869
3
votes
Good book/article that goes into depth about transversality conditions?
A textbook on dynamic optimization which treats transversality conditions at some length is Chiang Elements of Dynamic Optimization. See especially sections 7.2 - 7.4 and 9.1.
- 8,311
3
votes
Preference for consumption smoothing and actual smoothing
The argument relayed in the question as regards consumption smoothing is flawed. Consumption smoothing does not mean consumption equality over periods, but rather, tendency to avoid corner solutions, ...
- 33.4k
3
votes
Accepted
Paper where an integral in the constraint of an optimization problem is treated as infinite sum
Some papers that come to my mind are ;
-Calvo and Obstfeld (1988, Econometrica)
-Endres et al (2014, Resource and Energy Economics)
-Traeger et al. (2012, European Economic Review)
In these papers,...
- 2,115
3
votes
Accepted
Analytically tractable Ramsey model: how to solve ODE for optimal trajectories
The differential equation
$$\dot k = \frac{1}{\sigma} k^\alpha - \delta k$$
has the structure of a Bernoulli equation. We solve it by the following transformation steps:
1) Multiply throughout by $k^{...
- 33.4k
3
votes
Question about Euler condition
The Euler Equation typically refers to the interior optimal choice between consumption today and tomorrow (or some similar intertemporal choice). That is, it equalizes the marginal utility of ...
- 31
3
votes
Accepted
What is the result of the Bellman Equation
...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 ...
- 2,619
3
votes
Accepted
How can I formulate the following optimization problem?
If you want to determine how much carbon dioxide should be omitted by solving an optimization problem, then a constraint on the quantity of $CO_2$ isn't quite what you need. The normal constraint on ...
- 8,311
2
votes
Accepted
Update of value function in continuous time - HJB
You iterate towards a fixed point, so you want to reach a situation where plugging in your current iterated value produces itself. Now using your notation, we are told that we should calculate
$$V_{n+...
- 33.4k
2
votes
Accepted
Saddle path equilibrium on financial market with rational expectations
Suppose you have a dynamic system
$$ x_{t+1} = Ax_{t} $$
with a stationary point (or steady state as used in growth or RBC literature), say, $x^*$, i.e. $x^{*} = Ax^{*}$.
Now, consider the following ...
- 36
2
votes
Good book/article that goes into depth about transversality conditions?
I think that references in this field can be of interest and useful also at distance of time from the original question.
Transversality conditions are a complex subject in calculus of variations and ...
- 3,312
2
votes
Accepted
What is the difference between a perfect foresight equilibrium and a rational expections equilibrium?
This is not a formal definition, but a useful piece of intuition. I think that the best way to think about it is that when there is uncertainty in a model it arises mainly in two forms either there ...
- 4,188
2
votes
Accepted
Are overlapping generation (OLG) models extensions of a DSGE model?
You can find OLG models that do not classify as DSGE (in particular, the model might not be stochastic) as well as DSGE with overlapping generations (contrary to those with infinitely lived agents).
...
- 862
2
votes
Accepted
What is meant by the abbreviation 'MSV solution', used in the context of DSGE modeling?
Minimum state variable (MSV) solution is a special technique used to find an unique equilibrium with desirable properties in DSGE models. Often DSGE models can have multiple paths that will satisfy ...
- 54.4k
2
votes
Solving a HJB with a probability to transit to a new state
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 ...
- 151
2
votes
Introduction of an asset tax in the AK model
I think your math is mostly correct but I have to admit that I am not used to AK models.
A short answer for your main question: is it ok to introduce taxation in the model without including the ...
- 1,816
2
votes
Accepted
Resolution - Ramsey growth model with per capita variables
The reason why the $e^{nt}$ term is there is because you want to multiply the whole utility by the number of people. You are actually not substituting consumption into the utility but multiplying the ...
- 54.4k
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