5
votes
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Deriving the Euler equation from a Continuous Time Dynamic Programming Problem (HJB)
As already commented, the equation you probably meant is
$$
\rho V(k)= \sup_c \{\, u(c) + V'(k) ( f(k) -\delta k -c ) \,\}.
$$
I have never seen this equation called the HJB equation (probably missing ...
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5
votes
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What are the assumptions made about fixed points in the dynamics equations of Recursive macroeconomics?
On pages 53-55 of the Stokey, Lucas, with Prescott (1989) book they discuss the Contraction Mapping Theorem. This theorem guarantees existence and uniqueness of the solution (one fixed point).
The ...
- 487
5
votes
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can someone show more steps in one of Sargent's equations?
One confusion that might have arisen is the fact that the paper "overuses" the index $t$. It uses it both for the index of the variable $y_t$ to take the derivative with, and for the ...
- 10.3k
4
votes
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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
$$
...
- 1,869
4
votes
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Bellman Equation with Two Discount Factors
No. Consider the following problem: Each period, one total unit of consumption falls from the sky that can be distributed between the two agents in any way. Their per-period utility is simply how much ...
- 11.7k
4
votes
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What is a what is meant by a "workhorse" macroeconomic model
It is meant in a way that these models are highly influential and many papers are based on them.
From my own experience it is impossible to attend macroeconomic conference without having couple of ...
- 54.4k
3
votes
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The Cake Eating Problem with Depreciation (Modelling difficulties)
It would seem that the way you've formulated your production function/law of motion has introduced double counting into the problem. Note that substituting 1 and 2 into 3 gives:
$$k_{t+1}=(1-\delta)(...
- 605
2
votes
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Do policy functions exist for Finite Horizon Dynamic programming problems?
No, they cannot by definition.
To derive a time-invariant policy function, you need to have an infinite horizon problem. This is because the structure of the solution remains the same, no matter when ...
- 1,631
1
vote
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Understanding Duality between Individual and Collective Maximization in Macroeconomic Models
The proof of the first welfare theorem is almost the same as the one you are familiar with from MWG. The main difference is that if you have recursive budget constraints, you have to show that you can ...
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1
vote
Regarding the arbitrariness of states and controls
The change from $c$ as a decision variable towards $k'$ as a decision variable is by a simple change of variables.
In the original setup, $k$ is the state and $c$ is the control (decision) variable. ...
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1
vote
Accepted
When does it make sense to use variational methods, versus dynamic programming, versus nonlinear control methods so solve DSGE models
The main two tools for economists solving infinite-horizon constrained optimization problems in discrete-time, as in your example problem, are:
Karush–Kuhn–Tucker (KKT) conditions - which is a ...
- 487
1
vote
Proof monotonicity on Blackwell sufficient conditions
Blackwell's sufficiency theorem requires (1) Monotonicity and (2) Discounting.
Checked my notes from the first year macro, and this is what I have:
Theorem Blackwell's Sufficient Conditions for a ...
- 156
1
vote
More than one Bellman Equation
(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 ...
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