5
votes
Accepted
Prove the uniqueness of steady state
Rearranging the steady state equation
$$
\overline{p}^{\alpha}=\alpha y\overline{p}^{\alpha-1}-
\alpha\overline{p}^{\alpha}-\frac{a+1}{\sigma}
$$
we get
$$
(1 + \alpha)\overline{p}^{\alpha}=\alpha y\...
- 28.7k
4
votes
Looking for discussion on equilibrium vs dynamic models in econometrics
Economists (most of them) build their models assuming most of the time stochastic dynamic equilibrium. So Economics does not contrast "dynamic" with "equilibrium" - it synthesizes them.
It is ...
- 33.4k
4
votes
Accepted
System of differential equations in economics and areas of interest
One of the most fundamental distinctions in economics is that between stocks, measured at a point in time, and flows, measured over a period of time or as instantaneous rates. The construction of ...
- 8,311
4
votes
Accepted
Cross-section experiment with Differences-in-Diffrences estimation
Dif-in-dif (DiD) strategy relies on the identifying assumption of parallel trend. This essentially means that in the absence of the treatment, the control group and the supposedly treatment group ...
- 153
3
votes
Solving stochastic difference equation in New Keynesian model (FTPL textbook derivation)
Here is a revised derivation. Thanks! This is much clearer, I hope.
\begin{equation}
E_{t}\left[ (1-\lambda_{1}^{-1}L)(1-\lambda_{2}L^{-1})\pi_{t+1}\right]
=\sigma\kappa\lambda_{1}^{-1}i_{t}.
\label{...
3
votes
Accepted
What am I doing wrong in the derivation of Bass diffusion model?
You are missing an integration constant
$$
\log\left(\frac{p + qF(t)}{1 - F(t)}\right) = (p + q)t + \color{red}{\tilde{C}}
$$
This constant you can name it whatever you want, I'm going to name it as
...
- 1,216
3
votes
Accepted
Why is there a Walrasian Equilibrium if excess demand goes to infinity as price goes to 0?
Let $z_j(p)$ be the excess demand function for good $j$, where $p := \frac{p_2}{p_1}$ is the relative price between the two goods.
Note it is possible to express the excess demand functions as single ...
- 2,196
3
votes
How to linearize the following difference equation?
We have the recurrence relation
$$x_{k+1} = \frac{x_{k+8}}{x_{k+1}}$$
If the denominator is nonzero, this recurrence relation can be rewritten as follows
$$x_{k+7} = x_k^2$$
Assuming positivity ...
3
votes
Solving Leeper (1991) model
You have the government's flow budget constraint (re-written in real terms):
$b_{t} + m_{t} + \tau_{t} = g + \frac{m_{t-1}}{\pi_{t}} + R_{t-1}\frac{b_{t-1}}{\pi_{t}}$ (1)
Now all you need to do is ...
- 651
2
votes
Accepted
Deriving intertemporal budget constraint from flow constraint
Take your second equation, move it forward one period, and rearrange. You get:
$$ B_t = \frac{p_{t+1} s_{t+1} + \Delta M_{t+1}}{R_{t+1}} + \frac{B_{t+1}}{R_{t+1}} $$
Then, define the nominal ...
- 8,581
2
votes
Accepted
Difference equation in OLG framework
The following approach seems to work in this case:
look for the stead state formula of $T_t$. You can do this by taking (2.8) and the formula for $T_t$, and combine them. Then, get rid of all $t$ ...
- 8,581
2
votes
System of first order partial differential equation
I don't think your reasoning is correct. Some remarks:
If you have an optimization problem, then you assume that you know the objective function, which in your case contains the function $u(x,y)$. As ...
- 10.3k
2
votes
Accepted
Rigorous proof needed: Acemoglu (Intro Growth) Corollary $2.1.2$
The OP correctly identified a mistake here. Since the author claims monotonicity for a general function, let's disprove it for the simple linear case.
Consider
$$x_{t+1} = g(x_t) = -0.5x_t$$
This ...
- 33.4k
1
vote
Solving stochastic difference equation in New Keynesian model (FTPL textbook derivation)
I'll assume throughout that $|\lambda_1| > 1$ and $|\lambda_2| < 1$, since otherwise the sums would diverge.
Start from
$$
\mathrm{E}_t \left[(1 - \lambda_1^{-1} L)(1 - \lambda_2L^{-1})\pi_{t + ...
1
vote
Joint distribution from differential equations
From the fundamental theorem of calculus it follows that the relationship between a function and its partial derivative is given by:
$$ Z(a,b)= \int_0^a \frac{\partial Z}{\partial a}(x,b) dx + Z(0,b), ...
- 3,144
1
vote
Accepted
Solving differential equation in The Economics of Superstar (by Rosen)
The differential equation is of the form
$$y' + f(x)y = q(x)$$
The correct answer in our case is
$$p = -s$$
so that you know what you are targeting.
Namely, it does not depend on $z$. You can ...
- 33.4k
1
vote
Samuelson Acceleration Model Question
There is nothing more to it than the equation
$$Y_t - (c_y+v)Y_{t-1} = C^a + I^a$$
This is a linear non-homogeneous first-order difference equation, and it is non-homogeneous because there is a non-...
- 33.4k
1
vote
Solving rational expectations model - Sims form
If you are planning on using Dynare, you do not need to "solve" the model using Sim's method. Dynare takes care of the solution algorithm for you.
If you want to get to IRFs quickly, I suggest ...
- 41
1
vote
Solving rational expectations model - Sims form
I think I have managed to solve it. However, not the way I was initially hoping. I simplified the stacked matrices using the given conditions and some assumptions. Here is my solution:
Eq. (3) I ...
- 651
1
vote
Rigorous proof needed: Acemoglu (Intro Growth) Corollary $2.1.2$
Since the OP asked for a rigorous proof, here is one.
By Acemoglu's inequality in the first part of his proof, we can separate $\{x(t)\}_{t=0}^{\infty}$ into two subsequences, an increasing ...
- 86
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