<p>In general, a steady state in discrete time is that the first difference is zero at some $t=t_0$, and that it remains so: $$y_t-y_{t-1}=x_t-x_{t-1}=0$$ for all $t\geq t_0$.</p>

<p>Hence, in a steady state, $y_t$ and $x_t$ are constant at some values $y$ and $x$. Thus, $y=\beta(y+\gamma z)$, which implies $$y=\frac{\beta\gamma z}{1-\beta},$$ and $x=\rho x+y$, which implies $$x=\frac{y}{1-\rho}=\frac{\beta\gamma z}{(1-\rho)(1-\beta)}.$$</p>

CC BY-SA 3.0
2017-02-22T22:25:36.740Z
2017-02-23T07:43:23.933Z
2017-02-23T07:43:23.933Z
Elias
<blockquote>
  <p>Consider the system of two equations: $$y_t=\beta\mathbb{E}_t[y_{t+1}+\gamma\cdot z_{t+1}]$$ $$x_t=\rho x_{t+1}+y_t$$ $$z_t=(z_0-Z_t)e^{-at}+z_T$$
  Determine the steady state.</p>
</blockquote>

<p>The solution manual says: $$y=\frac{\beta\gamma z}{1-\beta}$$ $$x=\frac{\beta\gamma z}{(1-\beta)(1-\rho)}$$
Now I have no idea how to derive these values. The lecture sheets don't mention anything, except that the steady state is solved when we have the value of the shock $z_t$ (intial and terminal). Could anyone explain me the mathematical derivation of the steady state?</p>

CC BY-SA 3.0
2016-07-22T15:55:35.933Z
The system of two equations consists of three equations. It is also unclear what the variables are and whether time is discreet or continuous. Furthermore I do not know what you are asking. Do you know what a steady state is, just not in this system, or do you need the definition and its mathematical properties?
Giskard
CC BY-SA 3.0
2016-07-22T21:42:45.670Z
I assume it is a discrete model, although not explicitly given in the excercise. $z_t$ is the shock variable, where $T$ is the 'terminal' time. I am mostely interested in the mathematical definition. Although I know it is the derivative w.r.t. $t$, I can't seem to understand how we obtain the results here - taking the derivative w.r.t. $t$ is not exactely straight forward in this case.
Holograph
CC BY-SA 3.0
2016-07-22T23:54:57.407Z
Please edit this information into the body of the question. Also it may be wise to look up what a steady state is exactly and then rephrase the whole question because there seems to be some misunderstanding.
Giskard
CC BY-SA 3.0
2016-10-17T14:39:48.377Z
@Holograph which manual are you using?
An old man in the sea.
CC BY-SA 3.0
2017-02-24T07:48:09.297Z
This is a homework-type of question that shows no effort.
luchonacho
<mathematical-economics><steady-state>
How to determine the Steady State of this model?
Holograph
# How to determine the Steady State of this model?

(Asked by: username_0 on 2016-07-22)

> Consider the system of two equations: $$y\_t=\beta\mathbb{E}\_t[y\_{t+1}+\gamma\cdot z\_{t+1}]$$ $$x\_t=\rho x\_{t+1}+y\_t$$ $$z\_t=(z\_0-Z\_t)e^{-at}+z\_T$$
> Determine the steady state.

The solution manual says: $$y=\frac{\beta\gamma z}{1-\beta}$$ $$x=\frac{\beta\gamma z}{(1-\beta)(1-\rho)}$$
Now I have no idea how to derive these values. The lecture sheets don't mention anything, except that the steady state is solved when we have the value of the shock $z\_t$ (intial and terminal). Could anyone explain me the mathematical derivation of the steady state?

**Tags:** mathematical-economics, steady-state

> **Comments:**
> **username_1**: The system of two equations consists of three equations. It is also unclear what the variables are and whether time is discreet or continuous. Furthermore I do not know what you are asking. Do you know what a steady state is, just not in this system, or do you need the definition and its mathematical properties?  
> **username_0**: I assume it is a discrete model, although not explicitly given in the excercise. $z_t$ is the shock variable, where $T$ is the 'terminal' time. I am mostely interested in the mathematical definition. Although I know it is the derivative w.r.t. $t$, I can't seem to understand how we obtain the results here - taking the derivative w.r.t. $t$ is not exactely straight forward in this case.  
> **username_1**: Please edit this information into the body of the question. Also it may be wise to look up what a steady state is exactly and then rephrase the whole question because there seems to be some misunderstanding.  
> **username_2**: @username_0 which manual are you using?  
> **username_3**: This is a homework-type of question that shows no effort.  

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## 1 answers

### Answer by username_4 on 2017-02-22. Score: 1 | Accepted answer

In general, a steady state in discrete time is that the first difference is zero at some $t=t\_0$, and that it remains so: $$y\_t-y\_{t-1}=x\_t-x\_{t-1}=0$$ for all $t\geq t\_0$.

Hence, in a steady state, $y\_t$ and $x\_t$ are constant at some values $y$ and $x$. Thus, $y=\beta(y+\gamma z)$, which implies $$y=\frac{\beta\gamma z}{1-\beta},$$ and $x=\rho x+y$, which implies $$x=\frac{y}{1-\rho}=\frac{\beta\gamma z}{(1-\rho)(1-\beta)}.$$


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