15
votes
Accepted
Solow Model: Steady State v Balanced Growth Path
This is when the attempt at accuracy creates confusion and misunderstanding.
Back in the day, growth models were not incorporating technological progress, and led to a long-run equilibrium ...
- 32.9k
10
votes
Accepted
"If $\lambda$ is greater than than 1, the system explodes." Why does the system explode?
Eq. (2.64) can be written as (at first order)
$$
k_{t+1} - k^* = \lambda (k_t - k^*) \tag{1a}
$$
Define the quantity $\kappa_t$ as
$$
\kappa_t \stackrel{\rm def}{=} k_t - k^* \tag{2}
$$
So that
...
- 1,206
5
votes
"If $\lambda$ is greater than than 1, the system explodes." Why does the system explode?
A system is explosive if its coefficients are non-stationary.
Stationary is an important property to have in dynamic models as it tells us that an equilibrium value is obtainable (which is important ...
- 7,994
5
votes
Solow Model: Steady State v Balanced Growth Path
Following the conversation with user @denesp at the comments of my previous answer, I have to clarify the following: the usual graphical device we use related to the basic Solow growth model (see for ...
- 32.9k
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
$$
(1 + \alpha)\overline{p}^{\alpha}=\alpha y\...
- 28k
4
votes
An Optimal Control Model: A Ridiculous Result for a Steady State
One general issue I see is that you try to include uncertainty in a framework developed for a deterministic setup.
What you do is to use expected income in the equation of motion for human capital. ...
- 32.9k
4
votes
Accepted
Solution Method for Infinite-Horizon Maximization Problem
Your first question (regarding constraints on the parameters) can be answered through first and second derivative analysis. In order to satisfy strictly increasing, we need $u'>0$ and to satisfy ...
- 1,558
4
votes
Accepted
Solow Growth Model. Steady State. Can someone explain please?
An axis is characterized not by the variable it measures, but by the unit of measurement. So an axis can measure any number of variables, as long as they are measured in the same measurement unit.
In ...
- 32.9k
4
votes
Solow model with population growth - proof of steady state level of capital per worker
Part 1 of the solution
The fundamental equation of Solow model is (neglecting the $t$ subscripts):
$$ \Delta k= sf(k) -(n+d),$$
where $k= K/N$, $\Delta k= k_{t+1}-k_t$ and $f(k)$ is the intensive ...
- 2,217
3
votes
Accepted
Constant to the power of t in steady state
If the variables are constant, everything in the equation is time-invariant, while $a^t$ will still grow or fall over time (unless $a=1$). This is a contradiction and no steady state exists, unless $a=...
- 479
3
votes
Accepted
Log linearising EUler equation
As you say the first step is to take log of both sides after that you are just applying the rules for logarithms and rearrange.
For example:
$$\ln (XZ)=\ln X + \ln Z$$
$$\ln X/Z= \ln X - \ln Z$$
$$\...
- 50.3k
3
votes
Solution Method for Infinite-Horizon Maximization Problem
Your first question, if it's literally correct, is easy:
The only way for $u'$ to be positive for c=0 is for p=1. if p =1 then sign($\phi$)=sign($\theta$) so that the product is positive. But, since $...
- 2,628
3
votes
Multiple equilibria: which one to select?
I'm not sure I follow the logic on that equation having infinitely many solutions and steady states. In any case, in what follows are some guidelines for equilibrium selection.
It depends a lot on ...
- 10.6k
3
votes
Marginal Product of Capital in the Solow Model
According to your calculations MPK is not increasing in $K$. The Solow model assumes $0< \alpha < 1$, thus $\alpha - 1 < 0$ and $K^{\alpha - 1}$ is decreasing in $K$.
- 28k
3
votes
Accepted
Solow Model, Growth rate of K/L and Y/L in steady state
If $Y = C \cdot X$ where $C$ is constant and $\frac{\dot{X}}{X} = g$ then we can solve for $\frac{\dot{Y}}{Y}$ as follows:
$$ \frac{d}{dt} Y = \frac{d}{dt} C \cdot X = C \cdot \frac{dX}{dt} = C \cdot \...
- 16.2k
3
votes
Accepted
Derive the Demographic Structure in the Steady State
I don't know the paper nor the notation, so I am just guessing here. I gues $N(a,t)$ is the number of agents of age $a$ at time period $t$.
Let's follow the number of age $B$ accross generations:
$$
\...
- 8,692
3
votes
Steady-state savings rate
$\delta = 0.02$ is depreciation.
$p = 0.02$ is population growth.
$g = 0.03$ is technological growth.
$s = 0.14$ is the savings rate.
$Y=0.5\cdot K^{\frac{1}{3}}\left(AN\right)^{\frac{2}{3}}$ is the ...
- 16.2k
3
votes
Accepted
Steady state equilibrium in Solow model with a convex production function
You only provide partial information. E.g., this production function is unusual; is anything else unusual? Is depreciation still linear in $k$? Is the rate of population growth constant? etc.
If ...
- 28k
3
votes
The effect of saving rate on steady state
I suppose you are speaking of the standard Solow Growth model.
Yes, in this case, the economic system will come back to the original steady state equilibrium. This is a consequence of the stability of ...
- 2,217
3
votes
Accepted
Log-linearizing a second order term around the steady-state
If $\Pi_t$ is gross inflation then indeed $(\Pi_t-1)^2$ is a second-order term and is approximately zero. For example, for a reasonable quarterly steady state gross inflation rate of 1.005, the term ...
- 1,614
2
votes
Real Positive Eigenvalue, but Stable Dynamics
If you have are trying to discretize the continuous time model
$$
\dot{\textbf{x}} = A\textbf{x},
$$
then in discrete time you will have
$$
\textbf{x}_{t+1} = B \textbf{x}_t
$$
but $A\neq B$, since $A$...
- 28k
2
votes
Taylor Series Approximation around steady state in Solow
The steady-state value $k^*$ must be a fixed point :
$(1+g)(1+n)k^* = s(k^*)^{\alpha} +(1 - \delta) k^*$
Taking the difference between this equation and the dynamic one :
$(1+g)(1+n)(k_{t+1} - k^*) ...
- 170
2
votes
Solow model, time and steady state
In the model with technological progress the capital per effective worker remains constant, implies that capital per worker grows at the rate of exogenous rate of technological progress. See Barro and ...
2
votes
Accepted
Evaluation around steady state for a specific DSGE Model
I found the error in my derivation: I mistakenly supposet that the steady state of $p_t+1$ equals $\bar{\rho}$.
I did not recognize that p was different from $\rho$ because of the poor quality of my ...
- 153
2
votes
What does steady state mean?
Usually the term steady state is derived from the Solow Model and its derivatives that seek to explain long-term economic growth. The steady state is a state in which the growth rate of the economy is ...
- 1,272
2
votes
Would a zero-growth economy with zero-growth population still have the same GDP?
Smith pointed out that as wealth was growing in any nation, the rate
of profit would tend to fall and investment opportunities would
diminish.
Source
The first half of the sentence makes it ...
- 28k
2
votes
Accepted
Non-trivial steady state
Let's guess that the value function is of the form $a + b \ln(k)$.
Then substituting for $V(k) = a + b \ln(k)$ in the Bellman equation gives:
$$
a + b \ln(k) = \max_{k'}\left(\ln(k^\alpha - k') + \...
- 8,692
2
votes
Accepted
Solow model with population growth - proof of steady state level of capital per worker
Part 2 of the solution
I post a step by step solution, slow passages are the best way to avoid mistakes (I hope).
In the previuos answer we calculated the intensive production function:
$$f(k)=z[\...
- 2,217
1
vote
Accepted
Question on overlapping generations
Is that derivation for $k(t+1)$ correct? Technically, you never reach the steady state, but only asimptotically as $t\rightarrow\infty$, but at infinity the $A(t)$ will also be infinite because it ...
- 130
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