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 ...
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 ...
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. ...
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 ...
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 ...
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 ...
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