We assume that the intertemporal equilibrium exists and is unique. As noted when characterising the uniqueness of the temporary equilibrium, assuming that $\sigma>1$ is a sufficient condition. Therefore, the dynamics of capital are given by:

$$k_{t+1} = \frac{1}{1+n}s(\omega(k_{t}), f^{\prime}(k_{t+1})).$$

Hence, $k_{t+1}$ is an implicit function of $k_{t}$:

$$k_{t+1} = g(k_{t}).$$

The steady states of this economy can be found solving the previous equation for $k_{t+1} = k_{t}=\bar{k}.$

$$\bar{k} = \frac{1}{1+n}s\left(\omega(\bar{k}), f^{\prime}(\bar{k})\right).$$

### Steady state with $\bar{k}=0$

A steady state with $\bar{k}=0$ is only possible if $f(0)=0$ or, equivalently, $\omega(0)=0.$ We may call this steady state an autarky steady state. In that case, since $\omega(0)=0$ and we know that individuals save a fraction of their income:

$$s(\omega(0), R(0)) < \omega(0) = 0.$$

Hence, given zero capital, savings are equally zero and capital remains there, constituting a steady state.

Alternatively, if it is possible to produce without capital (CES production function with large enough substitutability), the autarkic steady state does not exist.

#### Example of the autarkic steady state

Take $F(K,L)=(\alpha K^{\rho} + (1-\alpha)L^{\rho})^{\frac{1}{\rho}}$ and $u( c ) = \log( c ).$ Furthermore, assume a low elasticity of substitution, $\rho « 0.$ In intensive form, production equals:

$$f(k_{t}) = (\alpha k_{t}^{\rho} + (1-\alpha))^{\frac{1}{\rho}}.$$ Wages and savings are:

$$\omega(k_{t}) = (1-\alpha)(\alpha k^\rho + (1-\alpha))^{\frac{1}{\rho}-1}$$ $$s_{t} = \frac{\beta}{1+\beta}w_{t}.$$

Then, capital accumulation becomes:

$$k_{t+1} = \frac{1}{1+n}\frac{\beta}{1+\beta}(1-\alpha)\left[\alpha k_{t}^\rho + 1 - \alpha \right]^\frac{1-\rho}{\rho}.$$

With low substitutability, $\rho < 0$, zero is always a possible steady state. There can be two additional, positive steady states depending on the curvature of $k_{t+1}.$

Similarly, the Cobb-Douglas case $\rho \rightarrow 0$ also has zero as a steady state. However, it is unstable.

The economy can also present interior steady states with $\bar{k}>0.$ Unlike the Ramsey model, here we can have stable and unstable steady states, as we shall see. Moreover, the configuration varies depending on the production and utility functions.

## Stability of the steady state

This model can feature one or several steady states. This depends on the characteristics of the production and utility functions.

First, the dynamics of $k$ are monotonic: either continuously increase or decrease. This follows from assuming a unique intertemporal equilibrium, see also. This extra assumption also implies that the intermporal elasticity of substitution is not too small. In particular, we operationalise (p. 25 in de la Croix and Michel.) it as:

$$\forall c>0, \quad u^\prime( c ) + c u^{\prime \prime}( c ) \geq 0 \Leftrightarrow \sigma( c ) \geq 1.$$

Due to the fact the monotonicity of the dynamics, capital can converge either to $0,, \bar{k}\in (0,+\infty),,$ or $+\infty.$

### The economy never goes to $k = +\infty$

This is equivalent to saying that starting with a a high level of capital $k_{0}$, the dynamics imply $k_{t+1} < k_{t}.$

First, we have that:

$$g(k) = \frac{1}{1+n}s(\omega(k),f^\prime(\underbrace{g(k)}_{k_{t+1}}))<\frac{\omega(k)}{1+n}.$$ This holds because savings $s$ are always a fraction of wages $\omega.$

Second, we show that:

$$\lim_{k \rightarrow +\infty}\frac{g(k)}{k} = 0.$$

To see the intuition for this limit, assume for a moment it holds. Then, it must be that for large enough $k,, g(k) = k_{t+1} < k_{t}$ and the dynamics are decreasing and converge to some $\bar{k}.$ In this case, $\bar{k}$ is the largest steady state, and because the dynamics are monotonic, the economy never goes to the other side of the steady state.

We show next that $\lim_{k \rightarrow +\infty}\frac{g(k)}{k} = 0.$ We do so by showing that $\lim_{k \rightarrow +\infty} \color{green}{\frac{\omega(k)}{k}} = 0.$

Detailed explanation We start by noting that:

$$\color{green}{\frac{\omega(k)}{k}} = \color{red}{\frac{f(k)}{k}} - \color{blue}{f^\prime (k)} \leftarrow \omega(k) = f(k) - f^\prime (k) k.$$

When $k \rightarrow +\infty$, the first term $\color{red}{\frac{f(k)}{k}}$ admits a positive limit, let’s call it $\mathcal{l}_{1}.$ This is because:

1. $\color{red}{\frac{f(k)}{k}} > 0.$
2. $\color{red}{\frac{f(k)}{k}}$ is decreasing: $$\frac{\partial \color{red}{\frac{f(k)}{k}}}{\partial k} = \frac{f^\prime(k)k-f(k)}{k^{2}} = - \frac{f(k) - f^\prime (k) k}{k^{2}} = - \frac{\omega(k)}{k^{2}} < 0.$$

The second term $\color{blue}{f^\prime (k)}$ is decreasing and positive, thus it admits a limit $\mathcal{l}_{2}$ when $k \rightarrow +\infty.$

Apply the mean value theorem evaluated at $k$ and $2k$:1

$$f(2k) - f(k) = (2k-k)f^\prime(k(1+\theta)), \quad \mathrm{with}, \theta \in (0,1).$$ Then, $$\frac{2f(2k)}{2k} - \frac{f(k)}{k} = f^\prime(k(1+\theta))$$ Taking the limit when $k \rightarrow +\infty$ we obtain $2\mathcal{l}_{1}-\mathcal{l}_{1} = \mathcal{l}_{2} \implies \mathcal{l}_{1} = \mathcal{l}_{2}.$

Finally,

$$\lim_{k \rightarrow + \infty}\color{green}{\frac{\omega(k)}{k}} =\lim_{k \rightarrow +\infty}\color{red}{\frac{f(k)}{k}} - \color{blue}{f^\prime (k)}= \mathcal{l}_{1} - \mathcal{l}_{2} = 0.$$

Since $\lim_{k \rightarrow +\infty} \color{green}{\frac{\omega(k)}{k}}=0$ and $\frac{g(k)}{k} < \color{green}{\frac{\omega(k)}{k}}$ then $\lim_{k \rightarrow +\infty}\frac{g(k)}{k} = 0.$ In conclusion, if the inital value of capital is large enough, the dynamics are decreasing and $k$ converges to an interior steady state $\bar{k}.$

## Examples

### Example 1: Log-utility and Cobb-Douglas

Suppose that $u(c,d) = \log ( c ) + \beta \log(d)$ and $f(k_t) = Ak_t^\alpha.$

The wage rate is given by $$w_t = f(k_{t}) - f^\prime (k_{t}) = (1-\alpha)Ak_{t}^\alpha$$

and the savings function is obtained solving $$u^\prime(w_{t} - s_{t}) = \beta R_{t+1} u^\prime (R_{t+1}s_{t}).$$

Rearranging and isolating $s$ we are left with the savings function

$$s_{t} = \frac{\beta}{1+\beta}w_{t}.$$

The dynamics are given by the capital accumulation law

$$k_{t+1} = \frac{1}{1+n}s_{t} = \frac{1}{1+n}\frac{\beta}{1+\beta}w_{t} = \frac{1}{1+n}\frac{\beta}{1+\beta}(1-\alpha)Ak_{t}^\alpha.$$

The model with log-utility and Cobb-Douglas production functions has two steady states: zero and a unique positive steady state. We can solve for them:

$$\bar{k} = \frac{1}{1+n}\frac{\beta}{1+\beta}(1-\alpha)A\bar{k}^\alpha \implies \bar{k} = \begin{cases} 0 \\\ \left(\frac{1}{1+n}\frac{\beta}{1+\beta}(1-\alpha)A\right)^{\frac{1}{1-\alpha}} \equiv \phi^\frac{1}{1-\alpha}\end{cases}.$$

Local dynamics: stability

We check the stability of the two steady states using the first derivative evaluated at the steady state. Note: in general we would use the Jacobian matrix, but the OLG model has only one equation in one variable.

Remember that, for a discrete-time system, a non-hyperbolic steady state is stable if it the determinant of the Jacobian (here the absolute value of the derivative) evaluated at the steady state is in the unit circle $(-1, 1).$

In our case:

$$g(k) = k_{t+1} = \frac{1}{1+n}\frac{\beta}{1+\beta}(1-\alpha)Ak_{t}^\alpha$$ $$g^\prime(k) = \frac{1}{1+n}\frac{\beta}{1+\beta}(1-\alpha)\alpha Ak_{t}^{\alpha-1}> 0.$$

Then

$$\bar{k}=0 \implies g^\prime (0) = + \infty \quad \rightarrow \mathrm{unstable}$$ $$\bar{k}=\phi^\frac{1}{1-\alpha} \implies g^\prime(\phi^\frac{1}{1-\alpha}) = \phi \alpha \phi^\frac{\alpha-1}{1-\alpha} = \alpha \in (0,1) \quad \rightarrow \mathrm{stable}.$$

Global dynamics

Suppose we have $k_{t+1} \geq k_{t}$. We show that the dynamics are bounded above by the steady state $\bar{k} = \phi^\frac{1}{1-\alpha}.$

$$k_{t+1} \geq k_{t} \Leftrightarrow g(k_t) \geq k_t \Leftrightarrow \phi k_t^\alpha \geq k_t \Leftrightarrow \phi \geq k_t^{1-\alpha} \Leftrightarrow \bar{k} \geq k_t.$$

### Log-utility and CES technology

Under a CES specification, $y_{t} = A \left[ \alpha k_{t}^\rho + (1-\alpha) \right]^\frac{1}{\rho}, \rho \leq 1$ wages are given by $$\omega_{t} = (1-\alpha)\left[\alpha k^\rho + 1 -\alpha \right]^{\frac{1-\rho}{\rho}}.$$ Hence, the dynamics of capital are governed by:

$$k_{t+1} = g(k_t) = \frac{1}{1+n}s_{t} = \frac{1}{1+n}\frac{\beta}{1+\beta}A(1-\alpha)\left[\alpha k_{t}^\rho + 1 - \alpha\right]^\frac{1-\rho}{\rho}.$$ The curve $g(k)$ is increasing:

$$g^\prime (k) = \frac{\beta A (1-\alpha)}{(1+n)(1+\beta )}(1-\rho)\left[ \alpha k^\rho + 1 - \alpha \right]^\frac{1-2 \rho}{\rho}k^{\rho -1} > 0.$$

Depending on the value of $\rho$ we can have different configurations regarding the steady states. Ultimately, what matters is the concavity of the function $g(k_{t}).$

$$g^{\prime \prime}(k_t) = -\frac{1}{1+n}\frac{\beta}{1+\beta}A(1-\rho)\alpha\left[\alpha k^\rho+1-\alpha\right]^\frac{1-3\rho}{\rho}k^{\rho-2}\left(\rho \alpha k^\rho +(1-\alpha)(1-\rho)\right).$$

#### Unique steady state, $\bar{k} > 0$

If $\rho \in (0,1)$, the second derivative is clearly negative, and $g$ is concave. Moreover, $g(0)>0$, so $\bar{k} = 0$ cannot be a steady state:

$$g(k_{t}) = \frac{\beta A(1-\alpha)}{(1+n)(1+\beta)}(\alpha k^\rho + 1 - \alpha)^\frac{1-\rho}{\rho} \implies g(0)=\frac{\beta A(1-\alpha)^\frac{1}{\rho}}{(1+n)(1+\beta)}> 0.$$ In that case, the economy displays a unique steady state, and it is globally stable.

#### Unique $\bar{k}=0$ or multiple steady states

If $\rho < 0,$ the function $g(k)$ changes concavity. In particular, there exists a level $\hat{k}$ such that: $$g^{\prime \prime}(k) > 0, \mathrm{if} , k < \hat{k}.$$ $$g^{\prime \prime}(k) < 0, \mathrm{if} , k > \hat{k}.$$ with $$\hat{k} = \left( - \frac{(1-\alpha)(1-\rho)}{\rho \alpha}\right)^\frac{1}{\rho} > 0 , \mathrm{since}, \rho <0.$$ Moreover, $g(0)=0$ so $\bar{k}=0$ can be a steady state.

In this case, there are two sub-cases:

1. If $k_{t+1} < k_{t} \implies \frac{\omega(k)}{k} < (1+n)\frac{1+\beta}{\beta}$ then the trajectory of $k_{t+1}$ is always below $k$. Then, there is one unique steady state: $\bar{k}=0.$ The economy converges towards this unique steady state. We can derive the condition above from: \begin{align} k_{t+1} < k_{t} & \implies \frac{1}{1+n}s\left(\omega(k_{t}),R_{t+1}\right) < k_{t} \implies \\\ \frac{1}{1+n}\frac{\beta}{1+\beta}w_t < k_{t} & \implies \frac{w_{t}}{k_{t}} < (1+n)\frac{1+\beta}{\beta}. \end{align}

2. If there is some $k_{t+1} > k_{t} \implies \frac{\omega(k)}{k} > (1+n)\frac{1+\beta}{\beta}$ then there are two positive steady states $\bar{k}_{a} < \bar{k}_{b}$ plus the zero steady state $\bar{k}=0.$

1. All trajectories starting at $k_0 < \bar{k}_{a}$ converge to $\bar{k}=0$ and $\bar{k}=0$ is locally stable in the range $[0,\bar{k}_{a}].$ The steady state with $\bar{k}=0$ is a poverty trap: whenever the economy starts with a low enough level of capital, it will always converge towards the $\bar{k}=0$ steady state.
2. Trajectories starting at $k_0=\bar{k}_{a}$ remain at $\bar{k}_{a}$. This steady state is unstable.
3. Finally, trajectories starting with $k_0 > \bar{k}_{a}$ converge towards $\bar{k}_{b}$, which is a locally stable steady state in $(\bar{k}_{a}, +\infty).$

1. The mean value theorem states that, for a continuous function on $[a,b],$ there is a point $c \in (a,b)$ such that the derivative a $c, f^\prime( c)$ coincides with the slope of the secant between $a$ and $b, \frac{f(b) - f(a)}{b-a}.$ More information. ↩︎

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