<p>In the model with technological progress the <em>capital per effective worker remains constant</em>, implies that capital per worker grows at the rate of exogenous rate of technological progress. See Barro and Martin book, Chapter 1.</p>

CC BY-SA 3.0
2017-02-02T13:37:14.647Z
2017-02-02T13:37:14.647Z
Kshitiz Mishra
<p>Kshitiz is right. If you define <em>effective capital per worker</em> as $\hat k(t) \equiv \dfrac{K(t)}{L(t)Z(t)}$, the equation for the dynamic of $\hat k(t)$ is:</p>

<p>$$ \dot {\hat k} (t) = s {\hat k}^a - (n+g+\delta){\hat k} $$</p>

<p>from where you get the constant effective capital per worker in the steady state:</p>

<p>$$ {\hat k} = \left(\frac{s}{n+g+\delta}\right)^{1/(1-a)} $$</p>

<p>Intuitively, every period the marginal product of capital is increasing because of the exogenous expansion in $Z(t)$. In the standard Solow diagram with <em>capital per worker</em> in the horizontal axis, this exogenous technical change means an outward expansion of the $y$ and $sy$ curves. This higher return to capital induces more investment and an expansion of capital per worker. As technical change occurs every period, this process continues forever.</p>

CC BY-SA 3.0
2017-09-19T12:20:45.693Z
2017-09-19T12:20:45.693Z
luchonacho
<p>Suppose we have a Solow model:</p>

<p>$$
Y(t)=C(t)+I(t)
$$</p>

<p>$$
I(t)=sY(t)
$$</p>

<p>$$
\dot K=I(t)-δK(t)
$$</p>

<p>With a given Cobb-Douglas:</p>

<p>$$
Y(t)=Z(t)K^aL^{1-a}
$$</p>

<p>$$
y(t)=Y(t)/L(t)
$$</p>

<p>$$
k(t)=K(t)/L(t)
$$</p>

<p>$$
y=Zk^a
$$</p>

<p>We also know these:</p>

<p>$$
L=L(t), \dot L/L=n
$$</p>

<p>$$
Z=Z(t), \dot Z/Z=g
$$</p>

<p>Therefore we have:</p>

<p>$$
L(t)=L(0)e^{nt}
$$</p>

<p>$$
Z(t)=Z(0)e^{gt}
$$</p>

<p>We get to this point:</p>

<p>$$
\dot k=sZk^a-(n+δ)k
$$</p>

<p>In the steady state:</p>

<p>$$
k=k^*
$$</p>

<p>$$
k^*=\left(\frac{sZ}{n+δ}\right)^{1/(1-a)}
$$ </p>

<p>The question has to do with the time dimension in Z. Time enters the final expression if we substitute Z with what we found above, but I'm not sure what to make of this.</p>

<p>$$
k^*=\left[\frac{sZ(0)e^{gt}}{n+δ}\right]^{1/(1-a)}
$$</p>

<p>Surely there's something this neophyte is mixing up. I thought that in a steady state capital per worker remains the same.</p>

CC BY-SA 3.0
2017-02-03T01:09:29.723Z
You can write $Z$ only in case where it grows "exogenously". This means that the differential equation $\dot{Z}$ should be an independent equation from the dynamic system. Equivalently, $Z$ evolves in an independant way from the other key variables of the model.
optimal control
<macroeconomics><steady-state><continuous-time><solow><capital>
Solow model, time and steady state
inquirius
# Solow model, time and steady state

(Asked by: username_0 on 2017-02-02)

Suppose we have a Solow model:

$$
Y(t)=C(t)+I(t)
$$

$$
I(t)=sY(t)
$$

$$
\dot K=I(t)-δK(t)
$$

With a given Cobb-Douglas:

$$
Y(t)=Z(t)K^aL^{1-a}
$$

$$
y(t)=Y(t)/L(t)
$$

$$
k(t)=K(t)/L(t)
$$

$$
y=Zk^a
$$

We also know these:

$$
L=L(t), \dot L/L=n
$$

$$
Z=Z(t), \dot Z/Z=g
$$

Therefore we have:

$$
L(t)=L(0)e^{nt}
$$

$$
Z(t)=Z(0)e^{gt}
$$

We get to this point:

$$
\dot k=sZk^a-(n+δ)k
$$

In the steady state:

$$
k=k^\*
$$

$$
k^\*=\left(\frac{sZ}{n+δ}\right)^{1/(1-a)}
$$

The question has to do with the time dimension in Z. Time enters the final expression if we substitute Z with what we found above, but I'm not sure what to make of this.

$$
k^\*=\left[\frac{sZ(0)e^{gt}}{n+δ}\right]^{1/(1-a)}
$$

Surely there's something this neophyte is mixing up. I thought that in a steady state capital per worker remains the same.

**Tags:** macroeconomics, steady-state, continuous-time, solow, capital

> **Comments:**
> **username_1**: You can write $Z$ only in case where it grows "exogenously". This means that the differential equation $\dot{Z}$ should be an independent equation from the dynamic system. Equivalently, $Z$ evolves in an independant way from the other key variables of the model.  

---

## 2 answers

### Answer by username_2 on 2017-02-02. Score: 2

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 Martin book, Chapter 1.


---

### Answer by username_3 on 2017-09-19. Score: 0

username_2 is right. If you define *effective capital per worker* as $\hat k(t) \equiv \dfrac{K(t)}{L(t)Z(t)}$, the equation for the dynamic of $\hat k(t)$ is:

$$ \dot {\hat k} (t) = s {\hat k}^a - (n+g+\delta){\hat k} $$

from where you get the constant effective capital per worker in the steady state:

$$ {\hat k} = \left(\frac{s}{n+g+\delta}\right)^{1/(1-a)} $$

Intuitively, every period the marginal product of capital is increasing because of the exogenous expansion in $Z(t)$. In the standard Solow diagram with *capital per worker* in the horizontal axis, this exogenous technical change means an outward expansion of the $y$ and $sy$ curves. This higher return to capital induces more investment and an expansion of capital per worker. As technical change occurs every period, this process continues forever.


---

