<p>You're right, since in basic Solow model (with population growth and no technological progress) macroeconomic closure condition (in aggregate terms) is:</p>
<p><span class="math-container">$$Y(t) = C(t) + I(t)$$</span></p>
<p>where <span class="math-container">$$I(t) = sY(t)$$</span></p>
<p>Now replacing the second in the first equation:</p>
<p><span class="math-container">$$Y(t) = C(t) + sY(t)$$</span></p>
<p>Factorizing we arrive at the equation you stated:</p>
<p><span class="math-container">$$C(t) = (1-s)Y(t)$$</span></p>
<p>Taking the first equation in per worker terms (multiplying both sides by <span class="math-container">$1/L_t$</span>) and differencing with respect to <span class="math-container">$t$</span>:</p>
<p><span class="math-container">$$\dot c = (1-s)\dot y$$</span></p>
<p>Using the identity <span class="math-container">$c(t) =(1-s)y(t)$</span>, dividing the previous expression (as you did) we get:</p>
<p><span class="math-container">$$\frac{\dot c}{c} = \frac{\dot y}{y}$$</span></p>
<p>I hope this is what you are looking for.</p>

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2021-01-04T22:52:34.837Z
2021-01-04T22:52:34.837Z
manifold
<p>I can write the transitional dynamics of output per capita as follows
<span class="math-container">$$y=f(k)$$</span></p>
<p>Take its derivative with respect to time t</p>
<p><span class="math-container">$$ \dot{y} =f’(k) \dot{k}$$</span></p>
<p>Divide it by <span class="math-container">$k/k$</span></p>
<p><span class="math-container">$$ \dot{y} =f’(k) \frac{\dot{k}}{k} k$$</span></p>
<p>And finally divide both side by <span class="math-container">$y=f(k)$</span></p>
<p><span class="math-container">$$\frac{\dot{y}}{y} = \frac{f’(k)}{f{k}}\frac{\dot{k}}{k}k$$</span></p>
<p>Now I want to derive this for consumption for per capita</p>
<p>I guess
<span class="math-container">$$c= (1-s)y$$</span></p>
<p><span class="math-container">$$\frac{\dot{c}}{c} =\frac{\dot{y}}{y} $$</span></p>
<p>So is this transition dynamics for consumption per capita correct? How can I obtain this?</p>

<economic-growth><self-study><solow>
Transitional Dynamics of consumption per capita in solow model
studentp
# Transitional Dynamics of consumption per capita in solow model

(Asked by: username_0 on 2021-01-04)

I can write the transitional dynamics of output per capita as follows
$$y=f(k)$$

Take its derivative with respect to time t

$$ \dot{y} =f’(k) \dot{k}$$

Divide it by $k/k$

$$ \dot{y} =f’(k) \frac{\dot{k}}{k} k$$

And finally divide both side by $y=f(k)$

$$\frac{\dot{y}}{y} = \frac{f’(k)}{f{k}}\frac{\dot{k}}{k}k$$

Now I want to derive this for consumption for per capita

I guess
$$c= (1-s)y$$

$$\frac{\dot{c}}{c} =\frac{\dot{y}}{y} $$

So is this transition dynamics for consumption per capita correct? How can I obtain this?

**Tags:** economic-growth, self-study, solow

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## 1 answers

### Answer by username_1 on 2021-01-04. Score: 2 | Accepted answer

You're right, since in basic Solow model (with population growth and no technological progress) macroeconomic closure condition (in aggregate terms) is:

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

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

Now replacing the second in the first equation:

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

Factorizing we arrive at the equation you stated:

$$C(t) = (1-s)Y(t)$$

Taking the first equation in per worker terms (multiplying both sides by $1/L\_t$) and differencing with respect to $t$:

$$\dot c = (1-s)\dot y$$

Using the identity $c(t) =(1-s)y(t)$, dividing the previous expression (as you did) we get:

$$\frac{\dot c}{c} = \frac{\dot y}{y}$$

I hope this is what you are looking for.


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