Transitional Dynamics of consumption per capita in solow model

Question

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?

nrivera · 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.

Transitional Dynamics of consumption per capita in solow model

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