Transitional Dynamics of consumption per capita in solow model 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?
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.