# 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.