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?


1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.