# The effect of saving rate on steady state

If there is a increase in saving rate and fall back to original level afterwards, what is the effect on steady state in the short run and long run? I am confused about this is because I thought increase in saving rate would cause a increase in gross investment and lead to a new steady state, but the saving rate falling back to original level and I am not sure whether will return to the original or not.

I suppose you are speaking of the standard Solow Growth model.

Yes, in this case, the economic system will come back to the original steady state equilibrium. This is a consequence of the stability of the steady state equilibria in the standard Solow model, that is, if the system is not on its steady state equilibrium, there are forces that lead it to equilibrium.

It can be shown graphically that a temporary increase of saving rate, with the saving rate first increasing and then falling back to the original level, causes the system to go back to the original steady state.

The fundamental equation in the standard Solow model (in continuous time) is:

$$\dot k= sf(k) - (n+d)k,$$

where $$k$$ is capital/labour ratio, $$f(k)$$ is the intensive production function, $$n$$ the rate of growth of population, $$d$$ depreciation, $$\dot k$$ the derivative of $$k$$ with respect to time.

When $$\dot k=0$$ the system is on its steady state path, where $$k$$ is constant and it can be shown that all per capita quantities are constant.

This dynamics can be represented graphically as usual:

Suppose the system begins at the steady state $$E$$, with a steady state of capital per worker of $$k^*$$. If $$s$$ increases to the new level $$s'$$, the $$sf(k)$$ curve shifts upward (the red curve), and we have the new steady state $$E'$$. The dynamics of $$k$$ is now represented by the red arrows, pointing to the new steady state level of $$k$$, that is $$k^{**}.$$ During the process of adjustment there is a transitional dynamics of $$k$$, that is $$k$$ increases, but this dynamics will stop when $$k$$ has reached its new steady, long run, constant value $$k^{**}$$.

But the saving rate, at a subsequent time, falls back to the previous value. The $$s'f(k)$$ curve shifts downward and becomes again the blue curve. So, the opposite situation occurs: now the steady state is again $$E$$, and the dynamics of $$k$$ is described by the blue arrows: the blue arrows point to $$k^*$$, there is a transitional dynamics toward it, which stops when the system has again reached the old steady state $$E$$.

The system will therefore come back and remain in the old steady state equilibrium.$$^1$$

$$^1$$ For a similar, more detailed, exposition, but with a different graph, see Acemoglu, Modern Economic Growth, chap. 2.