# How to prove dynamic inefficiency in the Solow model?

I know that if capital is below the golden rule level in the Solow model, then we are dynamically efficient, and vice versa.

How is that proven?

## 1 Answer

I give the intuitive explanation in words first, then below is the proof.

Define a Pareto improvement to be where consumption is no lower in every period and higher in at least one period. We're said to be dynamically efficient if there's no possible Pareto improvement and dynamically efficient if there's a possible Pareto improvement.

Assume we start off already in steady state.

Proposition 1. If the savings rate is below the golden-rule level (or equivalently, the steady-state capital stock is below the golden-rule level), then we're dynamically efficient.

Proof. If we switch to a lower savings rate, the new steady-state level of consumption will be lower, and so this cannot be a Pareto improvement.

Conversely, if we switch to a higher savings rate, then consumption in the initial period of the switch will be lower, so that again this cannot be a Pareto improvement.∎

Proposition 2. If the savings rate is above the golden-rule level (or equivalently, the steady-state capital stock is above the golden-rule level), then we're dynamically inefficient.

Proof. We claim that switching to the golden-rule savings rate is a Pareto improvement.

First, note that by switching, the new steady-state level of consumption will be higher.

Next, note that in the initial period (after we've switched), consumption is higher than before, because we've lowered our savings rate.

In fact, in the initial period, consumption will be higher than the new steady state. Consumption will then fall monotonically towards the new steady state level (which is higher than old steady state).

Altogether then, by switching to this lower savings rate, consumption in each period is higher than the old steady-state level (or in other words, than it would've been had we not made the switch). This is thus a Pareto improvement.∎

• Very good answer. The only thing you might want to specify wether those are your own notes and if not perhaps post a link / reference. 👍 – bbecon Jun 14 '18 at 17:35
• @br1: They aren't notes from anywhere. I wrote them myself on LyX (much less painful than writing math here), specially for this answer. I then took a screenshot of the output PDF then put the image here. – Kenny LJ Jun 15 '18 at 1:19