# Why is 10% the necessary upper bound for a negative interest rate?

Source: p 710, Economics, 3 Ed, 2014, by N G Mankiw, M P Taylor

... In that same article, Professor Mankiw recounted a discussion with one of his graduate students at Harvard about a scheme, put forward by the student, whereby the central bank announces that in one year’s time it would pick a digit from one to nine out of a hat and any currency ending in that number would cease to be legal tender. People would thus know that in one year’s time 10 per cent of the cash would cease to be legal tender, so what would they do? The logic is to spend it. The additional spending would increase aggregate demand and act as a boost to the economy. Such a policy might enable central banks to set negative interest rates provided the rate was less than 10% because there would then be an incentive to lend at (say) −4% rather than potentially losing 10%.

Please explain the last sentence (that I bolded). I don't understand the significance of the 10% and −4%. The foregoing discusses NEGATIVE interest rates, the maximum of which must be 0%.
So why bound the interest rate above, with a POSITIVE number (10%)?

## 2 Answers

You're right to ask; the sentence you bolded is unclear in its wording. The effect of randomly invalidating 1/10th of the currency outstanding in one year is that in expectation, cash would have an interest rate of -10%. So when Mankiw says the policy "might enable central banks to set negative interest rates provided the rate was less than 10%", he means "provided the rate was less negative than a -10% rate". This is equivalent to saying that a central bank could set negative rates just so long as the bank set the rate $r$ such that $-10\% \leq r < 0\%$.

Of course, the effects of such a policy in the real world would be much more complicated. If there were certainty about the date of the invalidation, for example, people could just hold cash as usual until just before the invalidation date, then deposit the cash in a bank for a couple days (thus not exposing themselves to negative deposit rates for long periods), and withdraw (valid) currency after the invalidation date.

Assuming your home is safe, would you keep your money in a bank that only gave 96% of it back? (-4% interest rate)

What if your home has mice which eat exactly 10% of your money? Would you then keep your money in the bank that only returns 96%?

And finally, what if every bill has a 10% chance of being eaten? This means that in expected value you lose 10% of your money.

In the above situations, what would be the lowest interest rate you would accept from the bank?