The Evolution of Small Change: Beliefs, Experiments, and ...

8 Introduction
These constraints capture how small denomination coins can be used to
purchase expensive items, but large denomination coins cannot be used
to buy cheap items. In this way, the second cash-in-advance constraint
embodies a demand for ‘change’. The addition of the cash-in-advance
constraint for small change (denoted as the penny-in-advance or p.i.a.
constraint) adds an occasionally binding constraint and an associated Lagrange
multiplier that plays a decisive role in characterizing ‘shortages’ of
small change. When the p.i.a. constraint is not binding, the model exhibits
a version of penny-versus-dollar exchange rate indeterminacy, a feature of
many models with inconvertible currencies. So long as pennies and dollars
bear the same rates of return, holders of currency are indifferent to the ratio
in which they hold pennies and dollars. ‘Shortages’ of pennies are manifested
in a binding p.i.a. constraint, and the emergence of the condition
that the rate of return on dollars exceeds that on pennies. Rate of return
dominance is the market signal that causes money holders to conserve on
pennies. Exchange rate indeterminacy stops during small change shortages.
The model makes contact with the ‘quantity theory of money’ in various
ways. The ranges between the melting and minting points for large and
small denomination coins creates a price level band within which the
ordinary quantity theory operates, cast in terms of the total quantity of
money. This operates during periods in which coins of both denominations
circulate and neither is being melted or minted. When the p. i. a. constraint
binds but neither coin is melted or minted and both circulate, the quantity
theory breaks in two, with one holding for ‘dollars’, another for ‘pennies’.
A more standard version of the quantity theory holds in a regime in which
the parameters of supply have been set to cause all large denomination
coins to disappear because they have been driven out by token small coins.
Analysis of the demand side reveals that, within the Medieval system,
shortages can occur away from the bounds of the intervals constraining the
price level, and that a shortage may fail to induce the price level movements
necessary to trigger production of more coins. Instead, by making the p.i.a.
constraint binding, the price signals induced by ‘shortages’ of small change
perversely hasten the day when small coins will eventually be melted, unless
there are appropriate adjustments in the parameters governing the melting
point for small coins. This feature of the model inspires our interpretation