# MONETARY UNION: Equilibrium with Local Currency Constraints

Given the conditions for optimization by the representative agent in each country and the market clearing conditions, the goal is to characterize the steady state equilibrium with valued fiat money in both economies given the two exogenous rates of money growth, о and <r*.8 We then turn to the determination of the equilibrium rates of inflation Source
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Steady States Given Rates of Money Creation

To characterize the equilibrium, we conjecture that agents on the home (foreign) island hold a fraction ф (ф*) of their local currency earnings in youth in domestic currency and the remainder is used to purchase foreign currency. Thus,

for generation t agents in the home economy. There is an analogous specification for foreign agents.

Let n and n* represent the steady state employment levels on the two islands. Then a steady state monetary equilibrium is given by (ф,п,ф*, n*) and a price system (pt,Pt>et)t£z 1 suc^ that the conditions for individual optimization and market clearing are met.

Our approach to characterizing the steady state is to use the market clearing conditions to substitute for the relative prices in (3) and (4). This will leave us with a set of first order conditions in terms of portfolio shares and employment levels, {ф%пуф*,n*). The result is summarized in the following proposition, using Z= (1 — 0/0).

Proposition 1 ; A steady state equilibrium is characterized by the following conditions:

Substituting for ф and using the definition of Z yields the second of the steady state conditions for the home country.

In terms of characterizing the existence of a steady state, we focus on equilibria in which fiat money is valued in both economies. That is, we look for equilibria in which citizens of both countries hold both currencies. We term these interior monetary equilibria and find:

Proof. : Note that both portfolio shares, ф and ф*, are given directly from the home and foreign conditions which relate these portfolio shares to Z and the rate of domestic money creation. The bounds on these rates of money creation guarantee that both ф and ф* lie in the interval (0,1), which is necessary for an interior monetary equilibrium.

Given the rates of domestic money creation, the labor supply choices, n and гг*, are immediately determined. Note that ng'{n) is monotonically increasing in n since p(*) is convex.

There are some interesting properties of the steady state. First, domestic output and employment depend on the rate of domestic money creation. In the usual manner, inflation brought about by lump-sum transfers creates a tax on domestic productive activity. Hence an increase in the rate of domestic inflation reduces employment and output. In a closed economy, domestic inflation would thus be welfare reducing.

Second, domestic output and employment do not depend on the foreign rate of money creation. In the same way, portfolio shares do not depend on foreign rates of money creation. These simplifications, of course, reflect the assumption of Cobb-Douglas preferences with the resulting implications for constant budget shares. However, as we shall see, this specification of preferences allows us to explicitly characterize the inflation game between countries as well as the gains to monetary union.