# MONETARY UNION: Equilibrium with Local Currency Constraints 2

Determination of Equilibrium Inflation Rates

Using the equilibrium from Proposition 2, let V{<!,&*) and V*(a*,<r) be the lifetime expected welfare of ал agent in the home and foreign islands respectively. Formally,

Again, there are analogous expressions for the foreign consumption levels. Note that the rate of money creation in foreign countries does effect the utility level of home agents through their equilibrium consumption levels of foreign goods. Further, in equilibrium, domestic money creation has three apparent influences: directly on the level of employment, through the transfer (the numerator of c*) and through the rate of price inflation (the denominator of ch).

The gains and losses from inflation are evident from these conditions. Given the employment levels, n and n*, home inflation increases the state contingent consumption of home agents while foreign inflation reduces it. From the definition of ch, in order for home consumption to increase with <r, ф must be less than one: i.e. not all of the domestic money supply is held by domestic citizens. In this model, this comes about because agents hold the currency of the other country in order to finance their consumption in old age. Thus, to emphasize an important point, the local currency requirement creates a demand for local currency by foreign agents and thus a basis for the inflation tax.9 This can be seen directly from the characterization of c*: if 0 = 1, then the only effect of cr would be to distort the labor supply decision More info.

Of course, the cost of inflation arises from the fact that it taxes the money holdings of all agents, domestic and foreign. This is a distortionary tax. Higher home inflation reduces the incentive for home agents to produce which ultimately reduces ch and thus lifetime utility. Still, starting at zero inflation, there is an incentive for countries to increase their money supplies. This observation leads to:

Using the condition characterizing n in the steady state (8) along with the definition of Z implies a = Z. A symmetric argument holds for the foreign country.

In fact, the equilibrium with positive inflation is a dominant strategy equilibrium. Even though the foreign rate of inflation has a (negative) effect on the welfare of home agents, it has no influence on the optimal level of domestic inflation. This simplification is again a consequence of the assumed preference structure.