# Equilibrium in the Foreign Exchange Market with Imperfect Asset Substitutability

This appendix develops a model of the foreign exchange market in which risk factors may make domestic currency and foreign currency assets imperfect substitutes. The model gives rise to a risk premium that can separate the expected rates of return on domestic and foreign assets.27

### Demand

Because individuals dislike risky situations in which their wealth may vary greatly from day to day, they decide how to allocate wealth among different assets by looking at the riskiness of the resulting portfolio as well as at the expected return it offers. Someone who puts her wealth entirely into British pounds, for example, may expect a high return but can be wiped out if the pound unexpectedly depreciates. A more sensible strategy is to invest in several currencies, even if some have lower expected returns than the pound, and thus reduce the impact on wealth of bad luck with any one currency. By spreading risk in this way among several currencies, an individual can reduce the variability of her wealth.

Considerations of risk make it reasonable to assume that an individual's demand for interest-bearing domestic currency assets increases when the interest they offer (R) rises relative to the domestic currency return on foreign currency assets [/?* + (£' — £)/£]. Put another way, an individual will be willing to increase the riskiness of her portfolio by investing more heavily in domestic currency assets only if she is compensated by an increase in the relative expected return on those assets.

We summarize this assumption by writing individual f s demand for domestic currency bonds, Bf, as an increasing function of the rate-of-return difference between domestic and foreign bonds,

Of course, Sfalso depends on other factors specific to individual i, such as her wealth and income. The demand for domestic currency bonds can be negative or positive, and in the former case individual i is a net borrower in the home currency, that is, a supplier of domestic currency bonds.

"The Mathematical Postscript to Chapter 21 develops a microecononiic model of individual demand for risky assets.

To find the aggregate private demand for domestic currency bonds, we need only add up individual demands Bf for all individuals / in the world. This summation gives the aggregate demand for domestic currency bonds, Bel, which is also an increasing function of the expected rate of return difference in favor of domestic currency assets. Therefore,

Since some private individuals may be borrowing, and therefore supplying bonds, B'1 should be interpreted as the private sector's net demand for domestic currency bonds.

### Supply

Since we are interpreting Bi! as the private sector's net demand for domestic currency bonds, the appropriate supply variable to define market equilibrium is the net supply of domestic currency bonds to the private sector, that is, the supply of bonds that are not the liability of any private individual. Net supply therefore equals the value of domestic currency government bonds held by the public, B, less the value of domestic currency assets held by the central bank, A:

A must be subtracted from B to find the net supply of bonds because purchases of bonds by the central bank reduce the supply available to private investors. (More generally, we would also subtract from B domestic currency assets held by foreign central banks.)

### Equilibrium

The risk premium, p, is determined by the interaction of supply and demand. The risk premium is defined as p = R - R* - (£' - £)/£, that is, as the expected return difference between domestic and foreign bonds. We can therefore write the private sector's net demand for domestic currency bonds as an increasing function of p. Figure 17AI-1 shows this relationship by drawing the demand curve for domestic currency bonds with a positive slope.

The bond supply curve is vertical at B — A* because the net supply of bonds to the market is determined by decisions of the government and central bank and is independent of the risk premium. Equilibrium occurs at point 1 (at a risk premium of p1), where the private sector's net demand for domestic currency bonds equals the net supply. Notice that for given values of R, /?*, and £, the equilibrium shown in the diagram can also be viewed as determining the exchange rate, since E = £7(1 + R - R* - p).

Figure 17AI-1 shows the effect of a central bank sale of domestic assets that lowers its domestic asset holdings to A2 < A}. This sale raises the net supply of domestic currency bonds to B - A2 and shifts the supply curve to the right. The new equilibrium occurs at

Figure 17AI-1

The Domestic Bond Supply and the Foreign Exchange Risk Premium Under Imperfect Asset Substitutability

An increase in the supply of domestic currency bonds that the private sector must hold raises the risk premium on domestic currency assets.

Risk premium on domestic bonds, p( = R - R* - {E0- E) IE)

Supply of Demand for domestic domestic bonds bonds, Bd