Homework's Mathematical Methods

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Exchange Rates II: The Asset

Approach in the Short Run

Exchange Rates and Interest Rate in Short Run: UIP and FX Market Equilibrium

Interest Rates in the Short Run: Money Market Equilibrium

The Asset Approach: Applications and Evidence

A Complete Theory: Unifying the Monetary and Asset Approaches

Fixed Exchange Rates and the Trilemma

Conclusions

Deviations from purchasing power parity (PPP) occur in the short run: the same basket of goods generally does not cost the same everywhere at all times.

Short-run failures of the monetary approach prompted economists to develop an alternative theory to explain exchange rates in the short run: the asset approach to exchange rates.

The asset approach is based on the idea that currencies are assets.

The price of the asset in this case is the spot exchange rate, the price of one unit of foreign exchange.

Introduction

1 Exchange Rates and Interest Rates in the Short Run: UIP and FX Market Equilibrium

Risky Arbitrage

The uncovered interest parity (UIP) equation is the fundamental equation of the asset approach to exchange rates.

(4-1)

TABLE 4-1

Interest Rates, Exchange Rates, Expected Returns, and FX Market Equilibrium: A Numerical Example The foreign exchange (FX) market is in equilibrium when the domestic and foreign returns are equal. In this example, the dollar interest rate is 5%, the euro interest rate is 3%, and the expected future exchange rate (one year ahead) is = 1.224 $/€. The equilibrium is highlighted in bold type.

Equilibrium in the FX Market: An Example

1 Exchange Rates and Interest Rates in the Short Run: UIP and FX Market Equilibrium

Equilibrium in the FX Market: An Example

FIGURE 4-2

FX Market Equilibrium: A Numerical Example

The returns calculated in Table 4-1 are plotted in this figure. The dollar interest rate is 5%, the euro interest rate is 3%, and the expected future exchange rate is 1.224 $/€.

The foreign exchange market is in equilibrium at point 1, where the domestic returns DR and expected foreign returns FR are equal at 5% and the spot exchange rate is 1.20 $/€.

1 Exchange Rates and Interest Rates in the Short Run: UIP and FX Market Equilibrium

Changes in Domestic and Foreign Returns and FX Market Equilibrium

To gain greater familiarity with the model, let’s see how the FX market example shown in Figure 4-2 responds to three separate shocks:

A higher domestic interest rate, i$ = 7%

A lower foreign interest rate, i€ = 1%

A lower expected future exchange rate, Ee$/€ = 1.20 $/€

1 Exchange Rates and Interest Rates in the Short Run: UIP and FX Market Equilibrium

FIGURE 4-3 (1 of 3)

A Change in the Home Interest Rate A rise in the dollar interest rate from 5% to 7% increases domestic returns, shifting the DR curve up from DR1 to DR2.

At the initial equilibrium exchange rate of 1.20 $/€ on DR2, domestic returns are above foreign returns at point 4. Dollar deposits are more attractive and the dollar appreciates from 1.20 $/€ to 1.177 $/€. The new equilibrium is at point 5.

Changes in Domestic and Foreign Returns and FX Market Equilibrium

A Change in the Domestic Interest Rate

1 Exchange Rates and Interest Rates in the Short Run: UIP and FX Market Equilibrium

FIGURE 4-3 (2 of 3)

(b) A Change in the Foreign Interest Rate A fall in the euro interest rate from 3% to 1% lowers foreign expected dollar returns, shifting the FR curve down from FR1 to FR2. At the initial equilibrium exchange rate of 1.20 $/€ on FR2, foreign returns are below domestic returns at point 6. Dollar deposits are more attractive and the dollar appreciates from 1.20 $/€ to 1.177 $/€. The new equilibrium is at point 7.

Changes in Domestic and Foreign Returns and FX Market Equilibrium

A Change in the Foreign Interest Rate

1 Exchange Rates and Interest Rates in the Short Run: UIP and FX Market Equilibrium

FIGURE 4-3 (3 of 3)

(c) A Change in the Expected Future Exchange Rate A fall in the expected future exchange rate from 1.224 to 1.20 lowers foreign expected dollar returns, shifting the FR curve down from FR1 to FR2. At the initial equilibrium exchange rate of 1.20 $/€ on FR2, foreign returns are below domestic returns at point 6. Dollar deposits are more attractive and the dollar appreciates from 1.20 $/€ to 1.177 $/€. The new equilibrium is at point 7.

Changes in Domestic and Foreign Returns and FX Market Equilibrium

A Change in the Expected Future Exchange Rate

1 Exchange Rates and Interest Rates in the Short Run: UIP and FX Market Equilibrium

The Assumptions

In this chapter, we make short-run assumptions that are quite different from the long-run assumptions:

In the short run, the price level is sticky; it is a known predetermined variable, fixed at P = P (the bar indicates a fixed value).

In the short run, the nominal interest rate i is fully flexible and adjusts to bring the money market to equilibrium.

The assumption of sticky prices, also called nominal rigidity, is common to the study of macroeconomics in the short run.

2 Interest Rates in the Short Run: Money Market Equilibrium

Money Market Equilibrium in the Short Run: How Nominal Interest Rates Are Determined

—

The Model

The expressions for money market equilibrium in the two countries are as follows:

Money Market Equilibrium in the Short Run: How Nominal Interest Rates Are Determined

(4-2)

(4-3)

2 Interest Rates in the Short Run: Money Market Equilibrium

FIGURE 4-4 (1 of 2)

The supply and demand for real money balances determine the nominal interest rate.

The money supply curve (MS) is vertical at M1US/PUS because the quantity of money supplied does not depend on the interest rate.

The money demand curve (MD) is downward-sloping because an increase in the interest rate raises the cost of holding money.

Money Market Equilibrium in the Short Run: Graphical Solution

2 Interest Rates in the Short Run: Money Market Equilibrium

Equilibrium in the Home Money Market

FIGURE 4-4 (2 of 2)

The money market is in equilibrium when the nominal interest rate i1$ is such that real money demand equals real money supply (point 1).

At points 2 and 3, demand does not equal supply and the interest rate will adjust until the money market returns to equilibrium.

Money Market Equilibrium in the Short Run: Graphical Solution

2 Interest Rates in the Short Run: Money Market Equilibrium

Equilibrium in the Home Money Market (continued)

FIGURE 4-6 (1 of 2)

In panel (a), with a fixed price level P1US, an increase in nominal money supply from M1US to M2US causes an increase in real money supply from M1US/P1US to M2US/P1US.

The nominal interest rate falls from i1$ to i2$ to restore equilibrium at point 2.

Changes in Money Supply and the Nominal Interest Rate

2 Interest Rates in the Short Run: Money Market Equilibrium

Home Money Market with Changes in Money Supply and Money Demand

FIGURE 4-6 (2 of 2)

Changes in Money Supply and the Nominal Interest Rate

In panel (b), with a fixed price level P1US, an increase in real income from Y1US to Y2US causes real money demand to increase from MD1 to MD2.

To restore equilibrium at point 2, the interest rate rises from i1$ to i2$.

2 Interest Rates in the Short Run: Money Market Equilibrium

Home Money Market with Changes in Money Supply and Money Demand (continued)

The Monetary Model: The Short Run Versus the Long Run

Consider the following: the home central bank that previously kept the money supply constant switches to an expansionary policy, allowing the money supply to grow at a rate of 5%.

If this expansion is expected to be permanent, the predictions of the long-run monetary approach and Fisher effect are clear. The Home interest rate rises in the long run.

If this expansion is expected to be temporary, all else equal, the immediate effect is an excess supply of real money balances. The home interest rate will then fall in the short run.

2 Interest Rates in the Short Run: Money Market Equilibrium

FIGURE 4-7 (1 of 2)

The figure summarizes the equilibria in the two asset markets in one diagram. In panel (a), in the home (U.S.) money market, the home nominal interest rate i1$ is determined by the levels of real money supply MS and demand MD with equilibrium at point 1.

3 The Asset Approach: Applications and Evidence

The Asset Approach to Exchange Rates: Graphical Solution

Equilibrium in the Money Market and the FX Market

In panel (b), in the dollar-euro FX market, the spot exchange rate E1$/€ is determined by foreign and domestic expected returns, with equilibrium at point 1′. Arbitrage forces the domestic and foreign returns in the FX market to be equal, a result that depends on capital mobility.

The Asset Approach to Exchange Rates: Graphical Solution

3 The Asset Approach: Applications and Evidence

FIGURE 4-7 (2 of 2)

Equilibrium in the Money Market and the FX Market (continued)

Capital Mobility Is Crucial

Our assumption that DR equals FR depends on capital mobility. If capital controls are imposed, there is no arbitrage and no reason why DR has to equal FR.

Putting the Model to Work

With this graphical apparatus in place, it is relatively straightforward to solve for the exchange rate given all the known (exogenous) variables we have specified previously.

3 The Asset Approach: Applications and Evidence

FIGURE 4-8 (1 of 2)

In panel (a), in the Home money market, an increase in Home money supply from M1US to M2US causes an increase in real money supply from M 1US/P1US to M 2US/P1US.To keep real money demand equal to real money supply, the interest rate falls from to i1$ to i2$, and the new money market equilibrium is at point 2.

—

Short-Run Policy Analysis

3 The Asset Approach: Applications and Evidence

Temporary Expansion of the Home Money Supply

FIGURE 4-8 (2 of 2)

In panel (b), in the FX market, to maintain the equality of domestic and foreign expected returns, the exchange rate rises (the dollar depreciates) from E1$/€ to E2$/€, and the new FX market equilibrium is at point 2′.

Short-Run Policy Analysis

3 The Asset Approach: Applications and Evidence

Temporary Expansion of the Home Money Supply (continued)

FIGURE 4-9 (1 of 2)

In panel (a), there is no change in the Home money market. In panel (b), an increase in the Foreign money supply causes the Foreign (euro) interest rate to fall from i1€ to i2€.

Short-Run Policy Analysis

3 The Asset Approach: Applications and Evidence

Temporary Expansion of the Foreign Money Supply

FIGURE 4-9 (2 of 2)

For a U.S. investor, this lowers the foreign return i€ + (Ee$/ € − E$/€)/E$/€, all else equal. To maintain the equality of domestic and foreign returns in the FX market, the exchange rate falls (the dollar appreciates) from E1$/€ to E2$/€, and the new FX market equilibrium is at point 2′.

Short-Run Policy Analysis

3 The Asset Approach: Applications and Evidence

Temporary Expansion of the Foreign Money Supply (continued)

For a complete theory of exchange rates:

We need the asset approach (this chapter)—short-run money market equilibrium and uncovered interest parity:

4 A Complete Theory: Unifying the Monetary and Asset Approaches

(4-4)

To forecast the future expected exchange rate, we also need the long-run monetary approach from the previous chapter—a long run monetary model and purchasing power parity:

Combining the asset and monetary approach, we can see how the two key mechanisms of expectations and arbitrage determine exchange rates in both the short run and the long run.

(4-5)

4 A Complete Theory: Unifying the Monetary and Asset Approaches

FIGURE 4-12 (1 of 4)

In panel (a), the home price level is fixed, but the supply of dollar balances increases and real money supply shifts out. To restore equilibrium at point 2, the interest rate falls from i1$ to i2$. In panel (b), in the FX market, the home interest rate falls, so the domestic return decreases and DR shifts down. In addition, the permanent change in the home money supply implies a permanent, long-run depreciation of the dollar.

4 A Complete Theory: Unifying the Monetary and Asset Approaches

Permanent Expansion of the Home Money Supply, Short-Run Impact

FIGURE 4-12 (2 of 4)

Hence, there is also a permanent rise in Ee$/€, which causes a permanent increase in the foreign return i€ + (Ee$/€ − E$/€)/E$/€, all else equal; FR shifts up from FR1 to FR2. The simultaneous fall in DR and rise in FR cause the home currency to depreciate steeply, leading to a new equilibrium at point 2′ (and not at 3′, which would be the equilibrium if the policy were temporary).

4 A Complete Theory: Unifying the Monetary and Asset Approaches

Permanent Expansion of the Home Money Supply, Short-Run Impact (continued)

FIGURE 4-12 (3 of 4)

Long-Run Adjustment: In panel (c), in the long run, prices are flexible, so the home price level and the exchange rate both rise in proportion with the money supply. Prices rise to P2US, and real money supply returns to its original level M1US/P1US.

The money market gradually shifts back to equilibrium at point 4 (the same as point 1).

4 A Complete Theory: Unifying the Monetary and Asset Approaches

Permanent Expansion of the Home Money Supply, Short-Run Impact (continued)

FIGURE 4-12 (4 of 4)

Long-Run Adjustment (continued): In panel (d), in the FX market, the domestic return DR, which equals the home interest rate, gradually shifts back to its original level. The foreign return curve FR does not move at all: there are no further changes in the Foreign interest rate or in the future expected exchange rate. The FX market equilibrium shifts gradually to point 4′. The exchange rate falls (and the dollar appreciates) from E2$/€ to E4$/€. Arrows in both graphs show the path of gradual adjustment.

4 A Complete Theory: Unifying the Monetary and Asset Approaches

Permanent Expansion of the Home Money Supply, Short-Run Impact (continued)

FIGURE 4-13 (1 of 2)

In panel (a), there is a one-time permanent increase in home (U.S.) nominal money supply at time T.

In panel (b), prices are sticky in the short run, so there is a short-run increase in the real money supply and a fall in the home interest rate.

Overshooting

4 A Complete Theory: Unifying the Monetary and Asset Approaches

Responses to a Permanent Expansion of the Home Money Supply

FIGURE 4-13 (2 of 2)

In panel (c), in the long run, prices rise in the same proportion as the money supply.

In panel (d), in the short run, the exchange rate overshoots its long-run value (the dollar depreciates by a large amount), but in the long run, the exchange rate will have risen only in proportion to changes in money and prices.

Overshooting

4 A Complete Theory: Unifying the Monetary and Asset Approaches

Responses to a Permanent Expansion of the Home Money Supply (continued)

Here we focus on the case of a fixed rate regime without controls so that capital is mobile (capital controls) and arbitrage is free to operate in the foreign exchange market.

Central banks buying and selling foreign currency at a fixed price, thus holding the market exchange rate at a fixed level denoted E.

We examine the implications of Denmark’s decision to peg its currency, the krone, to the euro at a fixed rate: EDKr/€

The Foreign country remains the Eurozone, and the Home country is now Denmark.

5 Fixed Exchange Rates and the Trilemma

What Is a Fixed Exchange Rate Regime?

—

What we now show is that a country with a fixed exchange rate faces monetary policy constraints not just in the long run but also in the short run.

What Is a Fixed Exchange Rate Regime?

5 Fixed Exchange Rates and the Trilemma

The Danish central bank must set its interest rate equal to i€, the rate set by the European Central Bank (ECB):

Denmark has lost control of its monetary policy: it cannot independently change its interest rate under a peg.

Pegging Sacrifices Monetary Policy Autonomy

in the Short Run: Example

5 Fixed Exchange Rates and the Trilemma

Our short-run theory still applies, but with a different chain of causality.

Under a float:

The home monetary authorities pick the money supply M.

In the short run, the choice of M determines the interest rate i in the money market; in turn, via UIP, the level of i determines the exchange rate E.

The money supply is an input in the model (an exogenous variable), and the exchange rate is an output of the model (an endogenous variable).

Pegging Sacrifices Monetary Policy Autonomy

in the Short Run: Example

5 Fixed Exchange Rates and the Trilemma

Our short-run theory still applies, but with a different chain of causality.

Under a fix, this logic is reversed:

Home monetary authorities pick the fixed level of the exchange rate E.

In the short run, a fixed E pins down the home interest rate i via UIP (forcing i =i*); in turn, the level of i determines the level of the money supply M necessary to meet money demand.

The exchange rate is an input in the model (an exogenous variable), and the money supply is an output of the model (an endogenous variable).

Pegging Sacrifices Monetary Policy Autonomy

in the Short Run: Example

5 Fixed Exchange Rates and the Trilemma

The price level in Denmark is determined in the long run by PPP. But if the exchange rate is pegged, we can write long-run PPP for Denmark as:

With the long-run nominal interest and price level outside of Danish control, monetary policy autonomy is impossible. We just substitute and into Denmark’s long-run money market equilibrium to obtain:

Pegging Sacrifices Monetary Policy Autonomy

in the Long Run: Example

5 Fixed Exchange Rates and the Trilemma

Consider the following three equations and parallel statements about desirable policy goals.

The Trilemma

A fixed exchange rate

May be desired as a means to promote stability in trade and investment

Represented here by zero expected depreciation

International capital mobility

May be desired as a means to promote integration, efficiency, and risk sharing

Represented here by uncovered interest parity, which results from arbitrage

5 Fixed Exchange Rates and the Trilemma

Consider the following three equations and parallel statements about desirable policy goals.

The Trilemma

Monetary policy autonomy

May be desired as a means to manage the Home economy’s business cycle

Represented here by the ability to set the Home interest rate independently of the foreign interest rate

5 Fixed Exchange Rates and the Trilemma

The Trilemma

1 and 2 imply not 3 (1 and 2 imply interest equality, contradicting 3).

2 and 3 imply not 1 (2 and 3 imply an expected change in E, contradicting 1).

3 and 1 imply not 2 (3 and 1 imply a difference between domestic and foreign returns, contradicting 2).

Formulae 1, 2, and 3 show that it is a mathematical impossibility as shown by the following statements:

This result, known as the trilemma, is one of the most important ideas in international macroeconomics.

5 Fixed Exchange Rates and the Trilemma

FIGURE 4-16

The Trilemma Each corner of the triangle represents a viable policy choice. The labels on the two adjacent edges of the triangle are the goals that can be attained; the label on the opposite edge is the goal that has to be sacrificed.

The Trilemma

5 Fixed Exchange Rates and the Trilemma

Intermediate Regimes

The lessons of the trilemma most clearly apply when the policies are at the ends of a spectrum:

a hard peg or a float,

perfect capital mobility or immobility,

complete autonomy or none at all.

But sometimes a country may not be fully in one of the three corners: the rigidity of the peg, the degree of capital mobility, and the independence of monetary policy could be partial rather than full.

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