Homework's Mathematical Methods

profileleon weng
Session4-chapter3.pptx

Exchange Rates I: The Monetary

Approach in the Long Run

3

Exchange Rates and Prices in the Long Run

Money, Prices, and Exchange Rates in the Long Run

The Monetary Approach

Money, Interest Rates, and Prices in the Long Run

1

2

The UIP approximation equation says that the home interest rate equals the foreign interest rate plus the expected rate of depreciation of the home currency.

Uncovered Interest Parity:

Recall

2

3

Absolute and Relative PPP:

(3-1)

Purchasing power parity implies that the exchange rate at which two currencies trade equals the relative price levels of the two countries.

Recall

3

4

The Measurement of Money

FIGURE 3-4

The Measurement of Money This figure shows the major kinds of monetary aggregates (currency, M0, M1, and M2) for the United States from 2004 to 2012. Normally, bank reserves are very close to zero, so M0 and currency are virtually identical, but reserves spiked up during the financial crisis in 2008, as private banks sold securities to the Fed and stored up the cash proceeds in their Fed reserve accounts.

The Supply of Money: In practice, a country’s central bank controls the money supply. We make the simplifying assumption that the central bank’s indirectly, but accurately, control the level of M1.

Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model

4

5

5

Money/Assets and Their Attributes

An investor’s entire portfolio of assets may include stocks, bonds, real estate, art, bank deposits in various currencies, and so on. All assets have three key attributes that influence demand: return, risk, and liquidity.

An asset’s rate of return is the total net increase in wealth resulting from holding the asset for a specified period of time, typically one year.

The risk of an asset refers to the volatility of its rate of return.

The liquidity of an asset refers to the ease and speed with which it can be liquidated, or sold.

We refer to the forecast of the rate of return as the expected rate of return.

The Demand for Money: What influences the demand?

2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model

6

6

Interest rates/expected rates of return: monetary assets pay little or no interest, so the interest rate on non-monetary assets like bonds, loans, and deposits is the opportunity cost of holding monetary assets.

A higher interest rate means a higher opportunity cost of holding monetary assets  lower demand of money.

Prices: the prices of goods and services bought in transactions will influence the willingness to hold money to conduct those transactions.

A higher level of average prices means a greater need for liquidity to buy the same amount of goods and services  higher demand of money.

Income: greater income implies more goods and services can be bought, so that more money is needed to conduct transactions.

A higher real national income (GNP) means more goods and services are being produced and bought in transactions, increasing the need for liquidity  higher demand of money.

What Influences Aggregate Demand of Money?

2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model

6

7

We assume money demand is motivated by the need to conduct transactions in proportion to an individual’s income and we infer that the aggregate money demand will behave similarly (known as the quantity theory of money).

All else equal, a rise in national dollar income (nominal income) will cause a proportional increase in transactions and in aggregate money demand.

The Demand for Money: A Simple Model

2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model

7

8

Dividing the previous equation by P, the price level, we can derive the demand for real money balances:

The Demand for Money: A Simple Model

Real money balances are simply a measure of the purchasing power of the stock of money in terms of goods and services. The demand for real money balances is strictly proportional to real income.

2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model

8

9

The condition for equilibrium in the money market is simple to state: the demand for money Md must equal the supply of money M, which we assume to be under the control of the central bank.

Imposing this condition on the last two equations, we find that nominal money supply equals nominal money demand:

Equilibrium in the Money Market

and, equivalently, that real money supply equals real money demand:

2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model

9

10

Expressions for the price levels in the U.S. and Europe are:

A Simple Monetary Model of Prices

These two equations are examples of the fundamental equation of the monetary model of the price level.

In the long run, we assume prices are flexible and will adjust to put the money market in equilibrium.

2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model

10

11

Plugging the expression for the price level in the monetary model to Equation (3-1), we can use absolute PPP to solve for the exchange rate:

A Simple Monetary Model of the Exchange Rate

This is the fundamental equation of the monetary approach to exchange rates.

(3-3)

2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model

11

12

The implications of the fundamental equation of the monetary approach to exchange rates are intuitive.

Suppose the U.S. money supply increases, all else equal. The right-hand side increases (the U.S. nominal money supply increases relative to Europe), causing the exchange rate to increase (the U.S. dollar depreciates against the euro).

Money Growth, Inflation, and Depreciation

2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model

12

13

Now suppose the U.S. real income level increases, all else equal. Then the right-hand side decreases (the U.S. real money demand increases relative to Europe), causing the exchange rate to decrease (the U.S. dollar appreciates against the euro).

Money Growth, Inflation, and Depreciation

2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model

13

14

The U.S. money supply is MUS, and its growth rate is μUS:

Money Growth, Inflation, and Depreciation

The growth rate of real income in the U.S. is gUS:

2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model

14

15

Therefore, the growth rate of PUS = MUS/LUSYUS equals the money supply growth rate μUS minus the real income growth rate gUS. The growth rate of PUS is the inflation rate πUS. Thus, we know that:

Money Growth, Inflation, and Depreciation

The rate of change of the European price level is calculated similarly:

(3-4)

(3-5)

When money growth is higher than income growth, we have “more money chasing fewer goods” and this leads to inflation.

2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model

15

16

Combining (3-4) and (3-5), we can now solve for the inflation differential in terms of monetary fundamentals and compute the rate of depreciation of the exchange rate:

Money Growth, Inflation, and Depreciation

(3-6)

2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model

16

17

The intuition behind Equation (3-6) is as follows:

If the United States runs a looser monetary policy in the long run measured by a faster money growth rate, the dollar will depreciate more rapidly, all else equal.

If the U.S. economy grows faster in the long run, the dollar will appreciate more rapidly, all else equal.

Money Growth, Inflation, and Depreciation

2 Money, Prices, and Exchange Rates in the Long Run: Money Market Equilibrium in a Simple Model

17

18

4 Money, Interest Rates, and Prices in the Long Run: A General Model

The trouble with the theory we studied earlier is that it assumes that the demand for money is stable, and this is implausible.

We will now explore a more general model that allows for money demand to vary with the nominal interest rate.

We consider the links between inflation and the nominal interest rate in an open economy.

We then return to the question of how best to understand what determines exchange rates in the long run.

18

19

A rise in national dollar income (nominal income) will cause a proportional increase in transactions and, hence, in aggregate money demand (as is true in the simple quantity theory).

A rise in the nominal interest rate will cause the aggregate demand for money to fall.

Dividing by P, we derive the demand for real money balances:

The Demand for Money: The General Model

4 Money, Interest Rates, and Prices in the Long Run: A General Model

19

20

The Demand for Money: The General Model

FIGURE 3-11

Panel (a) shows the real money demand function for the United States. The downward slope implies that the quantity of real money demand rises as the nominal interest rate i$ falls. Panel (b) shows that an increase in real income from Y1US to Y2US causes real money demand to rise at all levels of the nominal interest rate i$.

The Standard Model of Real Money Demand

4 Money, Interest Rates, and Prices in the Long Run: A General Model

20

21

With two relationships, PPP and UIP, we can derive a striking result concerning interest rates that has profound implications for our study of open economy macroeconomics. We use:

Long-Run Equilibrium in the Money Market

(3-7)

Inflation and Interest Rates in the Long Run

and

4 Money, Interest Rates, and Prices in the Long Run: A General Model

21

22

The nominal interest differential equals the expected inflation differential:

All else equal, a rise in the expected inflation rate in a country will lead to an equal rise in its nominal interest rate.

This result is known as the Fisher effect.

The Fisher effect predicts that the change in the opportunity cost of money is equal not just to the change in the nominal interest rate but also to the change in the inflation rate.

The Fisher Effect

(3-8)

4 Money, Interest Rates, and Prices in the Long Run: A General Model

22

23

Rearranging the last equation, we find

Subtracting the inflation rate (π) from the nominal interest rate (i), results in a real interest rate (r), the inflation-adjusted return on an interest-bearing asset.

This result states the following: If PPP and UIP hold, then expected real interest rates are equalized across countries. This powerful condition is called real interest parity.

Real interest parity implies the following: Arbitrage in goods and financial markets alone is sufficient, in the long run, to cause the equalization of real interest rates across countries.

Real Interest Parity

(3-9)

4 Money, Interest Rates, and Prices in the Long Run: A General Model

23

24

In the long run, all countries will share a common expected real interest rate, the long-run expected world real interest rate denoted r*, so

We treat r* as an exogenous variable, something outside the control of a policy maker in any particular country.

Under these conditions, the Fisher effect is even clearer, because, by definition,

Real Interest Parity

4 Money, Interest Rates, and Prices in the Long Run: A General Model

24

25

Evidence on the Fisher Effect

FIGURE 3-12

Inflation Rates and Nominal Interest Rates, 1995–2005 This scatterplot shows the relationship between the average annual nominal interest rate differential and the annual inflation differential relative to the United States over a ten-year period for a sample of 62 countries.

The correlation between the two variables is strong and bears a close resemblance to the theoretical prediction of the Fisher effect that all data points would appear on the 45-degree line.

APPLICATION

25

26

This model differs from the simple model (the quantity theory) by allowing L to vary as a function of the nominal interest rate i.

When nominal interest rates change the general model has different implications from the simple model.

We now examine the forecasting problem for an increase in the U.S. rate of money growth. We learn at time T that the United States is raising the rate of money supply growth from μ to a higher rate μ + Δμ.

The Fundamental Equation Under the General Model

(3-10)

4 Money, Interest Rates, and Prices in the Long Run: A General Model

26

27

Exchange Rate Forecasts Using the General Model

FIGURE 3-14 (1 of 4)

Before time T, money, prices, and the exchange rate all grow at rate μ. Foreign prices are constant. In panel (a), we suppose at time T there is an increase Δμ in the rate of growth of home money supply M.

4 Money, Interest Rates, and Prices in the Long Run: A General Model

An Increase in the Growth Rate of the Money Supply in the Standard Model

27

28

Exchange Rate Forecasts Using the General Model

FIGURE 3-14 (2 of 4)

This causes an increase Δμ in the rate of inflation; the Fisher effect means that there will be a Δμ increase in the nominal interest rate; as a result, as shown in panel (b), real money demand falls with a discrete jump at T.

4 Money, Interest Rates, and Prices in the Long Run: A General Model

An Increase in the Growth Rate of the Money Supply in the Standard Model (continued)

28

29

Exchange Rate Forecasts Using the General Model

FIGURE 3-14 (3 of 4)

If real money balances are to fall when the nominal money supply expands continuously, then the domestic price level must make a discrete jump up at time T, as shown in panel (c).

4 Money, Interest Rates, and Prices in the Long Run: A General Model

An Increase in the Growth Rate of the Money Supply in the Standard Model (continued)

29

30

Exchange Rate Forecasts Using the General Model

FIGURE 3-14 (4 of 4)

Subsequently, prices grow at the new higher rate of inflation; and given the stable foreign price level, PPP implies that the exchange rate follows a similar path to the domestic price level, as shown in panel (d).

4 Money, Interest Rates, and Prices in the Long Run: A General Model

An Increase in the Growth Rate of the Money Supply in the Standard Model (continued)

30

{

{

4

4

4

4

4

3

4

4

4

4

4

2

1

3

2

1

deposits

euro

on

return

of

rate

dollar

Expected

dollar

the

of

on

depreciati

of

rate

Expected

/

$

/

$

deposits

euro

on

rate

Interest

deposits

dollar

on

return

of

rate

Dollar

=

deposits

dollar

on

rate

Interest

$

E

E

i

i

e

D

+

=

{

4

3

4

2

1

levels

price

of

Ratio

rate

Exchange

/

$

/

EUR

US

P

P

E

=

4

3

4

2

1

3

2

1

al

differenti

Inflation

,

,

rate

exchange

nominal

the

of

on

depreciati

of

Rate

,

/

$

,

/

$

t

EUR

t

US

t

t

E

E

p

-

p

=

D

{

{

{

($)

income

Nominal

constant

A

($)

money

for

Demand

PY

L

M

d

´

=

M d

P Demand for real money

! = L

A constant ! × Y

Real income !

 

M

=

L

PY

 

M

P

=

L

Y

 

P

US

=

M

US

L

US

Y

US

 

P

EUR

=

M

EUR

L

EUR

Y

EUR

E$/€ Exchange rate !

= PUS PEUR

Ratio of price levels !

=

MUS LUSYUS

⎛

⎝ ⎜

⎞

⎠ ⎟

MEUR LEURYEUR

⎛

⎝ ⎜

⎞

⎠ ⎟

= MUS / MEUR( )

LUSYUS / LEURYEUR( ) Relative nominal money supplies

divided by relative real money demands

! "### $###

E$/EU Exchange rate !

= PUS PEUR

Ratio of price levels !

=

MUS LUSYUS

⎛

⎝ ⎜

⎞

⎠ ⎟

MEUR LEURYEUR

⎛

⎝ ⎜

⎞

⎠ ⎟

= MUS / MEUR( )

LUSYUS / LEURYEUR( ) Relative nominal money supplies

divided by relative real money demands

! "### $###

4

4

3

4

4

2

1

in U.S.

growth

supply

money

of

Rate

,

,

1

,

,

t

US

t

US

t

US

t

US

M

M

M

-

=

m

+

4

3

4

2

1

in U.S.

growth

income

real

of

Rate

,

,

1

,

,

t

US

t

US

t

US

t

US

Y

Y

Y

g

-

=

+

 

p

US

,

t

=

m

US

,

t

-

g

US

,

t

 

p

EUR

,

t

=

m

EUR

,

t

-

g

EUR

,

t

(

)

(

)

(

)

(

)

.

rates

growth

output

real

in

al

Differenti

,

,

rates

growth

supply

money

nominal

in

al

Differenti

,

,

,

,

,

,

al

differenti

Inflation

,

,

rate

exchange

nominal

the

of

on

depreciati

of

Rate

,

/

$

/

$

4

4

3

4

4

2

1

4

4

3

4

4

2

1

4

3

4

2

1

3

2

1

t

EUR

t

US

t

EUR

t

US

t

EUR

t

EUR

t

US

t

US

t

EUR

t

US

t

t

g

g

g

g

E

E

-

-

-

=

-

-

-

=

-

=

D

m

m

m

m

p

p

{

{

3

2

1

($)

income

Nominal

function

decreasing

A

($)

money

for

Demand

)

(

Y

P

i

L

M

d

´

´

=

{

{

{

income

Real

function

decreasing

A

money

real

for

Demand

)

(

Y

i

L

P

M

d

´

=

{

3

2

1

demand

money

Real

supply

money

Real

)

(

Y

i

L

P

M

=

4

3

4

2

1

3

2

1

al

differenti

inflation

Expected

on

depreciati

dollar

of

rate

Expected

/

$

/

$

e

EUR

e

US

e

E

E

p

-

p

=

D

{

{

rate

interest

euro

Net

rate

interest

dollar

Net

$

on

depreciati

dollar

of

rate

Expected

/

$

/

$

i

i

E

E

e

-

=

D

3

2

1

i$ −i € Nominal interest rate differential !

= πUS e −πEUR

e

Nominal inflation rate differential (expected)

! "# $#

e

EUR

e

US

i

i

p

-

=

p

-

$

 

r

US

e

=

r

EUR

e

 

r

US

e

=

r

EUR

e

=

r

*

.

,

*

*

$

e

EUR

e

EUR

e

EUR

e

US

e

US

e

US

r

r

i

r

r

i

p

+

=

p

+

=

p

+

=

p

+

=

{

{

(

)

(

)

4

4

4

4

3

4

4

4

4

2

1

demands

money

real

Relative

by

divided

supplies

money

nominal

Relative

$

$

levels

price

of

Ratio

rate

Exchange

/

$

)

(

/

)

(

/

)

(

)

(

EUR

EUR

US

US

EUR

US

EUR

EUR

EUR

US

US

US

EUR

US

Y

i

L

Y

i

L

M

M

Y

i

L

M

Y

i

L

M

P

P

E

=

÷

÷

ø

ö

ç

ç

è

æ

÷

÷

ø

ö

ç

ç

è

æ

=

=