The monetary theory of exchange rate determination modelA linear version of the modelHyperin⁄ation and de⁄ationTemporary and permanent changes The monetary theory of exchange rate

Basic model A linear version of the model Hyperinflation and deflation Temporary and permanent changes

The monetary theory of exchange ratedetermination4330 Lecture 7

Ragnar Nymoen

Department of Economics, University of Oslo

5 March 2012

The monetary theory of exchange rate determination Department of Economics, University of Oslo


References

I Ch 4 in OEM (this slide set)

I OR: Ch 8.2. In particular 8.2.7 where the monetary model ofthe exchange rate is presented as a varaint of the Caganmodel in Ch 8.2.1-8.2.5.

I Article by Rogoff: “Exchange rates in the modern floating era:what do we really know?”Accessible about the explanatoryand predictive power of theoretical exchange rate models



Model specification

Assumptions of the monetary theory of the exchange rate I

I Perfect mobility of goods; purchasing power parity, P = EP∗I Perfect mobility of capital; interest rate parity, i = i∗ + eeI Wage flexibility; output, Y , supply determined

I Exogenous money supply, M

I Model-consistent and exact expectations, ee = E/E



Model specification

The basic model

Money market equilibrium

MEP∗

= m(i ,Y ) (1)

Foreign exchange market equilibrium

EE= i − i∗ (2)

Endogenous: E and i .Exogenous: P∗, i∗, Y and M.



Solution if the basic model

Derive differential equation for E

Solve (1) for i :

i = i(MEP∗

,Y)

i1 < 0, i2 > 0

Insert in (2):EE= i

(MEP∗

,Y)− i∗ (3)

One differential equation in E (t)Given an initial value for E , (3) determines the whole future pathof E .E is free to take any value at any time– it is a “jump variable”How to determine the initial value of E?




Not global asymptotic stability




Equilibrium I

EE= i

(MEP∗

,Y)− i∗

dE/EdE

= −i1ME 2P∗

> 0

I In general, a predetermined E yields explosive path,accelerating depreciation or appreciation—not equilibriumdynamics

I However E is a jump variable, and we find the solution byfinding the singular value that E has to take for the evolutionof E to be non-explosive

I E is then on the equilibrium path




Equilibrium II

I Stationary point E = 0

I Solution: i = i∗

E =M

P∗m(i∗,Y )

I The equilibrium exchange rate is proportional to quantity ofmoney



Nominal anchor

Rational expectations and nominal anchorI Paper money has zero intrinsic valueI Its value depends entirely on beliefsI Usually there are several rational expectations paths for thevalue of money

I Several self-fulfilling beliefsI If money supply is exogenous, there is at most onenon-explosive rational expectations path for the value ofmoney.

I Assumption: Expectations coordinate on the non-explosivepath. If outside, E will jump up/down to the equilibrium path

I Confidence in the monetary systemI The exogenous money supply acts as nominal anchor



Linear model IMoney demand:

m− p = −ηi + κy (4)

Purchasing power parity

p = e + p∗ (5)

Interest rate parity

e = i − i∗ (6)

I m = lnM,p = lnP etcI η > 0, κ > 0, constantsI e = E/E etc



Linear model IIFrom money market equilibrium:

i = −(1/η)(m− p) + (κ/η)y

Insert this in e = i − i∗:

e = −(1/η)(m− e − p∗) + (κ/η)y − i∗ (7)

e = (1/η)e − z (8)

wherez = (1/η)(m− p∗ − κy) + i∗

First order linear differential equation



Linear model IIISydsæter:

x+ a(t)x = b(t)⇐⇒ x = x(t0)e−∫ tt0a(ξ)d ξ

+∫ t

t0b(τ)e−

∫ tτ a(ξ)d ξdτ

In our case x = e, a(t) = −1/η, b(t) = −z(t) and

−∫ t

τa(ξ)dξ =

1η(t − τ)



Linear model IV

Solution of linear model continued: Equation (8):

e = (1/η)e − z

has solution

e(t) =[e(t0)−

∫ t

t0z(τ)e−(1/η)(τ−t0)dτ

]e(1/η)(t−t0)

Expression explodes unless

e(t0) =∫ ∞

t0z(τ)e−(1/η)(τ−t0)dτ



Linear model VChoose this as the solution for all t:

e(t) =∫ ∞

t[(1/η)(m− p∗ − κy) + i∗]e−(1/η)(τ−t)dτ (9)

The exchange rate is determined by the whole future path of themoney supply and the other exogenous variables.



Linear model VI

More about the solution

e(t) =∫ ∞

t[(1/η)(m− p∗ − κy) + i∗]e−(1/η)(τ−t)dτ

Note that ∫ ∞

te−(1/η)(τ−t)dτ = η

Hence, if the exogenous variables are constant

e(t) = m− p∗ − κy − ηi∗

which is the log-linearized version the solution we found above.



Hyperinflation

Monetary model best when inflation very rapidHyperinflation: Inflation above 50 per cent per monthPossible causes:

I Explosive growth in money supplyI Expectations only

Cagan (1956), see OR Ch 8.2

I Studied 20th century European hyperinflationsI Caused by explosive growth in money supplyI Weak governments printing money to finance deficits



Hyperdeflation?

I Never observedI Nominal interest rate cannot be negativeI No rational expectations path with e < −i∗I (Perfect capital mobility)I With i = 0 money and bonds become equivalent



Cases where a nominal anchor is missing1. Exogenous interest rate e = i − i∗. No way to pin down thelevel of the exchange rate.

2. "Inflation targeting"

I i = i∗ + (π − p∗) + φ(p − π)

I π = inflation target, φ > 1I i∗ + (π − p∗) = normal interest rate

e = i − i∗ = i∗ + (π − p∗) + φ(e + p∗ − π)− i∗(1− φ)e = (1− φ)(π − p∗)

e = π − p∗No way to pin down the level of the exchange rate.



Four useful principles for thinking about this model

I The exchange rate is forward lookingI Begin with the distant future and look backwardsI There are never expected jumps in the exchange rateI The exchange rate jumps on news



An unexpected, temporary increase in the money supply

e(t) =∫ ∞

t[1η(m− p∗ − κy) + i∗]e−(1/η)(τ−t)dτ

I Money supply increased by ∆m at t0, reversed at t1.I Regressive expectations II Interest rate down at t0



Unexpected, temporary shock to money supply



An announced permanent increase in the money supply

e(t) =∫ ∞

t[1η(m− p∗ − κy) + i∗]e−(1/η)(τ−t)dτ

I At t0 it is announced that m will increase by ∆m from t1.I Initial depreciation followed by more. Interest rate up at t0



Anticipated, permanent shift in money supply



Using the term structure to derive exchange rateexpectations

Suppose

I Perfect capital mobility.I Term structure is equal to expected future interest rates

Since e = i − i∗, if everything works out as expected,

e(t)− et0 =∫ t

t0edτ =

∫ t

t0(i − i∗)dτ (10)

I The market expectations at t0 of e(t) can be calculated fromthe term structure of interest rates


Documents

The monetary theory of exchange rate determination modelA linear version of the modelHyperin⁄ation and de⁄ationTemporary and permanent changes The monetary theory of exchange rate