Lecture 3：Currency Basket & Effective Exchange Rate(国际金融-南开,胡春田)

Lecture 3：Currency Basket & Effective Exchange Rate

Currency basket ∑=i i N T B

$$

i H T i ≡ N i : quantity of i $

∵T i = e × J i

$

$

N H e ≡

$

$

i N J i ≡

∑?=i

i i J N e B

∑?=i

i i J N e B 000

If 0i it J J ≠ and 0e e =?∑≠=00B J N e B it i t Peg to a CB ? choosing e t so that B t = B 0 i.e. ∑=i it t N J e B 0

N i 、J i 、B 0 given, solve e t

Effective Exchange Rate Index $

￡ yen T 1 T 1k /T 10×100 T 2 T 3 T 4 …

1990 5 100 10 100 1/30 100 1995 10

200 15 150 1/20 150 2000 8

160

12

120

1/12

250

???=321321w w w T T T E

$

$

i H T i ≡

(1) ∏==N

i w i i T E 1

cross exchange rate ? T i = e × J i

$

$

N H e ≡

$

$i N J i ≡

(2)

∑∑∑=+=+==i i N

i i i i i J w e J e w T w E ??)??(??1

pegged to E ?0?=E ?∑-=i

i

i J w e ?? Questions: 1. unit ？

2. how many currencies; N =？

3. weights ？:

1. indexation, i.e. choose a base period

2. N = ? depends

3. weights: today ’s topic

權數的選擇

1. 雙邊出口權數

1 ()ij j n

ij

j X XW X ==

=

∑本國對 j 國的出口值

本國對主要貿易對手國之總出口值

i : 本國

j : 貿易對手國

意義僅代表雙邊關係，但未考慮到多邊關係。

2. 雙邊進口權數

1 ()ji j n

ji

j M j MW M ==

=

∑本國自國的進口值

本國對主要貿易對手國之總進口值

衡量匯率變動對本國進口能力與進口成本之影響

1

1

X ()ij ji j ij ji

j j X M j M W X M ==+=

=

∑+∑本國與國間的貿易值

本國與主要貿易對手國之總貿易值

我國與貿易對手國之外匯供需關係

4. 世界出口市場權數

**1()

ji j j n

ji j j X X XWW X X =-=

∑-=

各貿易對手國之出口值（不包括對我國之出口）各貿易對手國之出口值合計（不包括對我國之出口）

*j X j =國總出口值

5. 平均出口權數

**11X 0.50.50.5

0.5

()

j j j

ji ij j n

n

ij

ji j j j A W XW XWW X X X X X X ===+-=+∑∑-

6. 平均貿易權數

**1

1

1

T 0.250.250.50.250.250.5()j j j j

ji

ij

j ji n n n

ij ji ji

j j j j A W XW XWW MW X X X M X X X M ====++-=++∑∑-∑

1 n

jk

ik j j ik

k ik k j k

j k M X DXW X M M k k j k =?≠≠????=?

??∑-????=?????

∑∑我國出口至國的出口值國自國的進口我國出口總值國總進口（不包括自本國進口）

j , k ：我國主要貿易對手國

8. 多邊匯率模型（Multilateral Exchange Rate Model, MERM ）： See Artus and Rhomberg (IMF Staff Papers, 1973)

In theory, the approaches can be classified as follow ：

Branson & Katseli ’s model:

Based on objective function ? choose optimal weight, *i w

Every time when J i changes, if we pegged E which is built upon *i w , ? we can guarantee that the objective function is not affected by the changes of J i.

Branson & Katseli ’s model: (3)

∏=≡N

i w i i i

P P T I 1

)(

pegged to real effective ER subscript 0: home country

i i J e T ?=

pegged to I ? 0?=I

∑∑∑∑∑-++-+=-++-+=-++=-+=)???()???()???()???()????()???(?000

N

i i i N N i i i N i i i i i i i P P J w P P e P P J w P P e w P P J e w P P T w I

(4) ∑-+-=-+?=)???(???0?0N

i i i N P P J w P P e I Objective: 1. Balance of trade (BOT): P x X – P m M 2. Terms of trade (TOT): P x /P m

3. P x /P 0:

Suppose two countries only: Export market:

Supply: )/(0P P f X x = (5)

)ln (ln ln 0P P s X x x -=

or

s x x X s P P ln ln ln 1

0-+=

Demand: )/(q q g X x d = (6)

)ln (ln ln q q d X x x d -=

PPP ? x x P e q =? or e P q x x /=

e P q x x ln ln ln -=

)ln ln (ln ln q e P d X x x d --=

Equilibrium ? )ln (ln )ln ln (ln 0P P s q e P d x x x x -=--

0ln )ln (ln ln )(P s q e d P s d x x x x x -+=-

0?)??(?)(P s q e d P s d x x x x x -+=-

or 0

?)

()??()(?P s d s q e s d d P x x x

x x x x

--+-= (7) 0

?)1()??(P k q e k -++≡ k : market power of exports in world market

Small cty ? d x → ∞ k → 1

(8)

)???(]??)1()??([)??(?0

00

P q e

ks P P k q e k s P P s X x

x x x -+=--++=-= Import market: Supply: )ln (ln ln q q s M m m s -=

or

s m m M s q q ln ln ln 1

-+=

Demand: d m m M d P P ln ln ln 1

0-+=

m m P eq =

(9) equilibrium ? 0?)'1()??('?P k q e k P m -++= m

m m

d s s k -≡

'

mkt power in world import mkt

k ' → 1 if s m → ∞ (small cty)

(10) )???('?0

P e q d k M m -+= Disaggregation:

mi m q q ?

xi x q q ?

i q q ? i T e ?

i.e. export to many countries & import from many countries

Export market: Supply side is the same as eq.(5)

Demand side has to be disaggregated, (11) i.e. ∑=d i d X X x xi i P q T =

eq.(6) be modified as

(6’) )ln (ln ln i xi xi d i q q d X -=

from (11), d i i d X X ln ln ∑=α

i

i

i X X ∑≡

α )ln (ln i xi xi i q q d -=∑α

)ln ln (ln i i x xi i q T P d --=∑α (12)

)ln ln ln (ln i i x xi i q e J P d ---=∑α

(∵e J T i i =)

Equil. ? (5) = (12)

i.e. ∑---=-)ln ln ln (ln )ln (ln 0i i x xi i x x q e J P d P P s α

To simplify, assume i d d x xi ?=

000000()ln ln (ln ln ln )ln (ln ln ln ln ln ln ln )()ln (ln ln ln )(ln ln ln )

x x x x x i i i x x i i i N N x x x i i i N x N s d P s P d J e q s P d J q q e q P P s d P d J q q d e q P ααα-=-++=-+-++-+=--+--+-∑∑∑

∑-++-++=)ln ln (ln )ln ln (ln ln ln 00P e q k q q J k P P N N i i i x α

x

x x

s d d k -≡

(13)

∑-++-++=)???()???(??0

0P e q k q q J k P P N N i i i x α 1. k = 1 ? P 0 not affect P x

2. q i ↑? P x ↑ (∵q i ↑， demand for X ↑)

(14)

)]???(???[)??(?00

N i i

i N x x x q q J P e q ks P P s X -+∑+-+=-=α

By the same token,

Import market:

(15)

∑-++-++=)???(')???('??00P e q k q q J k P P N N i i i m β

m

m m

d s s k -≡

'

(16) )]???(???['?0N i i

i N m q q J P e q d k M -+∑+-+=β Target 1: dBOT = 0 BT = P x X – P m M

M

P M P X P X P P M dM P XdP dX P dBT m m x x m

d

m x x )??()??(+-+=--+=

代入(13)~(16) let P x = P m = 1

)???()1(' )???()1( )???)](1(')1([?)(0

0N i i

i m N

i

i

i

x

N m x q q J d Mk q q J

s Xk P q e d Mk s Xk P M X -+∑+--+∑++-++-++-=βα

dBT = 0 ?

00

?)]

1(')1([)(???P d Mk s Xk M X P q e m x N +-+--=-+

)???

()1(')1()1(')1(N i i m x i m i x q q

J d Mk s Xk d Mk s Xk -+??

????+-++-+-∑βα

)???(?)]1(')1([)(0N i i

i

BT i m x q q J w P d Mk s Xk M X -+-+-+--≡

∑

If X – M = 0, initially, dBT = 0 ?

=-+0

???P q e N )???(N i i i

BT i q q J w -+-∑

∴ if we peg I with BT i w as its weights, ?dBT = 0 even if 0?≠i

J Target 2: dTOT = d (P x /P m ) = 0 i.e. m x P P ??=

(13) = (15) ?

∑∑---≡-----=-+)???()???)(')'(???0N i i

TOT i N i i

i i i N q q J w q q J k k k k P q e βα

k = k ’ = 1 ? TOT i w undefined,

i.e. e

? exchange rate policy cannot change TOT Model features ：

1. partial: P 0, q i , q N exog, deterministic

2. deterministic: no stochastic terms

3. one policy instrument, one goal (e ?) (BT or TOT)

Turnovky ’s Model (JIE 1982) (1a) 1222

31113*121)()()]([u P E Q d P E Q d C C r d Y d Y t +-++-++---=+

101<d , 0 231

3>d d

(1-b) )()(2221110E Q E Q P C ++++=δδδ

1=∑i δ

(1c)

2*

1 ,221*1 ,11E E E E r t t -+Ω=-+Ω=++

(1d) V E E +=12

(1e) 2211E E E λλ+= 121=+λλ

(1f) 2*

1 ,2*1 ,10)()(u C P P P Y t t t t t +-+-+=--γγγ

(1g) 321u r Y C M t +-=-αα (1h) For all variables

)(* ,t s t s I X E X =

Derivation of (1f)

)(*1 ,*1 ,---+=t t t s t t C W b a N

N Y ε= 10<<ε

εN Y ~~=

N W N P N W Y P ~

~~~~~~~-=-=επ

max W N P ~

~~1*=?-εεπ

in logs *

1 ,*1 ,)1(ln ---=-+?t t d t t P W N εε

sub. *1 ,s t t N - into *1 ,d t t N -

*1 ,*1 ,)]()[1(ln ---=-+-+t t t t P W C W b a εε

)

1(1)1()1(ln *1

,*1 ,εεεε-++-+-+=

--b P bC a W t t t t W is set as above ? actual employment is demand determined

P W N -=-+)1(ln εε

)

()(]

)1(1)

()()1([1)1(1)ln (*

1 ,2*1 ,10*

1 ,*1 ,-----+-+≡-+-+---+-++==t t t t t t t t t C P P P b P P C P b b b a N Y γγγεεεεεεεε

from (1e)

V

E E V E E E 1222212

211)(λλλλλ-=+-=+=

Regarding disturbances

(2a) 0)()()(321===t t t u u u εεε

(2b) 22)(i it u σε=

3 ,2 ,1 =?i

(3a) it i it q Q Q += i = 1, 2

(3b) it i it w +Ω=Ω

(4a) ) ,0(~21qi t q σ

(4b) ) ,0(~21wi t w σ

uncorrelated w / each other

(4c) ) ,0(~2v t v σ

Log deviation from X X x -≡

(5a)

)]([*11*1 ,1121c c e e w d y d y t t t ---+-=++

1222

31113)()(u p e q d p e q d +-++-++

(5b) )()(2221110e q e q p c t ++++=δδδ (5c) v e e t +=12 (5d) 22110e e λλ+=

(5e) 2*1 ,2*1 ,1)()(u c p p p y t t t t +-+-=--γγ (5f) 31*1 ,1211)(u e e w y c m t t +-+-=-+αα

(6) 0*

** ,2* ,1====++++i t i t i t i t c p e e ,

,2 ,1 =?i

subs. (6) into (5a)、(5e)、(5f)

besides, 2*

,221* ,11e e w e e w r i t i t -+=-+=++

v e e w w -=-=-?2121

solving (5a)~(5f)

(10)

D z v d d d d d x y t t t /}]))(([{2202

3132223+++--+=λδδγ

A

where 21γγγ+=

)]([2211222

311312q q d q d q d w d x t δδ++---≡

12w w v t -=

22023131])[(u d d d u z t δγ+++≡

0)1(0223131>+++-≡d d d d D δγ

x : random fluctuations in the demand for y v t : interest rate differentials

z t : domestic disturbances

From (10), if v t ≠ 0 ? y is f (λ2)

v t = 0 ? y independent of λ2

optimal λ2 = ?

(11) min 22222

22/}]2[{D A A z xv v x y σσσσγσ+++=

(12) 1 )()(?2022313222232<>+++-=v xv v d d d d d σδσσδλ

trade share is not so important

special cases

1. PPP ? ∞==2

313d d

2. 0=xv σ

3. w 1 = w 2 = 0 (random disturbances in goods market only)

4. q 1 = q 2 = 0 (random disturbances in foreign interest rate only)

Optimality implies:

If we choose 32,12

? ,? ??y y y ?λ other 13121112 , ,y y y ?λ

23222122 , ,y y y ?λ

Compare 2

?y σ, 21y σ, 22y σ, 23y σ

→ 2

?y σ is the minimum