Kagawa Univer.sify EwnomκRevieω
Vo 6.l3， N o.2， September1990， 169-179
Notes
Exchange Rate Changes and
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Theory
本by
Takashi Shiomura .
.
I. Introduction As is stated in Kindleberger(1989)， we can observe the“ratchet effects" in prices. In depression， devaluation tends to be neutral for prices， while revaluation leads to a fall in prices. In prosperity， on the contrary， devaluation leads to a rise in prices， while revaluation tends to be neutral for prices Sometimes these 1) “ratchet effects" fail to occur As Kindleberger states日itis important to note that the initial conditions and economic environment play crucial roles on the effects of changes in exchange rate. He also states that it may be dangerous for an economist to adhere to only one model when he tries to investigate the effects of changes in exchanεe rateIn this note， however， we consider the“ratchet e任ects"by using only one model， that is， Negishi-Nakagome model which is an extended version of Na-kagome (1983， 1985)to a two-country economy川 Wewill find that the conjectural
equilibrium model is useful for the analysis of the“ratchet effects"
(
*
)
1 am deeply indebted to Professors Hideyuki Adachi， Kiyoshi Ikemoto (Kobe Univ.)， and Takateru Inoue (Kagawa Univ) Any shortcomings that may remain are口1me
170 Kagawa Universzty Ewnomzc Review
II.Basic model
The model used here is similar to Shiomura (1989) Notations are defined forz = 1， 2 as follows 1 ) ， : total output of country z
ρ
， : price of domestic good in country z q， p::rice of imported good in country z U1， ::wage rate in country z 368 e : exchange rate， e， price of country 2's currency in terms of country l's currency C，(y"ム， q，)::consumption of domestic good in country z 1，(1)i，ρ"qi):'import in countryz )(i(1)"ρi， q，)::export in countryZ， which is identically equal to1)i -C，(1)i，ム，q，)We assume that Cy， 1リ>0，C山 1，
q
<
0， and CiY<
1(z = 1， 2)， Here C叩denotes the partial derivative of C
，
with respect toYi，
and similarly definded for others， The lowercase letters of these represent corresponding elasticities. A hat ， on a variable represents its rate of change In the following， we ignore the scale difference between firms and industries， We suppose that there exsist two countries in the world.. Firms are p，、
_
_
_
Pi=A，(Yi;P"Yi) Pir-ー ー ー ー ー ー 守 、h O Yi Yi FigureI369 Exchange Rate Changes and Ratchet Effects 171
“monopolistically competitive" and maximize their profits under their subjective demand functions ρzニ A，(y，; ]5" Yi)..，5]and Yiare the current price and sale
respectively at the point where the subjective demand function has a kink (see，
2)
Figure 1) Here "monopolistically competitive" is used in the sense of Arrow
(1959)An individual competitive supplier might perceive an infinit1y elastic demand curveifthe desired supply is current1y realized at the going price When demand falls short of supply， however， a supplier who p巴rceivesan intnitly
elastic demand curve is out of conjectural equilibrium since he can not supply whatever he wishes in spite of the conjecture that he can. Thus， he must perceive， like a monopolist， an imperfect1y elastic demand curve for his supply
Under appropriate assumptions， we can derive a necessary and sufficient condition for the profit maximization problem :
Si(w ，) 三 p ，( l-~i) 一 ω ，/，-1'(y，)ミ0， St(w，)三
ρ
，(1-~t) -W'/i-1'(ぃ)壬0，(1) (2)
where
;
t
i andt
;
t
are respectively the left-hand sided and the right-hand sided elasticities ofρ， with respect toYiwhich we assume to b巴fixed..fi-l(y，) is theinverse production function which is assumed to be，/-1(0)
=
0， ，/-1'(y，)>
0， and八一1"(Yi)
>
O.Ifthe conditions (1) and (2) are satisfied at the current price andsale， the firm has no incentive to change the price. We assume that~t > 右 孟O
at (]5"
y
，)Wec丘n，then， draw Nakagome's
、
upplycoresspondence"， that is， the set of(ρi， 1ji) which satisfies the conditions (1) and (2) for given wages in two countries
To simplify the analysis， we assume that the money wage rates in two countries are fixed for some reasons.
2) Negishi states，“it is most natural to assume that firms respond to some partial equilibrium type of perceived functions" (Negishi(1972)， p.107)The reason for having a kink is due to an asymmetric reaction of customers For further details， see Negishi
-172- Kagawa University Ecoηomic Revzew
Now， equilibrium conditions for two goods are X，(y"九 q，)= ん(Y2，P2， q2，)
Xz(Y2，
ρ
2， q2)=
Ily"ρ
"
q，)370
(
3
)
(4)
We assume that domestic and foreign customers are identical from the firms' point of view in that there are no advantages or disadvantages associated with selling to a paticular customer This assumption makes it possible to use in this note the relationsq
，
=
ep2and q2=
p，
/eLet us solve the equations (3) and (4) forY"and denote it by
D，(九の~)
We call it "demand function for good t"Itstands for the equilibrium national income for countryt rather than the demand country I P
，
S
i(Wl)=O S;-(Wl)=O D，
(P"九;e) P2 Y，
y{S
i
(
地 )=0 Figure 2 S2(W2)=0 D2(P " P2; e) Y2 country 2 3) The consumption function and the import function indeed depend on the total amount of money371 Exchange Rate乙hangesand Ratchet Effects
-173-for good 1)
This demand function is not always downward sloping with respect to its own priceIfit is upward sloping， the observed price effects are not consistent with the conjectured price effects十 Theformer represents the effects which are observed on
the“upward" sloping demand function， while the latter the effects which are conjectured from the firm's“downward" sloping subjective demand function Let us sppose that the cross price e丘ects can be neglected. That is，
C
，q=
0 and J，p=
0， (z=
1， 2) Then， we can assume the demand function 5)to be downward sloping with respect to its own price
In Figure 2， where supply correspondences and demand functions are drawn on the same plane， (E" E2) represents a conjectural equilibrium and y{is the full employment income for country.zN ote that this equilibrium depends on the level of wages， exchange rate， initial values of prices or starting points in the sense of Sweezy (1939)， and demand elasticities.. The last ones crucially depend on the firms' expectations
II
I
.
Exchange rate changes and ratchet effectsFirst， let us consider the case where the world economy is in depression. In this case， a great deal of unemployment and an excess capacity exisit We sup -pose that the initial conjectural equilibrium is at， say， (A" A2) in Figure 3
When the exchange rate changes， demand functions D
，
and D2 can move in4) This“demand function" corresponds to Nikaido's“objective demand function" which takes into consideration the full effect on demand of a price change (see， Nikaido(1975)) 5) We assume thatX'yX2Y - Z'yZ却 >0 This is a sufficient condition for the follwing quantity adjustment process to be locally stabl巴・ れ=lz-X
，
Y2 = l，
- X2 where a dot“，"denotes the differentiation with respect to time Under this assump -tion， we can show that oD'/otj<
0 (Z， j = 1， 2)-174ー Kagawa Uη zve~szか Economκ Review 372
any direction according to the magnitude of income and price elasticities We confine our attention， for the time being， to the normal case where devaluation
6)
causes national income to increase
For simplicity， we assume in the following that the equilibrium point of country 1 remains in the interior of the supply correspondence in spite of the change in the exchange rate or the price of the domestic good in country 2..Ifthe price of country 2's curr巴ncyin terms of country l's currencye decreases， then the P. Figure 3 P2 Y2 6) By differentiating logarithmicalIy Eqs(3)and (4)， we can obtain
Y
l
= {(z2yiIQ-Z2qX2，
)!(X"X2，
-ZI.yi2，
)}e，
Y 2 =一{(z"i2Q-ZIQXl，
)!(Xl，
X2，
-Z
，
!
Z2，
)
}
e ーョ~ Y. Thus， under the assumption of equal price elasticities(ZIQ = Z叫，we can assume the normal case if two income elasticities are smalIer than one， (1>ι。
，Zり ;i = ，12) On the other hand， we can not assume it if these are greater than one373 Exchange Rate Changes and Ratchet Effects -175ー
equilibrium points move from (A1， A2)to， say， (A;， A~). Neither of the prices for
domestic goods changes by this movement But the national income of country 2 increases， while that of country 1 decreases. Note that the price rigidity is not assumed but is deduced from the optimal behavior of firms.
Let the price ind巴xfor country 2 be Cobb-Douglas type， that is，九=P2匂2βl
We assume thatα> 0，
s
> 0，α+
s
= 1， and thatβis much smaller than αbecause the weight for an imported good is usually smaller than that for a domestic goo
は
Under this assumption， th巴rateof change in the price index is一βewhich is
positve but may be sufficiently small
Turn to the case where exchange rate increases. Then， the equilibrium points move from (A1， A2) to， say， (Al，*A2
つ
Inthis case， the rate of change in theprice index isαP2一βewhich is definitly n巴gative
When the exchange rate changes， the price index changes discontinuously because of an asymmetric price movement of domestic goods Thus， we can verify the“ratchet effects" in depression
Next， w巴assumethat the world economy is in prosperity where excess
capacity and unemployment almost disappear..Suppose that the initial equilibrium points are (Bl， B2) in Figure 4， and confine our attention to the normal case
When the exchange rate decreases， the equilibrium points move from (Bl， B2) to， say， (B1， B2').Then， in country 1， the national income decreases while the
price of the domestic good does not change In country 2， the national income increases and the price of the domestic good increases sharply So， the price index of country 2 definitly increases 7) This assumption is pur巴lyfor convenience to explain the neutrality Indeed， if we introduce into our model the trade agencies who behave in a similar manner to our firms， we can show "the perfect neutra1ity" even though the weight for an imported good is not so small. Here "the perfect neutrality" means the downward-rigiditv of the price index In this case， however， the model is somewhat complicated
-176- Kagawa Unzverszty E.じonomuReviez刀 374
Pl
Figure 4
P2 Yl
Y2
When the exchang巴rateincreases， the equilibrium points move from (Bl， B2)
to， say， (B1*， B2 *).In this case， the price index of country 2 decreases slightly
because of the downwand price rigidity of the country 2's domestic good. Thus， we can verify the“ratchet effects" in prosperity
In a similar way， we can analyze other cases. The intermediate case of the above two extremes may be analyzed We can also analyze the case where the normal case is not valid.. We will easily find the case where the ratchet effects fail to occur
Lastly， we investigate the effects of changes in exchange rate on the balance of-trade.. Let B be the surplus of country 1 expressed in terms of country 2's currency:
375 Exchange Rate乙hang巴sand Ratchet Effects 7i ヶ '' 7
B =
号
ん
(1j2，P2，の)-PZ!I( 1，/1 ρ1，仇) (5)Let us differentiate Eq (5) logarithmically， by assuming the zero balance of trade in the initial state.
When we assume the prices of two countries' domestic goods to be rigid， we can obtain the following siri1ple formula though the general case is rather compli
似品;
dB/V=一(1十lJq+12q)e
+
Z2yi}2-l1yfh，where V is the value in terms of country 2's currency of imports and exports when the balance of trade is initially balanced Let us consider from the two points whether devaluation improves the balance -of-trade of the devaluating country or not; 1) the Marshall-Lemer's condition， e， I l1q十i2qI
>
1， and 2) the assumption of the normal case From a simple calculation， w巴canmaintain that devaluation improves the balance-of-trade if the former is valid while the latter is not On the contrary， devaluation deteriorates the balanc巴-of-tradeif the former is not valid while the latter is valid Other cases are not so simple.. So the detailed discussion is left for the inter -ested readers IV.Concluding remarks We explained theoretically the“ratchet etfects" by using Negishi-Nakagome model十 When the exchange rate changes， the price of the devaluating country's domestic good is likely to be neutral in depression but is likely to increase in prosperity for the normal case under the assumption of zero cross price effects十 On the other hand， the price of the revaluating country's domestic good is likely to 8) For general cases， refer to Benassy (1986)， pp 235-236， Appendix D-178- Kagawa Universiiy Economic Revzeω 376 decrease in depression but is likely to be neutral in prosperity This result crucially depends on the firms' expectations If a firm can not 巴xpecta sufficient increase in demand when he reduces his price， or if he fears a drastic decrease in demand when he raises his price， he will cope with the changes in demand by changing the quantity rather than the price.. This is the reason why we can observe the ratch巴teffects As is stated in Kindleberger (1989)， initial conditions and economic environ -ment play crucial roles on the effects of changes in the exchange rate Before concluding this paper， the following three points should be noted : 1) The direction of price movement or the price rigidity results from the firm's
optimal behavior
2) The “ratchet effect" is only one of the various properties which can be shown
9)
by the conjectural equilibrium model
3) As can be seen in the“ratchet effect"， the one-sided price change itself may follow without assuming the zero cross price effects and the normal case However， it crucially depends on the firms' expectations and the initial condi -tions Refere円ces Arrow， K J (1959)“Towards a Theory of Price Adjustment"， in Abramovitz， M.ed， The Alloωtzon0/Economic Resources， Stanford: Stanford Unversity Press， pp..41 51 Benassy， J P (1986) Macroeconomiω An Introduction to the N mト Walrasian Approach， N ew Y ork: Academic press
Bushaw， D.. W. and Clow巴r，R W (1957)Introduction to Mathematzwl Eヒonomics，
Home-wood， Illinois: Irwin
Gale， D (1978)“A Note on Conjectural Equilibria
ぺ
Review0/Economic Studies， Vol 45， pp 33-38377 Exchange Rate Changes and Ratchet Eff巴cts -179 Hahn， F H..(1978) “On Non-Walrasian Equilibria"， Revieωoj ew悶omzcStudies， Vol
45， pp. 1-8
Kindleberger， C P (1989) “Exchang巴RateChanges and Ratchet Effects: A Historical Perspective"， in Gerlach， S and Petri， P A. eds， Doru Hendo no Keizaigaku (The Economics of the Dollar Cyc1e)， Tokyo: Nihon Keizai Shinbunsha， pp 163-178 (tr in Japanese by Ishiumi， Y et al)
Nakagome， M. (1983) “Fukinkou Keizai niokeru Zaiseiseisaku no Kouka (The Eff巴ct of Fiscal Policy in Disequilibrium Economy'¥in Japanese， Aoyama ]ournal oj Eω日omics，V 0135， pp. 30-57
Nakagome， M (1985) Fukinko Riron to Keizaz Seisaku (Disequilibrium Theory and Economic Policy)， in Japan巴se，Tokyo: Sobunsha
Negishi， T. (1972) General Equi1zbrzum Theo~y and Internatzonal Trade， Amsterdam: North-Holland
Negishi， 1.(1979) Microeconomic Foundations oj K，りnesianMacroewnemzcs， Amster-dam: Norh-Holland
Nikaido， H (1975) Monotolistzc Comtetition and E.β~ctive Demand， 1975， New York: Princeton University Press
Shiomura， T (1989) “Negishi-Nakagome Moderu no Kaihoukeizai Moderu h巴 noKa-kutyou (An Extention of Negishi-Nakagome Model to an Open Economy)
ぺ
in Japanese， Kagawa Univers:ity Ewnomic Revzfw， Vol 61， pp.61-79Sweezy， P M.. (1939) “Demand under Condition of Oligopoly"， ]ournal oj PO似たal Eヒonomy，Vol 47， pp 568-573
