Consumption opportunities and the real value of the external debt

Journal of Development

Economics

CONSUMPTION

16 (1982) 93-iO1. North-Holland

Publishing

Company

OPPORTUNITIES AND THE REAL VALUE OF THE EXTERNAL DEBT Rudiger

DORNBUSCH*

Massachusetts lnstitute of Technolo~gy, Candvidge, MA 02139. USA

Received November

1980, final version received May 198 1

Developing countries have, in the period since the oil shock of 1973-- 1974, built up large external indebtedness. At the same time world inflation has in good part eroded the real value ofexisting debts. But the measurement of the inflation effects on real debt depend critically on which among a number of deflators is selected. The deflators proposed in this context have traditionally been export prices, import prices or prices in world trade. This paper argues that the correct deflator is the domestic consumer price index. Using the consumer price index as a debt deflator it is readily shown that conventional results in trade theory are recovered in the prcsenct. of external indebtedness: The income effect of an export price increase is proportional to the lzvcl of exports, the income effect of an import price increase is proportional to the level of imports. Re‘il income, using a comprehensive income measure, is equal to the value of domestic output less the real va!ue of real interest payments on external debt.

I. Introduction Repeated oil price increases have led developing countries to build up large external debts and world inflation has reduced the real value of these debts. Renewed shocks to the terms of trade will require adjustments in trade flows to finance the debt service obligations. In this setting the question of the appropriate deflator for foreign debt has arisen. Suppose import prices rise. As debtors we are better off because of the reduced real value of our debts, but as importers we are clearly worse off. What is the deflator that strikes the balance between these two effects and provides a meaningful measure of the welfare effects of price changes. Alternative deflators have been proposed. One suggestion is to deflate debt by export prices to show how much exports are required to pay off or service the external debt. With this measure it is clear that increased export prices reduce the real value oi‘debt and suggest all improvement in the debtor country’s welfare. Another view is to use import prices as the deflator, on the argument that this gives a measure of the reduction in imports required to pay off or service the debt. Of course, this procedure runs into the trouble of suggesting that higher import *I wish to acknowledge helpful comments from E. Cardoso and J. Willis mson. I-‘inancial support was provided by a grant from the National Science Foundation.

0304--3878/82/0000-0000,!$0:!.95

(!.J 1982 North-Holland

94

R. Dornbtuch., Consumption opportunities

prices reduce the burden of the debt. An intermediate position is to argue that the proper deflator is a weighted average of import and export prices where the weights are the marginal shares of imports and exports in the adjustment of current account imbalances. We argue here for a different measure, namely, the consumer expenditure deflator. It is shown that in combination with the comprehensive income measure, this deflator gives a very conventional statement about the welfare effects of price changes.

2. The concept of income The center of lthe analysis is the proper concept of income. We use the traditional Hicks--Bailey definition of comprehensive income as the value of consumption that can be maintained without impairing future consumption opportunities. Defining as c this value of sustainable consumption or real disposable income it is equal to c=(Y -i*ED-

ED(&+))/P,

where P

= home consumer price index, Y = home nominal value of output, D =external .iebt in dollars, E = exchange rate, cruzeiros/$, i* = world nominal rate of interest, J, d =rates of depreciation and home inflation.

The definition of real disposable income in (1) recognizes that debt service is a charge against resources and that capital losses, if consumption opportunities are to be maintained, likewise diminish the consumption level afforda by a given level of output. Eq. (1) is readily transformed into an alternative and suggestive form. For that purpose we define the home and foreign real interest rates, r = i - fi and r* = i* -@*. We also assume for the prc,sent that nominal interest rates, adjusted for expected depreciation, are eq\ -iliLzdor i = i* + u’.These relationB”are summarized in (2): r=_i-p,

p*=i*_@*,

j=

i”

$

&

(2)

Using these definition, adding and subtracting p* in (l), and rearranging terms ,yields ou* expression for sustainable consumption or real disposable income as c= Y/P-d,

(1’)

R. Dwnhusch, Consumption opporttmit its

95

where d = ED/P is the real value of the external debt. Real disposable income thus equals the real value of output less the real value of ieal interest payments on the external debt. Let x and m denote the real value of exports and imports both measured in terms of the consumer price index. We can then write consumption as equal

output, less exports, plus imports: c=Y/P-x+m

(3)

which in conjunction with (1’) implies that the sustainable current account must be balanced, with a trade surplus equal to the real debt service: x=m+rd.

(4)

Eqs. (1’) and (4) describe the average or long-term behavior of consumption opportunities and the current account. They are therefore the appropriate benchmark by which to judge the effects of changes in the levels of prices in world trade. We now proceed to that analysis. 3. Geometry Before moving to an algebraic statement of the effects of price changes we provide here a geometric interpretation. In fig. I we show cons1 mption of importables and exportables (C,, C,) as well as production (Q,,J,). Initial production equilibrium is at point A. (We take the case of a fixed >roduction point, although the algebra and the footnote below show that we coul li also look

Fig. 1

R. Dornbusch, Consumption opportunities

96

equilibrium on the transformation curve.) The line QQ through A indicates the value of output at thz initial relative prices. From the value of output we subtract the real value of debt service, being rD/Pr in terms of exportables and rD/P$ in terms of importables. Subtracting these quantities from point A yields points on the consumption frontier CC. Point B is the preferred consumption point with imports BF equal to exports FA less debt service GA, both measured in terms of importables at the prevailing relative price.

at an

Cm

Fig. 2

Suppose now a rise in the absolute world price of importables, In fig. 2 we show GE value of production changes as the budget line rota.tes through point A.’ To determine what happens to consumption opportunities we have to construct the=new budget line C’C’.It is apparent that the value of debt service measured in ter-mof exportables does not change. Hence point G remains a point on the new budget line CC’. Through point G we draw a line with the slope given by the new relative price ratio. The value of debt service in terms of importables of course, is seen to decline as shown by the reduced v&Cal distance between QQ and C’C’ at point A. The new consumption point will lie on the budget liae C’C’. Now we can ask holwmuch of an increase in income we would have to give the consumers to allow them to buy tile initial consumption basket B at the: new prices. Measured in telrmsof imy, \r\a bles the compensating variation in income is indicated by the vertical distance between C’C’ and a parallel through point B, that

‘The argument neglects movemt=nts along the transformation curve in response to price changes. For small price changes this effect is second order; for large price changes the geometry and the algebra below overestimate the income effect of price changes. The same applies to the treatmtnt of consumer adjustment below where the Slutzky approximation is exact for small changes.

k. Dornhusclt,

HB. It

is

Consmption

opportunitirs

97

readily shown that this compensating variation is equal to HB = BF - HF = P;(Q, - rD/P; - C,)/P:--- P,*(Q,- rD/Pz - C,)/P;’

where i;Tidenotes the physical level of imports and Pz and Pi’ are the initial and new levels of import prices. Thus the change in real income, measure importables, is equal to imports times the percentage change in import prices.2 As shown in the diagram a rise in import prices reduces welfare, a measure of the reduction in real income being the compensating variation H B. It is apparent that we can use the same technique to show that a rise in export prices raises welfare. (CC’ now rotates through a point on CC vertically below L, becoming steeper.) Finally. an equiproportionate change in export and impor prices will only reduce the real value of debt service thus shifting CC upward, coming closer to QQHaving established an intuitive interpretation for our results we nQw proceed to brief algebraic derivation of the change in real disposable income induced by world price changes.

4. The algebra of price changes

To determine the effect!: of price changes on real disposable income we start with the definitions of th;; value of output, the consumer price index and import prices, all measured in home currency:

Y = PxQx+ PmQw

p=

w,,

P,),

P,= EP?

(5)

Eqs. (5) reflect the assumption that the home country produces both importables and exportables. In the world market for importables we are price takers, but not necessarily in the export market. The price index P is a linear homogeneous function of import and export prices with elasticities equal to the expenditure shares of the two goods.3 Using (5) the change in real disposable income is given by AC=(&QJp)p,

-I-(P,,Q,/P)&,

- ( Y,‘P)p -- rd( I?- p)

(6)

2With P ha.c p rice index we can define the change in consumption measured in tern s of the indcx, rather than in terms of importable,:, as (Pzfi/P),“iz = rnPz. where m denotes the real va ue of imports. ‘For the case of Cobb-Douglas utility function the price index would be P = PXpk- ‘, where Q and I -a are the expenditure-shares of exportables and importables.

98

R. Dornhwch,

Consumptiotl

opportunities

which simplifies to the following expression:4 AC= x($: - P;) + rd P”,

(6’)

where x again equals the real value of exports. Price changes in world trade have two effects on real disposable income. A terms of trade improvement raises real consumption opportunities by the real v:ilue of exports times the terms of trade change. This is the conventional income effect of a terms of trade improvement. A rise in the world dollar price level. P*, raises real disposable income by reducing the real value of the debt and hence real debt service. This is represented by the second term in (6’). Another way of looking at th,z effects of price changes is to distinguish the role of world export and import prices. This is done in (7) where WChave substituted for the change in import price; pz= P* and rearranged terms to obtain Ac=xPz--rnjiz.

(7)

In eq. (7) the term pz measures the change in the dollar price of our exportables. The equation shows that a rise in the dollar price of exportables raises real income in proportion to exports. A rise in ;he dollar price of imports lowers real income in proportion to the level of imports. There is thus a complete symmetry that is assured by the fact that on average the inflation adjusted current account must balance or x = M+ rd. 5. Risk premium and home goods The analysis so far has made two assumptions that are easily relaxed. The first is that depreciation adjusted nominal interest rates are equalized. An alternative is to assume a risk premium ~1such that i = i* + t=-t- p. With this assumption c= Y/P-(r+p)d

(1”)

is our definition of real disposable income. The risk premium or ‘spread’ ~1reduces the level of real disposable income: for a given level of debt. It does not Ij though, affeci. the result derived in (7) above. A second generalization concerns the role of home goods. Suppose that in addition to exportables and import competing goods the home country produces non-traded goods. This, of course, changes the definitions of nominal income and 4From (6) we o btain the simplification by noting that (Y/P)p=o{ Y/P)p+( 1 -n)(Y/P)fi,. Next observe that a(Y/P)=a(c+ rd)=avd+ P&./P, where Ci denotes consumption of commodity i. Further note that the last term in (6) can be written as rd(EI-~=rdP*+nrrl(B,-B,\ and that on average the current account is balanced so that real exports, x, e:qual real imports, m, plus real iflterest payments: YF tn + ~1. Ilsing HII these substitutions in (6) yields the final form in (6’).

the price index. It is readily shown that the effects of price changes as derjved in (7) remain unaffected. This is so because the home country neither exports nor imports non-traded goods.s Having made a case for the consumer expenditure deflator as the pro with which to deflate the external debt consider now an application to Brazil. We show for the years 19X3--1980anindex ofthe value of the real debt in 1975 dollars using alternatively as deflators import prices, export prices and the consumer price index. The results are shown in table 1. Table I The real value of the external debt: Bra7il 1973 1980 ( I975 dollars).” -----__-____ Deflator

1972

I973

1974

1975

1976

1977

i978

Import prices Export prices Consumer prices _-

IO.2 8A 6.4

9.1 7.3 7.1

9.3 9.0 9.7

14.5 14.5 14.5

17.7 IS.9 16.9

20.6 15.6 18.8

24.7 25.X 21.7 25.3 22. I 27.5 ~-~-~

for price indises,

Boletim

“Sours: Internationd Finmcial for debt data.

Statistic-s

do

Bunco

1979

1980 n.ii. n.a. 36.0

Central

and

Conjrmtura

The estimate of the real value of external debt derived by using consumer prices as a deflator shows a substantially larger increase in real debt in the 1970-l 979 period than either of the other series. It suggests a substant:ally larger reduction in long-term consumption opportunities than is implied by either of the other series. 6. Canciuding remarks

This note has shown how to integrate the presence of an external debt in the conventional treatment of the income effects of a price chaage. The analysis has shown that a rise in export prices raises real income in proportion to exports and that it rise in world prices of our imports reduces real income in proportion to imports. The income effects of changes in the real value of the debt service liability ;Ire implicit in these results since the difference between exports and imports is ‘The value of home output and the price index now are defined as foilowc: Y = UK@,+ P,Qm + P,QA

and

I’ -=fyftP,f

’

‘,

wherethe subscript n denotes non-traded goods. The definition of sustainable consumpion remains c== Y,‘P-+d. A c hange in prices in world trade yields as an effect the terms developed in ( 7). This is so sin= dc = (PXQJP)& + (PnQ,JP)&, f (P,Q,JP}p” - a( Y/P)p, - hf Y/P)P, - ( I - a - h)( Y lP)pn - rd( I? -rd(a~X+b~,J+(l -a-b) (I;,-li)-rdE’=x(P,-E,-~~B,-E), where we have -q=x&--m~m used the definition of the expenditure share to note that C,P,iP=4 Y:P-- rl) and likeaNise for the other real commodity demands. We also use again the equation for curreLIt account balance. rd = .Y -nm, and the definition of the price index.

R.. Dornbusch, Consumption opportunities

100

equal to the debt service. As noted earlier these measures are only approximate. They are exact for small changes, but provide an overestimate when price changes are large Qnd hence substitution along the transformation curve and along the indifference curve become important. Bearing this qualification in mind our measure provides an upper estimate of the income effect. A more serious qualification concerns the presence of distortions such as tariffs. With distortions present changes in world prices no longer gi\,e a precise (in welfare terms) measure of the change in rl=alincome. An equiproportionate rise in world prices, by reducing real debt leads to an increase in real income and increased consumption of importables. Since these commodities lzre subject to a tariff and hence ‘under-consumed’ there is an extra welfare gain the extent of which depends on the size of the tariff and on the marginal propensity to import. Similarly :n the case of terms of trade changes there are extra costs or benefits associated with the induced changes in imports. Specifically in the case of a rise in export prices there are extra gains associated with the income and substitution effects that generate increases in the level of imports. Thus, again in parallel with the conventional measures of the real income effects of price changes we find that in the presence of distortions none of the conventional price indices gives an appropriate measure of real income changes? Appendix

Let the representative consumer have a utility function u = VQ, - X Q,,,+ Ml,

(A-1)

where we already have substitukd for consumption in terms of *Droductionand trade. The change in utility measured in terms of exportables iz dU/U,= -dXi-pdA4,

(A.2)

where p = P,Jf, and where we applied the envelop property dQ, -t-piSQ,=Q, Now from the economy’s budget constraint we have PZX = PzIw +rD, or for given nominal debt and a given real rate of interest: dX-p*dM=M*p:-

XI$,

p*=

PyP;,

M*rP~M/P_~.

(A.31

Adding and subtracting p*dM in (A.2) and using (A.3) yields dU,‘U,=(X~~--M*~~)ttna*~. ‘See appendix.

(A-4)

R. Dornhusch.

Consumption

opportunities

101

It is apparent from (A.4) that if there is no tariff (t =0) the formulation is identical to eq. (7) in the text. In the presence of a tariff though there are extra costs or benefits - the term M*fi - associated with changes in world trade prices that bring about, through income or substitution effects, changes in imports. With the deftnition of the marginal propensity to import, (p, and the compensated price elasticity of imports, (defined positive), e, we can write

k_=

1

> 1.

(A.3

1 - w/4 1 + t)

References Bailey, M., 1972, National income and the price level (McGraw-Hill. New York). Freeman,C., 1979, A note on interest payments to foreigners under inflationary conditions, Canadian Journal of Economics, 29 1-299. Hicks, I., 1946, Value and capital (Oxford University Press, Oxford). Siegel. J.. 1979. Inflation induced distortions in government and private statistics. Review of Fc~mmicc

JDE-

E

;Ind StiltiStic’S.

82 30.