On a Paradox in the Theory of International Capital Movements*

1. Introduction It is now well known (see Kemp [3] and [411) that in a one-

sector, two-country model with given capital endowments a country should tax the earnings of its capital invested abroad in order to exploit its monopolistic advantage. If it borrows capital it should tax the earnings of foreign capital.

In a model which explicitly allows for saving and capital depre- ciation, Negishi [6] shows that if saving in each country is a fixed fraction of its capital earnings the optimal policy in the steady state for a capital lending country is to subsidize its investment abroad (assuming an interior maximum2). Various attempts have been made to explain this ‘paradox’. Kemp [5] explains it in terms of the optimal balance between the ‘expansion effect’ of the subsidy and its (bad) ‘allocation effect’. Jones [2] attributes the curious result to the ‘second best’ nature of the problem. It is not clear, however, what Jones means by this.3 Francis [l] contradicts Negishi’s result by using an alterna- tive but (as will be seen later) unusual assumption about capital earnings.

This paper draws attention to a special feature of Negishi’s model that seems to have been unnoticed so far: Negishi implicitly assumes that while foreign capitalists save a fraction ( s * ) of their net capital earnings, domestic capitalists save a fraction (s) of their gross capital earnings, or equivalently, all tax revenue is distributed to home capitalists (who save a fraction s of it) and all subsidy is financed by taxing home capitalists. Negishi’s model can be generalized so that a fraction 0 < y < 1 of the tax-subsidy proceeds goes t o domestic capital- ists who save a fraction 0 < p ,< 1 of it.4 p is allowed to be different from s &s we make the distinction between capital earnings and

* I wish to thank Professor J. D. Pitchford, Professor L. R. Webb and the referee for helpful comments on an earlier version of this paper. Some proofs are omitted. These are available upon request to the author.

1 Figures in square brackets relate t o references listed a t the end of the article. 2By this we mean each country owns a positive amount of capital. 3 The meaning of his remark, quoted below, is not clear : ‘But note the “second

best” flavour of Negishi’s results: Interference . . . is profitable because the overall capital-labor ratio was not “optimal” to begin with.’ See Jones [Z, p. 381.

4 A general interpretation is possible: 0 < B-y < 1 is the fraction of the tax-subsidy proceeds saved no matter by whom.

440

SEPT., 1973 INTERNATIONAL CAPITAL MOVEMENTS 441 capitalists' transfer income. The proportion of the tax-subsidy proceeds saved is therefore 0 < fly < 1, the rest being consumed. Negishi's model is a special case for it implies py = s (e.g. y = 1, p = s) . I n our more general model an interior maximum may be consistent with tax or subsidy on borrowing or lending, depending on the relationships among parameters.

A second task of this paper is to examine boundary solutions. It will be demonstrated that the free flow of capital is optimal if the maximum is a point a t which the home country owns no capital. In particular when f l y = s (Negishi's case) one obtains a boundary solution if s > s* o r if s = 1, o r s+ = 1. When py = 0 (all tax- subsidy proceeds are consumed), one always obtains a boundary solution and both tax and subsidy are possible.

Finally i t mill be shown that in the first-best problem (i.e. when domestic saving is totally controllable) the Negishi paradox disappears. The optimal solution is that the home country exploits its monopoly power by setting marginal revenue equal t o replacement cost and at home marginal product equals the depreciation rate.

2. Negishi's Model Re-examined and Generalized Following Negishi, let us denote : k : capital owned by the home country z : capital lent (borrowed if z negative) by the home country f : home production function s :proportion of home capital earnings saved (0 < s < 1) a : rate of depreciation of capital k*, f", s" are the foreign counterparts of k, f and s respectively.

We do not rule out the case where s = s+ or f = f' identically. The Inada conditions are assumed for the production functions. This will ensure that production takes place in both countries.

In addition we generalize Negishi's model by assuming that a fraction 0 < y < 1 of the tax-subsidy proceeds goes to domestic capitalists who save a proportion 0 < f l < 1 of it.

In the foreign country the net (= gross) rate of return to foreign capital is f". By perfect competition, domestic capital lent abroad also earns the gross rate of return f"', while capital employed at home earns f'. T'he difference f"' - f' is the amount of tax (subsidy if negative) per unit of capital lent abroad. If the home country borrows,6 f*' - f ' is the per unit subsidy (tax if negative) on capital borrowed. (f*' - f')z represents the total tax-subsidy proceeds. In- come of domestic owners of capital is composed of their capital

6 Francis [ l ] argues that when the home country borrowr, interest payments to foreign capital should be lzlf (and not Izlf"). In the foreign country, how- ever, foreign capital earns f*' per unit. This implies some sort of imperfection in the competitive process. We adopt Negishi's more reasonable assumption that interest payments to foreign capital are Izlf*'.

442 THE ECONOMIC RECORD KEPT.

earnings kf’ and their transfer income y(f+’ - r)z. Their saving is assumed to beb :

(1)

(2 )

skf + /?y(f*’ - P > z = s ( k -zPy/s)f’ + z&ff‘.

s ( k - ~)f’ + S Z ~ + ’

Negishi’s expression for domestic saving is :

implying /3y/s = 1. Perhaps he had in mind y = 1 and /3 = s. Alterna- tively (2) corresponds to the case where domestic capitalists save out of gross capital earnings.

One wishes to maximize domestic consumption (o r net income) in the steady state where saving equals capital depreciation for both countries :

subject to : maximisey=f ( k - z ) + z f + ’ - a k (3)

(4) 8*k*f *’ = aE*. (5)

s(k - zpy /s ) f’ + $7 f *’ = ak

Assuming a maximum exists with k* > 0 and k > 0, one obtains’ from ( 5 ) :

f*’ = a/s*. (6) At an interior maximum, from ( 3 ) , (4) and (6 ) , there exists a

(7a) (7b)

(8)

(8) is obtained from (7) using the algebraic property that u / b = c / d implies u / b = (c - 6 u ) / ( d - Sb) .

constant A such that : f ’ - a = A[(k - zSy/s)f“ + (f‘ - a/s)l

f ’ - a/s* = A[(k - zSy/s) f + (f’ - a/s*)fly/sI.

f ‘ - a - - (k - zSrl4f” + (f’ - al-4 - (f‘ - a)8r/s f ’ - a/s*

Hence, provided f ’ # a/@:

(k - zb7le)f ‘I

Negishi’s implicit assumption Py = s reduces (8) to : f’ - ct - (k - 2)f” + a - a/s

f ‘ - a/s* -

(k - % I f ” For s < 1, the right-hand side of (9 ) is positive and greater than

(10) Hence the conclusion that the home country must tux foreign capital (if i t borrows) but it must subsidize its capital invested abroad (if it lends). The latter policy prescription is rather paradoxical.

Note that for s = 1 (9 ) holds if and only if S* = 1 but for this 8 In the following expressions, s is assumed to be positive so that the division

by 5 is meaningful. This will facilitate the analysis and computation and make our expressions comparable with those of Negishi and Francis. The results do not change for the rather trivial c u e s = 0.

7 The Inada conditions ensure that (K-8) and (k*-z) are positive but do not ensure that k* > 0, as asserted wrongly by Francis [l, p. 2551.

unity, implying : f ’ > a/s* = f *’.

1973 INTERNATIONAL CAF’ITAL MOVEXENTS 443

case it can be shown that global maximum is at k+ = 0, k > 0 and z > 0. One may conclude therefore that for Negishi’s case (py = 8 ) an interior maximum, if i t exists, is characterized by (10).

For the more general model (10) need not be true a t an interior maximum. We shall consider three cases s = s*, s > S* and s < s*.

Case 1 : s = s*. No interior maximum. At an interior maximum one would obtain from (4) and ( 6 ) :

In view of (7a) and (7b) this is true (at an interior maximum) only if s* = 1. It can be shown, however, that when s* = s = 1 the global maximum occurs on the boundary with k+ = 0. Therefore for s = s+, the global maximum is a boundary one.

f *’ = a/s* = ale = f ’. (11)

Case 2: s > s+. It will be shown that if an interior maximum exists it is charac-

From (4) and ( 6 ) , and noting that s > s+ : terized by domestic borrowing and a subsidy on capital borrowed.

. (k - zBr/e)f’ = ka/s - ( z ~ ~ Y / s ) ~ / s * < (k - Z / ~ Y / S ) ~ / S * . (12)

Hence: f‘ > a/$+ = f*‘ if and only if (k - z p y / s ) > 0 and f’ > a/s* = f*’ if and only if ( k - z p y / s ) < 0, which is possible only when f3y > s. The latter situation (f’ > f+‘ = U / S ) can be dismissed easily, for it implies the left-hand side of (8) is positive and greater than unity, and hence requires (f‘ - a / s ) - (f’ - a ) p y / s to be posi- tive, i.e. P y / s < (f’ - a / s ) /(f - a) < 1, contradicting p y > s.

We are left with f’< a/s* = f”’ and ( k - z p y / s ) > 0. Lending (2 > 0) can be ruled out by showing that f’ > a. Suppose f‘ < a < a/s. the left-hand side of (8) is non-negative and less than unity, and this requires that (f’- a / s ) - (f’- a ) P y / s > 0, i.e. P y / s > (f’ - a /s ) / f ’ - a ) 2 l/s. But py < 1, hence a contradiction. There- fore f‘ > a. The left-hand side of (8) is then negative. For this to hold it is necessary that (f‘ - a / s ) - (f’ - a ) p y / s > 0, hence f’ > a/s. Using this result together with (4) and ( 6 ) :

(z/$J/~)u/s’ = ka/s - (k - @~/s)f’ < (zSr/4a/s (13)

which is consistent with s > s* if and only if z < 0. Finally note that (f’ - a/s) - (f‘ - U) py /s > 0 and f‘ > a

imply P y / s < (f’ - a / s ) / ( f ’ - a ) < 1. Negishi’s implicit assumption @ y / ~ = 1) therefore rules out the existence of an interior maximum when o > so. In our general model an interior maximum is possible when s > s*, but it is only consistent with domestic borrowing (z < 0) and a subsidy on capital borrowed. This ‘paradoxical’ result is not preeent in Negishi’s model.

&S6 3: S < 8,.

In this case an interior maximum may be consistent with a tax on

444 THE ECONOMIC RECORD GEPT.

capital borrowed, a subsidy on capital lent (Negishi's paradox) or a taz on capital lent by the home country. The latter situation is possible only when py /s > 1 and is therefore ruled out by Negishi's assump- tion.

As s < s,, from (4) and ( 6 ) : (k - z / ~ Y / s ) f' = ka / s - z ( P y / s ) a/S*

Hence f' > a/S* = j"' if and only if (k - z p y / s ) > 0, and f' < a/s+ = f"' if and only if (k - z p y / s ) < 0.

When (k - z p y / s ) > 0, f' > a / S " 2 a. The left-hand side of (8) is positive and not less than unity. Hence (f' - a /s ) - (f' - o ) P y / s < 0; i.e. /3y > (f' - a / s ) / ( f ' - a ) . T'his conmtion is certainly met if f < a / S , in which case the home country borrowsa and taxes foreign capital, or if p y / s 2 1, which includes Negishi's case.

When (k - z&/s) < 0 (which is true only if z > 0 and p y / s > l), f' < f*' = a / S + < a/s, implying a tax on capital invested abroad.

The results obtained for the generalized Negishi model are complex but instructive. In this second-best model (i.e. where domestic saving is not totally controllable) one must expect that policy implications in general differ from those of a first-best model. Various possibilities can arise, depending on the relationships among parameters. It is possible to have tax or subsidy on borrowing or lending. Negishi's implicit restriction py = s rules out the possibility that f' < f" at an interior maximum. It is interesting to note that even in the generalized model the free flow of capital (f' = f"') is inconsistent with an interior maximum.

> (k - zPy/s)a/S".

We now turn our attention to boundary solutions.

3. Boundary Solutions A sufEcient condition f o r boundary solutions is that (7a), (7b),

( 6 ) and (4) are inconsistent. Thus, as shown in Section 2, when s = s" # 1 these four equations are inconsistent. When s = s* = 1 they are consistent but the global maximum can be shown to be a boundary one, with k* = 0. Two 'polar' cases deserve special'attention: (a) Negishi's case ( p y / s = 1) which rules out an interior maximum when s > s+ (as shown in case 2) or when s = 1 or s+ = 1 (as also shown earlier) and (b) py = 0 (all the tax-subsidy proceeds are con- sumed), in which case no interior maximum exists. For the latter case, if s > s+ it is optimal for the home country to be the sole supplier of capital and exploit its monopolistic advantage by setting marginal revenue equal to the rate of depreciation. In order to achieve this the home country may have to subsidize o r tax capital lent, depending on a, s and on the two production functions.

8 This is established by an argument similar to (13).

1973 INTERNATIONAL CAF’ITAL MOVEMENTS 445 For the general case the following results are obtained for boun-

dary solutions : If k = 0 a t the maximum, net income is obtained by maximizing:

which gives f‘ = a/S’ = f”, implying no tax or subsidy on capital borrowing.

y = f ( - 2 ) + Za/S’

If k’ = 0 at the maximum, there exists A such that : f’ - a = A [(k - pyz/s) f ” + (f’ - a / ~ ) ] f - ~ f ” ’ - f” = A [ (k - p y ~ / ~ ) f ” + (f’ - ~f’” - f ” ) / 3 ~ / ~ ] .

Not many results can be gleaned about the general case. If py > 0 and s = 1 = s’, then f’’ > a > f’, implying a taz on capital invested abroad. If, in addition By = s, then zf*” + f”’ = a (marginal revenue from lending equals replacement cost). When s = 1 and py = 0, f‘ = a = zf’” + f”, which is the familiar taxation rule found in Kemp [4]. If p y = 0 and s is ‘sufficiently’ small, the home country should subsidize capital lending.

While i t is unambiguous that the free flow of capital is optimal if the home country owns no capital, both tax and subsidy are possible when the home country owns all the world capital stock and lends capital abroad. A subsidy on capital lending is never optimal, however, if domestic saving is totally controllable.

4. Conclusion By generalizing Negishi ’s model we have shown that subsidizing

capital invested abroad is a possibility, not a necessity. This possibility does not arise, however, in a first-best optimization model, where the optimal relations is zf*” + f*’ = f’ = a. The first equality is the fam- iliar tax rule found in Kemp [$I. The second equality reminds us of the golden rule for a closed economy (see, for example, Phelps [7] ). The simple, unambiguous result of the first-best model sharply contrasts with complicated results of the second-best model. This paper can be regarded partly as an exercise in the theory of the second-best.

Two further points deserve attention. First, the possibility of a boundary solution is due to the assumption that only capitalists save. Boundary solutions cannot happen if a constant fraction of national income is saved. Second, there is no presumption that it is optimal to t ry to reach the steady state where home consumption is maximized, for even when it is possible to reach it the adjustment cost may be enormous.

Nao VAN LONQ Australian National University

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Trade’, Quarterly Journal of Economics, Vol. LXXXI, February 1967.

446 THE ECONOMIC RECORD sm., 1973 [3] Kemp, M. C., ‘The Benefits and Costs of Private Investment from Abroad:

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[7] Phelps, E. S., Golden Rules of Economic Growth (Norton, New York,

Comment‘, Economic Record, Vol. 38, March 1962.

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