-19-

occur until owners of capital in the home country earn the same rate, net

of taxes, at home and abroad. This is the case when

(32) (1 = tHyecty = eM(ey,
tet ut: = 1 - th and uF: V1 - af), Then (30) and (31) can be

rewritten as
(33) rF (ty = uPr(t) = uF Hey sul,

Proposition 6. There will always be non-negative capital flows from
country H and country F.

Proof. By contradiction. Assume that at some time a strictly positive
amount of country F capital is invested in country H. Let t be the first
time at which this happens. Then investors in country F must be

making the same return at home as abroad; hence heey

H F
to be true it must the case that g (t) = g (t). If t = 1 this implies

ae KT which is a contradiction. If t > 1 this implies s(t) < sF rey,

rF(ty. For this

For this to be true it must be the case that whet -1)< wrt - 1), which
implies we -1)< rt - 1), which is a contradiction.

Total taxes collected by country i at time t are
(34) H(t) = U(ey(u = ryeceyrk! (ty ~ ol (ty.

The governments are assumed to transfer tax revenue to the old of each

generation on a lump sum basis.


-36-

References
Bergsten, C. Fred, Horst, Thomas and Moran, Theodore H., American

Multinationals and American Interests. The Brookings Institution,

Washington, D.C., 1978.

Buiter, William H., "Time, Preference and International Lending and
Borrowing in an Overlapping Generations Model," Journal of
Political Economy 89:769-797, Aug., 1981.

Diamond, Peter A., "National Nebt in a Neoclassical Growth Model,"

American Economic Review 55:1126-50, Nec., 1964.

Hamada, Koichi, "Strategic Aspects of Taxation on Foreign Investment.

Income," Quarterly Journal of Economics 80:361-75, Aug., 1966.

Jones, Ronald W., “International Capital Movements and the Theory of

Tariffs and Trade," Quarterly Journal of Economics 81:1-38, Feb.,

1967.

Kemp, Murray C., The Pure Theory of International Trade, Prentice-Hall,

Inc., Englewood Cliffs, N.J., 1964. ch. 13.
Kindleberger, Charles P, International Economics, Richard D. Irwin,
Homewood, I11., 1973. 5th edition.

Lea, Sperry and Webley, Simon, Multinational Corporations in Developed

Countries. Technical Report, British-North American Committee,
1973,

, MacDougall, G.D.A, "The Benefits and Costs of Private Investment from
Abroad: A Theoretical Approach," Economic Record 36:13-35, Mar.,
1960.

Manning, R, “ Attitudes to International Capital movements in the Long-
Run: The Case with Neo-Classical Saving in the Foreign Country,"

Economic Record 51:242-48, Jun., 1975.


-13-

Proof. Totally differentiating equation (20) gives

[w! - n(gB)'Jdr = gBdn.

Stability requires

dr(t + 1)

arte) <1.
By (18), this implies w' - n(gB)' > 0. Thus £ > 0.

The intuition is that as n increases, the aggregate demand for
capital, which depends on the time t + 1 labor force, increases relative
to the supply of capital by the time t young population.

2.3.3 Optimality

It is a well-known result for growth models that a social planner
intending to maximize stationary welfare would set f' = n, the Golden
Rule rate.“/ In the fixed-coefficient technology overlapping generations
model it is possible to focus on the stationary equilibrium. In that
model setting f' > n cannot he considered inefficient because a lower f'
would require sacrifices from the current old. In this variable rate of
return model it is not possible to focus on a stationary equilibrium
because f' varies over time.

Proposition 2. An equilibrium is Pareto optimal if and only if r(t) an

for all but a finite number of times.


-23-

Proposition 8. Suppose a unique, stable, steady state equilibrium
exists. If r > (<) nm and a small tax is implemented by either country in
period ane then in the steady state, welfare at home is raised (lowered)
and welfare in the foreign country is lowered (raised).

Proof. Substituting (34) into (41) and (42) gives steady state utility

levels of

(a7) ul « res -1)twi - n(1- ult}, rutwl - n(1 - ula! ty,

ish, F
(Bh 4 u' -1) (B) + ul -1) ,
aul ipidr
By (47), —=> 0 if and only if (n - r)g B Gur > 0
the i

aut aut
Thus, —- > 0 if and only if n <r and => 0 if and only if n> r.
oT dt

i
By (47), 22 > 0 iF and only if (n - r) G >0
tha

au" aut
Thus, — > O if and only if n <r and | > 0 if and only if n>r.
che at

H

The imposition of a small tax by either country lowers r° and

raises r Hence, if rH is greater than the Golden Rule rate, n, a small
tax will lower rt improving matters in the home country in the steady
state. I[f rH is greater than the Golden Rule rate a small tax will raise
r, worsening matters in the foreign country in the steady state. If

rental rates are lower than n, the reverse occurs.


4.

In most of the international trade literature, unhindered capital
flows lead to efficiency. However, foreign investment taxation by both
capital importing and capital exporting countries represents a Nash
equilibrium in policy space. An examination of the stylized facts
reveals that neither the Pareto optimal nor the Nash solutions of
standards trade models are in fact. observed. There has not been a free
international capital market since 1930, and policies with respect to
foreign investment taxation are extremely inconsistent--not just across
countries, but within a given country over time.2/

The model presented here is not inconsistent with the stylized
facts. Allowing free trade in capital leads to factor-price
equalization, but not necessarily to efficiency and certainly not to a
regime which is Pareto superior to autarky. There will be no agreement
within a country as to what constitutes an optimal tax on foreign
investment. Thus, countries will not have an optimal level of foreign
investment taxation.

2. Capital Accumulation in an Open Economy
2.1 Production

In each period t > 0 firms use capital and labor to Produce output.
The capital is completely used up in the production Process and the
output is purchased by consumers. Part is consumed in period t and the
remainder is saved to he sold to the firms as capital for the period
t + 1 production process. There is a neoclassical production technology
which is assumed to be linearly homogenous. Thus, output per unit of
labor is a function only of the capital-labor ratio. The allocation of
factors is perfectly competitive; hence the wages of capital and labor

are equal to their respective marginal products. Capital mobility


-34-
proof is repeated. Proceed up to the formulation of d(t + 2). Here
0 A a
(A.11) L(t + 2)d(t + 2) = L(t + 1)feo(t +1)- colt + 1)] + F(t + 2)
- F(t + 2)
Thus, by (A.10)
A

d(t + 2) > [d(t +1) + a(t + 1)]{W(t +1) + ofd(t + 1) + a(t + 1)J}/n +

“ °

[F(t + 2) - F(t + 2)V/L(t + 2)

A A A
>od(t + 1)W(t + 1)/n + a(t + I)EW(t +1) - FL(t + 2)Y/n

by Euler's rule and the first-order conditions of the consumers' problem.

Thus
d(t + 2) > d(t + 1)r(t + 2)/n.

Hence {d(t)} is unbounded.


~20-
3.2 The Producers and Consumers
If taxes are not prohibitive the solutions to the producers
problems satisfy
(35) F(g' (t)) = gh (t)Ft(g'(t)) = w(t); f= Hy F
F(gi(t)) = r(t)s i= 4, Fe
There exist solutions

(36) wi(t) = wlu'r(t)]s 4

"
x
al

ai(t) = gfulr(t)]s 4 =H, F

The consumption of the old of generation zero is equal to their

rental earnings plus their transfers of tax revenues. Using (34) yields
(37) ¢5(9) = Er(1)u'ky + T(I)I/L,
= r(1)EK) + (ul = 1)6)(1)1/tg.

The choice problem of the young of generation t in country i is

_ ee ute! (t), ch(t))
{ey(t), cy(t), si(t +1) ae)


-26-

4 J ky = g(u'r(1)) a
(53) d) > "1 - —{— Vi,g = HF.j # i.
e -1 e g(u r(1))

Therefore the home old favor no tax or a tax smaller than the Kemp-
MacDougall optimal tax. The foreign old favor a prohibitive tax or a tax
larger than the Kemp-MacDougall optimal tax.

3.4.3 The Young

The opening of unrestricted trade will cause the young of the home
country to become identical to the young of the foreign country. All
advantage arising from belonging to the capital-abundant country is lost
with free rade and all benefits are captured by the old. All of the
benefits from trading with the capital-abundant country are captured by
the foreign young.

A tax imposed on foreign investment will have three effects.
First, it will enable the setter to act as a monopolist or a monopsonist,
as in the Kemp-MacNougall story. Second, it will help protect any
advantage the home young might have from belonging to the capital
abundant country and thirdly it will change the savings rate. Free trade
makes the home young better off than in autarky. Thus it would seem that
the advantages of acting as a monopolist are not as great for a young
home agent as for an agent in a capital-abundant country in the Kemp -
MacDougall world and the advantages of acting as a monopsonist are
greater for a young foreign agent than for an agent in a labor abundant
country in a Kemp-MacDougall world.
4, Conclusion

This paper presents a model which differs from the existing

literature in that it is dynamic and choice-theoretic. It is shown that


-22-

In equilibrium capital exported by the home country must equal

capital imported by the foreign country. Thus,

ultwl(ty = npi(t + 1g'(t +11,

| (0)
Bi(t+#1)+#u'-1

(45) Zu

Definition 3. An equilibrium with taxes is a sequence {r(t)} which

1?
is hounded away from zero and infinity such that (45), holds for every

t > 1 and r(1) is given by
(46) g(ur(1)) + a(uPr(1)) = KH + KE,

3.4 The Steady State Welfare Effects

Proposition 7. Suppose a unique, stable, steady state equilibrium
exists. Then the implementation of small taxes in period one by country
H causes the steady state values of r and rf to rise and rH to fall. The
implementation of small taxes by country F causes the steady state values
of r and rH to fall and rf to rise.

Proof. Totally differentiating (45) at the steady state and evaluating

at u' = 1, 1 =H, F yields

H
ar r ar r
a =5>O0 SH =-5 <0; i =H, F
at ytayFey 2 at futay ay @ , ,
ot =D 0 ar'| =5>0; 1-H, F
oT ufeuFey é at’ uty Fy YG


-35-

Footnotes
*/ 1 acknowledge the assistance of Nan Peled, Bob Hodrick, Marty
Eichenbaum and Matt Canzoneri. Errors were achieved independently. This
Paper represents the views of the author and should not be interpreted as
reflecting the views of the Board of Governors of the Federal Reserve
System or other members of its staff.
J/  MacNougall (1960), Kemp (1964), Hamada (1966), Jones (1967),
Pitchford (1970) and Manning (1974, 1975).
2/ For more on this issue see Shell (1977).
3/ Bergsten, Horst and Moran (1978), Chs. 1-6 and Kindleberger (1973),
p.269.
4/ Phelps (1966).

5/ Wallace (1980).


(25)

(26)

(27)

(28)

(29)

-17-

;
wet) = (1 = yy Qty

By definition

a(t) = 28 ,
By (18)
r(t +1) = Cr(t)’,

where C: = miet Bhnyt-y,

Equation (26) is a first-order difference equation with solution

qey*

t
rt #1) = chT yy

where by (19), r(1) is given by

H F

Ky +
nO) = yp

By (20) the steady rate is

1
re cy


~9-
Let s! (t +1) be the amount saved by the young of generation t > 1

in country i; i = H,F. Then the choice problem of a young agent in

country i is

5 (Max i i i
{cy (t), ea(t), si(t + 1))} Ulcz(t), ¢5(t))
subject to
(10) cl (t) + s¥(t +1) < wit)
(1) ep(t) <r(t + 1)si(t +1).
r(t + 1) can be regarded as one plus the interest rate earned on
savings because the capital is completely used up in the production

Process. Given the assumptions on U(+, +), the solution to the choice

Problem is given by (10) and (11) with equality and

"
2

Uy(eq(t), ep(t)) = a4(t)

r
2

(12) Upleq(t), ef(t)) - rg(t) =
- a, (t) tr(t + 1)rp(t) = 0,

where A, (t) and x(t) are non-negative multipliers associated with

consraints (10) and (11), respectively.


~32-

By (A.5) with equality, (A.7) and (A.8)
~ a A a
c(t +1) = c(t +1) + Ed(t +1) + a(t + V)AtWEe, (t +1), c(t +1)]
(A.9) + ofd(t +1) + a(t + 1)7}.
By colt +1) > p(t +1) and (A.9),
c(t +1) - Jt +1) > Ed(t +1) + a(t + Ince +1) +
(A.10) + o{d(t + 1) + a(t + 1)7}.

A a

d(t + 2) = cy(t + 2) + K(t + 3) - [cf(t + 2) + k(t + 3).

By (A.2) this implies

L(t + 2)d(t + 2) = L(t + Ieee +1) - See +1) -

C) A ‘ A
CFO (t + 2) = F(t + 2)Ja(t + 1)/CK°(t + 2) - k(t + 2)2.
This implies

d(t + 2) > [d(t + 1) + a(t + NTA + 1) + o[d(t + 1) + a(t + 1)%}/n


~29-

superior allocation, Let "A" over a variable denote an equilibrium

allocation. Feasibility requires
(A.2) L(t + T)fe, (t +1) + k(t + 2)J + L(t )cp(t) < F(t + 1)

If the potentially Pareto superior allocation does not satisfy
(A.2) with equality then it is not Pareto optimal. Thus, without loss of
generality, assume the potentially Pareto superior allocation satisfies

(A.2) with equality.

A
Case 1, tK(t + 1)} = {K(t + 1)} or at the first departure
A
K(t +1) > k(t +1),

By (A.2) c(t = 1) <¢,(t - 1) or ¢, (t) > et) or both.
0 A
Case la. co(t - 1) < Co(t - 1).

. A
(K°(i)} = {K(i)} for every i = 1, ..., t; hence by (A.2), for every

felsea, te
L(t - f)e\(t = i) # L(t - 1 - 1A (t 4-1) = L(t ~ ACME - 4)
L(t -i- ep tt -i-1).

By Pareto superiority f(t -1)> Jit - 1), and thus eplt -2)¢


(3)

~6-
f>+O0asg+, f' +e as q+, fi + Nas gr
Equating wages to marginal products implies

F(gi (t)) = gi(tyFi(g'(t)) = w(t); 4

HAF.

F'(gi (t)) = r(t)s i = WF,

where a prime denotes a derivative. System (3) is a set of four

F

H
equations in four unknowns: g (t), 9 (t), w(t), and r(t). By (2),

h(t), g(t), and w(t) can be solved as unique ‘functions of r(t). Thus,

(4)

(5)

ai(t) = a(r(t)); i = WF
w(t) = w(r(t)).

The desired capital-labor ratios are equalized across

countries, Totally differentiating (3) yields

w'(r(t))

a

- g(r(t)) <0

g'(r(t))

V/F"(afr(t)I) < 0.

Thus wages and the desired capital-labor ratio are decreasing in

the rental price of capital.


-5-

ensures that factor wages are equated across countries. The stock of
labor at time t is equal to the population born at time t. The

population is assumed to grow at a constant geometric rate.

Notation

a (t) : = the capital stock owned by citizens of country i at time t;
i= H,F.

gl (t) : = the capital stock located in country i at time t; i = H,F.

L(t) = the number of people born in each country at time t.

l(t) : = KU(tysL(t); i = HAF.

gi (t) : = Gi(t)/L(t); i = HAF.

r(t) : = the rental price of capital at time t.

w(t) : = the wage rate of labor at time t.

It is assumed that ory > iy; hence the home country is
relatively capital abundant.

Linear homogenity implies
(1) F(G'(t), L(t)) = L(t)F(g' (t))s 4 = HF,
where F(+,*) is the production function and f(+) is the average product
of labor function.

The production function is assumed to satisfy the usual

neoclassical assumptions

(2) fe C3; f> 0, f'> 0, f <0, for every g > 0


-18-

By (27), lim r(t +1) = 173 hence the equilibrium exhibits global

tro

stability.
3. Foreign Investment Taxation
3.1 Taxes

It has been shown that laissez-faire is not Pareto-superior to
capital mobility. Thus, restricting capital flows will raise the welfare
of some generations in a country. It will be assumed that each country
has one policy variable for curtailing foreign investment.

Country F may levy an ad valorem withholding tax 1, 0< af <1, on
the earnings of capital in country F owned by citizens in country H.

This leaves a rate of return at time t of
(30) r(t) = (1 - tf yet)

for the home country, where rF(t) is the foreign rental price. r(t) will
be referred to as the “world” rental price.
The government of the home country may tax the after tax earnings

of country H's capital exported to country F at the ad valorem rate a

H

Q<t <1. This leaves

(ary = thy eFyeFeey = c= cece)

available for investors in the home country. If (1 - y(t) < hey,
H .
where r (t) is the home country rental rate, the taxes are prohibitive

and no capital flows take place. If this is not the case, capital “lows


-8-
consumption. Gross substitutability ensures that an increase in the
return to saving, with wages held constant, increases saving.

The life-time utility function is

U= u(c;(t), co(t))s t>1,i=4H,F,
where ej (t) is the consumption of an agent of generation t of country i
in period t - 1+ 4; i = H,F, j = 1,2.

U is homothetic; hence Tet

Wey (t)/ep(t)) = Uy (ey (t), ep(t))MUpleq(t), ep(t))

be the marginal rate of substitution function. U is assumed to be such

that
Vet, U,> 0, (cy(t), o(t)) ¢ Rees j= 1,2
Jj 1 2
(8) WhO, cy(tI/ep(t) © 84, Woo as cy(t)/ep(t) > 0
W' + - © as cy(t)/co(t) +0, W'>+Oas cy(t)/e(t) +o,
U satisfies gross substitutability; hence

(9) W+ Wey (t)/ep(t) > 0, c,(t)/eg(t) B+.


-74-

The implication is that no small tax can make both countries better
off in the steady state, but any small tax will make one country better
off. Thus if r #n it will be to the benefit of one and only one country
to levy a small tax. There will be some tax which will be to the benefit
of at least one country in the steady state.

Suppose a social planner in country i is attempting to maximize the
steady-state utility in country i. If he can choose the tax rate and

savings rate he will solve.

Max

{g', t', cys cy} subject to
cy + chin + ng’ < f(g') + r(k’ -g'),

This amounts to simultaneously choosing a and gi so that r =n
and f(g!) +r(k - 9‘), the national income, is maximized. Suppose
however the economy is a market economy. Then the only policy variable
is a, and it may not be possible to simultaneously maximize the national
Product and achieve the optimal capital-labor ratio. Hence the optimal
tax may be larger or smaller than the one which maximizes national
Product, depending on the effects of taxation on r - n.

3.4.2 The old

The traditional static literature posits that each country
maximizes the value of national product plus the net after tax earnings
of foreign investment. Thus countries solve

(48) “8 erg! (rulyy + rtk! = g! (ru')3.


International Finance Discussion Papers
Number 218

February 1983

CAPITAL ACCUMULATION AND FOREIGN INVESTMENT TAXATION
by

ANNE SIBERT

NOTE: International Finance Discussion Papers are preliminary materials
circulated to stimulate discussion and critical comment. References in
publications to International Finance Discussion Papers (other than an
acknowledgment by a writer that he has had access to unpublished
material) should be cleared with the author or authors.


-ll-
(17) L(t)sMe #1) + U(t)sP(t +1) = L(t + Vote + 1) +
L(t + 1)gh(t #1); tod.
Substituting equations (4), (6), and (15) into equation (17) yields
(18) w(r(t)) = ngfr(t + 1)IBEr(t + 1)3; t> 1,
where B(r(t + 1)): = V[r(t + 1)Ir(t +1) +17,
In period 1

(19) ky + kt = 2gfr (1).

It can be seen that all capital flows take place in period one.
After period one owned capital-labor ratios, and thus neutral prices, are
equalized; hence no capital flows take place. This result differs from
Buiter (1981) and Ruffin (1979) where one country may run a countiual
current account dificit. Buiter's result is driven by the assumption
that the rate of time preference differs between countries. Ruffin's
result depends on differing savings rates and technologies between
countries and a depreciation rate between zero and one.
Def. 1 An equilibrium is a sequence, {r(t)}e ys which is bounded away
from zero and infinity, such that (18) and (19) are satisfied,

Nef. 2 A steady state is an reRe such that


-10-
By (8), W has an inverse function, V, such that
V(W(c, (t)/e4(t))) = cy (t)/e(t).

By the properties of U, (19) - (12) can be solved to yield

(13) cf (t) V(r(t + 1))r(t + T)w(t)/EV(r(t + 1))r(t +1) 4177

(14) ch (t) = r(t + 1)w(t)/Ev(r(t + 1))r(t +1) #17

(15) si(t +1) = w(t)/EV(r(t + 1))r(t #1) +17,

2.2.2 The old

The “old," or the agents of generation zero in each country are
endowned with their country's initial capital stocks. These stocks are
employed in the time t = 1 production process and the old consume the

rental earnings. Thus,
i _ i tae
(16) ¢5(0) = r(1)Ky/Lgs i = HAF,

where Ki is the original capital stock of country i.
2.3 Equilibrium
2.3.1 Definitions
In equilibrium the amount of capital supplied by individuals must

be equal to the amount of capital demanded by firms. Thus,


-31-
Min

cy tt +1), colt +1) colt + 1) subject to

a. * 0
(A. 4) cy(t + 1) + k(t + 2) - [cy (t +1) +k (t + 2)7 > d(t +1)

a Aa

(A.5) Ufc, (t +1), colt +1)] > Ufc, (t +1), Colt + 1)1

~ ~ + ti) 0 4

(cy (t +1), colt + 1)) exists because (cy (t +1), colt +1)) is
feasible, and satisfies (A.4) and (A.5) with equality.

Let

0 A

(A.6) A(t +1): = k(t + 2) - k(t + 2)
Then by (A.4) with equality,

A ~
(A.7) cy (t +1) - c(t +1) =d(t +1) + a(t +1)

“ A

Define g: R + R such that Ufc, (t), afc, (t)I} = Ufc, (t), c,(t)]. g
is continuously differentiable, with g' = W, the marginal rate of
substitution function. By the Immediate Value Theorem

A A Aa a a
(A-8) 9(cy) = a(cq) + (cy - cy) Loc, - ¢)) - Whey, 9(€,)7),

where ¢ is a continuously differentiable, monotonically strictly

increasing function and 9(0) = 0.


-30-

A Aa
co(t - 2). Proceeding backward we get £9(0) < (9) which is a
contradiction,

Case 1b.

Define the sequence {d(t)} such that

(A.3) d(t-+ 1) = ett #1) + k(t +2) - feO(e + 1) + kt + 2)3

It will be shown by induction that {d(t)} is positive and
A A
unbounded. q (t +1) + k(t + 2) is bounded; hence this will rule out

Case 1b. For the initial step we show d(t +1) > 0. Ry (A.2)
A 9 x 0

L(t + 1)d(t + 1) = F(t +1) - F(t +1) - L(t)c,(t) + L(t)co(t)

. 0 a 0 4 0

= L(t)£co(t) - Fy(t + V)ko(t + 1)] + F(t +1) - Fe(t +1) +

0 A A

L(t + 1)[k (t +1) - k(t + 1)IF, (t +1)
by Euler's rule and the first-order conditions of the consumers' problem,
where indices have been dropped for convenience. The first term is
strictly positive by strict quasiconcavity of the utility function and
the Pareto superiority of the “o" allocation. The rest of the expression

is. strictly positive by the strict concavity of the production function.

For the induction step let (< (t +1), E(t +1)) solve


-7-

It will also be assumed that

(6) e>1,

where e: = -g'r/g is the elasticity of demand for capital. Equation (6)
says that demand is elastic, and is true for all Cobb-Douglas functions.
Labor growth is given by

(7) L(t) = Lp’,

where Ly is the initial stock of labor in each’ country

2.2 The Consumers
2.2.1 The young agents

In period t > 1, L(t) individuals are born at home and abroad.
Individuals live for two periods. In the first period they are endowed
with one unit of labor which they supply inelastically. They save part
of their earnings in the form of capital, to he sold to firms, and
consume part. In the second period they retire and consume the value of
their capital earnings.

All individuals of generation t > 1 have identical tastes
represented by a utility function which satisfies the usual reqularity
conditions, is homothetic and displays gross substitutability.
Homotheticity is equivalent to assuming that the marginal rate of

substitution is soley a function of the ratio of first to second period


-21-
subject to
(38) h(t) +s'(t +1) < w(t)
(39) ep(t) sutr(t +1) +The + 1st).

The solutions are

(40) l(t) = re'(t +1) = 1tuir(t + tw (ty + T(t + T)/L(t))
1 u'r(t + 1)B'(t +1)

(a) htt) = uir(t + Uw! (t) + T(t + 1)/L(t)
B (t +1)

(42) si(t +1) = ulr(t_+ Twit) - pple +1) - rit 41)
u'r(t + 1)B'(t + 1)

3.3. Equilibrium
Equilibrium requires that in country i, i = H,F, owned capital
equals capital supplied by consumers. Thus,

(43) sl(t +1) =nki(t 41); 4-4, F

Substituting (43) into (42) yields

(4a) (41) = ol (e 1) = WO) = mle + DEE ea
nfp'(t +1) +ul - 174


-27-

all of these attributes are important in a model of capital accumulation
and foreign investment taxation. The imptications of the model are not
inconsistent with observed reality, as are the implications of the models
in the existing literature. The model does not, however, allow
prediction of the path of a countries taxes. The nonagreement between
different groups as to what taxes should be suggest that a model with
stronger positive results would contain a description of the political

process by which taxes are set.


-3-

winners could bribe the losers into accepting a policy change. This
notion may be vacuous here due to the difficulties in carrying out such a
bribe, Transactions costs and difficulties with the moral hazard problem
of ef*ecting a transfer between two competing groups at one period of
time may be formidable. Additional complications occur when competing
groups appear at different points in time.

The model employed here is a two-country, overlapping generations
(OLG) mode? with production. It is an international version of
Diamond's (1964) model, which combines a one-sector Solow growth model
with Samuelson's O1LG model. An OLG model in this setup is one where in
each period a given number of agents are born. The agents live two
periods and then die. In their first period of life agents born in
period t may trade only with agents born in t - 1 ort. In their second
period of life they may trade only with agents born int +1 or t. The
model begins at period one. At this time the young of generation one and
the old of generation zero are alive. It will be seen that in the first
period of life agents save and in the second period of life agents
consume the return on their savings. Thus, the abstraction of a two-
period life captures the essential feature of a life-cycle model: the
agents begin life by saving, and at some point start to dissave.

The possible non-optimality of both the autarky and laissez-faire
equilibria result from the structure of the OLG model. The double-
infinity of agents and goods violates the assumptions of the First
Theorem of Welfare Economics! The optimality theorem presented here
extends the results for an OLG model with constant returns on storage to

one with a variable rate of return.


-14-

Proof. See the appendix.

The result that a capital-stock steady state equilibrium in a model
with a fixed-coefficient technology is Pareto optimal if and only if
r >n is the standard result in the overlapping generations literature.2/
In this literature the nonoptimality of the Golden Rule rate follows from
the fact that individuals in these models are endowed with goods rather
than labor and r is the constant rate of return on storage. It is
possible to dominate the equilibrium allocation with one resulting from
no saving. If no savings occurred in the model presented here there
would be no second period consumption.

2.3.4 The gains from trade

In this section it will be assumed that a unique, stable steady
state equilibrium exists.

Under autarky the growth of rental rates in both economies is
governed by equation (18). Initial rates are found by equating domestic
endownments with domestic firms demand in the first period. Rental rates
in both economies approach the same steady state level monotonically,
with the path of home rental rates always below the path of foreign
rental rates. Thus, capital-labor ratios and wages in both countries
approach their steady state levels, which are identical across countries,
monotonically with home capital-labor ratios and wages always above
foreign levels. This is shown in propositions 3 and 4.

Proposition 3. In the absence of trade the home installed capital-labor

ratio will always be as great as the foreign capital-labor ratio. In the
steady state the two ratios will be the same.

Proof. Let ety and rt) represent the autarky home and foreign renta

prices at time t. mab) and rd) are determined by


-16-
Substituting (17) into (26) gives
(23) U= U(n[B(t + 1) - T]g(t +1), nr(t + T)g(t + 1)).

By (5), (6), and (9) n[B(t + 1) - 14g(t + 1) and nr(t + 1)g(t + 1)
are monotonically decreasing in r(t +1). An increase in rental rates
lowers consumption in both periods, thus the young at home are made worse
off and the young abroad are made better off.

it is seen from propositon 5 that all the gains from the opening of
trade are captured by the original capitalists in the relatively capital-
abundant country and the young agents in the relatively labor-abundant
country.

This result differs from that obtained in Buiter (1981). In
Buiter's model initial capital-labor ratios are identical and time t + 1
capital is installed at time t. The current old are not able to relocate
their capital in response to interest rate changes and thus their welfare
is unaffected by the opening of trade. This implicitly assumes that a
move from autarky to trade cannot be anticipated. The effects of capital
mobility on other generations is ambiguous in Buiter's model.

2.4 An Example

Suppose U(c,, C5) = cfc; a, 8 > O and F(K, L) = KY, Ly, O¢cy<

1. Then by (3)

1
(24) g(t) = CEYTT


-2-

The old in the relatively capital- (labor-) abundant country will prefer
a smaller (larger) tax than that which would maximize current national
income. In at least one country, a social planner trying to maximize
steady-state utility will choose to tax foreign investment earnings.

The above, unusual results follow from the assumptions of the
model, which is dynamic, choice-theoretic, has agents who are
heterogenous with respect to endownments, and countries which are
homogenous except for the size of the initial capital stock.

It is important that a model be dynamic because a policy
implemented in a given period will affect savings behavior in that
period, and hence, capital stocks in all future periods. A well-known
result is that a capital-exporting, or importing, country can act as a
monopolist, or monopsonist, and increase its current national income by
restricting capital flows ./ However, such a policy affects the growth
of the capital stock, representing a secondary burden or benefit to later
generations.

The agents in the model possess non-homogenous endownments. Some
agents are purely capitalists, endowed with their country's capital
stock. Others are laborers and capitalists, earning labor and rental
wages dependent upon the size of the capital stock during their lifetime.
Thus, there is disagreement among agents as to the optimal degree o*
capital mobility. For example, capital outflows tend to benefit current
capitalists at the expense of future laborers.

Much of the trade literature has ignored this problem of competing
groups by appealing to the Kaldor criterion of potential Pareto

optimality. Under this criterion a policy is deemed to he optimal if the


-37-

Manning, R., "A Nash-Cournot Model of Taxation of International Capital
Movements," Economic Record 50:113-118, Mar., 1974.

Pitchford, J.D., Foreign Investment and the National Advantage in a

Dynamic Context. North-Holland, Amsterdam, 1970.

Ruffin, Roy J., “Taxing International Capital Movements in a Growing
World." 1980,

Ruffin, Roy J., "Growth and the Long-Run Theory of International Capital

Movements," American Economic Review 49, Dec., 1979,

Shell, Karl., “Notes on the Economics of Infinity," Journal of Political
Economy 79:1002-11, Sept.-Oct., 1971.

Wallace, N., "Overlapping Generations Model of Fiat Money," in Models of
Monetary Economics, J. Karaken and N. Wallace, eds. Federal

Reserve Bank of Minneapolis, Minneapolis, 1980, 49-82.


-25-
In equilibrium
i i i
(49) j2y lk - 9 (ru )} = 05
hence,

c -rg' (ui 5
(sn) 20 = ara Ur) yea

7 i 3
au iBy pug! (u'r)

Solving (48), using equation (50), yields

. -i-. i
(51) ui=1-& 7 MU ij eH ae iG,
e-g(u r)
where epre u'rg'(u'r)/g(u'r), the elasticity of demand for capital in

country i; i = H,F. Equation (51) is the Kemp-Mac Dougall solution.
Consider now the current old. Maximizing c,(0), as given by

equation (37), with respect to ul yields
(52) fr(1) = (1 = ut PEO Da (ryg(uhe(ty = (1 ~ ut yr(1)
oru
i

[r(1) + wary, 1)g' up 1)) + «2rd > O with equality if + >
: au! Toul

Substituting (50) evaluated at t = 1 into (52) gives


Capital Accumulation and Foreign Investment Taxat ion~/
by Anne Sibert
1. Introduction

This paper provides a general equilibrim model of capital
accumulation in open economies. Capital movements between countries
result from differences in endowments. The paths over time of wages,
interest rates, and’ capital-labor ratios under autarky and laissez-faire
are described, welfare levels under autarky and laissez-faire are
compared, and the effect of restricting capital mobility by foreign
investment taxation is discussed.

The major results are that if two countries differ only in initial
capital abundance, then in autarky the initially more capital-abundant
country will always be more capital abundant and will always have higher
wages and lower interest rates than the initially less capital -abundant
country. However, both countries converge to the same steady state, if a
stable steady state exists. Neither an autarky nor a laissez-faire
equilibrium is necessarily Pareto optimal. The set of efficient
allocations is characterized and seen to depend on the relationship
between the path of interest rates and population growth. Autarky and
laissez-faire are shown to be Pareto non-comparable. A move from autarky
to capital mobility makes the original holders of capital in the
relatively capital-abundant country better off, the original holders of
capital in the labor-abundant country worse off, later generations in the
labor-abundant country better off and later generations in the capital-
abundant country worse off. There will be no unanimity as to the optimal
level of foreign investment taxation in either country. The optimal

taxes from the point-of-view of the original capital holders are derived.


-33-
- r(t + 2)a(t + 1)/n,

by (A.10), the first-order conditions of the firms' problem and strict

concavity of the production function. This is equal to
A A
d(t + 1)W(t + 1)/n + a(t + 1)[W(t +1) - r(t + 2)Iyn +

[d(t + 1) + a(t + 1)3fd(t +1) + a(t +1)Yn.

The second term is zero by the first-order conditions of the
consumers' problem and the last term is positive by the properties of be
Hence,

A
d(t +1) > d(t + 1)W(t + 1)/n = d(t + 1)r(t + 2)/n.
r(t) > for all but a finite number of times; hence {d(t)} is unbounded.

A A
Case 2. Min{t: K°(t +1) # K(t + 1)} = MintK°(t +1) < K(t + 1)}

By (A.2), E(t) > c8(t) or c(t +1) + kt + 2) « g(t +1) + ket +2)

or both.

a
Case 2a. co (t) < co(t). Work backwards as in Case la to show c9(0) <

©, (0).

0 t) a ” . .
Case 2b. Cy (t +1) +k (t + 2) < c(t +1) + k(t + 2). The induction


-28-

5. Appendix
5.1 The Proof of Proposition 5.

The strategy for the “only if" part is to construct a sequence
{s(t}, of feasible, Pareto improving transfers from the young of
generation t to the old of generation t + 1.

For every t > 1, Wc) (t), cp(t)] = r(t +1). Wis continuous;

hence for every e > 0, there exists 6 > 0,
(A.1) Ufe,(t) - 6, co(t) + 6fr(t + 1) + e]} > Uc, (t), cy(t)).

Let s(t + 1): = s(t){r(t + 1) + €] for a given s(1), Then for a

given s(1) the sequence of individual transfers is
s(1), {s(1at]Er(t +1) + em’ De,

The sequence is bounded if r(t) <n for all but a finite number of

times. Let
sup, (ttl er(t +1) + ent} = MH cm,

Choose {8(t)}e_, to satisfy (A.1) for every t > 1. Choose s(1) e
. (9, inf &(t)/M). Then {s(t}, makes the current old better off and the
young at least as well off. Hence the sequence is Pareto improving.
The "if" part is shown by contradiction and borrows heavily from

Wallace (1980). Let “o" over a variable denote a potentially Pareto


-12-

(20) w(r) = ng(r)B(r).

2.3.2 Existence, uniqueness, and stability

As in Diamond's (1965) model of a closed overlapping generation
economy with linearly homogenous production, it seems impossible to
guarantee existence, uniqueness, and stability without strong assumptions
on the relationships of parameters of the utility function to parameters
of the production function. An example in Section 2.4 will be given of a
utility and production function which yield a unique, stable steady
state. This will ensure nonvacuousness.

By equation (4) the demand curve for capital at time t > 1 is
unambiguously positive and downward sloping and it converges to both
axes. Given a strictly positive, finite r(t) there exists a strictly
positive, finite w(t). Then gross substitutability ensures that at time
t + 1 the supply curve slopes upward. Thus, for any strictly positive,
finite r(t) there exists a strictly positive, finite r(t +1). Hence,
there exists a sequence {r(t)} which satisfies equations (18) and (19)
and r(t + 1) is a monotonic function of r(t). If a unique steady state
exists, rental rates converge monotonically to the steady state. The
existence of a steady state, however, is not guaranteed. The difficulty
is that r(t) may approach zero or infinity as t goes to infinity.

If a unique, stable steady state equilibrium exists, it can be seen
from equation (20) that the equilibrium depends only upon the growth
rate, tastes, and technology. . It does not depend upon initial
endownments.

Proposition 1. If a unique, stable steady state equilibrium exists, an
increase in the rate of population growth must be associated with an

increase in the steady state rental rate.


-15-

(21) Ky = Late)
KP = L(a(r (1)),

Ki > Kt by hypothesis; hence by (5), ray < req), Convergence to

the steady state is monotonic and both economies are governed by equation
(18); hence ey < v(t) for every t > 1. Therefore, by (5), ety <
KF(t) for every t > 1.

Proposition 4. Capital mobility raises the wage rate in the capital
importing country and lowers the wage rate in the capital exporting
country.

Proof. This follows trivially from (5) and rMity < rit) for every
tel.

This is the same result as in MacNougall's (1964) and Kemp's (1960)
static models, Ruffin's (1979) model in the steady state and Buiter's
(1981) model.

Proposition 5. Capital mobility makes the old at home and the young
abroad better off and the young at home and the old abroad worse off.
Proof. The old are purely capitalists whose consumption levels are equal
to their earnings on capital. The opening of trade raises interst rates
at home and lowers them abroad, thus it makes the old at home better off

and the old abroad worse off.

By (13) and (14) the utility of a young agent of generation t is

[B(t + 1) - 1 w(t) r(t_+ T)w(t)

(22) U= UC ate Ty Rt eaT)