Journal
of International
Economics
9 (1979)
249-264.
CJ North-Holland
Publishing
Company
RANKING OPTIIMAL TARIFFS AND QUOTAS FOR A
LARGE COUNTRY UNDER UNCERTAINTY
Leslie YOUNG
University
of Cantcrbtq.
Christchurch I, New Zealand
Received November
1978
Optimal tariffs and quotas are comgared for a large country under uncertainty.
If the import
supply schedule has constant elastjcity and is subject to multiplicative uncertainty and domestic
demand is random then the optimal policy is a fixed +.d valorem tariff. If the supply schedule
has constant elasticity but this elasticity is random then the optimal tariff is superior to the
optimal quota. If the demaud and supply schedules are linear then the optimal quota is superior
to the optimal tariff if and only if the supply schedule is inelastic and the degree of uncertainty
in ;:A demand and supply schedules is small
1. htroduction
Bhagwati (1965, p. 53) initiated the analysis of the equivalence of quotas
and tariffs. He defined equivalence as obtaining when the replacement of a
quota (tariff) by a tariff (quota) at the level defined by the implicit tariff
(import level) in the quota (tariff) situation leads to an identical equilibrium.
He showed that equivalence obtains if there is perfect competition, certainty,
no retaliation or distribution effects and quotas are auctioned. However, as
Rhagwati (1978, ch. 2) has noted, if one of these conditions is violated then it
is important to compare the welfare effects of these two instruments wheG
they are ‘equivalent’ in the weaker sense that both satisfy some constraint,
e.p. on the resulting lecel of domestic production or imports.
Recent papers in the latter tradition have compared tariff< and quotas
under uncertainty. For a large country, Fish&on azd Flatters (1975)
compared the optimal quota with the aariff yielding the same expected
imports.’ For a small country, Pelcovits (‘1976) compared quotas with tariffs
yielding the same expected imports while Dasgupta and Stiglitz (1%‘7)
compared quotas with tar% yielding the same expected tariff revenue.
T?t% paper compares the optimal tariff with the optimal quota under
‘The title of their paper ‘The (non-) eyuivalerlce
of nptimul tariffs and quotas under
uncertainty’ (italics inserted) is therefore not completely accurate. However. as we shall see ill
sectio 1 :. their results; compartng
opttma’i quotas with mean eqwvaient tariffs lead immediately
to conclusrons about optimal quotas and optimal tariffs in two important
cases.
250
L. Young, Optimal
tar@
and quotas
uncertainty for a large country without imposing additional constraints, guch
as those above on expected imports or tariff revenue. We present mo-fels
where the optimal tariff is always superioj to the optimal quota and also
models where the reverse can hold. In a related paper, Qhta (1978) has
compared quotas and export price controls for a large country in terms of
expected export profits.
Our choice of models and method of analysis is best motivated by recalling
Fishelson and Flatters’ initial challenge to the widely held presumption that
tariffs are preferable to quotas under uncertainty because they permit
imports to respond to domestic and foreign disturbances. They pointed out,
very perceptively, that ‘what is ignored in this reasoning is the possibility
that under a given tariff structure the responsiveness of imports to a change
in demand may be such that the new level of imports overshoots the new
optimal level by an amount greater than the amount by which the new
optimum exceeds the old (quota) level of imports’. In their linear model this
point is illustrated by their proposition that if the demand schedule is fixed
but there is uncertainty about the height of the supply s<chedule then the
optimal quota is superior to the mean equivalent tarjr if and only if the
elasticity of the expected import supply is less than unity:
It is intuitively plausible that the ma.gnitude of the cEasticity of import
supply should be related to the possibility of ‘overshooting’ and hence to the
ranking of tariffs and quotas. We take up this suggestion in a model where
the elastici :y of supply is constant. Corresponding to Fishelson and Flatters’
analysis of stochastic shifts in the height and slope of a linear supply
schedule we analyse thle effects of multiplicative shifts in the supply function
and of shifts in the magnitude of the elasticity itself. (This corresponds to
stochastic shifts in t!le height and slope of the supply schedule when
expressed as a linear relationship between the logarithms of price and
quantity.) This model of supply u.ncertainty is of intrinsic interest because the
behaviour of supply over a wide price range is plausible, even given large
s:ochastic shifts in the parameters.2 As we shall see, this is not the case in the
linear model.
In section 2 we show that, in the case of multiplicative shifts of a constant
elasticity supply function, a fixeld ad valorem tariff is not only superior to a
fixed quota but is also the cspt.imal policy - even in the face of simultaneous
uncerrainty in demand. In the case of shifts in the elasticity of supply the
‘(lh~a I 1978) h:is used a special case of our model ef supply uncerrzinty. He assumes that
~rnp.3~1 4uppIy iz of constant elasticity rvlth :B multiniil.!~tive, lognornkly
distributed
random
term: that the d.omestic production
set is Ricardian;
and that the bcme country does not
consume the exported good. Thus, his objective function is essentially expected profits from
exrdjrts while our objective function is expected domestic welfare. Moreover, he considers
c~~trnk on export prices while vie consider the choice of a tariff rate -- which would still allow
rrat prlce5 70 F!uctuate.
L. Young, Optimul
rarjf1.i and
yuorlr.~
‘51
optimal tariff is again superior to the optimal quota, because a fixed tariff
can be chosen which always leads to imports which are closer to the optimal
imports than the optimal fixed quota: ‘overshooting’ never occurs. Our
analysis also identifies the general &-cumstances under which ‘overshooting’
can be excluded.
The constant elasticity model comprises an important class of cases where
the presumption that tariffs are superior to quotas under uncertainty is
vindicated despite the considerations
raised by Fishelson and Flatters.
Within it, a low expected elasticity of supply does root mean that the optimal
quota is superior to the optimal tariff. In contrasting this conclusion with
that of Fishelson and Flatters above, it should be noted that, not only do
they use a different model of supply, but they compare the optimal quota
with the mean equivalent tariff while we compare the optimal quota with the
optimal tariff. In section 3 we show that in their linear model, the optimal
quota is superior to the optimal tariff if and only if the expected supply is
inelastic and the degree of uncertainty in supply and demand is sufficiently
small. We shall relate this result to the importance of overshooting in the
linear model.
2. The model when the supply elasticity
is independent
of the quantity
imported
2.1. The general problem
The analysis of this section uses the standard two-country, two-good.
general equilibrium model of trade. We thereby avoid the use of consumerS
surplus and evaluate th,- welfare of the tariff-levying country (tt-me‘home’
country) directly in terms of its indifference map over the quantities s, (1 of
goods exported and imported. The home country ‘s indifference map depends
on a random variable 0 with a fixed distribution and is assumed to be
convex. We can suppose that the indifference curves arise from a utility
function V(x, 4,8). The supply curve of imports is given by
Ps =
p&q,a)*
where p, is supply price using domestic exports as numerairc. y iu domestic
imports and x is a random variable with a fixed distribution.
In the arguments in this seckm we show that, under the tariff ra:e
specified, domestic utility is higher than under the optimal quota for CM/I
value of x and 0. At no point shall we trade off gains and losses under
d@erent values of 01and 0. Thus, our conclusions on the ranking of tariffs
and quotas are valid not only for the expected utility criterion but also fog
anq’ welfare crilerion which is a nondecreasing function of the utility IevcI>
attained under each value of x and t9 and is an increasing function of tht.
32
L. Young, Optimal tur@
and quotas
utility level rrttainecl under at least one pair (a,@!,. F‘or example, our
conclusions hold mdier the maxmin criterion (i,pplied to utility levels) which
corresponds to extreme risk aversion. Note also that we begin by assuming
that there is no domestic production. The extension of our arguments to
include domestic production is eiementary and is given in Appendix A.
Since the supply curve of imports i;; given by ~&,a) the foreign offer curve
is given by x =qp,(q, ct), where x is domestic exports. The marginal cost of
lomestic imports expressed in terms of exports is
where &I, x) is the elasticity of the import supply function.
For ary given value of x and 8 the optimal choice for the home country is
the point Q* =(.X*(X,O), q*(r,O)) on the offer curve diagram where the home
country’s marginal rate of substitution equals the slope of the foreign offer
curve, i.e.
For the given r: 0 this could be achieved by a quota q*(a,O) or by a tariff
In general, there is no fixed tariff or quota which ensures that (2) will be
satisfied for all values of CCand 8. Hence, in general, a fixed tariff or quota
will lead to lower welfare than would be possible if the policies were
contingent on x and 0. However, the country is assumed to be constrained tc
adop; either a fixed tariff or a fixed quota. We must compare the welfare
levels resulting from these two policies, taking account of all possible
realisations of x and 8.
’ 7 Cc.xtan? suppl_re1asticit-yund multiplicatiw uncertaimy
0.e.
We first identify a case where
policy. The tariff t would lead
domestic relative price of imports
equals p,(+ a)(1 + l/q(q, 2)). Hence,
a fixed tariff would be the optimal
a fixed ad valorem tariff t is the optimal
to the optimal imports if the resulting
equals their marginal cost, which by (1)
if ~(CJ,OL)
were independent of 4 and Mthen
policy.
C. Zbunp, Opiimul
tuf$j.s
und yu0tu.s
on a rtrnriom rariuhle 0 \vith u jixed distribution.
supply schedule is ?j’ the jbrm
253
Suppose ulso that the import
where q. is a fixed positive number and fl is a positive random vuriable with a
fixed distribution. Then the optimal policy jirr improving the terms of trade jbr
the home country is a fixed ad valorem turijjf t= l/q,. ‘This policy is strictly
better than any fixed quota except in one special case.
ProoJ: When the world price of imports is p, the marginal cost of imports is
p&l + l!qo). If a tariff rate t= l/q, is set then the domestic relative price of
imports is p&l + l/qo), i.e. it equals the marginal cost of imports in terms of
home country exports or the slope of the foreign offer curve. At a
competitive equilibrium the home country’s marginal rate of substitution
between imports and exports equals the domestic price ratio. Hence, fcr any
value of fi and 0, equilibrium occurs at a point where a home counl.ry
indifference curve is tangential to the foreign offer curve.
The constant elasticity import supply schedule implies a foreign offer curve
Rich is concave to the home exports axis. Since the home indifference
Purve is convex the point of tangency represents the highest indifference
level attainable on the foreign offer curve for the current value of p and 0.
Thus, given a tariff t = l/qO, the equilibrium attained maximises the holme
country’s welfare for all values of fi and 8. Hence it constitutes the optimal
policy.
In general, the a.doption of a fixed quota in the face of uncertainty will lead
the home country to a point on the ex poste foreign offer curve which, for
some values of /? and 0, is not a point of tangency with an i.ndifference curve.
For such values the quota will yield strictly lower welfare than the tariff
above. The exception to this general conclusion clearly involves higlhly
specialised assumptions on the home country’s indifference map, the way in
which 6’ affects this map and the correlation between 13and 1). This completes
the ;XOL$of the theorem.
Notice that the foreign counlry’s import denland is flp,’ +‘11’so the elasticit!; fi
of its offer curve is 1 + qo. Thus, the optimal tariff rate is
t=l/($--1).
This is just the optimal tariff formula derived in textbooks [e.g. (Caves and
Jones, p. 23911. In general, fi depends on the quantity imported so application of the optimal tariff f,nrmula under certainty involves actually solbing
the problem of maximising t!le home country’s welfare given the foreign o!fcr
curve. The exception is whl:n 4 is independent of the quantity imported.
Theorem 1 simply involves noticing that, in this (case. the tariff remains
L.
254
Young, Optimal
rari&
and
quotas
optimal in the face of shifts in domestic preferences and multiplicative shifts
in the foreign o”Fer curve. No assumption, c are required about the stochastic
structure of domestic preferences apart from t;le convexity of the indifference
curves. Moreover. it is immaterial whether the supply curve is elastic or
inelastic.
2.3. Supply elasticity random but independent of quantity supplied
If the elasticity qfq, x) varies with q and Q then a fixed ad valorem tariff
provide the margin between the terms of trade and the
marginai cost of imports required to ensure an optimal choice by domestic
consun;e~ for all valules of a and 8. Hence, any fixed tariff will be inferior to
some policy contingent on a and 0. However, given the usual convexity
assumptions, a fixed tariff t would be superior to a fixed quota q provided
the following condition holds, for all Y.and 13:
cannot
always
fi*(r.0)>4
implies a*(r,8)2&~O,t)>~
and
y*(rYO)<cj: implies q*(a,8)5Zq(0z,O,t)<q,
(3)
where $r,f?.t) is the quantity imported under tariff t. it should be a simple
matter to choose a t so that 4*(x, 0)>4 implies @a, 0, r) ~4 and ‘q*(r, 0) <q
implies ij(r.0, t)-4, i.e. so that under the tariff imports respond in the right
direction to changes i-1 z. The addition4 requirements in (3) preclude the
possibility of ‘overshooting’ raised by Fishelson and Flatters.
Let us assume that domestic preferences are fixed so that 8 can be omitted
from the arguments of q* and 4’. Then q*(of)> q would imply that y*(a)
>$r. I)~ provided that a change in o! which increases q*(a) above q also
ensures that t ocerestimates the margin4 cost of imports. (Similar remarks
hold mutatis mudandis for the case q*(a‘;<g.) Our next theorem is based on
the obr-ervation that this wo!~.ldcertainly be the case if the random variable 01
were simply the eiasticity of import supply g and this elasticity is independent of CJ.The restriction S>fiO in the theorem ensures that all equilibria
occur ir! the region q>& where the more elastic supply curve results in
more domestic: imports for given exports. This ensures that an increase in
elasGcit_c-increases the quantity imported.
that the home country’s indifference
artd C~IIW.Y asd that rhe import supply function has the form
n1eortw1 2.
Suppose
Y =PoP’I,
curt’es are fixed
L. Young. Optimal turijjfs and yuoiu~
distribution. Zf the optimal
the optimal quota.
quota q>f10
255
then the optimal
Proof: The foreign offer curve OR,* corresponding
function 4 =&pz is given by
tnr$f‘ is superior
trr
to the import suppIy
where x is the exports to the foreign country required to pay for imports y.
This offer curve is concave to the home export axis. Choose a tariff rate t
such that the foreign offer curve corresponding to q = l/t is tangential to a
home indifference curve at q =[I. This tariff defines a tariff-ridden offer curve
OR, for the home country. Given tariff t, the point Q(q.r) chosen by the
home country when facing the import supply function with elasticity q is
given by the intersection of OR, with OR,*.
If II> l/t then the quantity imported satisfies
MRS(&(rl,t))=Ps(q’,~)(13-r)>P,(g,~)
(
1,1:
c?x(fj. q)
=-&---’
.’
)
Thus, if q> l/t then the home country’s indifference curve at Q(11,t) is less
steep than the foreign offer curve OR,* at Q(q,t). But the indifference curves
are convex and the foreign offer curve is concave from below. From fig. 1 it
can be seen that Q(u, t) lies on a higher indifference curve than any point Q
on OR,* which involves fewer imports than (?(tf,r). We now show that the
quota 4 leads to such a point Q(q) on OR;.
2%
L. Young, Optimal
lariJi?i and qtcolus
The tariff rate t is chosen so that OR, passes through the point &l/t,t)
with co-ordinates (j&J4//&,)’+‘, 4). Therefore O&l/t, t) has slope (PO/g)‘. This
is less than ! since y>/?“. In the regon 4 </I0 the foreign offer curves OR,*
all lie above the 45 hne and hence above the line OQ( 1 9, t) wihile the home
tariff.ridden offer curte OR, !ies below this line. Hem-e OR, must intersect
the foreign offer curves !n the region 4 >pO. In this region, if q >q’ then the
offer curve OR% lies abo;re OR;. Hence if q > l/t then Q(q,t) > q( l/t, t) = q.
We have shown that if 21> l/t then &, t) is superior to Q(q). If q’< l/t
then it can litqwise be shown that
2x(4. q’)
MlPS(Qbq’,
#+-----
84
and that #I’, t) ~4 so that @g’, t) is again superior to @g’). The theorem is
proved.
The hypothesis d>/&, of Theotern 2 will be satisfied provided that in the
region q-c&,
the home country’ s marginal rate of substituting imports for
exports is gre:ater than 1 + l/q, where q is the smallest value that q can take.
‘This condition, corresponding to a high marginal valuation of imports when
imports are low, ensures that in region q <PO all foreign offer curves are
steeper than all the home indifiz-ence curves so that the optimal quota 4
Go+
The argument of Theorem 2 also applies when the supply function is of
the form
4+c:,
where borlz B am: rl are random, provided that
This will be tlhe case if (a) the possible pairs (/3,r,1)are confined to a subset Z
of R: such that for any (pr,~+, ), (/&,PJ~)in Z,q: sq2 implie,s /3, $ppz and (b)
the optimal quota 4 is greater than any /!Isuch that (#?,q) is in Z for some q.
Let us see how far the argument of Theorem 2 can be adapted to the
general model of 2.1. We suppose that the supply price p,(q, r) depends on a
:andom variable z with i:p,/&<O.
Let q be the optimal quota. Define Cras
the value of z such that the optimal quota leads to a trade point Q on the
foreign offei curve associated with x satisfying:
L. bung,
Choose t -
1jq(@, E) so
Optimal
Cm-@
and
quotas
‘57
that
MRS(Q)=p,fg,dj(l
+t).
The quantity q(a, t) imported leads to a point &(cx,t) on the foreign offer
curve associated with Q such that
For CI> a, &a, t) > q. In order to conclude that
and hence to duplicate the reasoning of Theorem 2, we must have
r(Q, ct)> q(4, di), if and only if cc> &.
(4)
Thus, tariffs are superior to quotas in the general case provided that (4,
holds. This will certainly be the case if q is nondecreasing with respect I~C).i
and 4. Expresskon (4) can also hold if q is increasmg with respect to o! a&
decressing with respect to 4 - provided that the direct impact of !Yon v
overcomes its indirect impact through g(a,t). This requires assumption!; tin
domestic preferences as well as on the supply functic.n for imports.
3. Linear supply and demand functions
We now compare the optimal tariff and the optimal quota in the paLrtia1
equilibrium model of Fishelson and Flatters which assumes linear demand
and supply functions. In the linear model it can be shown that the optimal
quot.a q is determined by the intersection of the expected demand and
marginal cost schedules for imports. Fishelson and Flatters show that if the
supply schedule is fixed then, under stochastic shifts in the demand schedlule,
the quota q is inferior to the mean-equivalent tariff. They also show that if
the demand schedule is fixed then l:nder stochastic shifts in the height of the
supply schedule a quota q is inferior to the mean-equivalent tariff ;.f and only
if the expected supply schedule is elastic. Inspection of their arguments
reveals that the above results remain valid when the quantity 4 is replaced
by any 4 which leads to a binding quota.
If the optimal quota is, inferior to the mean-equivalent tariff then it is
inferior to the optimal tariff. However, if the oprimal quota is superior to fhe
mean-equivalent tariflf - a.s in the case of supply lncer:ainty :~nd inelastic
expected supply - then it (does not follow that it is superior to the optima!
L. Young, Optimal tan/j3 and quotas
258
This section characterises the circumstances
quota is superior to the optimal tarift.
We suppose ithat the supply curve has the form
tariff.
in which the optimal
(5)
p,=~a++ql,
where b rs a fixed positive number and o[ is a random variable with mean ~1.
The import demand function is
pD=c-dq,
V-9
where c and d are fixed positive numbers. The case where there is also
uncertainty in import demand is dealt with subsequently. We shall assume
that, in the absence of intervention, the expected equilibrium quantity
imported (c - u)/(b + d) is positive, i.e.
(7)
c-a>O,
and that the expected equilibrium price (bc+ad)/(h+d)
is positive, i.e.
bc i- ud > 0.
The welfare criterion is expected net consumers’ surplus. If quantity
imported at price p, then the consumers’ surplus is
(8)
9 is
kt’(q,3[)=cq-Q&f2 -p,q.
By (5!
W(c~,a)=(c-rA)q-((b+d/2)q2.
For a given X, 1Y(q, !x) is maximised by
Suppose a quota must be set before 01is known. The optimal quota @ is
that which maximises
BY (91
Now suppose that an ad valorem tariff t is set. By (7) and (6) the resulting
imports @= q(a, t) satisfy
(a+bq)(l+t)=c-dq.
Tl ierefore
$24 t) =
c-ct(1 +t)
d+b(l +t)’
The following result characterises the conditions under which the optimal
quota is inferior to some tariff t and hence: to the oprimd tariff. Let G = c/d,
the intercept of the demand schedule with the quantity axis; let F= --cl/b,
the intercept of the expected supply schedule with the quantity axis and let
rr2 be the variance of cc.
Theorem 3. If there is uncertainty about the height (3r the import supply
function then the optimal tariff is superior to the optimal ouoin if and only if
0’ 2G
->.._a2
F
1
’
(11)
Prooj:
The proof uses the theory of quadratic inequalities. The difference
between the expected surpluses under the tari@ rate t and the optimal quota
4 is E[W(q’(a, t), x) - W(q, (x)]. In Appendix B we show that this equals
-[(A2+D)t2+2A(A-B)t+(A-8)2-D~/2(2b+d)(d+bi1+I)12.
(12)
where Azab+bc+ad,
B~2bc+ad,
and Dsda2(2b+d).
It follows that the tariff t is superior to the optimal quot.a if and only if
The optitna! tariff is superior to the optimal quota if and only if L(r) is
negative for some values of t, i.e. the graph of y=L(t) crosses the t axis. This
will be the case if and only if the equation
(A2 + D)t2 + 2A(A - B)t + (,4 - B)2 --II =o
has real roots. By the theory of quadratic equations, this will t)c Irue if and
only if
This condition is eq&valent tc each of the following:
0<D[C-(A-B)2+A2~,
0 < Dlj9 - B(B - 24-J)
do2(2b + d) > (2bc + ad)( - 2ab - ad),
dcr’ > - Li(2bc+ ad),
CT2I_--2bc
T;i’ud
1
’
Thus, we have shown that the optimal tariff is superior to the optimal quota
if and only if the last inequality holds. The theorem is proved.
Thert are three points to note about this result. We first check: when the
optimal quota scan be dominated by a positive tariff. The coefficient of t in
L(r) is
2A(A-Z3)=2(ub+Ac+ad)b(a-c).
(13)
If the expected supply price of the optimal quota q is positive, i.e.
tb(c-a)
u+-Fb+d=
ub+ bc+ud>O
2b+d
’
then by (7) and (1.3) 244 -B) ~0. It follows that at least one of the zeros of
L(t) is positive. For some to ~0 sufficiently close to this zero, L&J <O. We
thus conclude that if the tariff implicit in the optimal quota q is positive then
the optimal qluota is inferior to some posit.ve tariff if (11) is satisfied.
Secondly, notice that if a>O, i.e. the expected supp1.v curve is elastic, then
F- -u/b ~0 and (I 1) is satisfied so that tariffs are superior to quotas.
However, if LICO., i.e. the expected supply cur\ re is inelastic, then (11) will still
be satisfied provided that 02/u2 >2G/F- 1, i.e. provided the proportional
uncertainty in the height of the supply schedule is suffkiently large. On the
other hand if u CO and a2/u2 is sufficiently small then (11) is violated and
quotas are superior to tariffs. We have t%us establushed that the optimal
quota can indeed be superior to the optrmal tariff in improving the terms of
trade: an important possibility raised, but not in fact demonstrated, by
Fishelson and Flatters. The following corollary to Theorem 3 identifies a
broad class of situations where this will be the case.
L Young, Qptimul tariffs and quotds
261
Corollary.
Zf a <O (except on u set of meusure zero), i.e. almost ill1 The
random supply curves are inelastic, then the optimal quotu is stiperior to the
optimal tariff:
Proof.
By assumption (8)
(-2bc/ad)-l>l.
Thus, if tariffs are superior to quotas then it is necessary that
a2/a2 > 1.
But this implies that the s.?t of o!1 a satisfying ix -al > la) is of positive
m2dSUre,
i.e. the set of ct>O is of positive measure. Therefore if Q~0 except
on a set of measure zero then the optimal quota is superior to the optimal
tariff.
Thirdly, suppose that the import demand curve is random and of the form
pd=;l-dq,
where y is a random variable, uncorrelated with a, with mean c and variance
s2. Then the argument of Theorem 3 can be extended to obtain:
Theorem 4. Zf there is uncertainty about the heights of both the demand and
the import supply functzons then the optimal tariff’ is superior to the optimal
quota vand only if
a2+s2 + s2 dA2 + 2bA( c-a)+b2(2b+d)2s2
-d2a2(2b + d)
a2
a2
,E_l
F
I
’
By (7) the term in the square brackets is positive. Therefore the optimal
quota will be superior to the optimal tariff only if expected supply is inelastic
und the degree of uncertainty in both demand and supply is small and the
variance of y is small compared to that of X.
We conclude by relating the results of this section to the analysis of
section 2. The tariff-augmented supply price (CI+ bq.)(l + t) will generally differ
from the marginal cost of imports or+2bq. Since the optimal tariff t must be
chosen taking account of all values of CYthere will always be values of x such
that q*(a) > 4’ and (0~+ bG(x, t))( 1 +-t) < SI+ 2bq’(z, t). Hence, the optimal tariff
will lead imports to overshoot for some values of CL
The ranking of tariffs and quotas in the linear model is hus a matter of
comparing the welfare losses from overshooting under taritfs with the losses
from the rigidity under quotas. Thi.s has been done in Theorem 3. From this
theorem we see that t.he sign of a (hence whether the expected supply curve
is elastic) is not the only factor: the degree of uncertainty in c1is also critical.
The inflexibility of quotas will always render them inferior to tariffs if the
degree of uncertainty is sufficiently large - despite the possibility of the
overshooting of imports under tariffs.
4. Conclusions
We have shown that optimal tariffs are superior to optimal quotas, if (1)
import supply demands on a random variable which affects the quantity
supplied and the supply elasticity in the same direction, and (2) the supply
elasticity is constant or increasing with the quantity supplied. No assumptions are required on demand apart from convexity of the home country’s
indifference map. The possibility, raised by Fishelson and Flatters, that
optimal quotas can be superior to optimal tariffs through ‘overshooting’ has
been established in the linear model when the degree of uncertainty is
sufficiently small.
In comparing the two models of import supply above we note that, for
large stochastic shifts of the type analysed, the assumption that the elasticity
of supply is independent of quantity is more plausible than the assumption
that the slope of the supply function is independent of quantity. With large
shifts in the height of the supply function the linear model yields implausible
behaviour for small prices and quantities. However, the possibility that
quotas can be superior to tariffs cannot be dismissed on this account since
the !inear model would be a good approximation to some supply situations
when stochastic shifts are small. It is in just such sit!:ations that we have
identified conditions under which quotas can be super;or to tariffs. It would
be interesting to establish general conditions on supply and demand under
which this would be true.
Appendix A: Theorems 1 and 2 when there is domestic production
Suppose production possibilities in the tariff-imposing
by the production function
country are given
where 0, is a random variable with a fixed distribution. Suppose also that
domestic preferences over the two goods are given by a utility function
L. Young, Optimal tur$fs and quotas
‘62
Lr(z,, z,,&). An indirect utility function over the quantities
exported and of good 2 imported is given by
of good
1
UX,~,~~,B~)= max (U(z,,z2,e2):gtzt+x,z2-q,et)~;o>
2,’
Iz
This defines a family of indifference maps over (x, 4) which are contingent on
O= (Or, 0,). The indifference curves will be convex provided that U is
quasiconcave and g is convex. The reasoning of Theorems 1 and 2 will hold
for the utility function V provided that when the domestic price ratio is
p&q, a)(1 +t) competitive choices lead to a trade point Q such that
av
dx
?V
-----=
i c?q
:W?S(Q)=p,(q,rx)(l
+t).
This in fact follows immediately from the fact that in the face of this price
ratio the representative consumer chooses a bundle such that
and the profit maximising firms choose output so that
Appendix B: Proof of
iv@, a) - W(&a) =
eq.
(12)
wyq*,
30 - wq,
r) -
cWq”, 3) -
w4
=f(26+d)[(q*-L#-(q*-lj)2].
=
(14)
AT-B+(X-u)[-~++T+Li’r]
(2b+d)(d+bT)
’
where Tr 1+ c, A z trh + hc + &, and BE 21%+ uJ. Also
q*--
(x - n)(d + bT)
__
Y’(2b+d)(d+bT).
@)I
L. Young, Optimal tariffs and yuottrs
264
Since E[ct -a]
= 0 and E[(a - CZ)~]= c2
(21 + d$2(d
+ t17-)~E[(q*
-q)2 - (q* - Q)2]
= - (AT-B)2+a2[(d+bT)2-(-di-bT)2-2dT(-dlibT)-d2T2]
= -(.4T-B)2+~2[4bdT+2d2T-2bdT2-dd2T2]
= -(AT-B)“+D(2T-T2)
= -(A2+D)T2+2(AB+D)T-B2
= -(A2+D)t2-2A(A-B)t-(A-B)2+D,
where D = da2(2b + d).
By (14) and (15) E[ W(@,cc)- W(& a)] equals
References
Bhagwati, Jagdish, 1965, On the equivalence of tariffs and iluotas, in: Robert E. Baldwin et al.,
eds., Trade, growth, and the balance of payments: Essays in honor of Gottfried Haberler
(Rand McNally, Chicago) 5347.
Rhagwati, Jagdish. 1978, The anatomy and consequences of exchange control regimes (Ballinger
& Cc., for N B.E.R., Cambridge, Mass.).
Caves, R.E. and R.W. Jones, 1973, World trade and payments: An introduction
(Little Brown.
Bostcn).
Dasguptd, Partha and Joseph Stiglitz, 1977, Taxes vs. quotas as rekcnue raising devices under
uncertainty, American Economic Review 67, 975-981.
Fishelsof., Gideon and Frank Flarters, 1975, The (non-) equivalence of optimal tariffs and quotas
under uncertainty, Journal of International
Economics 5. 385-393.
Ohta Hicoshi, 1978, On the ranking of price and quantity controls under uncertainty, Journal of
Interz#ational Economics 8, 543-550.
Pelcovits Michael G., 1976, Quotas versus tariUs, Journal of International
Economics 6, 363.-370.