- Email: [email protected]
The optimality of tariff quotas under uncertainty
Journal of International
Economics
13 (1982) 337-~351. North-Holland
Pull shine C_ompdny
THE OPTIMALITY OF TARIFF QUOTA!< UNDER UNCERTAINTY James E. ANDERSON Boston College, Chestnut Hill.
MA 02167,
L’SA
Leslie YOUNG University
of Canterbury and Uniuersity
of Texas, Ausrin. 7X ;‘.{ I ,‘2. USA
Received June 1981. revised version received February
I982
A tariff quota is a schedule of tariffs in which the tariff rate steps upward as imports exceed some critical quamrty or value. This has desirable properties in I$ face of uncertainty aboltt domestic demand and supply and about foreign supply. We show that a specific (ad valorem) tariff quota is the policy maximizing domestic expecteci surplus subject to constraints on average Imports (import expenditur:) and on the average amount by which imports (import expenditure.:) exceed some critical value.
1. Introduction Recently, Young and Anderson (1980) have examined the optimal policie!; for restricting trade under uncertainty. They obtained conditions under which the most common forms - the specific tariff and the ad valorem tariff - are optimal. However, trade restrictions ofren also take the form of tariff .scheduEes in which the tarif; rate depends on the quantity imported. A particularly common form is the tar$ qata in which the tariff rate step&; upward (perhaps from zero) when imports exceed a certain amount 01 value . rq2 There are a number of rationales for this form of trade restriction. For example, the rights to export at the lower tariff rate can be allocated tc.1 developing countries to foster their industries.” Alternatively, such rights carI be allocated to maintain customary tra.de flows for a time after the formation ‘These are sufficiently
common
to warrant
a book-length
treatment by Rom (!979), frocl:l section iChapter 6) Rom does not treat the questions raised in this paper but concentratszs on comparing :pecilic tariff quotas, a;1 valorem tariff quotas, fixed tariffs and quotas on quantity and on value in terms of their effec\.j on prices and on quantities imported. ‘The higher tariff rate is sometimes levied when imports excerd a certain percentage elf domestic production, e.g. the former U.S. tariff quota on woollen alld worsted products [Ror!,~ (1979, pp. 107-l 14)]. Our a.nalysis can easily be modilied to derive I he conditions under whicl-1 this form of tariff quota is optimal. ’ I.g the tariff quota granted by Australia to developing countries [ Rom I 1979. pp. 178 1X6)]. whom the subsequent examples are taken. In his analytical
00:!2-1996182100004KW/$O2.75 CQ 1982 North-Holland
of a customs union, thereby easing the adjustment.. process.4 In this paper, however, we are concerned with the desirable pro;zrties of unallocnted tariff quotas in the face of uncertainty ab(Dut domestic demand. and suyply and about foreign suppiy conditions. In section 2 we show tkai al specific tariff quotr.. Asthe policy maximizing domestic expeeted surplus whe:n the policy-maker wishes to restrict not only the average level of imlpolrlisbut also the average amount by which imports. exceed some critical Ilevel. This pair of constraints is a useful way of representing the policymakler’s trade-off between the interests of producers and those of ConsumeE’s and foreign exporters. Our analysis identifies the ways in which the tariff qu1ot.aqould be superior to both a fixed tariff and a fixed quota whenever such trade-offs are required. Section 3 discusses the optimality of an ad valorem tariff quota in similar terms. Sction 4 extends l.he argume:nt to the case of a large country and points out that if the representative ,domestic consumer ix risk-averse, then there is an additional role for tariff quotas in helping to buffer the domestic e.conomy against exogemow income shocks. 2. Specifictariffqwatas In this section we sulppose that the importing country is small and faces a world price P*(0) for lthe good in q t;estion. Thi.s price depends on 8, the random variable representing the stalk of the world. The domestic inverse excess demand function P(Q, 6) is asswned. to be decreasing function of imports Q and also depends on 0. In sta:.e 8, domestic surplus from imports Q(rQis me)
J {P(u, 9) - P*(!9)]du. 0
We shall suppose that the objective function is domestic expected surplus!, i.e. that the marginal utility of income of the representative consumer is constant. The implications of a non-constant marginal utility of income will ‘bediscussed in section 4. Any policy restricting trade is equivalent to Sl rule Q(0) for determining the equilibrium quantity of imports in, state 0. Young and Anderson (1980) showed that a fixed specific tariff is ,th: policy maximizing domestic expected surplus subject to a ceiling 11~on the average level of imports, i.e.
(1) ‘E.g. m Ine Benelux cwdcms unio:~ [Ram (1979. 1ic179. pp. 5 I I ).
p. 7)].
Other rationales
a,; listed :n Rom
J.E. rl~~tlrrson and L. bung,
Optimal
tar@” quota:s
339
Such a ‘non-economic’ constraint’ is a way of represl:nting, the political concessions by the government to domestic producers whc~ ar: damaged by impcrts. However, it Is often reasonable to suppose that the rate of damage is low when imports are low but rises as imports increase. The inierests of consumers and foreign exporters might then be better accommodated by pe.rmitting more imports on aver’age but restricting the probability weight at the upper tail of the import distribution. A simple way of representing this balance of interests is to suppose that the ceiling q1 on a.verage imports is set relatively highi, but that a low ceiling qz 3s imposed on the average amount by which implorts exceed the critical level Qr (at which domestic producers begin to ‘hurt’). Thus, we suppose that the government also imposes the constraint:
E C(QWQl) ‘I 312,
(2)
of this apparently ad hoc where 7+ =max(z, 0). The importance representation is that the optimal trade restriction is then the commonly used specific tariff quota, i.e. a schedule of specific tariffs con:ingt:nt upon the import level Q:
if
QIQ19
if
Q>Q1,
(3)
:trhere sr and s2 are non-negative constants. Trteorem I. The policy maximizing domestic expected szrrplt~ subject ‘:onstraints (1) and (2’ is a specific tariff quota of the form (3) wher
The problem is to max E IQW
Q(e)
[ (P(u,0)--P*(8)} dv b I
subject to
E CQtW 5 41 and
‘Bhagwati and Srinivasan (1969).
to
340
J.E. Anderson and L. Youqy, Optimal
takff’ yuotas
The Lagrangean for this problem is rQ(e)
E ; 1 {f(u, 0) -- P*(tl;~)dv - A.~j(O) -- /l(Q(O)-- Q1>+ + Aql + pq2, I
where i, and ,Yare the Lagrange multiplaers corresponding to constraints (1) and (2), respectively. A and ~1 satisfy the usual complementary slackness conditions: l(E
[(Q(0)-QI~.+
[Q(S)]-q,)=Q=jt(E
-q2).
(4)
We have: 8 __
4?(e)
Q(e)
1 (B(o..O)-P*(8))dv= f'(Q(@,O)-P*(t)),
0
and3
~(Q(@-Qlf’
= Q :if Q(MQl,
aQ!slj Therefore, the first-order problem are:
i 1, if Q(e)>QQ1. condiitions fblr Ihe constrained
maximization
(5) Suppose that the following scheldule of specific: tariffs is imposed:
if
QSQ1,
(6)
if Q>&. Then imports Q(O)in state 0 satisfy: .f(Q(e), 0)- f*(e) = am. “Appendix A confirms that the correct lirst-order condition! ,:!lO,--Q,;+ a? QWj=Q1 is interpreted as the left derivative.
(7) ;:re obtained if the derivative
of
J.E. Anderson and L. Young:, Optimal turiff’quoraJ
36.1
Hence, the tariff schedule (6) ensures that Q(0) always satisfies the first-order conditions (5). S’nce the d,3mestic excess demand function is downward-sloping, domestrc surplus is a concave function of imports. Moreover, ihe cc-nstraint functions in (1) and (2) are convex Functions of imports. Hence, the first-order conditions (5) are sufficient for a maximum.7 Q.E.D. E;y (4), if constraint (2) is not binding, then p =0 and the optimal policy is a fixed specific tariff. On the other hand, if constraint (1) is not binding, then R=O. Thus, if the government is concerned not with average imports at all but only with average imports beyond the critical le\-el Q,, then a positive tariff should be imposed only beyond Qr.This form of tariff quota is quite common. An alternative interpretation of theorem 1 is possible in the case where
If constraint (1) is binding, the:n constraint (2) can be v ritten as
Denoting the absolute value of z by 1~1,we have:
E EIQ(@--E CQU91!1=2E C{QW-W(W) ‘I. Hence, (2) is equivalent to the constraint
ECIQ(B)-ECQ(B)III~2q,,
(8)
i.e. 10 a ceiling on the mean absolute deviation of imports from their mean. Thus, theorem 1 shows that a specific tariff quota is optimal, given ceilings on [he mean and on the mean absolute deviation of imports. Chwmiy, it is also optimal if a ceiling were: placed on some weighted combination of the mean and the mean absolute deviation, with the weights preselected by the policy-maker. To bring out the intuition underlying theorem 1, let us suppose thal any imports of the good require a licence (called ‘type 1’1and th:it any. imports above the level Qi require an additional licence (calll:d ‘type 2’). Any policy Q(0) satisfying constraints (111and (2) is equivalent to an allocation of q1 stare-contingent type-l licences and q2 state-contine.ent type-2 liaences. A specific tariff quota ensure’s that the rents on type-l licences z.re equal in all states where imports are less than Q, and that the sum on’rerts paid for 1he ’ r.7~ formal proof is given in appendix A.
342
J.E. And~rsan and L Young, Optimal rar@qwta>.
two types of iicences are equal in all states where imports exceed Q1. Moreover, :he two tariff levels in the schedule 2a.n be chosen so that the rents on the two types of licences reflect their marginaS social value (j.and p, respectively). Hence, a tariff quota can ensure optimal arbitrage of the licences across states. By contrast, under a fixed specific tariff, the s?lrn of the rents paid for the relevant licences are equal in all states, whether or not imports exe-eedl Q1. This would be optimal if only constraint (1) were binding. However, if constraint (2) were also binding, then the required specific tariff would lie between i. and 2 -t ,X He,nce, if imports are less tha.n Q1 then the implicit rent on type-l licencles would exceed their marginal so&i1 value, while if exports exceed Q,? then the implicit rents paid on the two 1:lcenceswould be less than their marginal social value. It follo:vs that, under a specific tariff, the average level of imports would be too low and there wou!d not be sufficient incentive to rnctr+i La,..*. a&htional imports in states where imports tend to be high. A fixed quota may actually be superior to a fixed specific tariff in achieving con ztraints (1) ;and (12).Young and Anderson (1980) show that for the same average restrictiveness the tariff dominates, but (2) could be satisfied by a fixed quota without restricting,J imports by as much on average. However, under a quota., a fixed number of the licences are assigned to each state and their rents are determined by market equilibrium in that state alone. Hence, there remain we:lfare-improving arbitrage opportunities across states - opportunities which. are exploited under a tariff quota. The above discussion shc~ws why the tariff quota is an attractive instrument even when the policy-maker’s objectiives cannot be represented satisf%ctoriiy by constraints (I) and (2). Compared to a fixed tariff, it restricts more sharply the high import levels which are especially damaging to impert-competing producers -- without an excessive restriction of imports overall. Compared to a fixed quota, it encourages a shift of imports from states where their social value is low to states where rt is high. We can also derive the optimal form of trade restriction under alternative formulations of the government’s conc’:ssions to import-competing producers. Constraint (2) implies that marginal increases of imports are regarded as equally damaging in all states, so long; as the base level exceeds Qt. To model ai situation whlere the rate of d,ima;;e rises continuously with the impart level, we could replace (2) by a ceil!nk! o L>n the variance of imports: E [(Q(O) -- E [ Q(0)-J2] 5 1’. .,
J.E. Arzderwn
with the level
qf imports.
S(Q) = E.t Proof
and L. Young, Opti nal tarIf
911otu.s
343
Thus:
2b_(Q
- ql).
(10;
See appendix B.
Although there is no difficulty, in principle, in operating the optimal policy of theorem 2, administrative considerations might lead the government to approximate its effects using a tariff quota. 3. Ad valorem tariff quotas Young and Anderson (1980) showed that an ;d v:tlorem tariff is the poiicyrlaximiz ng domestic expected surplus subject !o a ceiling on foreign exchange expenditure on the good:
‘5uch a ‘non-economic restraint’ might be .mposed when political :onsiderltions prevent adjustment of the exchange Fate in f,ace of a persistent payments deficit. However, if the real concern is uith unusually large : xses of foreign exchange, t;ren the government mil;ht wish to restrict the probability weight at the upper tail of the expenditure distribution without tight restrictions on its mean. The former objective can be represented by a :eiling e, on the average amount by which expenditure exceeds t?c c!it,ical level El, i.e. E [(P*(@Q(O: - .E,}+] =e2. 4gpin, the interest of this representation is that thl: resulting optimal polic-:u is tf e commonly observed ;+I valorem tariff quota. Theorem 3. Thr polic_,t mwimizing domestic ex,7ectcd constraints (11) and (12) is tz schedule of ad vujorem [email protected]
surplus
stlhjec’t to
(131 where a, and a2 are non-negative corlstants and a, 5 az. ProoJ
See appendix B.
Sc‘hedule (13) is a first approximation to the case where increasing M.eight ‘::;pl,aced on 1a:rger deviations from El due to rking borrowing costs l-he
latter would call l’or more steps in the tariff schedule. Almost trivially, (12) could be replaced by .a.n exlpenditure variance con:;traint analogous to (9), implying a linear ad vaiorem tariff schedule as a function of expenditurt:. Each iitem in tht discussion on theorem I has an analogue here: if constraint (12) is not binding, then &heoptimal policy. is a fixed ad valorem tarifk if constraint (11) is not binding, then there should be a positive ad valorem tariff only when foreign exchange expenditure exceeds E,; if p’l= E,, then constraint (12) can be replaced by a ceiling on the mean absolute deviation of expenditure from its mean; there is an obvious; a.r!alogue of theorem 2 concerning the optimal plolicy given constraints on the mean and the variance of the expenditure distribution; finally, the optimal policy in theorem 3 can be interpreted as ensuring optimal arbitrage of the two types of stClte-ciontingent foreign exchange licenccs corresponding to the two constraints. This last mterpjretation brings out the similarity between the optimai policy here and some multiple exchange rate schemes. The optimal policy in theorem 3 involves a higher ad valorem tariff rate when import expenditure exceeds a certain u&e. Such ‘qualitative’ ad valorem kariff quotas exist ‘but often the higher tariff rate is applied when impDrts exceed a specified yuuntity, i.e. the ad valorem tariff schedule has the form: (14) where Q1, 3, andi a, are non-negative constants. We shall identify some important cases where the optimal policy in theorem 3 reduces to thk fc:rm. Suppose thut eitkr (a) there is uwertuimty only about domtlstic demand or (b) there is uncertainty only about the world price and the demand elasticity ulwu_ys exceeds 1. Then the policy ,marimizinf: domestic expected - surplus subjec,t to constraints (I 1) and (12) is an ud var’orem tu@ quota of’ the _fiw-m(14) wirh N, sa,.
TtlEOREXl4.
Prn~$
See aplpt’ndix B.
The intcitioni behind theorem 4(b) is that if demand is elastic, then. under a2 optim.ai policy.. high imp;)rt expenditure is associated with low domestic and world prices and hence with high &ports. lln order to prevent excessive weight at the upper tail of the expenditure distrilsution it is then ne:ecsary to raise the ad valorem tariff rate when iimports are high. However, if trzmand is inelastic.. t?ten, under an optimal policy, high import expenditure is associated v:ith high d..oz&c and wc~ld prices and hen:e with 101~ irnportc. We therefore 5a.ve the unexpected result tikt the higher tariff r.atc should bc imposed when imports are low
Suppose that {here is uncertainty only about the world price ~(1 thut the demand elasticity is always less than P. Ti’vn rhe policy* ma?-inti;:ir;g domestic expected surplus to rvnstraints (I I) and (12) is un r:ti ralorcnr ttrr$j quota:
THEOREM
5.
a(Q)
=
aI9
if
a,, if
Q
t,15)
Q.2QI.
where Q ,, a, and a, are non-negative constants Lrzd a , -;<11~.
Proof:
See appendix R.
Id the domestic demand elasticity exceeds unity oder one range of impc:rts and is less than unity over another range (as with a iinear demand function) then, under an optimal polic,y, the set of imports who:;e foreign exchange Ll~bi is less than E, need not comprise a comlectzd interl/al. The optimal policy cou.:d involve a low ad valorem tariff rate when implrl:ts are very low or very high but a high tariff rate for intermediate levels of ilnpc,rts. The above analysis also sho,ws that if there is unczrtdinty III both dcmznd arid supply, then a tariff quota in which the tar,ff rate ,;lepends on the q&an&y of impotts would generally fail to ensure that the first-or&t conditions behind theorem 3 are satisfied [see eqs. (13.2a) and (B2br in appendix B, which yield Cl_‘;)],It would be necessary to set the tariff r;itc according to the z?alue of imports. 4. Extensions The above analysis can readily be extended to the case whcrc the importing country is large and. the v:orld price of the good depends on the quantity imported as well as orit 0: P* = P’VQ, 0).
We shall only sketch the results: the proofs are an obvious amalgam of th:: arguments above and those in Young ar,d Anderson (19Sir) and Young ( 1980). Theorems 1 and 2 remain valid for a large country if we take tb : objective function to be world expected surplus: rQ(_B)
s, (P(v, O)- P4’(1;,0)) do
E
.
1
However, theorems 3, 4 and !; would be vahd for ihi:- objjec’:Ixz l’utl:ti; ~1 oni) if the inverse supply function
has consta,lt
elasticity
with ;,zspect to irlpor:s.
if the desired objectiv: function is dontestic expected surplus: QW
E
J ‘;f’(u.,0)- uik(Q(0), 0)) dv 0
1 ,
then the optimal policies are tariff schedules which are essentially those derived above plus an Edgeworth-‘Bickerdike ‘optimal tariff’ designed to exploit the importing country’s mono.ploly power.8
To examine zhc implicaticns of a non-constant marginai utifity of income, Ict the objective function be the expected value of the representative consumer’s indirect titility function V(P, I), where P is the price of the imported gctod in terms of the exportable and I is total production revenue @us tariff revenue. It can be shown’ that the first-order condition for a maximum or E [ L’(P,Z)] subject to constraints (1) and (2) is:
Hence. optimal arbitrage of the two types of import licences must noXv take account of the variation in the marginal utility of income aV/al. Iq effect, the optimal trade resTriction must be m,odified to buffer the domestic economy against the income changes resulting from random domestic and foreign shocks. This consideration implies that, although a tariff quota will no longer be oiltima!, it can be superior to both a pure tariff and a pure quota, even in the absence of a constraint like (2). F dr example, if the world price fluctuates, then it would be desirable to lower the tariff rate when imports are low in order to augment the country’s l gai ns from trade’ in sta:.es where it is being impoverished by a rise in the ~~;,rld price. On the other hand, if domestic production suffers random &oizks (such as weather) then it would be desirable I:, lower tine tariff rate \\hcn imports are high in order to augment the country’s ‘igains from trade’ in s?; tes where it is being impolverished by a reduction in domestic prrjductioil. A detailed analysis of these qulestiom is beyond the scope of this paper. but the above discussion at least indicates how the additional degree of freedom available in tariff qmo~as can be expl&ted to meet a number o; polic! objectives. Ll'o..mg 61980.p 429~ shows that if there is multiplicative uncertainty in the foreigr suppl) Irsncr~:~~,then this ‘optimal tariff is a deterministic schedule (If tasiffs depending only on the aorld prrrx. If Ihe wpply fun&m is linear or consta.nt elastic, t Iti: t.his schedule is A mixture of spe4c and ad vakrem tariffs. "l‘hc! prtsof i*, swilar to that in Young and knderson (1981, sel:tions 2 and 3).
J.E. Anderson nnd L. Yang.
Opthal
tarifl’quotay
147
References Bhagwati. J.N. and T.N. Srinil,asan. 1969, Optimal intervention to achieve non-ecxroxic objectrves, Review of. Economic Studies 36, 27-38. Rom, M., 1979, The role of tarifT quotas in commercial policy (Macmillan, London). Young, L., 1980, Optimal revenue-raising trade restrictions under uncertaintv, Journal of International Economics 10, 4.21:~~440. Young, L. and J.E. Anderson, 1980, The optimal policies for restricting trade under uncertaislty, Review of Economic Studies 46, 927.-932. Young, L. and J.E. Anderson, 1981. Risk aversion and c:ptimal trade restrictions, Universit!. of Canterbury, mimeo.
Appendix A We shall show that the first-orde;. conditions (3) and the complementary slackness conditions (4) are sufftcient for a constrained maximum in theorem 1. The Kuhn-Tucker theorem cannot be invoked because the constraint function (Q(O))- Q1} + is not differentiable everywhere and because there could be an infinity of states 0 and hence an infinity of decision vari,tbles
Q(e)* In state 6, lelt Q’(O) be imports under the policy defined by (3) and (4) and let Q(0) be imports under any other policy which satisfies constraints (1) and (2). Domestic surplus under Q’(C) exceeds, that under Q(#) by Q’(e)
Since P(Q, 0) is decreasing that @>
in Q, the Second
2 (P(Q’(@, 0)-- p*(@){Q'(O) -
Mean Value Theorem
implies
Q(Q).
BY (3):
i,(Q'(@ --Q(Ui),
if
Q(U)$QQ,
(A+~)(QG(ti) -Q(R)],,
if
Q((I)>Q,.
(A. I )
But: Q”( 0) > Q , impl.ies Q”( 0) - Q(O)
~(Q0(8)-Q,}+-~Qltl)--Q,3’. This obviously
holds if Q(6’)ggQ 1, while if Q(0)
{QW-Ql)+>Q’(@li-Q1,
(9.2)
J.E. ,4nderscn and I.. l’hng,
348
so agrain
Opimal tardy quoras
(A.2) holds. Moreover, E~(Q*rn,-Q(e)~.+(Q”(tl)~>.Q:]-~~C(Q(o)-Qlf+(QO(e)>Q1! (A. ;)
~E:I(Q’(O)-Q,t+]-E[(Q’(O)-(2,)+]. By (A.l), (A.2) and (A-3): E[d(OVJzi,E [Q”(O)-Q(O)] +p(E [{Q’(O)-.Q$] -E[{Q(@---Q,j+]). Since
Q"(Q)
satisfies the complementary slackness corditions (4): W(~~)l1~l~~,- E CQNI) + ,&I, - E C(Q(@l - Q1) ‘I).
Since Q(U)satisfies the constraints (1) a.nd (2):
Hence, Q’(O) is indeed the optimal policy. For all the other theorems, the proof that the first-order conditions and the complementary slackness conditicns are sufficient for optimality is similar.
Appendix B Proof
oj
2. The Lz:grangean of the constrained problem of theorem 2 is: theorem
QfjO
E
1 (P(c, 0) -P*(Q)
d u-nQ(e)-~{Q(e)-ECQ(e)!j’
where i and p are the Lagrange multipliers car:esponding anld (9). Hence, thle first-order condition is: P(Q(Oj,#)- P*(O)=i,+2~C1(Q(8)-EeQ(8,‘1).
maximization
I
+b
+P,
to constraints
( )
(EL1
This condition is satisfied if the following linear scheduie of specific tariffs is i m posed:
s(Q)= X t 2p(Q- q 11.
The first-order condition is sufficient for a maximum because the objeclive function is concave and ihe constraint functions are convex in Q. Q.E.D. Proof of theorem 3.
An argu,qent like that in theorem I shows that the firstorder conditions of the constrained maximization problem are: P(Q(~), e) -
p*(e) =
if*(e), (II.+ ppyej,
if P(O)Q(la) 5 E,, if P(o)Q(e) > El,
(B.24 (B.2b)
wilere 1*and p are the Lagrange multipliers associated with constraints (11) and (12), respectively. These first-order conditions will be satisfied if the schedule (13) of ad valorem tariffs is imposed with Q, = 1, and a2 = i, + 11.The first-order conditions a.r*: sufficient for a maximum since the objective function is concave and the constraint functions ar,: convex in Q. QED. In case (a) the world price is fixed at some level P* and the conclusion follows from theorem 3 with:
Proof of theorem 4.
Q.) = El/P*. In case l(b) let Q”(O) SC the optimal level of imports in state 8, let 0, be defined ilm,plicitly by
we,)Qom = E,,
(B.3)
Q1 = p(el).
WV
and let
We shall show thLat Q”(0)$Q1 implies P*(0)Q0(8)i$ktJ,.
(B.3
Since 3,20, (13.3)and (R.4) imply that P*(@QO(e)5 E, implies (1 9 n)P*(o)Q”[e) (B.6)
Combining this with thle first-order condition (B.2a) yields P*(@)QO(e) 2; E, implies P’(QO(@, @Q’(6) (B.7)
J.E. Anderson and L. Young, Optimal tariff quotas
3.50
We assumed that P(Q, 6) is independ.ent of 0 and fhaf the demand elasticity always exceeds 1, so P(Q”~(0),O)Q”(0)~P(Q,, #,)Q1 implks Q”(P)SQ1.
IB.8)
By (8.7) and (B.8): P*( S)Q’(#) 5 E 1 implies Q”(0) 5 Q 1.
(Es.9
Therefore Q’(H):>Q, implies P*(0)QD(O)> E,.
(R.lOt
Since i, 20 and /,rzO, (B.3) and (B.4) imply th;:,t .f *(@Q’(0)> E, implies (1-tI+ p)P*(O)Q(O)
>(1+4~*C~,~Q,.
(B.ll
/
Combining this with the first-order conditions (B.2a) and (B.2b) yields: P*(ll)Q”(@)> El implies .P(Q’(fl), f39Q”(@
We assumed that P(Q, 6~ IS independent of 8 and that the demand elasticity i:r always greater than 1, f;o PtQ”l @),@Q’(d) > P(Q 1, 01 )Q 1 implies Q”(@ > Q 1.
(BXJ)
By (B.12) and (EL13) P”(H)Q”(6)> E, implies l{!‘(e)> Q1_
(18.14)
Q"!i?) 5 Q, implies P*(0)Q”(6) 5 E,.
(B. 15)
Theref:Dre
(B.10) and (R.15) imply that the first-order conditions (B.2a) b,nd (B.2b) ;tre equfvafen? to the conditions P(QW), 0) - P*(o) =
I.E.
Andmon
md
L
Young,
Optlmcil
tarijjquotm
353
These conditions will be satisfied if the ad valorem tariff qu1ob.a(14j is imposed with a, = i, and ~1~= i. + p. Q.E.D. Proof
of theorem j.
Define Q1, as in (B.4). If the import demand function is deterministic and always has elasticity less than 1, then ,P(Q”(O),O)Q”(@ P(Q,, 6)Q1 implies Q”(O)$ Q1.
(B.16;
This means that thz ineqxalitles in the conclusions on (B.9) and (B.13) mus; be reversed and thist instead of (B. 10) and (B.14) we have: Q’(O)< Q1 implies P*(@Q0(8) > El,
[B.i7)
Q’(O)2 Q 1 implies P*(@Q’(O)5 E 1.
(B.18)
(B.17) and (B.18) imply that the first-order conditions (B.2a) and (B.2b) are equivalent to the conditions P( Q(O),0) - P*(Oj =
iP*(t?), f\(A -I p)P*(O),
if
Q(@2Ql,
if
Q(O)< Q1.
These conditions will be satisfied if the ad valorem txiff quota QED. imposed with a, =A+,u and a,==L
(15)
is
Economics
13 (1982) 337-~351. North-Holland
Pull shine C_ompdny
THE OPTIMALITY OF TARIFF QUOTA!< UNDER UNCERTAINTY James E. ANDERSON Boston College, Chestnut Hill.
MA 02167,
L’SA
Leslie YOUNG University
of Canterbury and Uniuersity
of Texas, Ausrin. 7X ;‘.{ I ,‘2. USA
Received June 1981. revised version received February
I982
A tariff quota is a schedule of tariffs in which the tariff rate steps upward as imports exceed some critical quamrty or value. This has desirable properties in I$ face of uncertainty aboltt domestic demand and supply and about foreign supply. We show that a specific (ad valorem) tariff quota is the policy maximizing domestic expecteci surplus subject to constraints on average Imports (import expenditur:) and on the average amount by which imports (import expenditure.:) exceed some critical value.
1. Introduction Recently, Young and Anderson (1980) have examined the optimal policie!; for restricting trade under uncertainty. They obtained conditions under which the most common forms - the specific tariff and the ad valorem tariff - are optimal. However, trade restrictions ofren also take the form of tariff .scheduEes in which the tarif; rate depends on the quantity imported. A particularly common form is the tar$ qata in which the tariff rate step&; upward (perhaps from zero) when imports exceed a certain amount 01 value . rq2 There are a number of rationales for this form of trade restriction. For example, the rights to export at the lower tariff rate can be allocated tc.1 developing countries to foster their industries.” Alternatively, such rights carI be allocated to maintain customary tra.de flows for a time after the formation ‘These are sufficiently
common
to warrant
a book-length
treatment by Rom (!979), frocl:l section iChapter 6) Rom does not treat the questions raised in this paper but concentratszs on comparing :pecilic tariff quotas, a;1 valorem tariff quotas, fixed tariffs and quotas on quantity and on value in terms of their effec\.j on prices and on quantities imported. ‘The higher tariff rate is sometimes levied when imports excerd a certain percentage elf domestic production, e.g. the former U.S. tariff quota on woollen alld worsted products [Ror!,~ (1979, pp. 107-l 14)]. Our a.nalysis can easily be modilied to derive I he conditions under whicl-1 this form of tariff quota is optimal. ’ I.g the tariff quota granted by Australia to developing countries [ Rom I 1979. pp. 178 1X6)]. whom the subsequent examples are taken. In his analytical
00:!2-1996182100004KW/$O2.75 CQ 1982 North-Holland
of a customs union, thereby easing the adjustment.. process.4 In this paper, however, we are concerned with the desirable pro;zrties of unallocnted tariff quotas in the face of uncertainty ab(Dut domestic demand. and suyply and about foreign suppiy conditions. In section 2 we show tkai al specific tariff quotr.. Asthe policy maximizing domestic expeeted surplus whe:n the policy-maker wishes to restrict not only the average level of imlpolrlisbut also the average amount by which imports. exceed some critical Ilevel. This pair of constraints is a useful way of representing the policymakler’s trade-off between the interests of producers and those of ConsumeE’s and foreign exporters. Our analysis identifies the ways in which the tariff qu1ot.aqould be superior to both a fixed tariff and a fixed quota whenever such trade-offs are required. Section 3 discusses the optimality of an ad valorem tariff quota in similar terms. Sction 4 extends l.he argume:nt to the case of a large country and points out that if the representative ,domestic consumer ix risk-averse, then there is an additional role for tariff quotas in helping to buffer the domestic e.conomy against exogemow income shocks. 2. Specifictariffqwatas In this section we sulppose that the importing country is small and faces a world price P*(0) for lthe good in q t;estion. Thi.s price depends on 8, the random variable representing the stalk of the world. The domestic inverse excess demand function P(Q, 6) is asswned. to be decreasing function of imports Q and also depends on 0. In sta:.e 8, domestic surplus from imports Q(rQis me)
J {P(u, 9) - P*(!9)]du. 0
We shall suppose that the objective function is domestic expected surplus!, i.e. that the marginal utility of income of the representative consumer is constant. The implications of a non-constant marginal utility of income will ‘bediscussed in section 4. Any policy restricting trade is equivalent to Sl rule Q(0) for determining the equilibrium quantity of imports in, state 0. Young and Anderson (1980) showed that a fixed specific tariff is ,th: policy maximizing domestic expected surplus subject to a ceiling 11~on the average level of imports, i.e.
(1) ‘E.g. m Ine Benelux cwdcms unio:~ [Ram (1979. 1ic179. pp. 5 I I ).
p. 7)].
Other rationales
a,; listed :n Rom
J.E. rl~~tlrrson and L. bung,
Optimal
tar@” quota:s
339
Such a ‘non-economic’ constraint’ is a way of represl:nting, the political concessions by the government to domestic producers whc~ ar: damaged by impcrts. However, it Is often reasonable to suppose that the rate of damage is low when imports are low but rises as imports increase. The inierests of consumers and foreign exporters might then be better accommodated by pe.rmitting more imports on aver’age but restricting the probability weight at the upper tail of the import distribution. A simple way of representing this balance of interests is to suppose that the ceiling q1 on a.verage imports is set relatively highi, but that a low ceiling qz 3s imposed on the average amount by which implorts exceed the critical level Qr (at which domestic producers begin to ‘hurt’). Thus, we suppose that the government also imposes the constraint:
E C(QWQl) ‘I 312,
(2)
of this apparently ad hoc where 7+ =max(z, 0). The importance representation is that the optimal trade restriction is then the commonly used specific tariff quota, i.e. a schedule of specific tariffs con:ingt:nt upon the import level Q:
if
QIQ19
if
Q>Q1,
(3)
:trhere sr and s2 are non-negative constants. Trteorem I. The policy maximizing domestic expected szrrplt~ subject ‘:onstraints (1) and (2’ is a specific tariff quota of the form (3) wher
The problem is to max E IQW
Q(e)
[ (P(u,0)--P*(8)} dv b I
subject to
E CQtW 5 41 and
‘Bhagwati and Srinivasan (1969).
to
340
J.E. Anderson and L. Youqy, Optimal
takff’ yuotas
The Lagrangean for this problem is rQ(e)
E ; 1 {f(u, 0) -- P*(tl;~)dv - A.~j(O) -- /l(Q(O)-- Q1>+ + Aql + pq2, I
where i, and ,Yare the Lagrange multiplaers corresponding to constraints (1) and (2), respectively. A and ~1 satisfy the usual complementary slackness conditions: l(E
[(Q(0)-QI~.+
[Q(S)]-q,)=Q=jt(E
-q2).
(4)
We have: 8 __
4?(e)
Q(e)
1 (B(o..O)-P*(8))dv= f'(Q(@,O)-P*(t)),
0
and3
~(Q(@-Qlf’
= Q :if Q(MQl,
aQ!slj Therefore, the first-order problem are:
i 1, if Q(e)>QQ1. condiitions fblr Ihe constrained
maximization
(5) Suppose that the following scheldule of specific: tariffs is imposed:
if
QSQ1,
(6)
if Q>&. Then imports Q(O)in state 0 satisfy: .f(Q(e), 0)- f*(e) = am. “Appendix A confirms that the correct lirst-order condition! ,:!lO,--Q,;+ a? QWj=Q1 is interpreted as the left derivative.
(7) ;:re obtained if the derivative
of
J.E. Anderson and L. Young:, Optimal turiff’quoraJ
36.1
Hence, the tariff schedule (6) ensures that Q(0) always satisfies the first-order conditions (5). S’nce the d,3mestic excess demand function is downward-sloping, domestrc surplus is a concave function of imports. Moreover, ihe cc-nstraint functions in (1) and (2) are convex Functions of imports. Hence, the first-order conditions (5) are sufficient for a maximum.7 Q.E.D. E;y (4), if constraint (2) is not binding, then p =0 and the optimal policy is a fixed specific tariff. On the other hand, if constraint (1) is not binding, then R=O. Thus, if the government is concerned not with average imports at all but only with average imports beyond the critical le\-el Q,, then a positive tariff should be imposed only beyond Qr.This form of tariff quota is quite common. An alternative interpretation of theorem 1 is possible in the case where
If constraint (1) is binding, the:n constraint (2) can be v ritten as
Denoting the absolute value of z by 1~1,we have:
E EIQ(@--E CQU91!1=2E C{QW-W(W) ‘I. Hence, (2) is equivalent to the constraint
ECIQ(B)-ECQ(B)III~2q,,
(8)
i.e. 10 a ceiling on the mean absolute deviation of imports from their mean. Thus, theorem 1 shows that a specific tariff quota is optimal, given ceilings on [he mean and on the mean absolute deviation of imports. Chwmiy, it is also optimal if a ceiling were: placed on some weighted combination of the mean and the mean absolute deviation, with the weights preselected by the policy-maker. To bring out the intuition underlying theorem 1, let us suppose thal any imports of the good require a licence (called ‘type 1’1and th:it any. imports above the level Qi require an additional licence (calll:d ‘type 2’). Any policy Q(0) satisfying constraints (111and (2) is equivalent to an allocation of q1 stare-contingent type-l licences and q2 state-contine.ent type-2 liaences. A specific tariff quota ensure’s that the rents on type-l licences z.re equal in all states where imports are less than Q, and that the sum on’rerts paid for 1he ’ r.7~ formal proof is given in appendix A.
342
J.E. And~rsan and L Young, Optimal rar@qwta>.
two types of iicences are equal in all states where imports exceed Q1. Moreover, :he two tariff levels in the schedule 2a.n be chosen so that the rents on the two types of licences reflect their marginaS social value (j.and p, respectively). Hence, a tariff quota can ensure optimal arbitrage of the licences across states. By contrast, under a fixed specific tariff, the s?lrn of the rents paid for the relevant licences are equal in all states, whether or not imports exe-eedl Q1. This would be optimal if only constraint (1) were binding. However, if constraint (2) were also binding, then the required specific tariff would lie between i. and 2 -t ,X He,nce, if imports are less tha.n Q1 then the implicit rent on type-l licencles would exceed their marginal so&i1 value, while if exports exceed Q,? then the implicit rents paid on the two 1:lcenceswould be less than their marginal social value. It follo:vs that, under a specific tariff, the average level of imports would be too low and there wou!d not be sufficient incentive to rnctr+i La,..*. a&htional imports in states where imports tend to be high. A fixed quota may actually be superior to a fixed specific tariff in achieving con ztraints (1) ;and (12).Young and Anderson (1980) show that for the same average restrictiveness the tariff dominates, but (2) could be satisfied by a fixed quota without restricting,J imports by as much on average. However, under a quota., a fixed number of the licences are assigned to each state and their rents are determined by market equilibrium in that state alone. Hence, there remain we:lfare-improving arbitrage opportunities across states - opportunities which. are exploited under a tariff quota. The above discussion shc~ws why the tariff quota is an attractive instrument even when the policy-maker’s objectiives cannot be represented satisf%ctoriiy by constraints (I) and (2). Compared to a fixed tariff, it restricts more sharply the high import levels which are especially damaging to impert-competing producers -- without an excessive restriction of imports overall. Compared to a fixed quota, it encourages a shift of imports from states where their social value is low to states where rt is high. We can also derive the optimal form of trade restriction under alternative formulations of the government’s conc’:ssions to import-competing producers. Constraint (2) implies that marginal increases of imports are regarded as equally damaging in all states, so long; as the base level exceeds Qt. To model ai situation whlere the rate of d,ima;;e rises continuously with the impart level, we could replace (2) by a ceil!nk! o L>n the variance of imports: E [(Q(O) -- E [ Q(0)-J2] 5 1’. .,
J.E. Arzderwn
with the level
qf imports.
S(Q) = E.t Proof
and L. Young, Opti nal tarIf
911otu.s
343
Thus:
2b_(Q
- ql).
(10;
See appendix B.
Although there is no difficulty, in principle, in operating the optimal policy of theorem 2, administrative considerations might lead the government to approximate its effects using a tariff quota. 3. Ad valorem tariff quotas Young and Anderson (1980) showed that an ;d v:tlorem tariff is the poiicyrlaximiz ng domestic expected surplus subject !o a ceiling on foreign exchange expenditure on the good:
‘5uch a ‘non-economic restraint’ might be .mposed when political :onsiderltions prevent adjustment of the exchange Fate in f,ace of a persistent payments deficit. However, if the real concern is uith unusually large : xses of foreign exchange, t;ren the government mil;ht wish to restrict the probability weight at the upper tail of the expenditure distribution without tight restrictions on its mean. The former objective can be represented by a :eiling e, on the average amount by which expenditure exceeds t?c c!it,ical level El, i.e. E [(P*(@Q(O: - .E,}+] =e2. 4gpin, the interest of this representation is that thl: resulting optimal polic-:u is tf e commonly observed ;+I valorem tariff quota. Theorem 3. Thr polic_,t mwimizing domestic ex,7ectcd constraints (11) and (12) is tz schedule of ad vujorem [email protected]
surplus
stlhjec’t to
(131 where a, and a2 are non-negative corlstants and a, 5 az. ProoJ
See appendix B.
Sc‘hedule (13) is a first approximation to the case where increasing M.eight ‘::;pl,aced on 1a:rger deviations from El due to rking borrowing costs l-he
latter would call l’or more steps in the tariff schedule. Almost trivially, (12) could be replaced by .a.n exlpenditure variance con:;traint analogous to (9), implying a linear ad vaiorem tariff schedule as a function of expenditurt:. Each iitem in tht discussion on theorem I has an analogue here: if constraint (12) is not binding, then &heoptimal policy. is a fixed ad valorem tarifk if constraint (11) is not binding, then there should be a positive ad valorem tariff only when foreign exchange expenditure exceeds E,; if p’l= E,, then constraint (12) can be replaced by a ceiling on the mean absolute deviation of expenditure from its mean; there is an obvious; a.r!alogue of theorem 2 concerning the optimal plolicy given constraints on the mean and the variance of the expenditure distribution; finally, the optimal policy in theorem 3 can be interpreted as ensuring optimal arbitrage of the two types of stClte-ciontingent foreign exchange licenccs corresponding to the two constraints. This last mterpjretation brings out the similarity between the optimai policy here and some multiple exchange rate schemes. The optimal policy in theorem 3 involves a higher ad valorem tariff rate when import expenditure exceeds a certain u&e. Such ‘qualitative’ ad valorem kariff quotas exist ‘but often the higher tariff rate is applied when impDrts exceed a specified yuuntity, i.e. the ad valorem tariff schedule has the form: (14) where Q1, 3, andi a, are non-negative constants. We shall identify some important cases where the optimal policy in theorem 3 reduces to thk fc:rm. Suppose thut eitkr (a) there is uwertuimty only about domtlstic demand or (b) there is uncertainty only about the world price and the demand elasticity ulwu_ys exceeds 1. Then the policy ,marimizinf: domestic expected - surplus subjec,t to constraints (I 1) and (12) is an ud var’orem tu@ quota of’ the _fiw-m(14) wirh N, sa,.
TtlEOREXl4.
Prn~$
See aplpt’ndix B.
The intcitioni behind theorem 4(b) is that if demand is elastic, then. under a2 optim.ai policy.. high imp;)rt expenditure is associated with low domestic and world prices and hence with high &ports. lln order to prevent excessive weight at the upper tail of the expenditure distrilsution it is then ne:ecsary to raise the ad valorem tariff rate when iimports are high. However, if trzmand is inelastic.. t?ten, under an optimal policy, high import expenditure is associated v:ith high d..oz&c and wc~ld prices and hen:e with 101~ irnportc. We therefore 5a.ve the unexpected result tikt the higher tariff r.atc should bc imposed when imports are low
Suppose that {here is uncertainty only about the world price ~(1 thut the demand elasticity is always less than P. Ti’vn rhe policy* ma?-inti;:ir;g domestic expected surplus to rvnstraints (I I) and (12) is un r:ti ralorcnr ttrr$j quota:
THEOREM
5.
a(Q)
=
aI9
if
a,, if
Q
t,15)
Q.2QI.
where Q ,, a, and a, are non-negative constants Lrzd a , -;<11~.
Proof:
See appendix R.
Id the domestic demand elasticity exceeds unity oder one range of impc:rts and is less than unity over another range (as with a iinear demand function) then, under an optimal polic,y, the set of imports who:;e foreign exchange Ll~bi is less than E, need not comprise a comlectzd interl/al. The optimal policy cou.:d involve a low ad valorem tariff rate when implrl:ts are very low or very high but a high tariff rate for intermediate levels of ilnpc,rts. The above analysis also sho,ws that if there is unczrtdinty III both dcmznd arid supply, then a tariff quota in which the tar,ff rate ,;lepends on the q&an&y of impotts would generally fail to ensure that the first-or&t conditions behind theorem 3 are satisfied [see eqs. (13.2a) and (B2br in appendix B, which yield Cl_‘;)],It would be necessary to set the tariff r;itc according to the z?alue of imports. 4. Extensions The above analysis can readily be extended to the case whcrc the importing country is large and. the v:orld price of the good depends on the quantity imported as well as orit 0: P* = P’VQ, 0).
We shall only sketch the results: the proofs are an obvious amalgam of th:: arguments above and those in Young ar,d Anderson (19Sir) and Young ( 1980). Theorems 1 and 2 remain valid for a large country if we take tb : objective function to be world expected surplus: rQ(_B)
s, (P(v, O)- P4’(1;,0)) do
E
.
1
However, theorems 3, 4 and !; would be vahd for ihi:- objjec’:Ixz l’utl:ti; ~1 oni) if the inverse supply function
has consta,lt
elasticity
with ;,zspect to irlpor:s.
if the desired objectiv: function is dontestic expected surplus: QW
E
J ‘;f’(u.,0)- uik(Q(0), 0)) dv 0
1 ,
then the optimal policies are tariff schedules which are essentially those derived above plus an Edgeworth-‘Bickerdike ‘optimal tariff’ designed to exploit the importing country’s mono.ploly power.8
To examine zhc implicaticns of a non-constant marginai utifity of income, Ict the objective function be the expected value of the representative consumer’s indirect titility function V(P, I), where P is the price of the imported gctod in terms of the exportable and I is total production revenue @us tariff revenue. It can be shown’ that the first-order condition for a maximum or E [ L’(P,Z)] subject to constraints (1) and (2) is:
Hence. optimal arbitrage of the two types of import licences must noXv take account of the variation in the marginal utility of income aV/al. Iq effect, the optimal trade resTriction must be m,odified to buffer the domestic economy against the income changes resulting from random domestic and foreign shocks. This consideration implies that, although a tariff quota will no longer be oiltima!, it can be superior to both a pure tariff and a pure quota, even in the absence of a constraint like (2). F dr example, if the world price fluctuates, then it would be desirable to lower the tariff rate when imports are low in order to augment the country’s l gai ns from trade’ in sta:.es where it is being impoverished by a rise in the ~~;,rld price. On the other hand, if domestic production suffers random &oizks (such as weather) then it would be desirable I:, lower tine tariff rate \\hcn imports are high in order to augment the country’s ‘igains from trade’ in s?; tes where it is being impolverished by a reduction in domestic prrjductioil. A detailed analysis of these qulestiom is beyond the scope of this paper. but the above discussion at least indicates how the additional degree of freedom available in tariff qmo~as can be expl&ted to meet a number o; polic! objectives. Ll'o..mg 61980.p 429~ shows that if there is multiplicative uncertainty in the foreigr suppl) Irsncr~:~~,then this ‘optimal tariff is a deterministic schedule (If tasiffs depending only on the aorld prrrx. If Ihe wpply fun&m is linear or consta.nt elastic, t Iti: t.his schedule is A mixture of spe4c and ad vakrem tariffs. "l‘hc! prtsof i*, swilar to that in Young and knderson (1981, sel:tions 2 and 3).
J.E. Anderson nnd L. Yang.
Opthal
tarifl’quotay
147
References Bhagwati. J.N. and T.N. Srinil,asan. 1969, Optimal intervention to achieve non-ecxroxic objectrves, Review of. Economic Studies 36, 27-38. Rom, M., 1979, The role of tarifT quotas in commercial policy (Macmillan, London). Young, L., 1980, Optimal revenue-raising trade restrictions under uncertaintv, Journal of International Economics 10, 4.21:~~440. Young, L. and J.E. Anderson, 1980, The optimal policies for restricting trade under uncertaislty, Review of Economic Studies 46, 927.-932. Young, L. and J.E. Anderson, 1981. Risk aversion and c:ptimal trade restrictions, Universit!. of Canterbury, mimeo.
Appendix A We shall show that the first-orde;. conditions (3) and the complementary slackness conditions (4) are sufftcient for a constrained maximum in theorem 1. The Kuhn-Tucker theorem cannot be invoked because the constraint function (Q(O))- Q1} + is not differentiable everywhere and because there could be an infinity of states 0 and hence an infinity of decision vari,tbles
Q(e)* In state 6, lelt Q’(O) be imports under the policy defined by (3) and (4) and let Q(0) be imports under any other policy which satisfies constraints (1) and (2). Domestic surplus under Q’(C) exceeds, that under Q(#) by Q’(e)
Since P(Q, 0) is decreasing that @>
in Q, the Second
2 (P(Q’(@, 0)-- p*(@){Q'(O) -
Mean Value Theorem
implies
Q(Q).
BY (3):
i,(Q'(@ --Q(Ui),
if
Q(U)$QQ,
(A+~)(QG(ti) -Q(R)],,
if
Q((I)>Q,.
(A. I )
But: Q”( 0) > Q , impl.ies Q”( 0) - Q(O)
~(Q0(8)-Q,}+-~Qltl)--Q,3’. This obviously
holds if Q(6’)ggQ 1, while if Q(0)
{QW-Ql)+>Q’(@li-Q1,
(9.2)
J.E. ,4nderscn and I.. l’hng,
348
so agrain
Opimal tardy quoras
(A.2) holds. Moreover, E~(Q*rn,-Q(e)~.+(Q”(tl)~>.Q:]-~~C(Q(o)-Qlf+(QO(e)>Q1! (A. ;)
~E:I(Q’(O)-Q,t+]-E[(Q’(O)-(2,)+]. By (A.l), (A.2) and (A-3): E[d(OVJzi,E [Q”(O)-Q(O)] +p(E [{Q’(O)-.Q$] -E[{Q(@---Q,j+]). Since
Q"(Q)
satisfies the complementary slackness corditions (4): W(~~)l1~l~~,- E CQNI) + ,&I, - E C(Q(@l - Q1) ‘I).
Since Q(U)satisfies the constraints (1) a.nd (2):
Hence, Q’(O) is indeed the optimal policy. For all the other theorems, the proof that the first-order conditions and the complementary slackness conditicns are sufficient for optimality is similar.
Appendix B Proof
oj
2. The Lz:grangean of the constrained problem of theorem 2 is: theorem
QfjO
E
1 (P(c, 0) -P*(Q)
d u-nQ(e)-~{Q(e)-ECQ(e)!j’
where i and p are the Lagrange multipliers car:esponding anld (9). Hence, thle first-order condition is: P(Q(Oj,#)- P*(O)=i,+2~C1(Q(8)-EeQ(8,‘1).
maximization
I
+b
+P,
to constraints
( )
(EL1
This condition is satisfied if the following linear scheduie of specific tariffs is i m posed:
s(Q)= X t 2p(Q- q 11.
The first-order condition is sufficient for a maximum because the objeclive function is concave and ihe constraint functions are convex in Q. Q.E.D. Proof of theorem 3.
An argu,qent like that in theorem I shows that the firstorder conditions of the constrained maximization problem are: P(Q(~), e) -
p*(e) =
if*(e), (II.+ ppyej,
if P(O)Q(la) 5 E,, if P(o)Q(e) > El,
(B.24 (B.2b)
wilere 1*and p are the Lagrange multipliers associated with constraints (11) and (12), respectively. These first-order conditions will be satisfied if the schedule (13) of ad valorem tariffs is imposed with Q, = 1, and a2 = i, + 11.The first-order conditions a.r*: sufficient for a maximum since the objective function is concave and the constraint functions ar,: convex in Q. QED. In case (a) the world price is fixed at some level P* and the conclusion follows from theorem 3 with:
Proof of theorem 4.
Q.) = El/P*. In case l(b) let Q”(O) SC the optimal level of imports in state 8, let 0, be defined ilm,plicitly by
we,)Qom = E,,
(B.3)
Q1 = p(el).
WV
and let
We shall show thLat Q”(0)$Q1 implies P*(0)Q0(8)i$ktJ,.
(B.3
Since 3,20, (13.3)and (R.4) imply that P*(@QO(e)5 E, implies (1 9 n)P*(o)Q”[e) (B.6)
Combining this with thle first-order condition (B.2a) yields P*(@)QO(e) 2; E, implies P’(QO(@, @Q’(6) (B.7)
J.E. Anderson and L. Young, Optimal tariff quotas
3.50
We assumed that P(Q, 6) is independ.ent of 0 and fhaf the demand elasticity always exceeds 1, so P(Q”~(0),O)Q”(0)~P(Q,, #,)Q1 implks Q”(P)SQ1.
IB.8)
By (8.7) and (B.8): P*( S)Q’(#) 5 E 1 implies Q”(0) 5 Q 1.
(Es.9
Therefore Q’(H):>Q, implies P*(0)QD(O)> E,.
(R.lOt
Since i, 20 and /,rzO, (B.3) and (B.4) imply th;:,t .f *(@Q’(0)> E, implies (1-tI+ p)P*(O)Q(O)
>(1+4~*C~,~Q,.
(B.ll
/
Combining this with the first-order conditions (B.2a) and (B.2b) yields: P*(ll)Q”(@)> El implies .P(Q’(fl), f39Q”(@
We assumed that P(Q, 6~ IS independent of 8 and that the demand elasticity i:r always greater than 1, f;o PtQ”l @),@Q’(d) > P(Q 1, 01 )Q 1 implies Q”(@ > Q 1.
(BXJ)
By (B.12) and (EL13) P”(H)Q”(6)> E, implies l{!‘(e)> Q1_
(18.14)
Q"!i?) 5 Q, implies P*(0)Q”(6) 5 E,.
(B. 15)
Theref:Dre
(B.10) and (R.15) imply that the first-order conditions (B.2a) b,nd (B.2b) ;tre equfvafen? to the conditions P(QW), 0) - P*(o) =
I.E.
Andmon
md
L
Young,
Optlmcil
tarijjquotm
353
These conditions will be satisfied if the ad valorem tariff qu1ob.a(14j is imposed with a, = i, and ~1~= i. + p. Q.E.D. Proof
of theorem j.
Define Q1, as in (B.4). If the import demand function is deterministic and always has elasticity less than 1, then ,P(Q”(O),O)Q”(@ P(Q,, 6)Q1 implies Q”(O)$ Q1.
(B.16;
This means that thz ineqxalitles in the conclusions on (B.9) and (B.13) mus; be reversed and thist instead of (B. 10) and (B.14) we have: Q’(O)< Q1 implies P*(@Q0(8) > El,
[B.i7)
Q’(O)2 Q 1 implies P*(@Q’(O)5 E 1.
(B.18)
(B.17) and (B.18) imply that the first-order conditions (B.2a) and (B.2b) are equivalent to the conditions P( Q(O),0) - P*(Oj =
iP*(t?), f\(A -I p)P*(O),
if
Q(@2Ql,
if
Q(O)< Q1.
These conditions will be satisfied if the ad valorem txiff quota QED. imposed with a, =A+,u and a,==L
(15)
is