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Optimal tariffs for trade in monopolistically competitive commodities
Journal of InternationalEconomics 12 (1982) 225-241. North-Holland Publishing Company
OPTIMAL. TARIFFS FOR TRADE IN MONOPOLl.STICALLY COMPETITIVE COMMODITIES
Anthony J. VENABLES University of Sussex, Brighton BNI YQN, U.K. Received February 1981, revised version received September 1981
The welfare implications of international trade are examined for an economy which contains a monopolistkally competitive industry and imports products directly competitive with this industry. Necessary and sufFkient conditions for such trade to increase welfare are found, and optimal import tariffs are derived and analysed.
1. Introlluction
This paper examines the welfare economics of internatiorlal trade for an etonomy which imports commodities that compete with a local monopolistically competitive industry. Production in the monopolisitically competitive industry takes place under conditions of increasing returns to scale, and the numbers and types of commodities produced by the industry are determined by profit maximisation and free entry. The welfare properties of closed economies of this type have been studied by Dixit and Stiglitz (1977) and Spence (1976). The analysis has been extended to international trade by the recent work of Krugman (1979) and Dixit and Norman (1980). From this work we know that international trade hbs essentially two possible effects on welfare: trade may permit fuller realisation of economies of scale, and may change the number and type of commodities available in an economy. This paper concentrates entirely on the effects of trade, and trade policy on the number and types of commodities available in the trading economy. In order to do this, analysis is focused on a small economy which is assumed to be a price-taker in world markets. The economy’s comparative advantage is such that it exports a homogeneous commodity produced competitively under condition(s of non-increasing returns to scale, and imports differentiated commodities that compete directly with a local monopolistically competitive industry. It will be assumed throughout that production conditions and foreign preferences make export of commodities produced by its monopolistically competitive industry unprofitable for the 0022-1996/82/OooO400 /$02.75 0 1982 North-Holland
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A.J. Venubles, Optimal tar@ for trade
economy under considerati0n.l The economy may perhaps be identified with a less-developed economy which exports primary commodities and imports manufactures which replace the output of the traditional manufacturing industry. The analysis may alternatively be construed as constituting a s detailed study of one side of a model of intra-industry trade [this is developed further in Venables (198O)J Section 2 of the paper sets out the model. In section 3 equilibria of the economy with and without trade are compared: trade generally has the effect of forcing some locally produced commodities out of production, and so changing both the type and the number of Commodities available to consumers. Conditions under which this change either increases or reduces social welfare are found. The effects of trade on welfare depends on the tax treatment of locally produced commodities, and of imports. Section 4 of the paper derives and analyses optimal tariffs for the import of monopolistically competitive commodities. It is established that tariffs should generally be non-zero, and different from tax rates on domesticaiiy produced commodities. There are essentially two reasons in this model why active tariff policy should be employed. The first is that the domestic economy is ‘imperfect’, so tariffs may be used as a second-best policy instrument to correct these imperfections. The second reason derives from the possibility that tar% may influence the number of commodity types imported. The ability to control the number of different ’ commodity types imported generates an argument for the use of tariffs, qualitatively different from arguments generally found in the literature. X Tbemodel
This section assumes a closed economy, and outlines the basic model which will be used throughout the paper. The equilibrium and social optimum of the model are character&d, and the use of domestic tax/subsidy policies to correct domestic imperfections is discussed. 2.1. Commodities The economy is divided into two sectors: a monopolistically competitive industry and the rest of the economy. The monopolistically competitive industry produces commodities from a set, which will be labelled S, of differentiated commodities. The commodities in the set S, of which there may be in6niteIy many, will be labelled i= 1,2,. . ., n,. . .; the quantity of the ith product will be denoted Xi with price pr. Firms will be identified with commodities, so each firm produces only one commodity, and. any %ez Lanawter (1980)for a discussion of asymmetricwade of this type.
A.J. Venables, Optimal tariQs for
trade
227
commodity is produced by a single firm. Prdduction of each of these commodities takes place under conditions of increasing returns to scale, which will be modelled by assuming that if the ith commodity is produced, it incurs a fixed cost, Fi, independently of the scale of output: operating costs will be denoted C&C,),ci >O, c; >O. The presence of increasing returns ensures that only some of the potentially available commodities from S will generally be produced: commodity labels will be arranged so that the equilibrium produces the first n commodities, where n is of course au endogenous variable. The rest of the economy is aggregated into a single commodity, labelled 0. This commodity will be taken as the numeraire, and the quantity of the commodity produced will be denoted x0. It will 1~ assumed that the supply of the numeraire commodity is perfectly competitive. If the economy has endowment M in terms of the numeraire, then XO=M-
i
(C,(XJ+ Fi).
ir.zl
vi
2.2. Demands Demands are derived from an individualistic social welfare funccioa defined on the aggregate quantities of the commodities consumed: lump-sum redistributions are assumed to have taken place to maxim& this function with respect to distribution. In the closed economy consumption of each commodity equals production so the social welfare function is defined on xc, and the Xi, itz S. The welfare function will be assumed to be separable between x0 and the products from the set S, so may be written
u = U(x0, V(x1, x2,.
.
.,x,. . .)),
CT4
where U is quasi-concave and twice differentiable, and it is assumed that xg and Yare everywhere normal in U. It will further bc assumed that the subutility function I/is additive separable, so if=
*&%w,
(31
with z+(O)=O,and OXXi!increasing, strictly concave and twice differentiable,. for all i 12S. Given the economy’s endowment, M, and denoting any transfer incomes r the budget constraint may be written
where n commodities from S are available.
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228
fortrade
Utility maximisation gives prices
where UxXi)is the derivative of Ui(xi)and q=4(x0, v) is the marginal rate of substitution between x0 and sub-utility Vin the function U. 2.3. Equilibrium Given the demand functions (5) the ith firm has profits ni=qXiU~Xi)(l--i)-CCXXi)-_i, Q=O,
if
Xi=O>
if Xi>03 (6)
where Fi is a tax rate on the ith commodity expressed as a proportion of markt price. Total profits and tax revenue constitute transfer incomes, i.e. f
(&+A?7i)=T
(7)
i=l
A Cournot-Nash equilibrium is a vector of output levels Xi,. . .,x,, . . ., where each xi maxim& (6) sabject to given Xj (i#j) and eqs. (l), (2), (3), (4), and (7). Commodity labels are arranged so that xi >O, for i = 1,. . ., n, and xi =0, for i=n+l.... The following three assumptions will be made for the rest of the paper. (I) Symmetry. All differentiated products which are produced in the country under consideration are symmetric. Symmetry requires that Fi=Fj, ti=tl, t+(x)=trj(x), ci(x)=cicx>, for a11x>O, and for all i, j which are produced domestically. Symmetry immediately implies that for a11 i, j which are produced xi = x1 and pi= pj. Although symmetric products have the same technologies and the same demand characteristics, they are not of course perfect substitutes. Strict concavity of the sub-utility functions Vi imply strictly convex indifference surfaces: indifference curves are however symmetric about the 45” ray through’ the origin. (2) ‘Large group case’. n is large enough for 1 to be regarded as a small increment.. Then (a) the output of any one commodity from the monopolistically competitive industry is small enough in relation to the industry’s total output for each firm to assume that a sma11 change in its output will not effect q, the marginal rate of substitution between x0 and K
A.J. Venabks,Optimal
tariffsfor trade
229
and (b) the entry condi?ion may be written with equality, so n,=O. (3) The profit functions, I7,, are strictly concave in xi, for xi>O.’ Assumptions (l)-(3) ensure that at the firm and industry equilibrium firms equate marginal revenue to marginal cost, and all firms in the industry earn zero profits. We therefore have the following two equations:
p,(l-t,){l+~]=C~XJ, i=l,...n,
(8)
i= 1,. . . n.
(9)
pi(l -ti)Xi-CXXJ-F*=O,
The equilibrium defined by eqs. (8) and (9), with (l)--(4) and (7), is not of course the social optimum, because of the presence of increasing returns to scale and imperfect competition. The social optimum may now be character&d, and the use of a tax/subsidy policy to decentralize the optimum discussed. 2.4. Social optimum For the closed economy, and with assumptions (l)-(3), social welfare is maxim&d when n, the number of firms, and xi, the output of a representative firm, are chosen to maximise welfare subject to the economy’s resource constraint. Using eqs. (1) and (3), the problem is to choose xi and n to maximise
UW -
nWi)
+
Fib
n~itxi)).
(10)
First-order conditions for this problem are, with respect to xi, qU;(Xi)- CXXJ= 0
and, with respect to n,
Pifxi)- Ci(Xi)-
Fi = 0.
(12)
Comparison of eqs. (11) and (12) with equilibrium conditions (8) and (9) indicate that, in general, 3 decentralisation of the optimum requires two 2This assumption implies that marginal revenue functions intersect marginal cosI functions from above, and is of course standard. Fur the implications of its failure, see Roberts and Sonncnsckin (1977). curves) arc iso-elastic, 3In the ispecial cabwhere the sub-utility functions v (and hence one policy instrumentis sulkient to daxntralisc the optimum.
230
A.J. Vmabks,
Optimd %[email protected] trade
policy instruments, one to ensure that, for each commodity, the optimal quantity is produced [eq. (ll)], and the other to ensure that the optimal number of commodities is produced [eq. (1211.For the purpose of this paper it may bc noted that if ti = tf, where ti*=]--
ui(xi)
XiUxXi)
’
(13)
then [comparing eqs. (9) and (12)], the equilibrium produces the optimal number of commodities, given Xi. Although the subsidy tr does not therefore generally take the domestic economy to the first-best optimum, it does ensure that, for given output levels, the optimal number of commodities is produced.
3. The import of eommodks In this section the model is extended to an open economy which is assumed to export its numeraire commodity, and import commodities from the group S. The welfare implications of such trade are examined. In the equilibrium with trade, products from the set S are supplied from two sources. Products from S are produced domestically and, as before, will be labelled i= 1,. . ., n. The prices, quantities and numbers of these commodities are determined as in the previous section, and assumptions (l)(3) will be taken to hold. A group of m products from the set S are imported: these m commodities will be labelled j=-s, . . ., s+m. The unit cost of the jth commodity to the economy will be denoted ci, and the quantity imported denoted xi. The 111imported commodities will be assumed to be symmetric to each other, and assumed to form a ‘large group’ (so that 1 is a small increment). Though both imported and domestically produced commodities form symmetric groups, it will not be assumed that the groups are necessarily symmetric to each other. The supply of the numeraire commodity available to the economy in the equilibrium with trade is X,=M-t&(XJ+FJ-??lCjXj,
(14)
where nr+xj is the total cost to the economy of its imports. Demands are derived, as before, from the social welfare function: this may now be written
AJ.
Venables, Optimal tari$s for trade
231
The possibility that the market price of imports differ from their resource cost to the economy is permitted: in particular, tariffs may be employed, in which case pJ(l -tj)'Cj,
W5)
where t, is the tariff expressed as a proportion of market price. If the total tariff revenue from the jth commodity is denoted q, the budget constraint may be written x0=
M-npixi-
mpfij+nT+mTj.
(17)
Utility maximisation gives prices for the locally produced commodities as before [eq. (S)], and for the imported commodities: pj'qUli(xi),
j=s,...,s+m.
(18)
The equilibrium under autarky and the equilibrium with trade may now :.e compared. The comparison is made straightforward by the following proposition. Proposition 3.1. With assumptions (l)-(3), and providing that n >O, the autarky equilibrium and the trade equilibrium have the following properties in common. (i) The prices and quantities of those products -from the monopolistically competitive industry which are produced domestically are the same in each equilibrium, (ii) q, the marginal rate of substitution between x0 and K is the same in each equilibrium (so the two equilibria lie on the same Engel curve of U). Proof: With assumptions (l)--(3) eqs. (8) and (9) hold in each equilibrium. For each i these are two equations in pi and Xi. Given strict concavity of the profit functions they have a unique solution: pi and Xi are therefore the same in each equilibrium. From eq. (5) pi =q t((Xi), i = 1,. . ., n. q must therefore be the same in each equilibrium, so each equilibrium lies on the same Engel curve of U. l
From proposition 3.1 we know that the effect of trade on the monopolistically competitive industry is only to change the number of commodities produced - the prices and output quantities of those firms that are active remain unchanged. Furthermore, the number of active firms must change in such a way as to hold q constant. The implications of trade for the arguments of the welfare function U may now be examined. If the d operator is used to denote the difference in the value of a variable between the
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AJ. Venables, Optimal tariKs ibr trade
equilibrium without trade, and the equilibrium with trade, we obtain: Ax0 = -
(CXXi) + FJAB- mcjxj,
A I’= uAXi)An + mt~j~j),
(19) (20)
where - dn is the number of domestically produced commodities ‘forced out’ by the presence of international competition. Using (19) in (20) gives Ax, = -_ tttcjxj - (ckxi) + Fi
(21)
We also know from proposition 3.1 that the equilibria with and without trade he on the same Engel curve of U. A’ particular Engel curve may be written
V=&o),
(22)
where, by normality, g’ >O. Along an Engel curve we therefore have A V=g$z) *Ax(-),
(23)
where QE [x(),x()+ Ax,]. Using l(21)in (23) gives
(24) Using eqs. (9), (16) and the price equations (5) and (18), eq. (24) may be rewritten:
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233
With positive income elasticities, social welfare is strictly increasing on both the arguments of U for movements along an Engel curve. Since the coefficient of A V in (25) is positive, the effects of trade on social welfare is determined by the sign of the left-hand side of (25). Proposition 3.2. The equilibrium with trade has a higher (lower) level of social welfare than the equilibrium without trade, if (l-ti)xivKxi)
vdxi)
--_
(l-fj)xjz’~xj)
,of
(26)
CkXj)
This is immediate from eq. (25). Several remarks may be made about the interpretation of this condition. First, suppose that ti= ti < 1. The quantity xiV~xi)/UAxi) may be interpreted as the ratio of the ith commodity’s revenue (and hence costs) to the ith commodity’s revenue plus consumer surplus. Proposition 3.2 establishes that, with identical taxation of imported and locally produced commodities, the introduction of new commodities (labelled j) raises welfare if and only if the ratio of consumer surplus to cost on these commodities is greater than the corresponding ratio on those commodities (labelled i) which are ‘forced out’ by the presence of imported commodities. If the sub-utility functions are iso-elastic so
then
a&*+!$ i
where si is the own-price elasticity of demand for commodity i (and similarly for imports, j). Proposition 3.2 then states that, with identical taxation of imported and locally produced commodities, the equilibrium with trade has a higher (lower) level of welfare than the equilibrium under autarky if the own-price elasticity of demand is lower (higher) for imported commodities than for domestically produti commodities. The role of taxation in proposition 3.2 has the following interpretation. Taxation, raises the marginal valuation of a commodity by consumers relative to its marginal cost, so raising the ratio of surplus to cost on a commodity. If the imported and locally produced commodities differ only in the proportional rates at which they are taxed, then the equilibrium with JIE-
0
234
AJ. Venables, Optimal tari@ for trade
trade has a higher (lower) level of social welfare than. the equilibrium without trade, if imported commodities are taxed more (less) heavily than locally produced commodities. The following corollary to proposition 3.2 may now be obtained. Proposition 3.3. lf taxation policy is used to optimise the number of domestic priucts produced (SO ti = tl), then free trade increases social welfare. Pro05
!Setting
[from eq. (13)] and tj = 0, condition (26) becomes xjvJxj)
1-->o.
vjfx j)
Since vl(xj) passes through the origin and is strictly concave, this condition holds. From the previous section, setting ti= tT is not generally sufficient to move the economy to a first-best optimum. However, from proposition 3.1, trade affects .the number of commodities produced, not the quantities in which each commodity is produced. The use of tax policy to optimally control the number of commodities produced is therefore sufficient to ensure that there are positive gains from free trade, although there remain imperfections in the domestic economy. The two propositions above provide simple conditions which characterise the welfare effects of the import ot*monopolistically competitive commodities. It is worth noting that these conditions-are independent of the costs to the economy of imported commodities, of the number of commodities imported, and of the number of domestic commodities forced out of production by trade. The welfare changes are due entirely to changes in consumer surplus associated with the change in product mix. The propositions are of course dependent upon the assumption that some commodities from the set S are produced locally in both the equilibrium with, and the equilibrium without, trade. It is possible that the group of imported commodi tics be sufficiently large to make any local production of commodities from the set S unprofitable. In this case 4 will be lower in the equilibrium with trade than in the autarky equilibrium. Negativity of the left-hand side of (25) is then a necessary, but no longer a sufficient, condition for the open economy to have a lower level of social welfare than the closed economy.
A.J. Venables, Optimal tarifi for trade
235
d Optimal import trriffs It is clear from the above analysis that the tariff rate is of importance in determining the welfare effects of trade. This section derives optimal tariffs for a country importing commodities from the set S, and exporting the numeraire commodity. In order to abstract from the possibility that tariffs may be used to change the terms of trade, it will be assumed that the country under consideration is a price-taker in international trade. 4 There are nevertheless two distinct reasons in this model why non-zero tariffs should be desirable. The first is that since the domestic economy is ‘imperfect’, tariffs may be used as a second-best policy instrument to correct domestic imperfections. The use of tariffs for this purpose depends of course on how other policy instruments have been used. In this model the use of alternative instruments to correct domestic imperfections is not a suffjcient condition for unrestricted free trade to be optimal. If the number of commodities imported to the economy can be influenced by the tariff rate, then this provides an additional argument for non-zero tariffs. The optimal tariff problem is to find a tariff t; which maximises social welfare subject to the allocation being a market equilibrium, and for given domestic taxes ti. Since the domestic supply price of imports, cj, is fixed. controlling tj is equivalent to controlling pi, the market price of imports. The problem will be analysed using pj as the control variable. As pi is varied both the number of commodities and the quantity of each commodity imported will generally change. The change in the quantity of each commodity imported is determined by the commodity’s demand function [eq. (1811. As quantities change, so do the revenues earned by foreign firms supplying the imports. If these firms face increasing returns to scale in either the production or export of their output, the number of commodities that they supply will in general depend on the quantities in which they are sold. Changes in xj then change the number of commodities imported. The response of the number of commodities imported, rr”,,to -Xjis exogenous to the importing country. Invoking the ‘large group’ assumption, the response of the number of commodities imported to changes in xj will bc summarised by an elasticity of the number of commodities imported with respect to the quantity of each imported, denoted q, and defined as (27) ‘This a~umption may be justified in two ways. First, if importers’marginalcost and marginal revenue functions are such that, over the appropriate range, the profit-maximising price is independent of quantity supplied (e.g. constant marginal cost, iso-elastic demand curves), then the supply price is constant. Secondly, if the country is small, and importers cannot price discriminatebetween countries, then the country under considerationis a price-taker.
AJ. Vmbks,
236
tariffsfor trade
Optimal
Proposition 3.1 provides information on the response of domestic production to changes in the tariff. From this proposition we know that Xi and pi* the prices and quantities of active firms, are unchanged by changes in pk The number of commodities produced domestically, n, will in general change as pi changes but, from proposition 3.1, we know that n must change in such a manner as to hold 4 constant, i.e. to hold the economy on the same Engel curve. Since along an Engel curve U(xO,v) is strictly increasing in both x0 and V, the optimal tariff problem reduces to the problem of maximising x0, subject to holding the economy on the same Engel curve. Formally, pi must be chosen to maximise ‘0 =
A4- n(CXXi) + Fi) - ?tlC jX
(28)
j
subject to n”Xxi)+ mtll(xi)=dM -
tiCi
+ FJ -
?WjXj),
(29
where n, m, and xj are allowed to vary.
Differentiating along the objective gives (30) Differentiating along the constraint @ves
v_ix jP?-
g”Ci(l +q)+V;(Xj)+F
X
[
l
I
-I
(31)
Using (30) in (31) to eliminate an/8pl gives
(32) For an optimum we require dxo/dpi =0, i.e.
(33)
AJ. Venables, Optimal tari#k for trade
237
Using the import demand equation C*=q*(l J
(18)
-tj)VxXj)
and the zero profit condition for domestic producers, CXXi)+
Fi=q *( 1- ti)XiUi(Xi)
(9)
gives the following first-order condition for the optimal tariff t?: 1-q
=l-ti
WXx3
l +tt
udxil
(
I+~. I[
Xju;(Xj) 1’ UJ4X.j)
(34)
It can be shown that second-order conditions for the problem are satisfied, so that eq. (34) defines a welfare maximum. This gives the following proposition. Proposition 4.1. Ifn >Q, the Dptimal import tar@ (expressed as a proportion of market price), t,* is given by (34)
Notice that since U~xi>lxiu;cxi>>1, we have XjVxAj)
(l- ti*)*--<(l
-ti)*-
r+J
XiVKxi) vXxi)
so that, from the analysis of section 3, imports which are subject to an optimal tariff never reduce social welfare. This is of course as it should be. The characteristics of the optimal tariff may now be examined. We shall look first at the relationship between the optimal tariff and the use of ti to control domestic production. When q = 0, we have the following proposition. Proposition 4.2. If q=O, and domestic production is subsidised such that the optimal number of domestic commodities are produced, i.e. ti= t?, then the optimal tarifl is zero. [f ti > tr, then tJ > 0. Proof:
Setting q = 0 in (34) gives t,*=l-(1.-rJ-
xi”;(xi) 4h)
(35)
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238
From eq. (13) (36) Use of (36) in (35) proves the proposition. Proposition 4.2 establishes that, for any economy which cannot influence the number of commodities it imports (q=O), optimal control of the number of domestic commodities produced is sufficient to ensure the optimality of free trade. There remain imperfections in the domestic economy, since the quamity in which each commodity is produced is not optimal. But, from proposition 3.1, trade does not affect these quantities, so this imperfection provides no argument for non-zero tariffs. If, however, domestic production is not subsidised to the level t?, then positive tariffs should be employed to increase the number of domestically produced commodities. We now turn to the case where q >O. Since tariffs now affect the number of imported commodities, non-zero tariffs are required, even though domestic tax instruments have been optimally used to correct domestic imperfections. Proposition 4.3. If q >O, and domestic taxes have been set to optimise the number of domestically produced commodities (that is, ti= tr), then the optimal tarifl is negative. Prm$
Using (1
_t+$+ i
1 i
[eq. (13)] in eq. (34) gives l-t+-[1+9(-$&)]*
Since
(37)
V,(Xj)/XjV;{Xj)>l, 5%
The intuition behind this is as follows . Since n is optimally controlled, a marginal change in n is worth nothing. Imports do, however, give positive consumer surplus, so that a negative tariff is desirable in order to increase the number of imported commodities, and thereby increase social welfare. An additional important feature of the problem may be pointed out at this stage. If tariffs are the only instrument of trade policy available, and policy-makers are constrained to set the same tariff rate for all symmetric imports, then policy-Lmakers only have one instrument for the control of trade. They are,
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Venabks, Optimal [email protected] trade
239
however, interested in two variables: the number of commodities imrzrted and the quantity in which each commodity is imported. Negative tariffs increase the number of commodities imported, so raising welfare (if the number of domestically produced commodities has been optimised), but also mean that imports are sold at prices below their marginal cost to the economy. The optimal tariff [eq. (37)] is determined as the outcome of these two forces. ’’ Some observations may now be made about the optimum tariff structure when domestic taxes have not been optimally set, so tariffs are being used as a second-best policy instrument to control the domestic economy. The following proposition provides conditions under which the tax/optimal tariff structure involves higher taxation of imports than of domestically produced commodities. Proposition 4.4.
(i)
Xju;(Xj)
-
vI(xj)
tj* > ti i$ ,
JriuxXi). vitxi)
’
Or (ii)
xj”;(xj) --=p
xiO~xi)
vjcxi,
f-+&i)
a*
vl <
oo, ,
0t
(iii) Proqf:
q=O.
By inspection of eq. (34), noting that
Part (i) of proposition 4.4 is as would be expected from the analysis of section 3 of the paper. If the ratio of surplus to cost is higher for domestically produced commodities than for imports, then imports should certainly be taxed more heavily than domestic commodities. Part (ii) is stronger: it establishes that even if imports and domestically produced commodities have the same ratio of surplus to cost (they may be symmetric to each other), then imports should be taxed more heavily than domestically produced commodities, providing q is bounded. The intuition behind this is as follows. Suppose imports and domestically produced goods are symmetric (i.e. have the same sub-utility functions u), and ti= tjs A small increase in tj has two effects. First, it may change the number of commodities produced and imported, but since condition (26) of section 3 holds with equality, this leaves welfare unchanged. Secondly, the increase in tj makes imports more valuable to the economy, by rais;ng their marginal valuation to consumers,
240
AJ. Vmables, Optimul
twi$sjiwtrade
and hence increasing the ratio of surplus to cost. Providing q is bounded, this effect may be exploited by taxing imports more heavily than domestically produced goods, even though the products are symmetric. Part (iii) of proposition 4.4 establishes that if the number of imported commodities is given, imports should always be taxed more heavily than domestic products. This is because the total number of commodities supplied to the economy can be increased by taxing imports (so increasing the number of commodities produced domestically), and additional commodities bring positive consumer surplus.
This paper has analysed the welfare effects of importing commodities which compete directly with a local monopolistically competitive industry. Assumptions of separability of utility functions, and of symmetry of groups of commodities have permitted a number of results concerning the welfare economics of such trade, and optimal tariff policies towards such trade, to be obtained. In section 3 of the paper it was established that necessary and sufficient conditions for trade to increase welfare were that the ratio of consumer surplus to cost on imported commodities be greater than that on locally produced commodities. For the case of iso-elastic demand functions, this reduces to the condition that imports increase welfare if and only if they are less elastically demanded than locally produced commodities. Section 4 of the paper derived an optimal tariff for imported commodities. It was established that if the number o? commodities imported was independent of the tariff rate, imports bhould UCsubjected to a positive tariff unless domestic production was subsidised to the point at which the number of commodities produced domestically was optimised. If, however, tariff rates affkct the number of commodities imported, then subsidisation of domestic production to optimise the number of commodities produced is not a sufficient condition for free trade to be desirable. Import subsidies should be employed in this case to exploit consumer surplus on imported commodities. It was also established that even if imported commodities were symmetric with locally produced commodities, an asymmetric tax treatment - taxing imports more highly than domestically produced commodities - was generally desirable.
AC
I &ouId like to thank P. Hammond, G. Heal, J. Stiglitz, a referee, and participantsin 8cmiMrsd the universities of Essex and !~ISSCX for useful comments on an earlier version of this paptr.
Al. Venables, Optimal tariffs for trade
Refwencea Dixit, AK. and V. Nosmran, 1980, Theory of international trade (Cambridge University Press). Dixit, A.K. and LE. Stiglitx, 1977, Monopolistic competition and optimum product diversity, American Economic Review, June, 297-308. Krugman, P.R., 1979, Increasing returns, monopolistic competition, and international trade, Journal of International Economics 9, -79. Lancaster, K., 19g0, Intra-industry trade under perfect monopolistic competition, Journal of International Economics 10,151. Roberts, J. and H. Sonnenschcin, 1977, On the foundations of the theory of monopolistic competition, Econometrica 45,101-l 13. Spence, A.M., 1976, Product selection, lixed costs and monopolistic competition, Review of Economic Studies 43,217-235. Venables, AJ., 1980, Monopolistic competiticn and the possible losses from international trade, University of Suss;ex Discussion Paper.
OPTIMAL. TARIFFS FOR TRADE IN MONOPOLl.STICALLY COMPETITIVE COMMODITIES
Anthony J. VENABLES University of Sussex, Brighton BNI YQN, U.K. Received February 1981, revised version received September 1981
The welfare implications of international trade are examined for an economy which contains a monopolistkally competitive industry and imports products directly competitive with this industry. Necessary and sufFkient conditions for such trade to increase welfare are found, and optimal import tariffs are derived and analysed.
1. Introlluction
This paper examines the welfare economics of internatiorlal trade for an etonomy which imports commodities that compete with a local monopolistically competitive industry. Production in the monopolisitically competitive industry takes place under conditions of increasing returns to scale, and the numbers and types of commodities produced by the industry are determined by profit maximisation and free entry. The welfare properties of closed economies of this type have been studied by Dixit and Stiglitz (1977) and Spence (1976). The analysis has been extended to international trade by the recent work of Krugman (1979) and Dixit and Norman (1980). From this work we know that international trade hbs essentially two possible effects on welfare: trade may permit fuller realisation of economies of scale, and may change the number and type of commodities available in an economy. This paper concentrates entirely on the effects of trade, and trade policy on the number and types of commodities available in the trading economy. In order to do this, analysis is focused on a small economy which is assumed to be a price-taker in world markets. The economy’s comparative advantage is such that it exports a homogeneous commodity produced competitively under condition(s of non-increasing returns to scale, and imports differentiated commodities that compete directly with a local monopolistically competitive industry. It will be assumed throughout that production conditions and foreign preferences make export of commodities produced by its monopolistically competitive industry unprofitable for the 0022-1996/82/OooO400 /$02.75 0 1982 North-Holland
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A.J. Venubles, Optimal tar@ for trade
economy under considerati0n.l The economy may perhaps be identified with a less-developed economy which exports primary commodities and imports manufactures which replace the output of the traditional manufacturing industry. The analysis may alternatively be construed as constituting a s detailed study of one side of a model of intra-industry trade [this is developed further in Venables (198O)J Section 2 of the paper sets out the model. In section 3 equilibria of the economy with and without trade are compared: trade generally has the effect of forcing some locally produced commodities out of production, and so changing both the type and the number of Commodities available to consumers. Conditions under which this change either increases or reduces social welfare are found. The effects of trade on welfare depends on the tax treatment of locally produced commodities, and of imports. Section 4 of the paper derives and analyses optimal tariffs for the import of monopolistically competitive commodities. It is established that tariffs should generally be non-zero, and different from tax rates on domesticaiiy produced commodities. There are essentially two reasons in this model why active tariff policy should be employed. The first is that the domestic economy is ‘imperfect’, so tariffs may be used as a second-best policy instrument to correct these imperfections. The second reason derives from the possibility that tar% may influence the number of commodity types imported. The ability to control the number of different ’ commodity types imported generates an argument for the use of tariffs, qualitatively different from arguments generally found in the literature. X Tbemodel
This section assumes a closed economy, and outlines the basic model which will be used throughout the paper. The equilibrium and social optimum of the model are character&d, and the use of domestic tax/subsidy policies to correct domestic imperfections is discussed. 2.1. Commodities The economy is divided into two sectors: a monopolistically competitive industry and the rest of the economy. The monopolistically competitive industry produces commodities from a set, which will be labelled S, of differentiated commodities. The commodities in the set S, of which there may be in6niteIy many, will be labelled i= 1,2,. . ., n,. . .; the quantity of the ith product will be denoted Xi with price pr. Firms will be identified with commodities, so each firm produces only one commodity, and. any %ez Lanawter (1980)for a discussion of asymmetricwade of this type.
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227
commodity is produced by a single firm. Prdduction of each of these commodities takes place under conditions of increasing returns to scale, which will be modelled by assuming that if the ith commodity is produced, it incurs a fixed cost, Fi, independently of the scale of output: operating costs will be denoted C&C,),ci >O, c; >O. The presence of increasing returns ensures that only some of the potentially available commodities from S will generally be produced: commodity labels will be arranged so that the equilibrium produces the first n commodities, where n is of course au endogenous variable. The rest of the economy is aggregated into a single commodity, labelled 0. This commodity will be taken as the numeraire, and the quantity of the commodity produced will be denoted x0. It will 1~ assumed that the supply of the numeraire commodity is perfectly competitive. If the economy has endowment M in terms of the numeraire, then XO=M-
i
(C,(XJ+ Fi).
ir.zl
vi
2.2. Demands Demands are derived from an individualistic social welfare funccioa defined on the aggregate quantities of the commodities consumed: lump-sum redistributions are assumed to have taken place to maxim& this function with respect to distribution. In the closed economy consumption of each commodity equals production so the social welfare function is defined on xc, and the Xi, itz S. The welfare function will be assumed to be separable between x0 and the products from the set S, so may be written
u = U(x0, V(x1, x2,.
.
.,x,. . .)),
CT4
where U is quasi-concave and twice differentiable, and it is assumed that xg and Yare everywhere normal in U. It will further bc assumed that the subutility function I/is additive separable, so if=
*&%w,
(31
with z+(O)=O,and OXXi!increasing, strictly concave and twice differentiable,. for all i 12S. Given the economy’s endowment, M, and denoting any transfer incomes r the budget constraint may be written
where n commodities from S are available.
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228
fortrade
Utility maximisation gives prices
where UxXi)is the derivative of Ui(xi)and q=4(x0, v) is the marginal rate of substitution between x0 and sub-utility Vin the function U. 2.3. Equilibrium Given the demand functions (5) the ith firm has profits ni=qXiU~Xi)(l--i)-CCXXi)-_i, Q=O,
if
Xi=O>
if Xi>03 (6)
where Fi is a tax rate on the ith commodity expressed as a proportion of markt price. Total profits and tax revenue constitute transfer incomes, i.e. f
(&+A?7i)=T
(7)
i=l
A Cournot-Nash equilibrium is a vector of output levels Xi,. . .,x,, . . ., where each xi maxim& (6) sabject to given Xj (i#j) and eqs. (l), (2), (3), (4), and (7). Commodity labels are arranged so that xi >O, for i = 1,. . ., n, and xi =0, for i=n+l.... The following three assumptions will be made for the rest of the paper. (I) Symmetry. All differentiated products which are produced in the country under consideration are symmetric. Symmetry requires that Fi=Fj, ti=tl, t+(x)=trj(x), ci(x)=cicx>, for a11x>O, and for all i, j which are produced domestically. Symmetry immediately implies that for a11 i, j which are produced xi = x1 and pi= pj. Although symmetric products have the same technologies and the same demand characteristics, they are not of course perfect substitutes. Strict concavity of the sub-utility functions Vi imply strictly convex indifference surfaces: indifference curves are however symmetric about the 45” ray through’ the origin. (2) ‘Large group case’. n is large enough for 1 to be regarded as a small increment.. Then (a) the output of any one commodity from the monopolistically competitive industry is small enough in relation to the industry’s total output for each firm to assume that a sma11 change in its output will not effect q, the marginal rate of substitution between x0 and K
A.J. Venabks,Optimal
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229
and (b) the entry condi?ion may be written with equality, so n,=O. (3) The profit functions, I7,, are strictly concave in xi, for xi>O.’ Assumptions (l)-(3) ensure that at the firm and industry equilibrium firms equate marginal revenue to marginal cost, and all firms in the industry earn zero profits. We therefore have the following two equations:
p,(l-t,){l+~]=C~XJ, i=l,...n,
(8)
i= 1,. . . n.
(9)
pi(l -ti)Xi-CXXJ-F*=O,
The equilibrium defined by eqs. (8) and (9), with (l)--(4) and (7), is not of course the social optimum, because of the presence of increasing returns to scale and imperfect competition. The social optimum may now be character&d, and the use of a tax/subsidy policy to decentralize the optimum discussed. 2.4. Social optimum For the closed economy, and with assumptions (l)-(3), social welfare is maxim&d when n, the number of firms, and xi, the output of a representative firm, are chosen to maximise welfare subject to the economy’s resource constraint. Using eqs. (1) and (3), the problem is to choose xi and n to maximise
UW -
nWi)
+
Fib
n~itxi)).
(10)
First-order conditions for this problem are, with respect to xi, qU;(Xi)- CXXJ= 0
and, with respect to n,
Pifxi)- Ci(Xi)-
Fi = 0.
(12)
Comparison of eqs. (11) and (12) with equilibrium conditions (8) and (9) indicate that, in general, 3 decentralisation of the optimum requires two 2This assumption implies that marginal revenue functions intersect marginal cosI functions from above, and is of course standard. Fur the implications of its failure, see Roberts and Sonncnsckin (1977). curves) arc iso-elastic, 3In the ispecial cabwhere the sub-utility functions v (and hence one policy instrumentis sulkient to daxntralisc the optimum.
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A.J. Vmabks,
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policy instruments, one to ensure that, for each commodity, the optimal quantity is produced [eq. (ll)], and the other to ensure that the optimal number of commodities is produced [eq. (1211.For the purpose of this paper it may bc noted that if ti = tf, where ti*=]--
ui(xi)
XiUxXi)
’
(13)
then [comparing eqs. (9) and (12)], the equilibrium produces the optimal number of commodities, given Xi. Although the subsidy tr does not therefore generally take the domestic economy to the first-best optimum, it does ensure that, for given output levels, the optimal number of commodities is produced.
3. The import of eommodks In this section the model is extended to an open economy which is assumed to export its numeraire commodity, and import commodities from the group S. The welfare implications of such trade are examined. In the equilibrium with trade, products from the set S are supplied from two sources. Products from S are produced domestically and, as before, will be labelled i= 1,. . ., n. The prices, quantities and numbers of these commodities are determined as in the previous section, and assumptions (l)(3) will be taken to hold. A group of m products from the set S are imported: these m commodities will be labelled j=-s, . . ., s+m. The unit cost of the jth commodity to the economy will be denoted ci, and the quantity imported denoted xi. The 111imported commodities will be assumed to be symmetric to each other, and assumed to form a ‘large group’ (so that 1 is a small increment). Though both imported and domestically produced commodities form symmetric groups, it will not be assumed that the groups are necessarily symmetric to each other. The supply of the numeraire commodity available to the economy in the equilibrium with trade is X,=M-t&(XJ+FJ-??lCjXj,
(14)
where nr+xj is the total cost to the economy of its imports. Demands are derived, as before, from the social welfare function: this may now be written
AJ.
Venables, Optimal tari$s for trade
231
The possibility that the market price of imports differ from their resource cost to the economy is permitted: in particular, tariffs may be employed, in which case pJ(l -tj)'Cj,
W5)
where t, is the tariff expressed as a proportion of market price. If the total tariff revenue from the jth commodity is denoted q, the budget constraint may be written x0=
M-npixi-
mpfij+nT+mTj.
(17)
Utility maximisation gives prices for the locally produced commodities as before [eq. (S)], and for the imported commodities: pj'qUli(xi),
j=s,...,s+m.
(18)
The equilibrium under autarky and the equilibrium with trade may now :.e compared. The comparison is made straightforward by the following proposition. Proposition 3.1. With assumptions (l)-(3), and providing that n >O, the autarky equilibrium and the trade equilibrium have the following properties in common. (i) The prices and quantities of those products -from the monopolistically competitive industry which are produced domestically are the same in each equilibrium, (ii) q, the marginal rate of substitution between x0 and K is the same in each equilibrium (so the two equilibria lie on the same Engel curve of U). Proof: With assumptions (l)--(3) eqs. (8) and (9) hold in each equilibrium. For each i these are two equations in pi and Xi. Given strict concavity of the profit functions they have a unique solution: pi and Xi are therefore the same in each equilibrium. From eq. (5) pi =q t((Xi), i = 1,. . ., n. q must therefore be the same in each equilibrium, so each equilibrium lies on the same Engel curve of U. l
From proposition 3.1 we know that the effect of trade on the monopolistically competitive industry is only to change the number of commodities produced - the prices and output quantities of those firms that are active remain unchanged. Furthermore, the number of active firms must change in such a way as to hold q constant. The implications of trade for the arguments of the welfare function U may now be examined. If the d operator is used to denote the difference in the value of a variable between the
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AJ. Venables, Optimal tariKs ibr trade
equilibrium without trade, and the equilibrium with trade, we obtain: Ax0 = -
(CXXi) + FJAB- mcjxj,
A I’= uAXi)An + mt~j~j),
(19) (20)
where - dn is the number of domestically produced commodities ‘forced out’ by the presence of international competition. Using (19) in (20) gives Ax, = -_ tttcjxj - (ckxi) + Fi
(21)
We also know from proposition 3.1 that the equilibria with and without trade he on the same Engel curve of U. A’ particular Engel curve may be written
V=&o),
(22)
where, by normality, g’ >O. Along an Engel curve we therefore have A V=g$z) *Ax(-),
(23)
where QE [x(),x()+ Ax,]. Using l(21)in (23) gives
(24) Using eqs. (9), (16) and the price equations (5) and (18), eq. (24) may be rewritten:
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233
With positive income elasticities, social welfare is strictly increasing on both the arguments of U for movements along an Engel curve. Since the coefficient of A V in (25) is positive, the effects of trade on social welfare is determined by the sign of the left-hand side of (25). Proposition 3.2. The equilibrium with trade has a higher (lower) level of social welfare than the equilibrium without trade, if (l-ti)xivKxi)
vdxi)
--_
(l-fj)xjz’~xj)
,of
(26)
CkXj)
This is immediate from eq. (25). Several remarks may be made about the interpretation of this condition. First, suppose that ti= ti < 1. The quantity xiV~xi)/UAxi) may be interpreted as the ratio of the ith commodity’s revenue (and hence costs) to the ith commodity’s revenue plus consumer surplus. Proposition 3.2 establishes that, with identical taxation of imported and locally produced commodities, the introduction of new commodities (labelled j) raises welfare if and only if the ratio of consumer surplus to cost on these commodities is greater than the corresponding ratio on those commodities (labelled i) which are ‘forced out’ by the presence of imported commodities. If the sub-utility functions are iso-elastic so
then
a&*+!$ i
where si is the own-price elasticity of demand for commodity i (and similarly for imports, j). Proposition 3.2 then states that, with identical taxation of imported and locally produced commodities, the equilibrium with trade has a higher (lower) level of welfare than the equilibrium under autarky if the own-price elasticity of demand is lower (higher) for imported commodities than for domestically produti commodities. The role of taxation in proposition 3.2 has the following interpretation. Taxation, raises the marginal valuation of a commodity by consumers relative to its marginal cost, so raising the ratio of surplus to cost on a commodity. If the imported and locally produced commodities differ only in the proportional rates at which they are taxed, then the equilibrium with JIE-
0
234
AJ. Venables, Optimal tari@ for trade
trade has a higher (lower) level of social welfare than. the equilibrium without trade, if imported commodities are taxed more (less) heavily than locally produced commodities. The following corollary to proposition 3.2 may now be obtained. Proposition 3.3. lf taxation policy is used to optimise the number of domestic priucts produced (SO ti = tl), then free trade increases social welfare. Pro05
!Setting
[from eq. (13)] and tj = 0, condition (26) becomes xjvJxj)
1-->o.
vjfx j)
Since vl(xj) passes through the origin and is strictly concave, this condition holds. From the previous section, setting ti= tT is not generally sufficient to move the economy to a first-best optimum. However, from proposition 3.1, trade affects .the number of commodities produced, not the quantities in which each commodity is produced. The use of tax policy to optimally control the number of commodities produced is therefore sufficient to ensure that there are positive gains from free trade, although there remain imperfections in the domestic economy. The two propositions above provide simple conditions which characterise the welfare effects of the import ot*monopolistically competitive commodities. It is worth noting that these conditions-are independent of the costs to the economy of imported commodities, of the number of commodities imported, and of the number of domestic commodities forced out of production by trade. The welfare changes are due entirely to changes in consumer surplus associated with the change in product mix. The propositions are of course dependent upon the assumption that some commodities from the set S are produced locally in both the equilibrium with, and the equilibrium without, trade. It is possible that the group of imported commodi tics be sufficiently large to make any local production of commodities from the set S unprofitable. In this case 4 will be lower in the equilibrium with trade than in the autarky equilibrium. Negativity of the left-hand side of (25) is then a necessary, but no longer a sufficient, condition for the open economy to have a lower level of social welfare than the closed economy.
A.J. Venables, Optimal tarifi for trade
235
d Optimal import trriffs It is clear from the above analysis that the tariff rate is of importance in determining the welfare effects of trade. This section derives optimal tariffs for a country importing commodities from the set S, and exporting the numeraire commodity. In order to abstract from the possibility that tariffs may be used to change the terms of trade, it will be assumed that the country under consideration is a price-taker in international trade. 4 There are nevertheless two distinct reasons in this model why non-zero tariffs should be desirable. The first is that since the domestic economy is ‘imperfect’, tariffs may be used as a second-best policy instrument to correct domestic imperfections. The use of tariffs for this purpose depends of course on how other policy instruments have been used. In this model the use of alternative instruments to correct domestic imperfections is not a suffjcient condition for unrestricted free trade to be optimal. If the number of commodities imported to the economy can be influenced by the tariff rate, then this provides an additional argument for non-zero tariffs. The optimal tariff problem is to find a tariff t; which maximises social welfare subject to the allocation being a market equilibrium, and for given domestic taxes ti. Since the domestic supply price of imports, cj, is fixed. controlling tj is equivalent to controlling pi, the market price of imports. The problem will be analysed using pj as the control variable. As pi is varied both the number of commodities and the quantity of each commodity imported will generally change. The change in the quantity of each commodity imported is determined by the commodity’s demand function [eq. (1811. As quantities change, so do the revenues earned by foreign firms supplying the imports. If these firms face increasing returns to scale in either the production or export of their output, the number of commodities that they supply will in general depend on the quantities in which they are sold. Changes in xj then change the number of commodities imported. The response of the number of commodities imported, rr”,,to -Xjis exogenous to the importing country. Invoking the ‘large group’ assumption, the response of the number of commodities imported to changes in xj will bc summarised by an elasticity of the number of commodities imported with respect to the quantity of each imported, denoted q, and defined as (27) ‘This a~umption may be justified in two ways. First, if importers’marginalcost and marginal revenue functions are such that, over the appropriate range, the profit-maximising price is independent of quantity supplied (e.g. constant marginal cost, iso-elastic demand curves), then the supply price is constant. Secondly, if the country is small, and importers cannot price discriminatebetween countries, then the country under considerationis a price-taker.
AJ. Vmbks,
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tariffsfor trade
Optimal
Proposition 3.1 provides information on the response of domestic production to changes in the tariff. From this proposition we know that Xi and pi* the prices and quantities of active firms, are unchanged by changes in pk The number of commodities produced domestically, n, will in general change as pi changes but, from proposition 3.1, we know that n must change in such a manner as to hold 4 constant, i.e. to hold the economy on the same Engel curve. Since along an Engel curve U(xO,v) is strictly increasing in both x0 and V, the optimal tariff problem reduces to the problem of maximising x0, subject to holding the economy on the same Engel curve. Formally, pi must be chosen to maximise ‘0 =
A4- n(CXXi) + Fi) - ?tlC jX
(28)
j
subject to n”Xxi)+ mtll(xi)=dM -
tiCi
+ FJ -
?WjXj),
(29
where n, m, and xj are allowed to vary.
Differentiating along the objective gives (30) Differentiating along the constraint @ves
v_ix jP?-
g”Ci(l +q)+V;(Xj)+F
X
[
l
I
-I
(31)
Using (30) in (31) to eliminate an/8pl gives
(32) For an optimum we require dxo/dpi =0, i.e.
(33)
AJ. Venables, Optimal tari#k for trade
237
Using the import demand equation C*=q*(l J
(18)
-tj)VxXj)
and the zero profit condition for domestic producers, CXXi)+
Fi=q *( 1- ti)XiUi(Xi)
(9)
gives the following first-order condition for the optimal tariff t?: 1-q
=l-ti
WXx3
l +tt
udxil
(
I+~. I[
Xju;(Xj) 1’ UJ4X.j)
(34)
It can be shown that second-order conditions for the problem are satisfied, so that eq. (34) defines a welfare maximum. This gives the following proposition. Proposition 4.1. Ifn >Q, the Dptimal import tar@ (expressed as a proportion of market price), t,* is given by (34)
Notice that since U~xi>lxiu;cxi>>1, we have XjVxAj)
(l- ti*)*--<(l
-ti)*-
r+J
XiVKxi) vXxi)
so that, from the analysis of section 3, imports which are subject to an optimal tariff never reduce social welfare. This is of course as it should be. The characteristics of the optimal tariff may now be examined. We shall look first at the relationship between the optimal tariff and the use of ti to control domestic production. When q = 0, we have the following proposition. Proposition 4.2. If q=O, and domestic production is subsidised such that the optimal number of domestic commodities are produced, i.e. ti= t?, then the optimal tarifl is zero. [f ti > tr, then tJ > 0. Proof:
Setting q = 0 in (34) gives t,*=l-(1.-rJ-
xi”;(xi) 4h)
(35)
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238
From eq. (13) (36) Use of (36) in (35) proves the proposition. Proposition 4.2 establishes that, for any economy which cannot influence the number of commodities it imports (q=O), optimal control of the number of domestic commodities produced is sufficient to ensure the optimality of free trade. There remain imperfections in the domestic economy, since the quamity in which each commodity is produced is not optimal. But, from proposition 3.1, trade does not affect these quantities, so this imperfection provides no argument for non-zero tariffs. If, however, domestic production is not subsidised to the level t?, then positive tariffs should be employed to increase the number of domestically produced commodities. We now turn to the case where q >O. Since tariffs now affect the number of imported commodities, non-zero tariffs are required, even though domestic tax instruments have been optimally used to correct domestic imperfections. Proposition 4.3. If q >O, and domestic taxes have been set to optimise the number of domestically produced commodities (that is, ti= tr), then the optimal tarifl is negative. Prm$
Using (1
_t+$+ i
1 i
[eq. (13)] in eq. (34) gives l-t+-[1+9(-$&)]*
Since
(37)
V,(Xj)/XjV;{Xj)>l, 5%
The intuition behind this is as follows . Since n is optimally controlled, a marginal change in n is worth nothing. Imports do, however, give positive consumer surplus, so that a negative tariff is desirable in order to increase the number of imported commodities, and thereby increase social welfare. An additional important feature of the problem may be pointed out at this stage. If tariffs are the only instrument of trade policy available, and policy-makers are constrained to set the same tariff rate for all symmetric imports, then policy-Lmakers only have one instrument for the control of trade. They are,
Al.
Venabks, Optimal [email protected] trade
239
however, interested in two variables: the number of commodities imrzrted and the quantity in which each commodity is imported. Negative tariffs increase the number of commodities imported, so raising welfare (if the number of domestically produced commodities has been optimised), but also mean that imports are sold at prices below their marginal cost to the economy. The optimal tariff [eq. (37)] is determined as the outcome of these two forces. ’’ Some observations may now be made about the optimum tariff structure when domestic taxes have not been optimally set, so tariffs are being used as a second-best policy instrument to control the domestic economy. The following proposition provides conditions under which the tax/optimal tariff structure involves higher taxation of imports than of domestically produced commodities. Proposition 4.4.
(i)
Xju;(Xj)
-
vI(xj)
tj* > ti i$ ,
JriuxXi). vitxi)
’
Or (ii)
xj”;(xj) --=p
xiO~xi)
vjcxi,
f-+&i)
a*
vl <
oo, ,
0t
(iii) Proqf:
q=O.
By inspection of eq. (34), noting that
Part (i) of proposition 4.4 is as would be expected from the analysis of section 3 of the paper. If the ratio of surplus to cost is higher for domestically produced commodities than for imports, then imports should certainly be taxed more heavily than domestic commodities. Part (ii) is stronger: it establishes that even if imports and domestically produced commodities have the same ratio of surplus to cost (they may be symmetric to each other), then imports should be taxed more heavily than domestically produced commodities, providing q is bounded. The intuition behind this is as follows. Suppose imports and domestically produced goods are symmetric (i.e. have the same sub-utility functions u), and ti= tjs A small increase in tj has two effects. First, it may change the number of commodities produced and imported, but since condition (26) of section 3 holds with equality, this leaves welfare unchanged. Secondly, the increase in tj makes imports more valuable to the economy, by rais;ng their marginal valuation to consumers,
240
AJ. Vmables, Optimul
twi$sjiwtrade
and hence increasing the ratio of surplus to cost. Providing q is bounded, this effect may be exploited by taxing imports more heavily than domestically produced goods, even though the products are symmetric. Part (iii) of proposition 4.4 establishes that if the number of imported commodities is given, imports should always be taxed more heavily than domestic products. This is because the total number of commodities supplied to the economy can be increased by taxing imports (so increasing the number of commodities produced domestically), and additional commodities bring positive consumer surplus.
This paper has analysed the welfare effects of importing commodities which compete directly with a local monopolistically competitive industry. Assumptions of separability of utility functions, and of symmetry of groups of commodities have permitted a number of results concerning the welfare economics of such trade, and optimal tariff policies towards such trade, to be obtained. In section 3 of the paper it was established that necessary and sufficient conditions for trade to increase welfare were that the ratio of consumer surplus to cost on imported commodities be greater than that on locally produced commodities. For the case of iso-elastic demand functions, this reduces to the condition that imports increase welfare if and only if they are less elastically demanded than locally produced commodities. Section 4 of the paper derived an optimal tariff for imported commodities. It was established that if the number o? commodities imported was independent of the tariff rate, imports bhould UCsubjected to a positive tariff unless domestic production was subsidised to the point at which the number of commodities produced domestically was optimised. If, however, tariff rates affkct the number of commodities imported, then subsidisation of domestic production to optimise the number of commodities produced is not a sufficient condition for free trade to be desirable. Import subsidies should be employed in this case to exploit consumer surplus on imported commodities. It was also established that even if imported commodities were symmetric with locally produced commodities, an asymmetric tax treatment - taxing imports more highly than domestically produced commodities - was generally desirable.
AC
I &ouId like to thank P. Hammond, G. Heal, J. Stiglitz, a referee, and participantsin 8cmiMrsd the universities of Essex and !~ISSCX for useful comments on an earlier version of this paptr.
Al. Venables, Optimal tariffs for trade
Refwencea Dixit, AK. and V. Nosmran, 1980, Theory of international trade (Cambridge University Press). Dixit, A.K. and LE. Stiglitx, 1977, Monopolistic competition and optimum product diversity, American Economic Review, June, 297-308. Krugman, P.R., 1979, Increasing returns, monopolistic competition, and international trade, Journal of International Economics 9, -79. Lancaster, K., 19g0, Intra-industry trade under perfect monopolistic competition, Journal of International Economics 10,151. Roberts, J. and H. Sonnenschcin, 1977, On the foundations of the theory of monopolistic competition, Econometrica 45,101-l 13. Spence, A.M., 1976, Product selection, lixed costs and monopolistic competition, Review of Economic Studies 43,217-235. Venables, AJ., 1980, Monopolistic competiticn and the possible losses from international trade, University of Suss;ex Discussion Paper.