metadata: authors: 0: author: Anderson,James E.
docna: 0: docna: Tariff index theory
docty: Policy Research Working Paper
subtopic: Trade Policy,International Trade and Trade Rules,Rules of Origin,Macroeconomic Management,Trade and Multilateral Issues
teratopic: International Economics and Trade
abstracts: cdata!: For a single tariff, the height of the
            tariff is an unambiguous measure of the policy's
            restrictiveness. With more than one tariff, theory has not
            provided an extension that captures the idea of the
            tariff's height, so analysts have used index numbers
            such as the mean and the coefficient of variation (standard
            deviation divided by the mean) of tariffs. By contrast, the
            theoretical literature on the piecemeal reform of tariffs
            shows that efficiency gains from tariff reform depend on
            complex conditions that have little relation to the mean or
            variance of tariffs. But in the absence of a connection
            between theory and empirical measures, it is difficult to
            know whether to discard the measures. Moreover, the
            piecemeal reform question of measuring the welfare gain from
            a tariff is not directly related to the problem of
            evaluating the height of restrictiveness. The problem of
            finding a single number analogous to the height of tariffs
            is the tariff index number problem. The authors have
            developed a solution: the Trade Restrictiveness Index, which
            they define as the uniform tariff factor that is equivalent
            in trade restrictiveness (equivalent in the balance of
            trade) to the actual differentiated tariff structure. Here,
            the author develops the Trade Restrictiveness Index in terms
            of mean and variance-covariance indexes of the tariff
            schedule. There are two payoffs. First, the Trade
            Restrictiveness Index can be decomposed into expressions
            that rescue the commonsense idea that lower mean and lower
            variance of tariffs are both efficient. Second, a special
            case is offered in which the proper weights in the mean and
            variance of tariffs are the observed trade weights. Thus,
            the Trade Restrictiveness Index is superior to traditional
            summary measures such as the average tariff rate and the
            coefficient variation for the tariff schedule. It requires
            only limited additional information on the structure of the
            economy to yield a measure that is preferable on both
            theoretical and practical grounds.
repnme: repnme: Tariff index theory
display_title: Tariff index theory
listing_relative_url: /research/1992/11/699514/tariff-index-theory
topicv3: Trade m1326924 698,Balance of Payments m1326965 441,Trade Policy and Integration m1327407 775,Trade Policy m1327640 776,Trade Balance m1327679 1136
publishtoextweb_dt: 2010-07-01T00:00:00Z
historic_topic: International Economics and Trade
origu: UNKNOWN
no_of_pages: 38
author_keys: 0: author_key: 797593
model_extractions: <!-- Start of picture text --> Policy Research|<br>International Trade<br><!-- End of picture text -->

**WPS 1023**

- This paper- a product of the Intemational Trade Division, International Economics Department is part of a larger effort in the department to analyze the effects of trade distortions on developing countries. The study was funded by the Bank's Research Support Budget under research project "Cost of Protection Index" (RPO 676-49). Copies of the paper are available free from the World Bank, 1818 H Street NW, Washington, DC 20433. Please contact Maria Teresa Sanchez, ioom S7-025, extension 33731 (November 1992, 32 pages).

For a single tariff, the "height" of the tariff is an unambiguous measure of the policy's restrictiveness. With more than one tariff, theory has not provided an extension that captures the idea of the tariff's height, so analysts have used index numbers such as the mean and the coefficient of variation (standard deviadon divided by the mean) of tariffs.

By contrast, the theoretical literature on the "piecemeal reform" of tariffs shows that ef.iciency gains from tariff reforn depend on complex conditions that have little relation to the mean or variance of tariffs. But in the absence of a connection between theory and empirical measures, it is difficult to know whether to discard tLhe measures. Moreover, the piecemeal reform question of measuring the welfare gain from a tariff reform is not directly related to the problem of evaluating the height of restrictiveness.

The problem of finding a single number analogous to the "height" of tariffs is the tariff index **iiumber** problem. Anderson and Neary have developed a solution: the Trade Restrictiveness Index, which they define as the _uniform_ tariff factor that is equivalent in trade rstrictiveness (equivalent in the balance of trade) to the actual differentiated tariff structure.

Here, Anderson develops the Trade Restrictiveness Index in terns of mean and variancecovariance indexes of the tariff schedule. There are two payoffs. First, the Trade Restrictiveness Index can be decomposed into expressions that rescue the commonsense idea that lower mean and lower variance of tariffs are both efficient. Second, a special case is offered in which the proper weights in the mean and variance of tariffs are the observed trade weights.

Thus, the Trade Restrictiveness Index is superior to traditional summary measures such as the average tariff rate and the coefficient variaton for the tariff schedule. It requires only limited additional information on the structure of the economy to yield a measure that is preferable on both theoretical and practical grounds.

**'MThe Policy Research** Working **PaperoSeriesdissemiinates the** ridinp **of work** unda way in **therBank. Anobectiveof the** series **is to get these findings out quickly, even if presentations are less than fully polished. The** Fmdings, **interpretations, and conclusions** in **these papers do not necessarily represent of ficial Bank policy.**

**Produced by the Policy Research** Dissetnination Center
# **Tariff Index Theory**

James **E.** Anderson Boston College
# Table of Contents

|**I.**Index Numbers for**Tariffs ...........................................**<br>**4**|
|---|
|1. 1 The Consumer Price Index and the Average Tariff**.......................................4**|
|I.2 The Trade Restrictiveness Index ...........................................<br>6|
|The Balance cf Payments Function ......................<br>6...|
|Derivation of die Trade Restrictiveness Index ........................................8|
|1.3 A Diagrammatic Exposition ..........................................<br>10|
|II.**The TRI and the Standard Indices ..........................................**<br>13|
|The TRI in Terns of Tariff Moments .............................<br>............. 14|
|The TRI and Piecemeal Reform Theorems .......................................... 16|
|tII.The Cobb-Douglas Example**..........................................**<br>19|
|IV. Lessons**..........................................**<br>26|
|**Appendix 1. Proof of Proposition 3 ...........................................**<br>28|
|**Appendix 2. Cobb-Douglas Simulations ..........................................**<br>29|
|References**..........................................**<br>30|

|6....................6|
|---|
# **Tariff Index Theory**

## James E. Anderson

Tariff index numbers are widely used for policy evaluation, despite the deficiencies which stem from their lack of theoretical roots. The restrictiveness of tariff structures is routinely compared over space and time with import-weighted mean tariffs. Another practically important tariff index is the import-weighted coefficient of variation of tariffs (the square root of the variance divided by the mean). The World Bank operations staff use the coefficient of variation along with the mean tariff to measure progress in trade liberalization as a condition for structural adjustment loans.

For a single tariff, the height of the tariff is an unambiguous measure of the restrictiveness of policy. With more than one tariff, theory has not provided an extension capturing the idea of the 'height' of the tariff, hence analysts have used the mean and coefficient of variation. For uniform tariffs, a higher mean tariff is trivially more restrictive, which is the intuitive sense behind the import weighted mean tariff index. But with a differentiated tariff structure, high tariffs will get low weight and low tariffs will get high weight, due to the substitution effect. This seems to be wrong. Concerning the variance, on the one hand, the reasoning behind the World Bank's promotion of lower dispersion is that uniform ad valorem tariffs do not distort relative prices among tariffridden goods. This appears to suggest that if two tariff schedules have equal importweighted mean tariffs, hence in some sense equal distortion relative to other goods, the schedule having less variance is more efficient. On the other hand, the theoretical literature on piecemeal reform of tariffs (Fukushima (1979), Hatta (1977), and BertrandVanek (1971)) has very stringent sufficient conditions for efficient reform which warn that this is a dangerous procedure. Because the condidons are over-sufficient, and not lSee for example any undergradua **te** xt in intemadonal economics.
Tariff Index Theory J.E. Anderson 2 directly related to tariff moments, they provide little practical guidance. That is :he subject of this paper.

The problem of finding a single number analogous to the 'height' of tariffs is the tariff index number problem. Anderson and Neary (1991) develop a solution, the trade restrictiveness index. It is defined to be the _unifonn_ tariff factor which is equivalert in trade restrictiveness (equivalent in the balance of trade) to the actual differentiated tariff structure. In rates of change it is equal to the familiar marginal dead-weight loss of changes in tariffs normalized by the 'shadow value of distorted trade'. The normalization is based on the distance function formalization of index number theory. In contrast, the standard assessment of the welfare effect of tariff changes uses other normalizations, such as national income. For the purpose of comparing changes in the restrictiveness of trade policy internationally, normalizing by national income has the fatal defect that 'naural' openness and economy size interact with the 'height' of the tariff.<sup>2</sup> This paper develops the trade restrictiveness index in terms of mean and variancecovariance indices of the tariff schedule. There are two payoffs. First, the trade restrictiveness index can be decomposed into expressions which rescue the common sense idea that lower mean and lower variance of tariffs are both efficient. Second, a special case is offered in which the proper weights in the mean and variance of tariffs are the observed trade weights.

Section I reviews the theory of the trade restrictiveness index developed in more detail in Anderson and Neary (1991). Its interpretation and relation to the standard measure of efficiency are set out. Section II develops rate of change of the trade 2Consider the uniform tariff case, where the height of the tariff is unambiguous. The welfare measure normalized by national income is the Harberger tiangle divided by national income, which works out to r2eq 1for the move to free trade, where T is the height of the ad valorem tariff (on the domestic price base), pZ/I is the ratio of the value of tariff-ridden imports to national income, and e is the elasticity of composite import demand. Natural openness and economy size affect both pZ/I and e , hence make this index useless for the purpose of comparing the height of the tariff across nations . For intertemporal comparisons over short time periods, normalizing by income is less objectionable.
Tariff Index Theory J.E. Anderson 3 restrictiveness index in terms of changes in the mean and 'generalized variancecovariance' functions of the tariff schedule. Efficiency is indeed decreasing in both of these 'moments'. The coefficient of variation index used by the World Bank varies monotonically with the trade restricdveness index when: (i) the only form of distortion is tariffs, (ii) the two indices are used to compare a given tariff schedule with the uniform mean-equivalent tariff schedule, (iii) the import-weights in the index are "marginal" import weights, and (iv) the variance concept used in the coefficient oi variation is a 'generalized variance'.

Section III shows that when the trade preferences for imported final goods are CES, the ordinary variance concept may be used in place of the generalized variance, and marginal trade weights may be replaced with the ordinary trade weights. Thus mean preserving eliminations of trade weighted variance in tariffs are efficiency-increasing. The rate of change of the trade restrictiveness index in the Cobb-Douglas case is generally a function of the trade weighted first and second moments of the tariff schedule, although it is not the coefficient of variation function. This form of the trade restrctiveness index highlights an intuitiv sense in which the trade weighted mean and variance of tariffs were both 'partially' legitimate. Finally, a Cobb-Douglas example is offered in which variance reductions are efficiency improving subject to the constraint that the trade-weighted mean tariff be the same. That is, comparisons can be made in which variance need not be eliminated. However, even in the Cobb-Douglas model, when the mean and variance both change there are always welfare-improving tariff reforms which coincide with a rise in the import weighted coefficient of variation.

Sectior IV concludes with some suggestions for future practice. Future work (underway at the World Bank as part of the project on trade policy evaluation) will reveal whether the refinement of the trade restrictiveness index yields different results in practice. However, in related work Anderson and Bannister (1992) show that the trade
Tariff Index Theory 4

# J.E. Anderson

restrictiveness index for Mexican agrici<sup>1</sup> 'jre yields results very different from the standard producer subsidy and consumer subsidy equivalent indices.
Tariff Index Theory J.E. Anderson _5_

# **1. Index Numbers for Ta.iffs**

Changes in index numbers are generally weighted averages of changes in the components of the index. To be consistent with economic theory, the weights must arise ftom a fundamental economic structure. It is helpful to begin with reviewing the consumer price index, or CPI where the weights are familiar. Subsection 1. 1 derives the consumer price index based on the consumer's expenditure function and relates it to the average tariff. In Subsection 1.2, the trade restrictiveness index, or TRI, is derived based on the econ3my's trade balance function. The latter is defined in Anderson and Neary. Here, the weights are less familiar, but the same logic girds the construction of the index. Subsection 1.3 relates this welfare-consistent index to the trade-weighted mean and variance of the tariff structure.

## 1.1 The Consumer Price Index and the Average Tariff

The basis for the CPI is the consumer's expenditure function, e(q,u), where q is the vector of prices and u is the reference level of utility. e is the minimum level of income required to achieve u when the consumer faces q. The derivative of e with respect to q is equal to the vector of the consumees demands, X, owing to the minimum value property of e(.).

The vector of consumer prices changes by dq. The consumer price index (CPI) in rates of change measures the (hypothetical) _uniform_ rate of change in all prices which produces an equivalent change in the expenditure required to maintain welfare. The effect of an arbitrary set of price changes on the level of income required to support u is X'dq. The effect of a _uniform proportional_ set of changes is X'qcda, where a is the (scalar) proportionality factor (q = aqO, where q° is the initial level of prices). Solving for the uniform proportional change in q which creates the same rise in required expenditure as the arbitrary change in q implies dca = Xdq/X'q, or (1.1) da = £X3q)qi.
6 J.E. Anderson Tariff Index Theory Initially a is equal to one, so dt is a percentage change. This familiar expression weights the proportionate change (denoted by a **A)** in each qi by the consumption share of good .

One final step is required to link the development of the CPI fimly to the trade restrictiveness index developed below. Suppose that the means of compensation is through (hypothetical) price reductions rather than through (hypothetical) income transfers. The compensating change in expenditure is replaced by a (hypothetical) uniform proportionate change in prices which compensates for (i.e., offsets) the actual change in prices dq. This procedure defines a version of (1.1) which has opposite sign: (1 1') da' = X(tX)qi) The opposite sign denotes the direction of the compensatory change in hypothetical prices. This latter form is the one used in the trade restrictiveness index.

The import-weighted average tariff index is based on the false analogy of the small trading country with the price taking consumer model which is the basis of (1.1), with X being the import quantity. A set of tariffs raises expenditure of the consumer relative to free trade, and the index computes the uniform tariff which raises expenditure by an equivalent amount. Thus the proportionate change in qj is set equal to the ad valoreM tariff rate and the value of trade in each category is used as the weight: (1.2) = The mean tariff defined in (1.2) appears to have the index number problem that the' current level of imports reflects the tariff-ridden level of prices. An appropriate adjustment might seem to involve using the struc=e of substitution effects to obtain the 'right' weights.

It is well-known, however, that the economic structure of a tariff-ridden small trading economy is not equivalent to that of the price-taking consumer. The external budget constraint links consumer expenditure requirements to the level of production income and the government revenue. A change in the tariff vector causes a change in the
7 J.E. Anderson Tariff Index Theory expenditure required to support u equal to X'dq. It also causes a change in government tariff revenue equal to Xdq + (q-p*)'dX, where p* is the extemal price vector. Assuming redistribudon, the net effect is equal to (q-p*)'dX, the marginal dead weight loss. This suggest that the weights in a properly based tariff index must somehow be linked to marginal dead weight loss. The trade restrictiveness index formaliz&. this insight. 1.2 The Trade Restrictiveness Index Depending on the analyst's objective, there am a number of possible ways to define an index of tariffs. Govempent bureaus and international agencies which use tariff indices for comparison are interested in potential welfare, so welfare-equivalent indices are useful. Mercantilists are interested in the trade balance, so trade-balance equivalent indices are useful. The trade restrictiveness index is the gwifon tariff which is equivalent in its (utility-constant) trade balance impact to the differentiated tariff structure. Thus the trade restrictiveness index is both equivalent in welfare and in the trade balance.

The first step of this section is to develop a counterpart to the expenditure function in the general equilibrium context of a small tariff-ridden trading economy. Based on it, the trade restrictiveness index may be defined. The Balance of Payments Function The _balance of payments fuwdon_ gives the net foreign exchange required to maintain the utility of a representative consumer facing given levels of tariffs. The balance of payments funcdon is built up from the consumer's expenditure function and the gross domestic product funcdon.

The gross domestic product function g(p,v) is the maximum national vs-hie added (output less imported inputs) when the economy faces a vector p of prices of final output or imported inputs and a vector v of primary factor supplies. The vectors p and v are assumed to be fixed, the former by the small country assumption combined with a tax policy; the latter by institutional constraints on employment. The derivative of g with respect to p, gp (in what follows, a subscript denotes differentiation save for the index i or
8 J.E. Anderson Tariff Index Theory j), is equal to the output supply or input demand vector, due to the maximum value property.

The trade balance is equal under the balance of payments constraint to the net ex ernal borrowing of the economy. This means that the value of tradoe in terms of external prices q* and p* is equal to the value of expenditure evaluated at prices q, less national product evaluated at prices p, less (q-q*)'eq plus (p-p*)'gp: q*'eq - p*'gp = qeq - p'gp - (q-q*)'eq + (p-p*)'gp. Government revenue is equal to the final consumption tax revenue (q-q*)eq plus the net revenue from taxation of final output or imported inputs, - (p-p*)'gp. In balance of payments equilibrium, the trade balance is equal to the amount of external borrowing, I8. The balance of payments function is equal to (1.3) B(qp,u;q*,p*,v,) = e(q,u) - g(p,v) -,

- (q-q*)'eq(q,u) + (p-p*)'gp(p,v).

B(.) gives the net foreign exchange required to support the initial level of utility u when the vector of final consumption taxes is equal to qaq*, the vector of output or input taxes is equal to -(p-p*), external prices are equal to (q*,p*), and the net external borrowing is equal to P. The standard case of international trade theory is when imported inputs are not taxed and when domestic output is a perfect substitute for imports. Then consumers and producers face the same price vector p, and an import tariff (p-p*) is equivalent to a tax on consumption plus a subsidy on production at the same rate. The balance of payments function may be reduced to: (1.3') B(p,u;p*,v,) = e(p,u) - g(p,v)-,B

- - (p-p*)'(ep(p,u) gp(pv)).

The last term on the right hand side of (1.3) is equal to the tariff revenue, since Z, the import vector, is equal to ep-gp (negative for exports).
riff Index Theory J.E. Anderson 9 The marginal dead weight loss from a chiange in the tariff vector is equal to minus change in the foreign exchange required to support u, or -Bp'dp. f4ere, the marginal t of tariffs -Bp, may be obtained from differentiating (1.3') as:

-B' = (p-p*)'(epp - gpp) , (p,-p*)'Zp. )taining (1.4) the envelope theorem is used along with cancellation of like terms. nbining (1.4) with the vector of tariff changes, the marginal dead weight loss from the ff change is equal to

-Bp'dp (p-p*)'(epp - gpp)'dp

(p-p*)'Zpdp = (p-p*)'dZ, dZ is the change in the import demand vector. See Anderson and Neary (1991) for *e on the properties of B(.). erivation of the Trade Restrictiveness Index The utility-constant change in the foreign exchange requirement (the trade deficit isured in terms of foreign prices) due to a change in p is equal to Bp'dp. All vectors vhich result in equal changes in Bp'dp are termed equivalent in trade restrictiveness. ,'dp is also a compensating variation measure of welfare change. The trade rictiveness index is based on first noting that the _uniform proportional_ change in p offsets the change in B is -Bp'pOdA. Solving for the uniform proportional change which is equivalent in trade restrictiveness to the arbitrary change dp, dA = B(P-) A.

W the similarity of structure between (1.1') for the consumer price index and (1.5) for ariff index. Index number construction has two steps. The first is to uncover the xct marginal real income effect (Xi in the case of the consumer problem, Bpi in the of the trading economy). The second step is to find the appropriate nornalization of of marginal changes (q'X in the consumer problem and Bpp in the trading omy). The normalization Bp'p in (1.5) is called 'the shadow value of distorted trade'
Tariff Index Theory J.E. Anderson 10 by Anderson and Neary (1991). It is equal to the net foreign exchange required to compensate the representadve consumer for a 1% uniform proportionate rise in all distorted prices.

A more general definition is used in index theory to define an index even when the values of pl are far from the inidal point p°. ThUS:3 (1.6) A(pl,u,<sup>0</sup> 0) (l B(plA,uO) =. 0) Here, f0 is the initial foreign exchange requirement (trade deficit), and the constraint requires A to change as p changes so as to maintain a constant trade deficit at the given u°. The most intuitive interpretation arises when the new value of p is equal to p*. Then A is equal to one plus the uniform ad valorem tariff rate which is equivalent in restrictiveness to the initial tariff structure. Elsewhere, A is equal to the uniform tariff factor _surcharge_ which compensates for (offsets the change in trade restricdtiveness implied by) the move to a new tariff structure. Note that under definition (1.6), the compensadng change in foreign exchange induced by A is just sufficient to maintain u<sup>0</sup> when prices shift to pl. Thus A is a compensating variation measure of the welfare effect of the change, and A is interpreted as the uniform tariff surcharge which is equivalent in welfare to the change to the new prices.

The rate of change of _A_ is obtained from implicit differentiation of (1.6):

> dA X **B** PP) A

The structure of the TRI combines the standard welfare economics of the marginal dead weight loss due to a tax change, which is the numerator of (1.5'), with a normalization (division) by Bp'p, the 'shadow value of distorted trade'. In contrast to other normalizations which have little or no theoretical foundation, scaling by the shadow value of distorted trade leads to a well-founded and intuitive index.

3 The analog for theCPI is: a(q,u,) = (a l e(qI u) = OJ.
11 J.E. Anderon Tariff Index Theory Equation _(1.5')_ is made operetional using the partial derivatives Bpj in (1.4) and informadon on substitudon effects. The elements of Bp are expected to be posidve, but cross effects can make some elements negadve. This index can be integrated to obtain levels of A, using an inidal level of A equal to one to tie down the constant of integration. Alternatively, a computable general equilibrium model c n solve A in (1.6).

A.
J.E. Anderson 12 Tariff Index Theory

# 1.3 A Diagrammatic Exposition

The balance of payments function and the trade restrictiveness index are both defined above in ternis of the domesdc price vector p. The instruments of protection are tariffs, however, and A is interpreted as a uniform tariff factor surcharge. To build intuition it is helpful to rewrite the balance of payments and trade restrictiveness index functions in terms of the tariff factor vector T such that pi = P*iTi. Based on this, (1.7) BT = Bpdiag(p*) = (T-l)'diag(p*)Zpdiag(p*), where diag(p*) is a diagonal matrix with the elements of the vector p* on the principal diagonal, and I is the vector of ones. Then: BT'dT = Bp'dp.

Figure 1 plots balance of payments contours in tariff factor space (TI,T2) for a level of utility u. The level of the trade balance is constant along the contours, and if u° is an equilibrium utility associated with a tariff setting on the contour, the balance of trade value B must be zero. For a given u, B, the foreign exchange requirement of consumption, increases as tariffs rise. The curves are drawn as convex, and will ordinarily but not necessarily be so.

Suppose that the initial equilibrium setting of tariffs is at A with equilibrium udlity u°. The purpose of any compensating variation<sup>4</sup> index number of tariffs is to consistendy map some alternative setting of the tariffs such as point B into a tariff setting which supports the base level of utility u°. Consistency means that the index should be monotonically related to efficiency for all tariff comparisons.

41t is also possible to defne equivalent variation indices. The present development focusses on compensating variation measures since they are more practicaL The equivalent variation concept is illusted with a plot of isoudlity contours in tariff factor space. Set B equal to zero and solve (1.3') for the equilibrium level of utility. The isoutility contours will look like trade balance contours of Figure **1, with** lower tariff setings required to achieve higher levels of utility.
Tariff Index Theory J.E. Anderson 13 Figure 1. The Trade Restrictiveness Index <!-- Start of picture text --> T 2<br>.450<br>/ , / ~~~~~~~~B(,ut<br><!-- End of picture text --> Trhe curve labelled t illustrates the locus of tariff factors along which the imported-weighted average tariff remains constant. Its shape depends on the substitution properties of the economy, but it is likely to be downward sloping. The trade restrictiveness index associated with the move from A to B is equal to OC/OB. The interpretation is that the tariff reform to point B reduces the foreign exchange required to support u°. The compensating uniform rise in tariff factors is equal to OC/OB. In contrast, the mean tariff index **Xt** registers a rise in protection in the move from A to B, and is thus an inconsistent index. In either its level form or its rate of change form, A is a tariff index which consistently ranks values of T relative to u. If the transitivity
Tariff Index Theory J.E. Anderson 14 qualification of compensating varation measures is mets, A ranks values of T relative to each other.

Now consider the application of these concepts to the evaluation of a proposed tariff reform which preserves the trade-weighted mean tariff, such as point D at the intersection of the constant-mean tariff locus and the ray OC. Is it possible that the coefficient of variation of tariff rates will properly register an increase in efficiency? The coefficient of variation of tariff factors is equal to the square root of the variance divided by the mean. Isovariance contours are positively sloped and increasing in variance with distance from the _450_ line.<sup>6</sup> As drawn, the move firom A to D is a move toward the _450_ line and is associated with a reduction in dispersion, as mandated by the World Bank. However, point D could lie northeast rather than southwest of point C. Thus it appears that the dispersion index cannot safely be used even when the mean does not shift. The remainder of this paper is an attempt to be more specific.

5It is well-known that compensating variation measures need not yield transitive comparisons. A sufficient condition for transitivity is homothetic preferences.

> 6 For given weights wi, the parLial derivative of the variance with respect to tariff rate i is equal to

> 2wi(r, - I). The slope of the isovariance contour is equal to -w2(r2 -1)/wl(Cl - -i). One of the partial derivatives must be positive and the other negative; hence the slope is positive.
15 Tariff Index Theory

## J.E. Anderson

# 11. The TRI and the Standard Indices

The relation of the the trade restrictiveness index to the mean and coefficient of variation of tariffs indices has two separate elements. The first is the issue of the proper weights, and the second is the issue of the proper formula given the proper weights. In this section, the proper marginal trade weights are defined, based on the concept of the marginal cost of tariffs. The trade restrictiveness index is then expressed as a function of the mean and 'generalized variance' of the tariff schedule using these weights. Next, the trade restrictiveness index is used to restate the two main theorems on piecemeal tariff reform in terms of tariff 'moments'. Finally, assuming the use of the proper weights, and also the generalized variance concept in the coefficient of variation function, changes in the mean or coefficient of variation are related to change in the trade restrictiveness index.

First, define the usual tariff indices, the mean, the variance and the coefficient of variation of tariffs for some set of weights (wi)<sup>7</sup> . The mean ad valorem tariff is: (2. 1) M() = wi(Tt-l) = 1'diag(w)(T-1). i The variance of the tariff schedule is equal to 2 (2.2) V(T) = £wi(-MM) = (T-M(r)lydiag(w)(T-M(T)l). i The coefficient of variation of the tariff schedule is (2.3) W(T) = V(T1/2 Here, diag(w) is the diagonal matrix with wj as the ith diagonal element, and 1 is the vector of ones. The mean tariff and the coefficient of variation of tariffs are usually measured with weights wi equal to the observed relative trade weights, wj = pijfYpjZj.

7Normally, wi>O and Ewi=l are imposed. The first restriction is relaxed here.
Tariff Index Theory J.E. Anderson 16 Two tariff vectors are to be compared with the various index functions, the initial tariff factor vector being TO and the new tariff factor vector being Ti. The difference in tariff factors equals the difference in ad valorem tariffs. The TRI in Terms of Tariff Moments The relative change form of the trade restrictiveness index is given by (1.5'). It is va'id only for local changes. For discrete changes, using the intermediate value theorem and evaluating at T, an intermediate value of T. (1.5') becomes: (2.4) A = -£ B rI) Here, _(2.5)_ T = _k_ + (1-X)TI for 1Ž>ŽO.

The Divisia index approximation to (2.4), often used as a practical expedient, replaces the weights evaluated at T in (2.4) with the arithmetic average of the weights evaluated at TI and at _TO:_ the Divisia index.<sup>8</sup> Using (1.4) and (1.7) in (2.4), the numerator of (2.4) is equal to (2.6) YBTiCD(T1 ' - T?) = £(Ta - l 1)diag(p*)Zjpi(Cdiag(p*)(Ti - TO)The denominator of (2.42 is equal to: (2.7) £BTiMT = 1 -(Tj - 1)diag(p*)Zjpi(T)diag(p*)T i J Next, divide both the numerator and the denominator of A by the double sum of the elements p*jZjp;p*i. The resulting weights will be termed _marginal tMade weights._ Formally, the marginal trade weights (evaluated at the intermediate value T) are defined as: (2x8 zij P* j jpip*i ( "p*jzjpip*i i j The marginal trade weights are symmetric and add to one. Then A becomes:

> 8 The justification is that with a convex function A, the weights must be intermediate between the endpoint weights. Convex A is assumed here by imposing convexity on b(T,u) in T.
Tariff Index Theory JE. Anderson 17

# - (2.4') A = - (-1), -1 (Zot **Izij4(Tl** -'I)

The quadratic expressions in the numerator and denominator of (2.4') resemble the generalized variance expressions which arise in generalized least squares.<sup>9</sup> The mean M(T) is equal to 1'(zij)T = z.j'T, which is the standard form for a mean with the 'probabilities' wj being the column sums of (z;;). (Note, however, that while l'zj is equal to 1, not all elements of zj are necessarily positive.) The generalized variance V*(M) is equal to (T-M(Tl)z)'(zj)(T-M(T)l). In the trade restrictiveness index structure, the matrix (zij) is positive definite and plays the role of the positive definite matrix flQ in generalized least squares. The quadratic expression in the denominator of (2.4'), using the algebra of covariance, is equal to -(V*(T) + McI)<sup>2</sup> - M(T)). Similarly, the numerator expression is equal to

- - (Cov*(tTI TO) + M(bI)(M(<sup>1</sup> ) - M(TO))) + (M(TI) - M(TO))

= XV*(TO) -(1-X)V*C(l) - (2 - 1)Cov*(Tl,TO) + **(1** - M(T))(M(TI) - M(TO)).

Here, Cov* stands for the generalized covariance.<sup>10</sup> The second line follows from the algebra of covariance, using equation (2.5) for T.

Now A may be written in terms of the 'moments' of the 'distribution' of tariffs by collecting terms and substituting back into (2.4'). Thus: Proposition 1

- (2.9) XV*(T) (1-x)V*(Tl) (2X-l)Cov(T<sup>1</sup> ,) V*(I) - (1 - M(T))MCb V*(l) - (1 - M(C)McI

+ - ( M_b - (M(TI)-( ). V*() - (1 -M(l))M(T

> 9The standard linear regression model is Y = X, + e. If the variance-covariance matrix of e is equal to n, where Q is positive definite, then the generalized least squares estimator of , P*, is equal to _(X'f X)'X'n'_ Y. The matrix (WX) is a generalized variance-covariance matrix of X. (X'D'X)' is the variance-covaiance matrix of the estimator P, and so forth.

> lOCov*(T0,Tl = C<sup>0</sup> -McO)'zijl(T<sup>1</sup> -M(Tl)).
18 JE. Anderson Tariff Index Theory

# The TRI and Piecemeal EReform Theorems

Several intuitive propositions on tariff reforn can be stated using Proposition 1. Note from the first term on the right hand side of (2.9) that (i) mean-tariff-preserving generalized variance reducdon is efficiency improving provided the influence of the covariance termn is small. The influence of the covariance term goes to zero (and X is equal to 1/2) as tariff changes are small. Intuitively, mean-preserving variance reducdon implies cutting high tariffs and raising low tariffs. In the case of generalized variance, however, this implies cutdng highly weighted tariffs and raising lowly weighted tariffs. Formally, mean-preserving tariff reform is efficient if £ (Ti - M(T))zijdTj < 0, for each j, and v zijdT = 0. i&i Next, (ii) mean tariff reduction with constant variance is efficiency improving. Thus the rule Tl = TO - a for all tariffs i is efficient. That is, a constant ad valorem rate cut is efficient. For tariff -eforms of this type, the mean tariff is a valid index.

The three main piecemeal tariff reform theorems in the literature may be stated in terms of (2.9): Uniform radial cut dT = -aWT is efficient because it lowers both the mean and the generalized varianceI<sup>1</sup> , Concertina cut If all goods are net substitutes, cutting the highest tariff will reduce both the generalized variance and the mean, and<sup>12</sup> 1This theorem is mos **t** horoughly developed inHatta (1977a,b). 12Under the condition **fzi)** has negative off-diagon **a** l nd positive diagonal elements. Moreover, due to the homogeneity property of excess demand functions combined with substitutability of taxed with untaxed goods, trzij}>O. This means that Tjzjj> - £Tizij If tariffj is ctt, i*j (r-MMn)'tz.j)dTj mmTZiej The term in brackets is positive if Tj is the maximae ement **l** of the vector T, noting that (T-M)/T is increasing inT. QED. This proposition is due to Bertand and Vanekl(1971).
19 J.E. Anderson Tariff Index Theory Dispersion u dT = - (TO -M(I)1) is efficient because it lowers variance.<sup>13</sup> Thus both the marginal trade weighted mean and variance of the tariff structure have the influence on efficiency andcipated from simple intuition based on the trade weighted mean and variance. For large changes in tariffs, the influence of the covariance in (2.9) is always zero if ; is equal to 1/2. While this is a 'natural' value, the intermediate value theorem does not guarantee it for large changes unless import demand is linear.

Notice that the normalization term in the denominator of (2.9) is not simply the square of the mean tariff, as in the (square of) the coefficient of variation. The normalization of the trade balance effect of tariff changes in (2.9) is in fact the main contribudon of Anderson and Neary. Using it to form the rate of change of the TRI yields the proportional change in the uniform tariff factor surcharge which compensates for the tariff changes.

Finally, what can be said of the coefficient of variation index relative to (2.9)? If the mean is preserved and in additon the schedule T<sup>1</sup> is uniform, when the coefficient of variation is formed with the same marginal trade weights, the difference in the coefficient of variation is equal to V*(T0)1/2/(M(TO1-).

# Then:

# Corollary

_Let aUl indices be based on marginal trade weights. Then for the mean-preserving changes in the schedule of tariffs which eliminates variance, both 3 and the change in the coefficient of variation index conwecdy sign efficiency changes._ The germ of reasonable intuition behind analysts' use of the coefficient of variatdon is revealed in Proposition 1 and its corollary. The marginal dead weight loss of tariff changes is equal to (p-p*)'Zpdp, which is essentially quadratic in p. Thus 13This rather under-appreciated proposiion isis due to Fuushima (1979).
20 Tariff Index Theory J.E. Anderson something like a variance must be involved in correctly signing the efficiency change due to a tariff shift. But even with marginal trade weights, the (change in the) generalized variance alone is not a consistent measure. The trade restrictiveness index, which is consistent, is not monotonically related to the change in the coefficient of variation of tariffs, even when the latter uses generalized variance based on marginal trade weights. The change in the coefficient of variadon is equal to V*(Tl)l/<sup>2</sup> V*(TO)1/2 (2.10) W(Tl) - W(TO) = M(Tl) M(TOj The first problem is that the numerator of (2.9), which signs A, may differ in sign from V*(TI) 1/2V*(TO)1/<sup>2</sup> , which signs the change in W under a mean-preserving reduction in variation. Covariation ofT<sup>0</sup> and T<sup>1</sup> matters to A, as does the magnitude of ). However, in Section Im a Cobb-Douglas example is presented in which covariation and the size of X do not interfere with the validity of inference from variance reduction subject to a constant mean. The second problem is that when the mean shifts, A is shifted through both the numerator and denominator. In contrast, it acts through the denominator alone in W. No definite reladon can be established between A and the change in W given by (2.10).

The problem with the mean shift is worth emphasizing. The coefficient of variation normalizes the standard deviation of a random variable by the mean, which is a useful procedure in descriptive statistics. In the context of an index of trade policy, however, the proper normalization of the efficiency change, the numerator of (2.4), is the 'shadow value of distorted trade' (see Anderson and Neary (1991) for more discussion) which appears in the denominator of (2.4).
21 J.E. Anderson Tariff Index Theory

# llI.The Cobb-Douglas Example

Under Cobb-Douglas preferences for imported final goods, the trade restrictiveness index can be written as a function of the trade-weighted moments of the tariff schedule. Mean-preserving eliminations of variance are then efficiency improving. The sufficient condition can be relaxed to the CES case, but it does not appear possible to go beyond this: for example, strong separability is not sufficient. Subsection 11.1 gives the details. Subsection 111.2 further develops the Cobb-Douglas structure to show that all mean preserving reductions in tariff variance are efficiency improving.

Import preferences which are Cobb-Douglas or CES are very restrictive, of course, but perhaps not quite as restrictive as they appear at first. If the Armington assumption is made, then no inport is a perfect substitute for a domestically supplied good. The reasoning is that packaging, advertising, and safety design segment all national markets. Moreover, undistorted nontraded goods consumer prices are invariant to trade taxes if a constant returns technology has more traded goods than nontraded goods and factors. Then imports are based on the representative consumer's expenditure function. A In contrast, it is not possible to extend the Cobb-Douglas model to tax-ridden exports and imported inputs. A Cobb-Douglas gross domestic product function arises only under joint production of a very special kind. Moreover, Lopez and Panagariya (1992) show that for tariffs on imported inputs, complementarity necessarily arises somewhere in the substitution effects structure. Thus the special case is offered mainly to illuminate how restrictive the circumstances are which yield the validity of the World Bank dispersion index. A second use of this section is that in offering a fully worked out example of the trade restrictiveness index, it provides more intuidon as to the general structure.
Tariff Index Theoiy J.E. Anderson 22

# 111.1 Marginal Trade Weights and Trade Weights

The relevant portion of the trade expenditure function is assumed to be the consumer's expenditure function e(q,u). With Cobb-Douglas preferences, this implies a demand system: Ilqaiiu (3.1) eqi(q,u) = aieai/qi = Zi. qi Each internal price qi is equal to the external price q*j times the tariff factor Ti, possibly different from unity.

The marginal trade weights are based on reladve substitudon effects. The substitution effects system is: (3.2) q*jeqiqjq*i e(-8ij+ aj)ai/TliTj, where<sup>8</sup> 1j is the Kronecker delta. The **Zij** weights are forned by the division of the elements of (3.2) by lEq*jeqiqjq*i.

The trade restrctiveness index in rate of change form is (3 3) **A** MS =BT(T1 BfTI _ 1r) -TO) = -(T-1)'(Zij)(T' -AO ___________________ZijJ (3.3 A =BT-(T-1)Y( **zjj)T** For discrete changes, A is formed by substituting T for T. The simplification of this expression begins by removing the normalization of the Zij weights, multiplying numerator and denominator by ZZq*jeqiqjq*i. Now substitute the elements of (3.2) evaluated at T for **zij** on the right hand side of (3.3). The structure of (3.2) substituted into (3.3) suggests defining tariffs on the domestic price base: Xi= (T1-l)/Ti T (TCi-1)/T; gi 0l)i =
imiff Index Theory J.E. Anderson 23 The numerator is then equal to denominator of (3.3) is evaluated at T using (3.2). Note that post-multiplying by T els one of the denominator terms in (3.2). Thus:

> BT'T e i£aiti e(ji);axii = (1 - oai)aici.

**A**

Finally, form a simplified expression for A by substituting (3.5) and (3.4) into and dividing numerator and denominator by Zcxq to form weights wi = aj/ai. The .ights wi are the domestic value of (tariff-ridden) trade share weights, qiZiIZq Zj (based 3.1)). The numerator term (3.4) becomes covc,r1;40) + MC*(l-Zai)(M6&<sup>1</sup> ) - M(tO)) ,w the ratio in (3.3) can be simplified to yield: gosition 2 _A_ **A** = _;V(tO)_ - _(]-A) V(t(Ia)_<sup>_1_</sup> _)_ **+** _(2A-1)Cov(, ?) _-_ - _(Wci')_ - _Mcro)). ( I -zi) M(r)_ For the Cobb-Douglas case, Proposition 2 shows that the _variance and covariance 7ns of the TRI are the usual trade weighted forms._ The term (l-laci) is equal to one nus the tariff-ridden-import share of total expenditure. Subjec **t** o a given erpolation procedure, the trade restrictiveness index can be expressed in terms of trade hted moments of the interpolated tariff schedule. The problem of properly erpolating between c and l is generically the same for any discrete approximation, t it does mean that the covariance may matter. Thus the coefficient of variation of iffs can be a consistent index when variance is eliminated (so covariance is zero) and mean tariff does not rise. Formally
Tariff Index Theory J.E. Anderson 24

# Corollarv

_Letfinal good importpreferences be Cobb-Douglas. Then formean-preserving changes in the schedule of tariffs which eliminate variance, both_ d _and the change in the trade-_ _weighted coefficient of variation of aiffs index colTecdy sign welfare._ For CES expenditure functions, the shares, the a's, are endogenous, and Zij is equal to the elasticity of substitution times expression (3.2). Since this is a constant (which moreover appears in numerator and denominator, hence cancels) it does not alter the results. Thus Proposition 2 can be extended to the CES case.

The main conclusion which should be drawn from (3.6) is that if the preferences for imports really are CES, so that the coefficient of variation is valid for mean- **AA** preserving eliminations of tariff variance, it is also feasible to calculate from (3.6) subject to the interpolation error. There was some intuitive sense to the earlier procedure, but the trade restrictiveness index makes optimal use of the moments of the tariff schedule. More generally, using any available information on substitution effects, it is A possible to measure A using the marginal trade weights identified in Section II. The trade restrictiveness index can then consistently measure all tariff changes. Moreover, it has the interpretadon of a uniform tariff surcharge which compensates for the tariff change.
25 Tariff Index Theory

# J.E. Anderson

# 111.2 Mean-preserving Reductions In Variance

A full development of the Cobb-Douglas example leads to a proposition in which variance need only be reduced, not eliminated. It also further illustrates the properties of the trade restrictiveness index and the issues of Section II.

Let the representative consumer's expenditure function with two tariff-ridden goods be e(T1,T2,u) = '-'T'2u 1 2 where al+a2<1, 0k(a;l,2)<l and the remaining goods (nontraded goods and other imports and exportables), are untaxed, hence have prices equal to unity.

Under the Armington assumption, there is no production of the tariff-ridden goods and with no taxation of exports, or n :itraded goods and a constant returns technology with more exportables than nontraded goods and factors, the value of producdon is fixed in world prices at y. This becomes a parameter with respect to the trade policy analysis.

Under these circumstances, the balance of trade function is ra1T!2u T0alT% (3.7) B(T,u) = Tl12Tu - y - (T1 - l)aI_ T2 1 (T2 - 1) 2 Assuming no external borrowing for simplicity and setting B(TO,u<sup>0</sup> ) equal to zero, the trade balance equation may be solved for the initial level of utility u°. The solution utility level u is related to T by the _distorted trade udility fimction:_<sup>_14_</sup> (3.8) v_ ____T__ __=_ _____ - l_aT7 (3.8) v(T) = 1 - Xl - a2 + al/Tl + a2/T2" 1 2 The trade restricdveness index is implicidy defined as (3.9) A(T,uO) = (A I B(TA,uO) = 0). With u° defined by v(TO) in (3.8), this implies that A solves: 14The distorted trade utility function is the basis for an equivalent variation version of the trade restrictiveness index.
Tariff Index Theory J.E. Anderson 26 Tij~ i Al~i-X (3.10) 1t _-_l a2_ F _a,/°_ 10 _t° a( al - a2) + al;Tl + a2/T2_ Now consider the standard measures, the mean and variance of the tariff schedule.

The import-weighted mean tariff is: The import-weighted variance of the tariff is: The purpose of this subsection is to show that variance reducing reforms which preserve the mean tariff are also welfare-improving. Mean-preserving tariff changes imply, from (3.11), that dT2 _ a dT1 a2 Under the constant mean-tariff constraint, the variance changes with a rise in Ti according to (3.13) dV(T) - 2 " (T - T2)The analysis is completed by showing that dA/dT1 under the constraint has the opposite sign to (3.13); i.e., if the variance decreases, efficiency measured by A increases. See the Appendix for the technical details. This means that Proposition 3 _In the Cobb-Douglas case, mean-preserving local reductons in the tradeweighted varance of tariffs are welfare-improving._ Discrete changes in the tariff structure appear to allow for a possible reversal of implications. But if the distorted trade utility function is quasi-concave (the tariff factor indifference curve bounds a convex set of tariffs preferred to the initial tariff), local superiority implies superiority over a range of changes on the mean tariff line. The Cobb-Douglas model has a concave distorted trade utility function for non-negative tariffs. Moreover, for the Cobb-Douglas model, for a constant mean tariff, it is readily
Tariff Index Theory J.E. Anderson 27 shown that the _welfare-maximizing tariff structure is a uniform tariff,_ which agrees with the variance-minimizing solution. This means that each step of variance reducdon is welfare-improving, so that subject to a constant mean, the coefficient of variation of tariffs is a legitimate welfare index. Figure 2 illustrates.

Figure 2. Tariff Reform in the Cobb-Douglas Model <!-- Start of picture text --> Ti WMT<br>wX ~~~~~~Ti= T2<br>u ~~uO<br>0 T<br><!-- End of picture text --> The positively sloped curves are coefficient of variation isoquants, more variation implied by more distance from the equal tariff factors ray. For the Cobb-Douglas model, the slope of the coefficient of variation function is <!-- Start of picture text --> T-T VMI)/ 2<br>dT1 -2 Tx2 -T<br>dT2 - VT)/<br>Ti-T<br><!-- End of picture text --> The "isovariation" curves, labelled W (for the World Bank) necessarily have positive slope in the two good case. The local properties of A(T,u) at A show that A rises as variance falls in the inove along the constant tariff line ~I from A to C. Both imply a rise in welfare. The global properties of v(T show that variance falls all along AE to the maximal utility under the constraint, so that the variance measure is globally valid under the constant mean restriction. In contrast, when the mean and variance can both shift, the
28 J.E. Anderson Tariff Index Theory diagram shows that there is a set of tariff reforms which are welfare-superior, but which the coefficient of variation would measure as welfare-inferior. This is depicted as the cross-hatched region to the left of point A. When the mean shifts, the intuitive scaling of the standard deviation by the square of the mean does not provide a legitimate welfare index.

It would be nice if more general preference structures, which would require different weights, still had the property that variance reduction subject to a constant mean was welfare increasing. If so, (i) the optimal tariff would be uniform and (ii) each reduction in marginally weighted variance would improve welfare. The results of Section II show that (ii) cannot be guaranteed. Regarding (i), uniforrnity of the optimal tariff subject to a mean constraint unfortunately cannot be guaranteed for more general preference structures. The constant-mean tariff constraint is similar to the revenue constraint of public finance theory, which shows that a uniform tax on the set of goods available for taxation is possible only under very strong assumptions.

# **IV.** Lessons

Section II lays out the foundation of the trade restrictiveness index in relation to standard tariff indices. The trade restrictiveness index is linked to the marginal trade weighted mean and the generalized variance of the tariff schedule, which fulfills the intuition behind the World Bank's use of trade weighted mean and coefficient of variation indices. Section Im gives a strong sufficient condition for marginal trade weights to be equal to trade weights, and it does not appear to be possible to weaken the substitution conditions beyond the (TS. For example, strong separability does not suffice. A correct version generally requires the use of marginal rather than average trade weights, but even so, covariance and "curvature" also matter outside the Cobb-Douglas case.

It is simple to construct examples where welfare moves in the opposite direction from either the mean or the variance or coefficient of vaiation, so long as these move in
29 J.E. Anderson Tariff Index Theory the opposite direction. Appendix 2 presents four such simulations for the Cobb-Douglas case. Thus, the trade restrictiveness index should be used to replace tariff averages or coefficients of variation, while recognizing the limits of any such measure due to having to identify or assume a substitution effects structure. This problem is met with severe substitution structure assumptions in Computable General Equilibrium (CGE) models, and sensitivity analysis on substitution parameters. A similar procedure in calculation of the trade restrictiveness index should be usable in principle. Here, a potentially great advantage is that the TRI can feasibly aggregate consistently from the fine structure of protection, a task which is currently not possible with CGE models.

Future work should experiment with various substitution effects assumptions and approximative vs. numerical integration, in order to see how much difference the refinements make, and how the trade restrictiveness index compares with the standard indices. A current World Bank project is attempting to implement the index in a pilot study of a dozen developing countries. Early results<sup>15</sup> indicate that (i) the trade restrictiveness index often gives results opposite to the standard methods, and (ii) the trade restrictiveness index is not very sensitive to the substitution elasticity values.

15See Anderson and Bannister (1991) and Anderson (1991).
Tariff Index Theory 30

## J.E. Anderson

# **Appendix 1. Proof of Proposition 3.**

Under the constant mean-tariff constraint, the index A changes with a rise in T1 in a rather complex fashion. Fortunately, the issue of the relation of tariff variance to A can be illustrated by evaluadng in the neighborhood of 1<sup>0</sup> , where A is equal to one. Let (3.10) be written as: Al-al -42 - (3.10') AB 0) B C(OX) At T equal to TO, A is equal to one, B is equal to C, and A is equal to one. Differentiating (3.10') under the constant-mean-tariff constmint, <!-- Start of picture text --> A lQal 2 1<br><!-- End of picture text --> O - MD-a al dA. -Xal **- c c** Using the local properties at T equal to TO, this implies: dIA I_I I I 1 (3.14) T _a (T -T)CT) O -i - T2i-C 2At T equal to T0, C is equal to 1 - a T1 -1a2 T2 - which is less than one for nonnegative tariffs. Thus the first ratio is negative. The large bracketed term factors into <!-- Start of picture text --> IT I2 1- al Txi*+a;F1 T2 2 O<br>T 111 T(-1<br>(*- i)((l~1VT + T2<br>(;F1 -;(-F2) - 1<br><!-- End of picture text --> where the right bracket is negative. Then: sign dA/dTl = sign (T- jT)for non-negative tariffs.
Tariff Index Theory J.E. Anderson 31

# ARpendix 2. **CobbtDouglas** Simulations

This appendix presents the results of the simulation of the Cobb-Douglas model. The two tariff case is chosen with a tariff iniially equal to .5 for good 2 with 10% of expenditure on it and .2 for good 1 with 5% of expenditure on it. The initial tradeweighted mean tariff is equal to 40%. In the Cobb-Douglas model, so long as the mean and variance of tariffs move in the same direction, the TRI must also move in that direction. The interest is in the cases where this does not hold. The simulations show four cases of tariff changes for which the mean and variance (and coefficient of variation) move in opposite directions. The first two rows show the relative tariff changes (from the initial base of `z = .2 and r2 = .5). The next four rows of the table illustrate that any possible combination of TRI change with mean and offsetting variance or coefficient of variation change may be found. For tariff changes of the type used here, the mean or variance (or coefficient of variation) index would be wrong half the dme.

|cases|I|II|III|IV|
|---|---|---|---|---|
|relative tariff 1<br>|1.458333333<br>|<br>1.125<br>|0.91666667<br>|1.18333333<br>|
|relative tariff 2|0.8|0.96666667|0.96666667|0.93333333|
|relative mean tariff<br>|0.958333333<br>|1.04166667<br>|0.83333333<br>|1.01666667<br>|
|relative variance<br>|1.007631258<br>|0.11111111<br> <br>|1.36111111<br>|0.00444444<br>|
|relcoeffvar|1.047452219<br>|<br>0.32<br>|1.4<br>|0.06557377<br>|
|TRI|0.99192921|0.99836569|1.04294631|1.00648972|