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The effects of excise taxes on non-homogeneous populations.


In a fascinating and challenging article, Martin Feldstein (1997) calls upon public finance economists to address the important issue of measuring the effect of tax rate changes (and the costs of incremental tax revenue) on the tax revenue and/ or on the level of government spending.

A by product of a tax rate increase is the deadweight loss associated with changes in tax revenue, which according to Feldstein is often likely to be equal to or greater than the direct revenue cost itself. In other words, a dollar of government outlay (equivalent to a dollar of tax revenue) may have a total cost including the deadweight loss (or marginal excess burden) that exceeds two dollars.

Feldstein comments that despite the important contributions of economists such as Harberger (1964), Stuart (1984), Ballard Shoven and Whalley (1985), Browning (1987), Shemrod and Yitzhaki (1996), much more work still needs to be done. He concludes by saying that "I hope that my remarks will convince you as well, and will persuade many of you to devote some of your own research efforts to this important task".

In some sense a recent paper of Holcombe (2002) took up Feldstein's challenge by calling for a reconsideration of Ramsey's rule for optimal excise taxation. Holcombe was somewhat critical of the rule that in order to minimize the excess burden of an excise tax, goods should be taxed in inverse proportion to their elasticity of demand. In the same year, Coady and Dreze (2002) developed what they termed "the generalized Ramsey rule" which also takes into account interpersonal redistribution and resource allocation, issues that lie at the heart of our paper.

In this paper we revisit the very basic concepts of tax revenue maximization, the "optimal" tax rate (we wish to emphasize that optimal tax in this paper is used in the narrow sense of tax revenue maximization and not the socially efficient tax) required to achieve it, and the resulting deadweight loss. For the sake of clarity and simplicity, as well as for the sake of having a simple and standard point of reference for our later discussions, we begin by assuming a standard downward sloping linear market demand curve. We briefly develop the solution for this simple case and show that under these assumptions the tax revenues generated by the optimal tax are exactly double the remaining consumer surplus and the deadweight losses. But all this is predicated on the assumption that the market demand is a simple summation of linear demand curves of identical consumers. The question that we raise is how would these results be affected if we assume that these linear demand curves are not identical but rather are distributed among consumers in proportion to their income. How would this assumption affect tax revenues and deadweight losses (in both absolute and relative terms)? Furthermore, how would this affect consumer surplus? Who "pays" more and who "pays" less in terms of lost consumer surplus? This issue of a "fair" or "unfair" tax burden doesn't arise in the case of identical consumers, but is very relevant and significant in the case of consumers who differ in income and tastes.

To the best of our knowledge this basic but important analysis has been ignored by the public finance literature. Of course the issue of fairness has long been a major theme in public finance, but this issue in the very important and realistic context of non-homogeneous consumers, has tended to be overlooked.

We are hopeful that our paper, due to both its simplicity and its robust conclusions will shed light on this very important public finance issue. The structure of this paper is simple. We first discuss the case of non-homogeneous consumers with a rectangularly (uniformly) distributed linear demand curve for which we derive the basic results of tax revenues, consumer surplus, and deadweight loss. We then compare the results obtained with the case of an ordinary downward sloping linear demand curve of homogeneous consumers, and end with implications and conclusions.

Optimal Excise Tax Policy with Non Homogenous Consumers

We discuss the case of non-homogenous consumers who exhibit a rectangularly (uniform) distributed demand function. For simplicity of exposition, we assume a marginal cost of production of zero, although the forthcoming analysis could be undertaken independent of this assumption. The assumption of MC = 0 implies that the market price is equal to the tax per unit, i.e. t = P. We assume further a market consisting of A non-homogenous consumers. The demand function is linear and given by: [d.sub.i]: [x.sub.i] = RP - [p.sub.i], where R[P.sub.i] represents the reservation price of customer i.

The demand of the consumer with the highest possible reservation price (RP) of A is d: x = A - p, and he therefore purchases a quantity of A units at a price of zero. The demand curve of the customer with the second highest reservation price given by A - 1 is: x - A - 1 - p, and so on down the line for further customers. The last consumer (consumer A) purchases a quantity of zero even at a price of zero: x = A - A - 0 = 0.

If a government that faces the demand of A non-homogeneous consumers tries to maximize tax revenue by imposing a tax of t per unit, total tax

revenue (T) is given by: Tax = T = t x [n.summation over (i=1)] [x.sub.i] where

N is the last customer (i.e. he purchases only one unit of the product). Therefore if for example a tax of t per unit is imposed, the number of customers who continue to purchase the product (and thus pay the tax) is A -t. The customer with the highest RP purchases A - t units, while the last customer, whose reservation price is A - t + 1, purchases one unit of the product. From the above we can determine that the total number of units purchased is:

[A - t.summation over (i=1)][x.sub.i] = [A - t.summation over (i=1)](A - t) + (A - t- 1) + ... + 1 = (A - t + 1) x (A - t)/2 (1)

and the total tax revenues are given by:

T = t (A = t + 1) (A - t)/2 (2)

where, t represents the unit tax rate, the left-side parenthesis represents the average consumption of those consumers who actually purchase some positive quantity of the product, and the right-side parenthesis represents the number of consumers who purchase some positive quantity of the product and therefore participate in paying the excise tax.

Since we are assuming discrete demand curves such that there exists a singular interval between a given consumer and the one following behind him, we wrote equation (2) in the form given above. For the purpose of maximizing equation (2) we will take its derivative by assuming that the demand function is continuous. We find the t that yields maximum tax revenues for the continuous case as:


where the first customer consumes (A - t) units, and the last customer, who has a reservation price of exactly (A - t), consumes 0 units. The middle expression therefore represents the average consumption of those consumers who actually purchase the product (and also includes the customer at the very border who does not make a purchase). We can therefore rewrite equation (2') as follows:


This can be differentiated and yields: (1)

dT/dt = [(A - t).sup.2] - 2t (A - t)/2 = 0 (3)

from this we derive:

(A - t) = 2t (3')

therefore the optimal [??] that maximizes tax revenue is given by:

[??] = A/3 (4)

We now substitute this optimal value for t-into equation (2") which yields the maximum possible level of tax revenues [??] follows:


Now that we have determined the maximum possible tax revenue for the case of the rectangularly distributed demand functions, we proceed to determine the associated deadweight social loss.

This deadweight loss is generated from the following:

(a) A - [??] consumers who continue to buy the product but have reduced their purchases as the result of the imposition of the tax at the level of [??] = A/3.

(b) The remainder of the [??] consumers who have now stopped buying any of the product, although prior to the imposition of the tax they purchased quantities ranging from one unit (for the consumer with a reservation price of $1) up to [??] units.

We open the analysis by focusing on group (a) above, the consumers who continue to purchase the product, but at reduced levels due to the imposition of the tax. These consumers purchased prior to the imposition of the tax a quantity equal to the value [x.sub.i] = R[P.sub.i]. After the tax imposition the quantity purchased is equal to the value [x.sub.i] = R[P.sub.i] - t, i.e. the level of consumption of each one of the consumers fell by t units, and therefore the loss to each consumer is given by the triangle L, where:

[L.sub.i] = t x [DELTA]x/2 = [??] x [??]/2 = [[tt].sup.2]/2

for: [??] = A/3, we get [L.sub.i] = [A.sup.2]/18.

Since there are (A - [??]) such customers, we get total loss of

[A - [??].summation over (i = 1)] [L.sub.i] = (A - A/3) x [L.sub.i] = 2A/3 x [A.sup.2]/18 = [A.sup.3]/27

Now we turn our attention to group (b), those customers who have stopped purchasing the product altogether as a result of the imposition of the tax. The first of those consumers (with an RP of [??]) will lose consumer surplus with a value of:

[D.sup.[??].sub.j] = [[??].sup.2]/2

The next consumer loses:

[D.sup.[??] - 1.sub.j] = ([[??] - 1.sup.2]/2

The following consumer loses:

[D.sup.[??] - 2.sub.j] = ([[??] - 2.sup.2]/2

The final consumer, who originally purchased one unit loses:

[D.sup.1.sub.j] = [1.sup.2]/2 = 1/2

and the total loss is given by:


In order to determine the value of the term [[??].summation over (j=1)][D.sub.j], we use an arithmetic progression of the second order (see Appendix A). The value obtained is:

[[??].summation over (j=1)][D.sub.j] =[A.sup.3]/162 + [A.sup.2]/36 + A/36 (7)

The total social loss of all A customers, Loss, is:


Revenue Maximizing Excise Tax Policy with Homogeneous Consumers

At this point we return to the case of A consumers, but this time all the consumers are identical in income and tastes. Their demand curves are therefore identical and are exactly equal to the average that we derived for the rectangularly distributed demand, i.e. all A customers have a RP equal to A/2 and each demand curve is identical and equal to: x = RP - p = A/2 - p

The total quantity purchased by the A identical customers for t = p (as marginal cost, MC, is assumed equal zero)is: X = A (A/2 - t)

Maximization of tax revenue is accomplished as follows:

Max Tax = t x X = t x A (A/2 - t) = t[A.sup.2]/2 - [t.sup.2] x A (9)

Tax revenues are maximized at [t.sup.*]:

dTax/dt = [A.sup.2]/2 - 2tA = 0, or, [t.sup.*] = A/4 (10)

The loss per consumer can be derived as follows. When free of the tax the consumer purchases A/2 units, and in the presence of the tax this declines to A/4 units.

The loss per consumer is therefore given by:

[Loss.sup.*.sub.i] = [A.sup.2]/32 (11)

and the total loss to all A consumers is:

[Loss.sup.*] = A x [Loss.sub.*.sub.i] = [A.sup.3]/32 (12)

The total tax revenue from all A consumers is:


The ratio of the welfare loss per dollar tax revenue is:

[Loss.sup.*]/[Tax.sup.*] = 1/2 (14)

In the case of the rectangular demand distribution the value of the above ratio is:


Therefore for the case of A > 0, we can conclude that

Loss/Tax > 1/2 = [Loss.sup.*]/[Tax.sup.*] (16)

We now compare the optimal tax rates, optimal tax revenues, Losses, and the ratio of Loss per dollar tax for the two cases, i.e.:

[??] with [t.sup.*], [??] and [T.sup.*], Loss and [Loss.sup.*],

and Loss/[??] with [Loss.sup.*]/[T.sup.*]

We can summarize the results up to this point as follows:

(a) From (16) we conclude that the deadweight loss per dollar tax revenue for the case of a rectangular demand distribution of non-homogenous consumers is greater than that of the case of identical homogenous consumers (i.e., identical in taste, income, etc.). Although this gap decreases with A, it is always correct to say that the deadweight loss per dollar tax revenue is greater for the rectangular distribution. Or, if applying Ramsey's rules, a tax imposed on a rectangular demand distribution (i.e., on a population with non-homogeneous tastes and income) is less efficient than a tax imposed on a homogeneous population.

(b) From (4) and (11) we find that [??] = A/3 > .~ = t*, the level of the optimal per unit tax is greater for the rectangular distribution than for the identical distribution, and this gap increases proportionally with A. (c) From (5) and (13) we find that: [??] = 2[A.sup.3]/27 > [A.sup.3]/16 = [T.sup.*].

The total revenue generated by the optimal tax under identical consumers is lower than under a rectangular demand distribution, and this gap increases with the cube of A. Thus for tax revenue purposes, a rectangular demand distribution is preferable to an identical demand, and the gap in revenues increases more than in proportion to A.

(d) From (a) and (c) it is clear that the total deadweight loss to society is always much greater under a rectangular distribution than under an identical (homogenous population) distribution.

A final issue that needs addressing is how the tax burden is shared between the rich and poor in the case of non-homogeneous consumers.

We focus on the tax burden and deadweight loss of a representative consumer (for the case of homogeneous demand) versus that of an "average" customer, a "wealthy" customer (i.e. the one with the highest reservation price), and a "poor" customer (who still purchases one unit after the tax is imposed), for the case of heterogeneous demand. The above are looked at under the assumption that the government has imposed the optimal tax that yields maximum tax revenue.

The Case of Identical Customers (Homogeneous Demand)

In the case of A identical customers the deadweight loss per customer is, as previously discussed, equal to:

Loss per consumer = [A.sup.2]/32

The tax collected from each customer is:

Tax per consumer = [A.sup.2]/16

Therefore the proportion of loss to tax for each customer is equal to:

Loss/Tax per consumer = 1/2

The Case of Heterogeneous Customers

In the case of heterogeneous customers with a rectangular distribution, the average or median customer is "located" at A/2, and thus has a reservation price of A/2.

His quantity demanded is given by:

x = RP - [??] = [P.sub.max] - [??] = A/2 - A/3 = A/6,


Tax from the mediam consumer = [??] x x = A/3 x A/6 = [A.sup.2]/18.

His deadweight loss is given by:

Loss of mediam consumer = [??] [DELTA]x/2

= (Xbeforetax - Xaftertax)/2 = [??] (A/2 - A/6)/2 = [A.sup.2]/18

Thus the Loss/Tax ratio for the median consumer is equal to 1 (vs. 1/2 for the case of identical customers), indicating that when customers are heterogeneous in tastes each dollar of tax burden imposed carries with it an additional excess burden over and above that for the case of homogeneous consumers.

As to the "wealthy" customer, with a reservation price of A, his demand is given by:

x = (A - [??]) = A - A/3 = 2A/3.

His tax payments are given by:

Tax = [??] x x = A/3 x 2A/3 = 4[A.sup.2]/18

His deadweight loss is given by:

Loss = [??][DELTA]x/2 = A/3 x A/3/2 = [A.sup.2]/18

i.e., the loss of the "wealthy" customer is identical to that of the median customer. Therefore the Loss/Tax ratio of the "wealthy" customer is equal to 1/4.

As to the "poorest" customer, i.e. the last one to still purchase one unit of the product, his reservation price is:

[??] + 1 = A/3 + 1

After a tax of A/3 is imposed the "poor" customer purchases one unit x = 1. Thus, the tax revenue derived from the poor is:

Tax = [??] x x = A/3

The loss of the poor is:

Loss = [??] x [DELTA]x/2 = [??] x [??]/2 = [(A/3).sup.2]/2 = [A.sup.2]/18

It is clear from this comparison between the three representative consumers that each one suffers an identical total deadweight loss equal to [A.sup.2]/18. But on the other hand, it is also clear that due to the assumption of a rectangular distribution of customers, tax revenues derived from the wealthy customers are greater than those derived from the median customer, and they in turn are greater than those derived from the poorest customer who purchases just one unit. Specifically, tax revenues derived from the poorest customer are equal to A/3, and are smaller than those derived from the median customer, which is equal to [A.sup.2]/18. This holds true for A > 6, therefore [A.sup.2]/18 > A/3.

We now order the ratios of deadweight loss to tax revenue as follows:

(Loss / Tax) poor = A / 6 > (Loss / Tax) median

= 1 > (Loss / Tax) rich = 1/4

Not surprisingly we find that as A increases each tax dollar of revenue derived from the poor generates increasing levels of deadweight loss both in absolute terms, and in relative terms, in comparison to both the median income earner and to the rich. This can also be understood as follows: the deadweight losses of the poor, the median, and the rich are assumed equal, but the total tax paid by the rich is given by (4/18)[A.sup.2]. Comparing this to the total tax paid by the poor, given by A/3, we get [for the minimum possible value of A (i.e. A = 6)]:

Tax(rich) = 4/18 x [6.sup.2] = 8, Tax(poor) = 6/3 = 2.

In other words the taxes paid by the rich are at least four times that of the poor, and that is only the minimum possible multiple, holding true only for the borderline case (of A = 6) where the median income earner pays the same tax as the poor. Clearly as A increases and the median tax payer starts paying more taxes than the poor, then the taxes paid by the wealthy will increase far beyond this fourfold ratio.

Furthermore, as A increases the "damage" (as measured by the Loss/Tax ratio) to the poor far exceeds the possible benefits derived from his tax payments, which is not the case for the median or wealthy tax payer.

The welfare inequality implications are straightforward. Not only are there higher levels of deadweight losses under a heterogeneous population, but also the share of the burden borne by the poor is higher. We believe that it is important to strongly argue this point, since there have been recent attempts to portray some highly regressive taxes as being in effect less regressive and perhaps possibly even progressive (see for example Chernick, H. and Reschovsky, A (1997) with regard to the gasoline tax, and Remler (2004) with regard to the cigarette tax).

Summarizing the above in aggregate terms, we can say as follows. On the one hand, when looking at what situation would enable the tax authorities to squeeze the maximum possible tax revenues out of the population, clearly a heterogeneous distribution with large variability would be preferred to a unified population with similar tastes and demand functions. On the other hand, this same heterogeneous demand distribution comes with a very serious drawback in that its tax generating benefits come with a price tag in the form of far greater deadweight loss for every income level in comparison to that same income level under a homogeneous demand distribution. As A increases, i.e. as the population becomes more heterogeneous, the gap between the deadweight losses of the various income groups increases and deepens, although not proportionally (see Table 1). Increasing the tax burden on a heterogeneous population, therefore, comes at a heavy cost, both in terms of the total burden on the heterogeneous population and on the loss within the various income levels. As A increases the proportion of Loss/Tax for the rich declines with respect to that of the poor, something that would not take place if the population distribution were homogeneous.

In this section we analyze the distribution of total social welfare A for the two cases of homogeneous consumers and rectangularly distributed heterogeneous consumers. It is clear that the total social welfare W [equivalent to] [DELTA] can be broken down into three components for each of the two types of consumers, as follows:


The values for the "representative customer" in the case of homogeneous customers can be seen in the graph (Fig. 1):

Thus for the representative customer:

[[DELTA].sup.*.sub.i] = [A.sup.2]/8

and for all the customers:

[A.summation over (i=l)] [[DELTA].sup.*.sub.i] = A[[DELTA].sup.*.sub.i] = [A.sup.3]/8







See Table 2 for the breakdown by component.

In the event that we are faced with non-A homogeneous customers, the [A.summation over (i=l)] [??] can be calculated via an arithmetic progression as follows:

[DELTA] Consumer 1 = 2 x 2/2 = 1/2

[DELTA] Consumer 2 = 2 x 2/2 = 2

[DELTA] Consumer 3 = 3 x 3/2 = 4.5

[DELTA] Consumer A = A x A/2 = [A.sup.2]



For [??] = A/3 we get:


This is summarized in Table 3.


For two parameters (i.e. that of tax revenues and consumer surplus) we find that non-homogeneous customers, who are dispersed around some average consumption, are "preferable" to the case of homogeneous customers with an identical demand. On the other hand, homogeneous customers also take the lead in terms of inefficiency and social loss (both in absolute and relative terms). Thus a government desiring to raise additional tax revenues would prefer to be faced with a case of as wide a dispersion of demand distribution as possible. Although a homogeneous population may be technically more efficient in that it leads to less deadweight loss, in terms of tax revenues generated the non-homogeneous case yields far better results. Those results continue to improve as A increases, since increasing A means a higher degree of dispersion with an increasing gap between the highest reservation price, A, and the lowest one (zero).

Implications and Conclusions

When considering the imposition of an excise tax, policy makers should consider such elements as who bears the tax burden, tax effectiveness and tax efficiency, and should of course always attempt to minimize the inefficiencies generated by the imposition of a tax.

These issues are well known and have been rigorously analyzed for the case of an "optimal" (in the sense of maximum revenue generating) excise tax imposed upon a population homogeneous in terms of their demand for the item being taxed. But in truth most populations are not homogeneous in income, tastes and preferences. Our goal therefore has been to compare the well-known results of the homogenous population with the parallel results obtained by assuming a non-homogeneous population. To this end we assumed a population uniform in income and tastes and therefore possessing a uniform demand for the taxed product, and compared the results obtained to those obtained under the assumption that the population is heterogeneous and exhibits a rectangular linear demand function.The results obtained can be summarized as follows:

A rectangular demand function resulting from a heterogeneous population enables the tax authorities to impose an optimal tax that will yield higher revenues than under the homogeneous case.

Moreover, we have shown above that for any given tax rate that is imposed on a heterogeneous population with the same average demand as the demand of a representative customer of a homogeneous population, the former yields higher revenues than the latter. Furthermore we have shown that the tax yield can be increased even further by imposing a new optimal higher tax rate in the case of a heterogeneous population (versus the lower optimal tax rate derived for the homogeneous population). Thus a higher degree of diversity (dispersion) of population demands enables us to impose a higher optimal tax which increases tax revenues even further. Therefore the Leviathan government will prefer the case of a rectangulary distributed demand, since it yields higher revenue for a given tax rate, and its optimal tax rate is even higher, thus increasing tax revenues even more in comparison to a homogeneous population with identical demand.

From the point of view of lost consumer surplus, the heterogeneous population will remain with more consumer surplus (i.e. will lose less) than in the homogeneous case. Nevertheless, it should be pointed out that it is primarily the rich (i.e. those with a high reservation price) who capture the bulk of this consumer surplus. If we assume that the item being taxed is a normal good, this means that although the tax authorities do indeed collect more tax revenues from the rich than from the poor, they still leave the rich with more consumer surplus than the poor. This has important welfare aspects. Not only do we face the usual well-known regressive effects of an excise tax, but in the case of a heterogeneous population we have a further source of regressivity that is absent in the case of a homogeneous population. This arises from the higher excess burden on the poor when compared to the homogeneous case, and even more so when we look at the excess burden per dollar tax revenue. These issues don't arise under the assumption of a homogeneous population, but in reality they could be of major importance, since in practice populations don't tend to be homogeneous.

We see therefore that in reality (since real life consists of heterogeneous populations) the gap between the poor and the rich in terms of both total and relative excess burden increases with population dispersion and is more favorable to the rich and more penalizing towards the poor than under the standard assumption of homogeneous populations. Although on the one hand the tax authorities would prefer facing a more heterogeneous population with a larger dispersion (since tax collections are more effective (i.e. higher)), on the other hand this same population dispersion results in less equality and fairness in distributing the burden of a given level of tax revenues.

This paper has added another dimension to the often-discussed conflict between efficiency and equality in the public finance literature by suggesting that these issues be analyzed in the context of the kind of populations that exist in real life, i.e. populations that are heterogeneous in tastes and income. We hope that this basic discussion will motivate others in the spirit of Feldstein (1997) to undertake further research in this area.

Appendix A

Since S is an arithmetic progression of the second order, we can apply the rule that says:


Solving for optimal tax rate [??]:





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Browning, E. 1987. "On the marginal welfare cost of taxation", American Economic Review 77, no. 1, pp. 11-23.

Chemick, H. and Reschovsky, A. 1997. "Who pays the gasoline tax?", National Tax Journal 50, no. 2, pp. 233-259.

Coady, D. and Dreze, J. 2002. "Commodity Taxation and Social Welfare: The Generalized Ramsey Rule", International Tax and Public Finance May; 9, no. 3, pp. 295-316.

Feldstein, M. 1997. "How big should government be?", National Tax Journal 50, no. 2, pp. 197-213.

Harberger, A. 1964. "Taxation, resource allocation, and welfare", in the Role of Direct and Indirect Taxes in the Federal Reserve System, edited by John Due. Princeton: Princeton University Press.

Holcombe, R.G. 2002. "The Ramsey rule reconsidered", Public Finance Review, 30, no. 6, pp. 562-578.

Remler, D.K. 2004. "Poor smokers, poor quitters, and cigarette tax regressivity", American Journal of Public Health 94, no.2, pp. 225-229.

Slemrod, J. and Yitzhaki, S. 1996. "The costs of taxation and the marginal cost of funds", International Monetary Fund Staff Papers 43, no. 1, pp. 172-198.

Stuart, C. 1984. "Welfare cost per dollar of additional tax revenue in the United States", American Economic Review 74, no. 3, pp. 352-362.


(1.) The derivation of (2") is more accurate and convenient than of (2). When A is larger and/or the difference of RP between consumers is smaller the maximization solution of (2) approaches that of (2").

by Uriel Spiegel, * Joseph Templeman, ** and Tchai Tavor ***

* Uriel Spiegel, Department of Management, Bar-Ilan University, and Visiting Professor, University of Pennsylvania, Email:

** Joseph Templeman, The College of Business Administration, Rishon LiTzion, Israel. Email:

*** Tchai Tavor, Department of Economics, Yisrael Valley College, Email:
The Social Welfare Distribution of Homogeneous and
Non-Homogeneous Groups

                      [??] /
A       [??]     L *     L *

6        10.5    6.75   1.555
10     46.265   31.25  1.4805
100     43490   31250  1.3917
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Author:Spiegel, Uriel; Templeman, Joseph; Tavor, Tchai
Publication:American Economist
Geographic Code:1USA
Date:Sep 22, 2010
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