Ramsey Tax 2

Ramsey Tax in Imperfect Competition Jim Y. Jin School of Economics and Finance, University of St. Andrews, U.K. Laurence Lasselle* School of Economics and Finance, University of St. Andrews, U.K. March 2005 Abstract We study the effects of the Ramsey tax in an imperfectly competitive environment. Our results are as follows. (1) The expression of the tax revenue maximization tax rate is independent of the market structure. (2) While the Ramsey tax leads to a minimum damage to social welfare in perfect competition, we show that this does not hold under imperfect competition. Instead, we demonstrate that it maximizes the potential tax revenue. (3) In all markets, we provide a common expression of the optimal tax rate and prove that there exists a simple tax adjustment process converging to this optimal tax. JEL Classification Number: H20, D40 Key Words: Ramsey Tax, Perfect Competition, Imperfect Competition.

*

Corresponding author: University of St. Andrews, School of Economics and Finance, St. Andrews, Fife, KY16 9AL, U.K. E-mail: [email protected], Tel: 00 44 1334 462 451, Fax: 00 44 1334 462 444.

Ramsey Tax in Imperfect Competition

1.

Introduction

The aim of this paper is to analyze the effects of the Ramsey tax in terms of equilibrium prices and output and to derive some welfare results in imperfectly competitive frameworks.

In his seminal paper, Ramsey (1927) explained how to minimize consumers’ utility loss when the government has to collect a certain amount of tax revenue in a perfectly competitive environment. Given the assumption of a quadratic utility function with n differentiated products and of a unit tax rate levied on each of those products, he obtained two elegant results in terms of optimal tax rate and of tax revenue maximization. First, the optimal tax rate should reduce all outputs by a same proportion. Second, the tax revenue is maximized when this proportion is equal to one half. This paper questions these results in a more contemporary framework allowing imperfect competition. This extension to imperfect competition is essential. Perfect competition is rare in modern economies and Ramsey’s model with differentiated goods is more likely to be associated with imperfectly competitive markets. This generalization may make the welfare analysis more appealing as the tax implementation could become less harmful under imperfect

2

competition.

One could then wonder why this analysis has not been offered earlier. Let us first remind that Ramsey’s contribution seems to have been overlooked for almost forty years. Ramsey’s results were rediscovered by Samuelson (1986) in a 1951 memo to the U.S. treasury.1 Second, Ramsey’s framework seems to be too restrictive on the demand side, becoming even too simple as noted by Atkinson-Stiglitz (1980). But it found an advocate in Mirrlees (1976), when the latter strongly argued that what was really relevant was to obtain the real effect of the tax system upon the equilibrium quantity of each good. Finally, economists often remind Ramsey’s results in terms of demand elasticities for small changes in taxation. However, research on Ramsey tax has gained momentum recently. Generally speaking, over the past decades, the literature on the topic has not only given a deeper theoretical analysis with more general frameworks (assumptions on the demand side have been notably relaxed), but it has also presented, through experiments, the implementation of the different kinds of tax. More recently, research has more focused on the impact of ad valorem and excise taxes on consumer surplus and firm profits on the one hand and of tax incidence on social welfare in theoretical and applied frameworks on the other hand.

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Skeath and Trandel (1994) showed that ad valorem tax Pareto dominates a unit tax in all Cournot oligopolies. Anderson et al. (2001) analyzed the incidence of these taxes in a Bertrand oligopoly with differentiated products. Denicolo and Matteuzzi (2000) considered other specific taxes and ad valorem tax in Cournot oligopolies. They demonstrated that if the tax rates are sufficiently high, ad valorem taxation welfare dominates specific taxation in the sense that for any given specific tax, one can find an ad valorem tax that leads to greater tax revenue, consumer surplus, and industry profits. The tax impact on the products sold in Cournot oligopoly with differentiated products was also studied by Fershtman et al. (1999). Finally, Gabszewicz and Grazzini (1999) and Coady and Drèze (2002) introduce a policy dimension in these frameworks. The former investigated the effectiveness of tax and transfer policies in correcting market failures in imperfect competition settings. The latter explained the role of taxes in a second-best equilibrium framework. They then gave a generalized Ramsey rule for optimum taxation.

But it seems that the effects of the Ramsey tax in terms of prices and output and of welfare in simple imperfectly competitive frameworks have not been analyzed.

1

See Myles (1995).

4

Our paper attempts to fill this gap by asking several questions. (1) Does Ramsey’s tax revenue maximization tax rate remain valid in imperfect competition? (2) Does the Ramsey tax lead to a minimum damage to social welfare in an imperfectly competitive environment? If not, one can wonder twofold. (3) What does it maximize? (4) Does the optimal tax rate still have a general form independent of the market structure?

We follow Ramsey’s simple assumptions on the utility function and study their implications in four models. These models differ by the nature of the competition in the product market. We respectively consider perfect competition, monopoly, Cournot oligopoly, and Bertrand oligopoly. For each variant of competition, we compare the quantity of output produced, the price, and different welfare measures before the tax is implemented with those after the tax on each good is levied.

Intuitively, we would expect that the Ramsey’s results in terms of optimal tax and tax revenue maximization would be more difficult to obtain and to interpret (even maybe disappear) because of the imperfect competition setting. Indeed, as previously mentioned, the literature has usually shown that tax effects vary according to the nature of competition. But as we shall see, we prove that Ramsey’s result of tax revenue maximization is independent of the nature of the

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competition. It allows us to rank the maximum amounts of the tax according to the nature of the market structure.

However, the Ramsey tax, which reduces all output proportionally, is no longer optimal in imperfect competition. In fact, it maximizes the potential tax revenue. As we shall see it does not yield to a minimum damage to social welfare in imperfectly competitive frameworks. But we provide a uniform expression of the optimal tax rate for all markets and show that there is a simple adjustment process converging to this optimal tax.

The paper is organized as follows. The next section extends Ramsey’s model. As well as in perfect competition, we use Ramsey’s quadratic utility function in three different markets of imperfect competition: monopoly, Cournot and Bertrand oligopolies. It allows us to compare Ramsey’s results with these obtained in imperfectly competitive markets. Section 3 gives uniform results of tax revenue maximization in the four markets. We rank the amounts of maximum tax and associated welfare losses in the four market structures in Section 4. In Section 5, we find a general objective function which Ramsey tax always maximizes in the four markets. In Section 6, we provide a general form of the optimal tax and present a simple converging process towards this optimal tax. We discuss our findings in the last section.

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2.

The Ramsey’s Model in Imperfect Competition

There are n + 1 goods: a numeraire good x0 with a price normalized to 1 and n differentiated goods grouped in an n × 1 product vector x. The representative consumer has a quadratic utility function u such as:

u ( x0 , x ) = x0 + a′x − 0.5 x′ B x where a is a n × 1 positive vector and B is a symmetric n × n matrix. We assume that u ( x0 , x ) is strictly concave in x, so B is positive definite and u ( x0 , x ) is strictly quasi-concave in x0 and x. The price of the representative

firm i is denoted by pi for i = 1,… , n and the price vector is p. The consumer has a fixed income W and chooses a consumption bundle x to maximize her utility given her budget constraint x0 + p′ x ≤ W . We assume that W is sufficiently high so that an interior solution exists. Since u ( x0 , x ) is strictly quasi-concave in both arguments, the optimal demand vector x can be solved from the first-order condition: a − B x − p = 0 . If there is quantity competition in the product market, from the above condition we deduce the inverse demand function for n-firm Cournot oligopoly: p ( x ) = a − Bx

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(1)

By denoting the elements of B by bij 's for all i and j, we can write this n

inverse demand function for firm i as: pi ( x ) = ai − ∑ bij x j . j =1

If there is price competition in the product market, from (1) we deduce the demand function for n-firm Bertrand oligopoly: x ( p ) = α − B −1p

(2)

where α = B −1a . By denoting the elements of B −1 by β ij 's , we can write this n

direct demand function for firm i as: xi ( p ) = α i − ∑ β ij p j . j =1

We assume that each firm i has a constant marginal cost ci and we denote the n ×1 cost vector by c . The profit of firm i is then given by π i = xi ( pi − ci ) .

We consider three types of imperfect competition. In Cournot oligopoly, every n ⎛ ⎞ firm chooses its output to maximize its profit xi ⎜ ai − ci − ∑ bij x j ⎟ . In j =1 ⎝ ⎠

Bertrand oligopoly, every firm chooses its price pi to maximize its profit

⎛

n

⎝

j =1

( pi − ci ) ⎜ α i − ∑ βij

⎞ p j ⎟ . In monopoly, the prices or outputs are jointly chosen ⎠

to maximize the total profit ( p − c )′ x .

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Ramsey’s framework is made to study how to maintain the highest possible social welfare given a fixed amount of tax revenue to be collected when each unit of product is taxed by ti .2 The output market is assumed to be perfectly competitive.3 Ramsey obtained two elegant results. First, the optimal tax rate for output i, which maximizes the social welfare subject to a fix amount of tax revenue, is ti* = k ( ai − ci ) , where k > 0 . Consequently, the after-tax equilibrium output i is given by xi*,T = (1 − k ) xi* where xi* is the pre-tax equilibrium output. Second, the tax revenue is maximized when k = 1 2 . Our paper examines whether these two results remain valid in the three different market structures with imperfect competition.

3.

Uniformity in Tax, Quantity and Price

In this section, we assess Ramsey’s results in models with imperfectly competitive product markets. We derive the tax revenue maximization tax rate and compare its effects on the equilibrium output and prices in monopoly, Cournot and Bertrand oligopolies. 2

“The problem I propose to tackle is this: a given revenue is to be raised by proportionate taxes on some or all uses of income, the taxes on different uses being possibly at different rates; how should these rates be adjusted in order that the decrement of utility may be minimum?” Ramsey (1927, p. 47). 3 “I shall deal only with a purely competitive system with no foreign trade.” Ramsey

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Proposition 1. Given our assumptions, whatever the nature of the competition (perfect competition, monopoly, Cournot oligopoly, and Bertrand oligopoly) in the product market, we have for all i: ti* = 0.5 ( ai − ci ) , xi*,T = 0.5 xi* , and

pi*,T = 0.5 ( ai + pi* ) . Proof: see Appendix A.

The scope of validity of Ramsey’s results is wider than previously thought. It does not depend on the nature of competition in the product markets. There exist a common expression of the tax revenue maximization tax rate, a common proportional relationship in output between the amount produced without tax and the level of output once the tax is implemented, and a common relationship between pre-tax level of the equilibrium price and that after-tax level. The latter relationship is slightly more complex than the former as it is a linear transformation.

3.

Uniformity in Welfare

In this section, we study the effects of the tax revenue maximization tax in terms of social welfare and agents’ burdens. We show that the welfare

(1927, p. 47).

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expressions do not depend on the nature of the competition considered and simple relationships between pre-tax expressions and after-tax expressions exist. Finally, we compare the levels of tax revenue associated with each kind of competition and we are able to rank them.

From our framework presented in Section 2, we derive the general expressions of the social welfare (SW) function, the consumer surplus (CS), and the total profit (TP) and obtain: SW = u ( x ) − cost = ( a − c )′ x − 0.5 x′ B x , CS = u ( x ) − p′ x = 0.5 x′ B x , and TP = ( p − c )′ x = ( a − c )′ x − x′ B x . We can then compare the pre-tax level of each welfare measure with that aftertax level. As our next proposition reveals, simple relationships between pre- and after-tax levels exist in the different welfare measures whatever the nature of the competition.

Proposition 2. Given our assumptions, in the four market structures considered, we have CST* = 0.25 CS * and π i*,T = 0.25 π i* .

Proof: It is easy to show: CST* = 0.5 xT* ′ B x* = 0.5 ( x* 2 )′ B ( x* 2 ) = 0.25 CS *

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and

(

π i*,T = ( pi*,T − ci − ti ) xi*,T = 0.5 ( ai + pi* ) − 0.5 ( ai + ci

) ) ( 0.5x ) = 0.25 ( p * i

* i

− ci ) xi* = 0.25 π i*

Proposition 3. Given our assumptions, in the four market structures considered, we have R* = 0.5 CS * + 0.25 TP* . Proof: see Appendix B.

These expressions allow us to compute the value of the social loss. We conclude that its value does not depend on the nature of competition considered, but that it is always equal to one half of the tax revenue collected.

Proposition 4. Given our assumptions, in the four market structures considered, we have SW * − SWT* = 0.25 CS * + 0.5 TP* . Proof: see Appendix B.

This result is not a surprise. Intuitively, we believed that the implementation of the tax would make the social welfare drop. We also believed that this drop should be positively related to the degree of market power of the oligopolistic agents. When this market power increases, the level of the total profits is known to enlarge, becoming more and more important in the evaluation of the social welfare, leading to a higher value of the efficiency loss (cf. Section 7).

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Let us now turn to the incidence of the tax for the agents in the different variants of the model, i.e. the consumer and producer burdens and the tax revenue for the government. Not surprisingly, tax revenue maximization harms both consumers and producers.

Proposition 5. Given our assumptions, tax revenue maximization implies that the consumer and producer burdens are both maximized. Proof: see Appendix C.

Proposition 6. Given our assumptions, we always have 2 RM = RPC > RB > RC . If, in addition, the goods are substitute, we have: RB > RM . If they are complements, RM > RC . Proof: see Appendix D.

Ranking the tax revenues according to the nature of competition is a natural extension of our work. This extension is sustained by results already provided by the literature and a simple observation. Indeed, equilibrium quantities are the highest in a perfect competition framework. Bertrand, Cournot and monopoly quantities can be compared once the nature of the goods in terms of

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substitutability or complementary is defined. As the expression of the tax rate is common to all forms of competition, ranking the tax revenues according to the nature of the competition is not too difficult to obtain.

5.

The Potential Tax Revenue Maximization Tax Rate

In this section, we show that the Ramsey tax does not yield to a minimum damage to social welfare in an imperfectly competitive environment, but maximizes the potential tax revenue.

Proposition 7. Whatever the nature of the competition considered, the Ramsey tax, t* = r ( a − c ) , leading to x*T = (1 − r ) x* and p*T = r a + (1 − r ) p* , maximizes the potential tax revenue given the current tax revenue.

If we consider that the government wants to maximize the potential tax revenue, RTpot , for a given tax income, its objective function4 should be as follows:

Max R + α RTpot t

4

(3)

Note the maximization program is not written with a constraint. Instead, we have preferred its evaluation from a combination of the direct tax revenue and the potential additional tax revenue, in other words once the current tax revenue has been fixed.

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As R = 0.25 TP* + 0.5 CS * , we have RTpot = 0.25 TPT + 0.5 CST . (3) can be rewritten as: Max R + α (TPT + 2 CST ) t

Denote R + α (TPT + 2 CST ) by L. Recall that R = t′ x*T , CST = 0.5xT* ′ B xT* ,

TPT = ( pT* − cT )′ xT* , cT = c + t and x*T = B −1 ( a − pT* ) . Therefore, L can be rewritten as:

⎡ ⎤ L = t′ x*T + α ⎢( pT* − cT )′ xT* + xT* ′ B xT* ⎥ ⎣ ⎦

α 1 ⎡ ⎤ = ⎢ t′ + ( a − c − t′ )′ ⎥ x*T = ⎡( 4 − α ) t′ + α ( a − c )′ ⎤ B −1 ( a − pT* ) ⎥⎦ 4 4 ⎢⎣ ⎣ ⎦ ∂L ∂2 L is negative definite. = 0 and L is maximized when ∂t ∂ t∂ t′

In the case of perfect competition, recall that we have pT = c + t . Therefore L can be rewritten as LPC =

1⎡ ( 4 − α ) t′ + α ( a − c )′ ⎥⎤ B −1 ( a − c − t ) . Let us ⎢ ⎦ 4⎣

evaluate the first order and the second order conditions. ∂ LPC 1 −1 ⎡ = B −2 ( 4 − α ) t′ + ( 4 − 2α )( a − c )′ ⎤ ⎢⎣ ⎥⎦ ∂t 4

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∂ 2 LPC 1 = − B −1 ( 4 − α ) which is negative definite when α < 4 . ∂ t∂ t′ 2

Let us evaluate the Ramsey tax. ∂ LPC 2 −α = 0 ⇔ t* = ( a − c ) which is definite positive when α < 2 . We can 4 −α ∂t then easily deduce the expressions of the after-tax equilibrium price and of the after-tax equilibrium quantity: p*T = r a + (1 − r ) p* and x*T = (1 − r ) x* where

r=

2 −α and α < 2 . 4 −α

The rest of the proof is left to the reader. Note that in case of monopoly, Cournot competition and Bertrand competition, L becomes respectively:

1⎡ ⎛a−c−t ⎞ 4 − α ) t′ + α ( a − c )′ ⎤ B −1 ⎜ ( ⎟ ⎦⎥ 4 ⎣⎢ ⎝ 2 ⎠ 1 −1 LC = ⎡( 4 − α ) t′ + α ( a − c )′ ⎤ ( B + D ) ( a − c − t ) ⎢ ⎥ ⎦ 4⎣ −1 1 LB = ⎡( 4 − α ) t′ + α ( a − c )′ ⎤ ( B + Λ −1 ) ( a − c − t ) ⎥⎦ 4 ⎢⎣

LM =

The Ramsey tax, which reduces all output proportionally, is therefore no longer optimal in imperfectly competitive frameworks, i.e. it does not minimize the damage to social welfare in all markets. Nevertheless, the expression of the

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Ramsey tax is still common in all markets. In the next section, we wonder whether there exists a common expression for the optimal tax rate.

6.

The Optimal Tax Rate in All Markets

In this section, we then find the expression of the optimal tax and prove that there exists a simple tax adjustment process converging to this optimal tax whatever the nature of the competition considered.

6.1

The Optimal Tax

Proposition 8. Whatever the nature of the competition considered, the optimal tax is t * = ( I − S )( a − c ) , leading to x*T = S′ x* and pT = ( I − S )( a ) + S p .

If, as in Ramsey (1927), the government wants to minimize the damage to social welfare, its objective function should be as follows:

Max R + γ SW t

(4)

where R = t′ xT and SW = ( a − c )′ xT − 0.5 xT B xT . If we denote t′ xT + γ ( a − c ) xT − 0.5 γ xT B xT

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by LO and we recall

xT = H ( a − c − t ) , LO can be rewritten as:

LO = t′ H ( a − c − t ) + γ ( a − c )′ H ( a − c − t ) − 0.5 γ ( a − c − t )′ H B H ( a − c − t ) ∂ LO ∂ 2 LO is negative definite. Let us = 0 and L is maximized when ∂t ∂ t∂ t′ O

evaluate the first order and second order conditions. ∂ LO = H ( a − c − 2t ) − γ H ( a − c ) + γ H B H ( a − c − t ) ∂t ∂ 2 LO = −2 H − γ H B H which is always negative definite. ∂ t∂ t′ Let us evaluate the optimal tax. ∂ LO =0 ∂t

⇔ ( a − c − 2t * ) − γ ( a − c ) + γ B H ( a − c − t* ) = 0

(5)

⇔ ( 2 I + γ B H ) t* = ( 2 I + γ B H − (1 + γ ) I ) ( a − c )

(

⇔ t * = I − (1 + γ )( 2 I + γ B H )

−1

) (a − c)

−1 1+ γ ) ⎛ ( γ ⎞ I + B H , then we have t * = If we let S =

2

⎜ ⎝

2

⎟ ⎠

In perfect competition, we have H = B

t* =

−1

(5)

( I − S )( a − c ) .

−1 1+ γ ) ⎛ ( γ⎞ 1+ , so S = I . We obtain

1+ γ (a − c) . 2+γ

18

2

⎜ ⎝

⎟ 2⎠

−1

In monopoly, we have H = 0.5 B , so

t* =

In

S=

−1 1+ γ ) ⎛ ( γ⎞ 1+ S= I.

⎜ ⎝

2

⎟ 4⎠

We obtain

2−γ ( a − c ) . Note that γ needs to be less than 2. 4+γ Cournot

oligopoly,

(1 + γ ) ⎛ I + γ B

−1 ( B + D ) ⎞⎟ 4 ⎠

⎜ ⎝

2

H = (B + Λ

)

−1 −1

we

−1

,

so

we

have

−1

.

In

(1 + γ ) ⎛ I + γ B , so S = 2

H = ( B + D)

have

⎜ ⎝

4

Bertrand

(( B + Λ ) ) −1 −1

oligopoly, −1

⎞ ⎟ . ⎠

We leave to the reader to demonstrate that x*T = S′ x* and pT = ( I − S )( a ) + S p .

6.2

The Tax Adjustment Process

In this sub-section, we are going to show that there exists a tax adjustment process common in all markets converging towards the optimal tax rate.

First note that (5) can be rewritten as:

( a − c − 2t ) − γ ( a − c − B x ) = 0 ⇔ ( a − c − 2t ) − γ ( p *

*

T

⇔ t* =

T

− B xT ) = 0

( a − c ) − γ ( pT − c ) 2

So the tax levied on each good is equal to ti =

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ai − ci − γ ( pi ,T − ci )

2

. Let us

evaluate the different tax rates in periods 0, 1 and 2. At period 0, no tax is implemented, so the price of the good is simply denoted by 0 pi . At period 1, the tax is implemented and we have ti = 1

becomes ti = 2

ai − ci − γ ( 1 pT ,i − ci )

2

ai − ci − γ ( 0 pi − ci )

2

. At period 2, it

. So we can wonder whether this procedure

convergence towards the optimal tax rate, t = * i

ai − ci − γ ( pi* − ci )

2

.

Let us evaluate the gap between the period tax and the optimal tax, i.e.

∆ τ +1ti ≡ τ +1ti − ti* =

−γ 2

(

τ

pT ,i − pi* ) . Let us work in matrix form from now on.

From (1), we deduce τ pT = a − B H ( a − c − t ) . The gap between the after-tax period tax and the optimal after- tax price is

τ

Combining the two gap expressions, we obtain: ∆ τ +1 t =

p T − p* = B H ( τ t − t ) .

γ 2

B H ∆ τ t . Let us pre-

and after-multiply a symmetric positive definite matrix Ω by this expression. It yields: ∆ τ +1 t′ Ω ∆ τ +1 t =

γ2 4

∆ τ t′ H B Ω B H ∆ τ t

If we are able to demonstrate that

γ2 4

∆ τ t′ H B Ω B H ∆ τ t − ∆ τ t′ Ω ∆ τ t < 0 , we

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will have shown that the tax gap is decreasing.

γ2 4 ⇔

∆ τ t′ H B Ω B H ∆ τ t − ∆ τ t′ Ω ∆ τ t < 0

⎛γ2 ⎞ ∆ τ t′ ⎜ H B Ω B H − Ω ⎟ ∆ τ t < 0 ⎝ 4 ⎠

Is the matrix Κ = Ω −

γ2 4

H B Ω B H positive definite?

Perfect Competition H PC = B −1 . So we have Ω −

γ2

⎛ γ2 ⎞ Ω = ⎜1 − ⎟ Ω . If γ < 2 , then K is positive 4 4 ⎠ ⎝

definite. Monopoly ⎛ γ2 ⎞ H = 0.5 B . So we have Ω − Ω = ⎜1 − ⎟ Ω . As γ < 2 , then K is 16 ⎝ 16 ⎠ M

γ2

−1

positive definite. Cournot Competition

H C = ( B + D ) . Let us pre- and after-multiply Ω by ( H C ) . So we have −1

−1

( B + D) Ω ( B + D) −

γ2

⎛ γ2 ⎞ B Ω B = ⎜1 − ⎟ B Ω B + D Ω B + B Ω D + D Ω D . The 4 4 ⎠ ⎝

first term of the RHS expression is definite positive. If we let Ω = D−1 , the last three terms of the RHS expressions are definite positive. So K is positive

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definite. Bertrand Competition

H B = ( B + Λ −1 ) . The proof is left to the reader. −1

We have just shown that throughout time, the tax gap is decreasing. The question is then to wonder what happens to the limit. Recall that we have just demonstrate

that

the

expression

between

brackets

in

⎛γ2 ⎞ ′ ∆ t ⎜ H B Ω B H − Ω ⎟ ∆ τ t is always positive definite. So towards the ⎝ 4 ⎠ τ

infinite, the tax gap is nil. Whatever the initial tax rate levied on the output, it will always converge towards the optimal tax. Let us finally note that the government can define the correct ratio between the social welfare and the tax revenue as γ =

7.

ai − ci − 2 ti . pT ,i − ci

Discussion

We have shown that Ramsey’s first result on revenue tax maximization is compatible with all simple forms of imperfect competition in the product market. From this statement, several points need to be stressed. It is first important to note that the expression of the revenue tax maximization tax rate of each good is independent of the market structure. Indeed, it is always

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equal to one half the difference between the maximum of the after-tax marginal utility of that good and its cost. No element of imperfect competition, such as for instance a measure of the market-power of oligopolistic agents, is part of this expression. Furthermore, the simplicity of the optimal tax rate formulae is remarkable, allowing us to derive several identities. These all deal with the real effect of this tax rate on the prices and output and its incidence on agents’ welfare, burdens or tax revenue. Overall the real effect does not depend on the nature of the competition. Ramsey’s simple relationships still hold in the monopoly, in the Cournot and in the Bertrand frameworks. The after-tax price of good i is always the average of the maximum of the after-tax marginal utility of that good and its cost. At the after-tax equilibrium, the quantity of good i always drops by half. Note however that the more competitive the product market is, the larger the absolute value of the drop between the pre-tax level and the after-tax level is.

The introduction of the imperfect competition allows us to have a more relevant discussion on the normative incidence of the tax. Our results are twofold. First, the idea of simple relationships remains true in terms of welfare. Indeed, the after-tax values of consumer surplus and profit must be a quarter of its pretax level and the efficiency loss of taxation is equal to the quarter of the pre-tax consumer surplus and half of the pre-tax total profits. As the expression of the

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optimal tax rate is common to the four variants of imperfect competition, we can rank the total tax revenue according to the nature of the competition. Generally speaking, as the degree of market-power of the oligopolistic agents is getting smaller, the tax revenue is larger. But ranking the four expressions of the tax revenue require additional information about the substitutability or complementary of the goods. Second, our intuition about the social loss is validated. As the total profits are higher in less competitive markets, the relative social loss is getting proportionally higher. We can even show that its value varies between 1 4 and 5 12 . Indeed, following Proposition 3, the relative social loss can be computed

as follows:

⎛ ⎞ SW * − SWT* 0.25 CS * + 0.5 TP* 1 . = = 0.25 + 0.25 ⎜ * * * * ⎟ SW SW ⎝ 1 + CS TP ⎠

When there is perfect competition, we have TP* = 0 , and we obtain

SW * − SWT* 1 = . 4 SW * In

a

monopoly

environment,

we

CS * = 0.5 x ′ B x = 0.125 ( a − c ) B -1 B B -1 ( a − c ) = 0.5 TP* . TP* = 2 CS * = 2 3 SW * , we then obtain

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SW * − SWT* 5 . = * 12 SW

can So

compute we

have

Ramsey’s second result in terms of a minimum damage to social welfare still hold but for an alternative objective function for the government. Indeed Ramsey’s result derived from utility maximization does not hold in imperfectly competitive environment. We showed that the Ramsey tax rate maximizes the potential tax revenue in all markets. Utility maximization is not always good in social welfare, especially in frameworks with imperfect competition. Our specification has two advantages: the government preserves its interest as it still maximizes its tax revenue, it is not detrimental to the consumers as the consumer surplus has a higher weight than the total profits. Finally, we show that there exists a common expression of the optimum tax rate when the government wishes to minimize the damage to social welfare. To reach this optimal tax rate, the government can first fix the amount of the taxes it needs to collect, then there exists a simple adjustment process towards this tax rate.

8.

Concluding Comments

In this paper, we have studied the effects of the Ramsey tax in terms of equilibrium prices and output on the one hand and of welfare on the other hand in a contemporary framework. First, we have assessed the scope of validity of the well-known Ramsey’s

25

results and their possible extensions by considering different market structures in the product market: perfect competition, monopoly, Cournot and Bertrand oligopolies respectively. Indeed, we have shown that unit tax rates on products maximize the total tax revenue and the relationships between the pre-tax levels of the price and quantity and of different welfare measures and their after-tax levels are independent of the nature of competition considered in the product market. Second, we have analyzed the incidence of the Ramsey tax in imperfectly competitive frameworks. We have been able to demonstrate that as the degree of market power of the oligopolistic agents increases, the social loss for the economy is getting proportionally higher. Finally, we have demonstrated the Ramsey tax maximizes the potential tax revenue whatever the nature of competition considered and there exists a common expression of the optimum tax rate in all markets when the government wishes to minimize the damage to social welfare. To reach this tax rate, the government can fix the amount of the taxes it needs to collect. Then, a simple adjustment process converges towards this tax rate.

Acknowledgements We thank Gerald Pech for his comments on a preliminary version of the paper. The responsibility for any remaining errors is solely ours.

26

This work was partly completed while Laurence Lasselle was visiting the Economics Department of the European University Institute, Florence, Italy. She is thankful to the Royal Society of Edinburgh for the funding of her visit.

Appendices Appendix A Perfect Competition At the social optimum, the price is equal to the marginal cost for each non numeraire good, i.e. for all i pi* = ci . The corresponding equilibrium output is n

derived from the demand function and is equal to x = xi ( c ) = α i − ∑ β ij ci . * i

j =1

We denote the after-tax equilibrium output and price for good i by xi*,T and pi*,T . The after-tax marginal cost and the after-tax equilibrium price become respectively ci ,T = ci + ti and pi*,T = ci + ti* . The total tax revenue is

R = t′ x ( c + t ) is maximized for a tax rate of ti* = 0.5 ( ai − ci ) . Therefore the after-tax price is pi*,T = 0.5 ( ai + ci ) = 0.5 ( ai + pi* ) , the after-tax equilibrium output

⎛ a − cT ⎞ x⎜ ⎟ ⎝ 2 ⎠

.

Using

27

(2),

we

find

⎛ a − cT xT* = x ⎜ ⎝ 2

⎞ −1 −1 * * ⎟ = B ( a − c − t ) = 0.5 B ( a − c ) = 0.5 x . ⎠

Monopoly When the monopoly evaluates its profit, π M = ( p − c )′ x M , she evaluates the demand from x M = B -1 ( a − p ) . She sets p* = ( a + c ) 2 to maximize her profit computed as π M = ( p − c )′ B -1 ( a − p ) .

The marginal cost after the tax is

implemented is again equal to ci + ti . The optimal price and the corresponding output

for

given

t

are

respectively:

p*T = 0.5 ( a + c + t )

and

x*T = 0.5 B −1 ( a − c − t ) . The total tax revenue is R = t ′ B −1 [a − c − t ] and it is maximized for a tax rate of t* = 0.5 ( a − c ) . The optimal after-tax price and after-tax

output

become

pi*,T = ( 3 ai + ci ) 4 = 0.5 ( pi* + ai )

and

x*T = 0.25 B −1 ( a − c ) = 0.5 x* . Cournot Oligopoly Let us recall the first-order condition of the representative firm i when there is quantity competition in the product market: pi − ci − bii xi = 0 . Let D be an n × n positive diagonal matrix whose i-th diagonal element is bii , x * and p *

be the output and price vector in a Cournot equilibrium. The Cournot

28

equilibrium condition in a vector form then becomes:

p* = c + D x *

(5)

Equations (5) and (1) imply a - c = ( B + D ) x * . As ( B + D ) is positive definite, its inverse ( B + D )

−1

exists and is also positive definite. So we can compute the

unique Cournot equilibrium output vector:

x*T = ( B + D )

−1

(a − c)

(6)

Let denote by I the identity matrix. Substituting (6) into (1), we obtain the unique Cournot equilibrium price vector as:

p* = ( I + B D−1 )

−1

(a − c) + c

(7)

We denote by t the vector of the unit tax rate. The marginal cost vector becomes c + t . We can substitute this latter expression in (7), we then obtain the Cournot equilibrium output vector x*T = ( B + D ) The total tax revenue is then R = t′ ( B + D ) expression is concave in t as ( B + D )

−1

−1

-1

( a - c - t ) for a given t.

(a − c − t )

. Note that this

is positive definite. It is maximized

when t* = 0.5 ( a − c ) . The optimal after-tax price and after-tax output become

p*T = 0.5 ( p* + a ) and x*T = 0.5 x* . Bertrand oligopoly Let us recall the first-order condition of the representative firm i when there is

29

price competition in the product market: xi − β ii ( pi − ci ) = 0 . Let Λ be an n × n positive diagonal matrix whose i-th diagonal element is β ii , x * and p *

be the output and price vector in a Bertrand equilibrium. The Bertrand equilibrium condition in a vector form then becomes:

x* = Λ ( p - c )

(8)

Equations (8) and (2) imply α + Λ c = ( Λ + B -1 ) p* . As Λ + B −1 is positive definite, its inverse ( Λ + B −1 )

−1

exists and is also positive definite. So we can

compute the unique Bertrand equilibrium price vector:

p* = ( I + B Λ −1 )

−1

(a − c) + c

(9)

Substituting (9) into (8), we obtain the unique Bertrand equilibrium output vector as:

x* = ( B + Λ )

−1

(a − c)

(10)

Once the unit tax is implemented, the marginal cost vector becomes c + t . We can substitute this latter expression in (10), we then obtain the Bertrand equilibrium output vector xT* = ( B + Λ −1 )

−1

( a − c − t ) for a given t.

The total tax revenue is then R = t′ ( B + Λ −1 )

−1

(a − c − t ) .

It is maximized

when t* = 0.5 ( a − c ) . The optimal after-tax price and after-tax output become

p*T = 0.5 ( p * +a ) and x*T = 0.5 x * .

30

Appendix B In all models, the maximum tax revenue R* = t *′ x*T = 0.25 ( a − c )′ x * . As the social welfare before tax is SW * = ( a − c )′ x * −0.5 x *′ B x * and the consumer surplus

before

tax

is

CS * = 0.5 x *′ B x *

,

we

have

R* = 0.25 ( SW * + CS * ) = 0.25 TP* + 0.5 CS * . The after-tax social welfare SWT* = CST* + TPT* + R* = 0.25 CS * + 0.25 TP* + 0.5 CS * + 0.25 TP*

,

i.e.

SWT* = 0.75 CS * + 0.5 TP* . Hence the efficiency loss of taxation is SW * − SWT* = 0.25 CS * + 0.5 TP* . Q.E.D.

Appendix C The

consumer

burden:

CB = ( p*T − p* )′ xT*

with

x*T = B −1 ( a − pT* ) ,

p*T = 0.5 ( a + p* ) and t* = 0.5 ( a − c ) . Therefore CB = ( p*T − p* )′ B −1 ( a − pT* ) Perfect Competition

CB PC = ( c + t − c )′ B −1 ( a − c − t ) = t′ xT* = R PC Monopoly

31

* ⎛ a + c + t a + c ⎞′ * t′ xT 1 PC x CB M = ⎜ − = R ⎟ T = 2 ⎠ 2 2 ⎝ 2

In both cases, the consumer burdens are always maximized.

Cournot Oligopoly As

(p

* T

p* = ( I + B D−1 )

−1

(a − c) + c

and

p*T = ( I + B D−1 )

−1

(a − c − t ) + c + t

,

′ −1 − p )′ = ⎡⎢( I + B D−1 ) ( −t ) + t ⎤⎥ . Therefore ⎣ ⎦

(

)

′ −1 −1 CB C = ⎡I − ( I + B D−1 ) ⎤ t B −1 ⎡I − ( I + B D−1 ) ⎤ ( a − c − t ) . Let us denote ⎢⎣ ⎣⎢ ⎦⎥ ⎦⎥

I − ( I + B D−1 )

−1

by Α C . So we obtain CB C = t′A C′ B −1A C ( a − c − t ) .

∂ CB C ∂ 2 CB C The consumer burden is maximized when <0. = 0 and ∂ t ∂ t′ ∂t ∂ CB C = A C′ ( a − c − t )′ − A C t = 0 ⇔ t * = 0.5 ( a − c ) ∂t ∂ 2 CB C = −2 A C ∂ t ∂ t′ −1 ′ −1 The matrix A C is definite positive if ⎡ I − ( I + B D−1 ) ⎤ B −1 ⎡ I − ( I + B D−1 ) ⎤ ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥

is definite positive.

⎡ I − ( I + B D−1 )−1 ⎤′ B −1 ⎡ I − ( I + B D−1 )−1 ⎤ = ⎡ I − ( I + B D−1 )−1 ⎤′ B −1 B B −1 ⎡ I − ( I + B D−1 )−1 ⎤ ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ −1 ′ −1 = ⎡⎢B −1 − ( B + B D−1B ) ⎤⎥ B ⎡⎢ B −1 − ( B + B D−1 B ) ⎤⎥ which is definite positive ⎣ ⎦ ⎣ ⎦

32

if B −1 − ( B + B D−1 B )

−1

is definite positive.

Pre and after-multiply B −1 − ( B + B D−1 B )

by ( B + B D−1 B ) , we obtain:

−1

( B + B D B ) B ( B + B D B ) − ( B + B D B )( B + B D = (B + B D B) B (B + B D B) − (B + B D B) −1

−1

−1

−1

−1

−1

−1

−1

B)

−1

(B + B D

−1

B)

−1

= ( I + B D−1 )( B + B D−1B ) − ( B + B D−1B ) = B D−1B + B D−1BD−1B

which

is

definite positive.

Bertrand oligopoly As p* = ( I + B Λ −1 )

(p

* T

−1

(a − c) + c

and p*T = ( I + B Λ −1 )

−1

(a − c − t ) + c + t

,

′ −1 − p )′ = ⎡⎢( I + B Λ −1 ) ( −t ) + t ⎤⎥ . Therefore ⎣ ⎦

(

)

′ −1 −1 CB B = ⎡I − ( I + B Λ −1 ) ⎤ t B −1 ⎡ I − ( I + B Λ −1 ) ⎤ ( a − c − t ) . Let us denote ⎢⎣ ⎣⎢ ⎦⎥ ⎦⎥

I − ( I + B Λ −1 ) by Α B . So we obtain CB B = t′A B′ B −1A B ( a − c − t ) . −1

The consumer burden is maximized when

∂ CB B ∂ 2 CB B <0. = 0 and ∂ t∂ t′ ∂t

∂ CB B = A B′ ( a − c − t )′ − A B t = 0 ⇔ t * = 0.5 ( a − c ) ∂t ∂ 2 CB B = −2 A B ∂ t∂ t′

Q.E.D.

33

Whatever the nature of competition in the product market, the producer burden is equal to PB = ( p* − p*T + t )′ xT* .

Perfect Competition

PB PC = ( c − c − t + t )′ xT* = 0 Monopoly

t′ xT* 1 PC ⎛ t ⎞′ PB M = ⎜ − + t ⎟ xT* = = R 2 2 ⎝ 2 ⎠ In both cases, the producer burdens are always maximized.

Cournot Oligopoly

(

)

′ −1 −1 PB C = −t ⎡I − ( I + B D−1 ) ⎤ + t B −1 ⎡I − ( I + B D−1 ) ⎤ ( a − c − t ) ⎥⎦ ⎢⎣ ⎣⎢ ⎦⎥ −1 −1 = t ( I + B D−1 ) B −1 ⎡⎢ I − ( I + B D−1 ) ⎤⎥ ( a − c − t ) = t H C ( a − c − t ) . ⎣ ⎦

∂ PB C ∂ 2 PB C The producer burden is maximized when < 0. = 0 and ∂ t ∂ t′ ∂t ∂ PB C = H C ( a − c − t ) − t H C = 0 ⇔ t * = 0.5 ( a − c ) ∂t ∂ 2 PB C = −2 H C ∂ t ∂ t′

The matrix H C is definite positive. Indeed:

34

−1 −1 −1 −1 −1 H C = ( I + B D−1 ) B −1 ⎡⎢I − ( I + B D−1 ) ⎤⎥ = ( I + B D−1 ) B −1 − ( I + B D−1 ) B −1 ( I + B D−1 ) ⎣ ⎦

= B −1I + B −1 B D−1 − B −1 = D−1

Bertrand oligopoly

(

)

′ −1 −1 PB B = −t ⎡I − ( I + B Λ −1 ) ⎤ + t B −1 ⎡ I − ( I + B Λ −1 ) ⎤ ( a − c − t ) ⎥⎦ ⎢⎣ ⎣⎢ ⎦⎥ −1 −1 = t ( I + B Λ −1 ) B −1 ⎡⎢I − ( I + B Λ −1 ) ⎤⎥ ( a − c − t ) = t H B ( a − c − t ) . ⎣ ⎦

The producer burden is maximized when

∂ PB B ∂ 2 PB B <0. = 0 and ∂ t ∂ t′ ∂t

∂ PB B = H B ( a − c − t ) − t H B = 0 ⇔ t * = 0.5 ( a − c ) ∂t ∂ 2 PB C = −2 H B ′ ∂t∂t

The matrix H B is definite positive. Indeed: −1 −1 −1 −1 −1 H B = ( I + B Λ −1 ) B −1 ⎡⎢I − ( I + B Λ −1 ) ⎤⎥ = ( I + B Λ −1 ) B −1 − ( I + B D−1 ) B −1 ( I + B Λ −1 ) ⎣ ⎦

= B −1I + B −1 B Λ −1 − B −1 = Λ −1

Appendix D Whatever the nature of competition in the product market, the tax revenue is

(

)

equal to R = t′ x*T = t′ B −1 ( a − pT* ) = 0.5 ( a − c )′ B −1 a − 0.5 ( a + p* ) .

Perfect Competition

35

⎛a−c⎞ R PC = 0.5 ( a − c )′ B −1 ( a − 0.5 ( a + c ) ) = 0.5 ( a − c )′ B −1 ⎜ ⎟ ⎝ 2 ⎠ Monopoly

⎛ a−c ⎞ PC R M = 0.5 ( a − c )′ B −1 a − 0.5 ( a + c + 0.5 ( a + c ) ) = 0.5 ( a − c )′ B −1 ⎜ ⎟ = 0.5 R 4 ⎝ ⎠

(

)

Cournot Oligopoly C −1 −1 R C = t ′ xT* = 0.5 ( a − c )′ ( B + D ) ( a − c − t ) = 0.5 ( a − c )′ ( B + D ) ( a − c − 0.5 ( a − c ) )

−1 R C = 0.25 ( a − c )′ ( B + D ) ( a − c )

Bertrand oligopoly −1 −1 R B = 0.5 ( a − c )′ ( B + Λ −1 ) ( a − c − 0.5 ( a − c ) ) = 0.25 ( a − c )′ ( B + Λ −1 ) ( a − c )

Ranking the tax revenues: R B > RC

(B + Λ )

if

−1 −1

( a − c )′ ⎡⎣⎢( B + Λ −1 )

− ( B + D)

−1

is

−1

−1 − ( B + D ) ⎤⎥ ( a − c ) > 0 ⎦

definite

positive.

This

which

is

holds

if

equivalent

to

−1 ( B + D ) ( B + Λ −1 ) ( B + D ) − ( B + D )( B + D ) ( B + D ) to be definite positive. −1

This expression is equal to

( B + D ) ( B + Λ −1 ) ( B + D ) − B − D −1

= ( B + Λ −1 + D − Λ −1 )( B + Λ −1 )

−1

( B + D) − B − D

36

(

= I + ( D − Λ −1 )( B + Λ −1 )

−1

) ( B + D) − B − D

= ( D − Λ −1 ) + ( D − Λ −1 )( B + Λ −1 )

−1

(D − Λ ) −1

The matrix ( D − Λ −1 ) is positive definite if bii > 1 β ii for every i. Prove that bii βii > 1 . Let b i and fi be ( n − 1) × 1 sub-vectors of the ith column of

B and B −1 , without the ith element bii and β ii , respectively. Let B − i be an

( n − 1) × ( n − 1) sub-matrix of B without its ith row and ith column. Then we have:

bii β ii + fi bi = 1 bii βii + B − i fi = 0 Pre-multiplying the second equation by fi , we get bii fi β ii + fi′B − i fi = 0 . As B − i is positive definite, fi′B − i fi > 0 . So fi b i < 0 . Then the first equation implies

bii βii > 1 .

−1 R PC > R B if 0.25 ( a − c )′ ⎡B −1 − ( B + Λ −1 ) ⎤ ( a − c ) > 0 ⎢⎣ ⎦⎥

which holds if

B −1 − ( B + Λ −1 ) is positive definite. −1

This

expression

into

brackets

is

definite

positive

( B + Λ ) B ( B + Λ ) − ( B + Λ )( B + Λ ) ( B + Λ ) is definite positive. −1

−1

−1

−1

−1 −1

−1

( B + Λ ) B ( B + Λ ) − ( B + Λ )( B + Λ ) ( B + Λ ) = Λ −1

−1

−1

−1

−1 −1

37

−1

−1

+ Λ −1B −1Λ −1

if

Λ −1 + Λ −1B −1Λ −1 is definite positive.

−1 R B > R M if ( a − c )′ ( B + Λ −1 ) ( a − c ) − 0.5 ( a − c )′ B −1 ( a − c ) > 0 , which holds

if ( B + Λ −1 ) − 0.5 B −1 is positive definite. −1

( a − c )′ ( B + Λ −1 ) ( a − c ) − 0.5 ( a − c )′ B −1 ( a − c ) = −1

( a − c )′ ( B + Λ −1 )

−1

{(B + Λ

−1

) ⎡⎣⎢ 2 ( B + Λ )

−1 −1

}

− B −1 ⎤⎥ ( B + Λ −1 ) ( B + Λ −1 ) ⎦

−1

(a − c)

So we can consider: 2 ( B + Λ −1 ) − B −1 > 2 ( B + Λ −1 )( B + Λ −1 ) −1

−1

(B + Λ ) − (B + Λ ) B (B + Λ ) −1

−1

−1

−1

= B − Λ −1 B Λ −1

This expression is definite positive if the goods are substitutes, i.e. for all j ≠ i , β ij < 0 and bij > 0 , for i we know that bii βii > 1 .

Q.E.D.

−1 R C > R M if ( a − c )′ ( B + D ) ( a − c ) − 0.5 ( a − c )′ B −1 ( a − c ) > 0 , which holds if

( B + D)

−1

− 0.5 B −1 is positive definite.

( a − c )′ ( B + D ) ( a − c ) − 0.5 ( a − c )′ B −1 ( a − c ) = −1

( a − c )′ ( B + D ) {( B + D ) ⎡⎣ 2 ( B + D ) −1

−1

}

− B −1 ⎤ ( B + D ) ( B + D ) ⎦

38

−1

(a − c)

So we can consider:

2 ( B + D ) − B −1 > 2 ( B + D )( B + D ) −1

−1

( B + D ) − ( B + D ) B −1 ( B + D )

= B − D B −1 D

This expression is definite negative when the goods are complements. So we have R C < R M Q.E.D.

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