Ramsey Tax

Ramsey Tax in Imperfect Competition Jim Y. Jin Laurence Lasselle1 University of St. Andrews

October 2005

Abstract: In a competitive market with a linear demand and cost structure, Ramsey (1927) shows that unit taxes which reduce all product quantities by the same proportion cause the least social loss for any given tax revenue. This paper explores this result in imperfectly competitive markets: monopoly, Cournot oligopoly and Bertrand oligopoly. Our main findings are: (1) The impact of Ramsey’s proportional tax is robust in all markets, e.g. the tax revenue maximization tax rate is the same whatever the market structure. (2) If Ramsey’s proportional tax only remains efficient in monopoly, it maximizes the potential tax revenue in all frameworks. (3) We provide the efficient taxes in all markets and demonstrate that they can always be approached by the same simple adjustment process. JEL Classification Number: H20, D40 Key Words: Ramsey Tax, Revenue Maximization, Efficient tax

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Corresponding author: University of St. Andrews, School of Economics & Finance, St. Andrews, Fife, KY16 9AL, U.K. E-mail: [email protected], Tel: 00 44 1334 462 451, Fax: 00 44 1334 462 444. We thank Gerald Pech, Rahab Amir, and participants of seminars at Heriot-Watt University, the University of St. Andrews, the PET2005 Conference and the CEPET2005 Workshop for their constructive comments. The responsibility for any remaining errors is solely ours. 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.

1. Introduction In a competitive market with a linear demand and cost structure, Ramsey (1927) explained how to minimize social loss when the government has to collect a certain amount of tax revenue. He considered a quadratic utility function with n differentiated products upon which unit taxes were levied. He obtained an elegant result: the efficient tax reduces all outputs by a same proportion. For the sake of brevity, the tax issued from Ramsey’s minimization problem is labeled in this paper “Ramsey tax”. One of Ramsey’s tax features is its proportionality (…). Ramsey’s outcome has become one of the guiding principles in public economics and taxation theory in particular. However, the reality of imperfect competition in most markets today raises a question about its robustness. This paper is an attempt to answer this question by exploring his result in imperfectly competitive markets, i.e. monopoly, Cournot oligopoly and Bertrand oligopoly. Our extension is essential not only because perfect competition is increasingly rare in modern economies and Ramsey’s models with differentiated goods are more likely to be associated with imperfectly competitive markets, but also because the welfare analysis is more relevant and important under imperfect competition. As we shall see, Ramsey’s result is quite robust in all three markets. For instance, the tax revenue maximization tax rate is the same whatever the market structure. However, if the Ramsey tax only remains efficient in monopoly, it maximizes the potential tax revenue in all markets. One could wonder why this extension has not been carried out earlier. We believe that

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there are at least three reasons. First, let us remind that Ramsey’s contribution seems to have been overlooked for more than forty years. Sandmo (1976, p. 38) recalls that “in spite of its exposure to the profession the analysis seems to have fallen into oblivion for many years. It was hardly mentioned in textbooks on public finance, nor did it have any impact on the analysis of the welfare economics of the second best.” Pigou, who had suggested the research to Ramsey [see Ramsey (1927, p. 47)] did mention Ramsey’s work in the first two editions of his famous book “A study in Public Finance” (1928 p. 126, 1929 p. 130 and in a footnote p. 128) and discussed it more extensively in the third edition (1947, pp. 100-5). He stressed that “the optimum system of proportionate taxes yielding a given revenue will cut down the production of all commodities and services in equal proportions” (1929, p. 130). Ramsey’s result was rediscovered by Samuelson in a 1951 memo to the U.S. treasury. Second, Ramsey’s framework seems to be too restrictive on the demand side [see Myles (1995)], or even too simple as noted by Atkinson-Stiglitz (1980). Samuelson (1982, p. 177) called it even “too ambiguous, too ill defined”. But it found an advocate in Mirrlees (1976), where 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. Nevertheless, “it was around 1970 that there began a general revival” of the optimal taxation topic [Sandmo (1976, p. 39)] “with publication of articles by Baumol and Bradford (1970), Lerner (1970), Dixit (1970) and Diamond-Mirrlees (1971); of these, the Diamond-Mirrlees article in particular represents a major generalization and extension of Ramsey formulation”. The field was then well established and gained its recognition in top journals and textbooks. Third,

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economists often remind Ramsey’s generalized result in terms of demand elasticity for small changes in taxation, as these are often exposed in textbooks. Over the last three 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 led to, through experiments, the effects of the implementation of the different taxes. It has also moved from the strict efficiency criterion to that of optimality which is concerned with both efficiency and fairness. More recently, research has more focused on the impact of ad valorem and excise taxes on market outcomes (that is to say prices and output) on the one hand and on welfare on the other hand. 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 two taxes in a Bertrand oligopoly with differentiated products. They demonstrated that an increase in taxation could damage the consumer welfare (the consumer burden could be more than 100%) but might, under certain conditions, enlarge firms’ short profits. Denicolo and Matteuzzi (2000) considered ad valorem and other specific taxes in Cournot oligopoly. They proved that if the tax rates are sufficiently high, ad valorem tax welfare dominates specific tax in the sense that the former leads to greater tax revenue, consumer surplus, and industry profits. Fershtman et al. (1999) estimated the effects of changing tax regimes in Cournot oligopoly with differentiated products, using empirical data from the automobile market in Israel. In oligopolistic industries taxation affects the profile of goods that are sold as well as relative prices in a way that depends on the elasticity of

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demand of all products and the degree of competition in the market. Finally, Gabszewicz and Grazzini (1999) and Coady and Drèze (2002) introduced a policy dimension. 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. However, it seems that the study of the effects of the Ramsey tax on market outcomes and welfare have not been examined in imperfectly competitive markets. One obvious reason is the intractability of oligopoly markets, especially when their demand and/or costs are non-linear. In the present paper, we adapt Ramsey’s original models in four different market structures: competitive market, monopoly, Bertrand oligopoly and Cournot oligopoly. As he did, we assume a quadratic utility function which generates linear demands. This linearity restriction is necessary for our analysis of the Ramsey tax’s incidence in imperfect competition which aims at answering the following questions. When a Ramsey tax is imposed, what common results in all markets in terms of market outcomes and welfare can be obtained? Is a Ramsey tax still efficient for any given tax revenue in imperfect competition? If not, what does it maximize? In this latter case, what is then the efficient tax in imperfect competition? How can it be easily reached? Our main findings are: (1) The impact of the Ramsey tax is quite robust in all markets. (2) The Ramsey tax is efficient only in competitive market and in monopoly. It always maximizes the potential tax revenue. (3) We can evaluate the efficient taxes in all markets and prove they can always be approached by the same

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simple adjustment process. The paper is organized as follows. The next section presents our models. Section 3 investigates the common implications of the Ramsey tax in terms of market outcomes and welfare of the Ramsey tax in all markets. In Section 4, we show twofold. The Ramsey tax is always efficient in competitive market and in monopoly. It maximizes the potential tax revenue in all markets. In Section 5, we provide the efficient taxes in four markets and present an adjustment process converging to them.

2. The Models 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. Each good i is produced at a constant marginal cost ci and we denote the n × 1 cost vector by c . 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 (with c < a ) 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 good 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 . W is sufficiently high so that an interior solution exists. Since u ( x0 , x ) is strictly quasiconcave, the efficient demand vector x can be solved from the first-order condition:

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a−Bx−p = 0 . In a competitive market as considered by Ramsey, p = c . In monopoly, the prices or outputs are jointly chosen to maximize the total profit ( p − c )′ x . In quantity competition firms face an inverse demand function for a n-firm Cournot oligopoly, p ( x ) = a − Bx . By denoting the elements of B by bij 's for all i and j, we n

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

n ⎛ ⎞ i chooses its output xi to maximize its profit xi ⎜ ai − ci − ∑ bij x j ⎟ . Its first-order j =1 ⎝ ⎠

condition is pi − ci − bii xi = 0 . For all firms, we have p − c − D x = 0 , where D is an n × n positive diagonal matrix whose i-th diagonal element is bii .

In price competition the demand function for n-firm Bertrand oligopoly is

x ( p ) = α − B −1p , where α = B −1a . By denoting the elements of B −1 by β ij 's , we can n

write the demand function for firm i as: xi ( p ) = α i − ∑ βij p j . Every firm chooses its j =1

n ⎛ ⎞ − − p c α price pi to maximize its profit ( i i ) ⎜ i ∑ β ij p j ⎟ . Its first-order condition is j =1 ⎝ ⎠

xi − β ii ( pi − ci ) = 0 . For all firms, we have x − Λ ( p − c ) = 0 , where Λ is an n × n

positive diagonal matrix whose i-th diagonal element is β ii . In all four markets, each good i is taxed at a unit rate ti and we denote the n × 1 tax

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vector by t, the after-tax output being denoted by xi ,t . We define a Ramsey tax as

t = r ( a − c ) . It reduces all outputs by a same proportion k, i.e. xi*,t xi* = 1 − r , where 0 ≤ r ≤ 1 . Ramsey (1927) showed that unit taxes reducing output proportionally are efficient in competitive markets. This paper will assess whether this result still hold in three imperfectly competitive markets. For that purpose, we will first evaluate the impact of the Ramsey tax on the market outcomes and welfare. Then we will see whether this tax is still efficient in imperfect competition. If not, we will establish what it maximizes, and finally compute the efficient tax.

3. The Ramsey Tax In this section, we examine the Ramsey tax in our three imperfectly competitive markets. We analyze its consequences on the market outcomes and the welfare in four markets: a competitive market, monopoly, Cournot and Bertrand oligopolies. First of all, we solve the equilibrium output in all markets.

Proposition 1: Given a unit tax t, the equilibrium output can be written as: x*t = H ( a − c − t )

(1)

where H is a symmetric and positive matrix, equal to B −1 , 0.5 B −1 , ( B + D )

(B + Λ )

−1 −1

−1

and

in a competitive market, monopoly, Cournot and Bertrand oligopolies,

respectively. Proof: Appendix A.

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Given (1), it is easy to compute the Ramsey tax and evaluate its impact on prices and outputs.

Proposition 2: In all four markets, the Ramsey tax, ti* = r ( ai − ci ) , leads to xi*,t = (1 − r ) xi* and pi*,t = r ai + (1 − r ) pi* . The tax revenue maximization tax rate is a

Ramsey tax when r = 0.5 . Proof: see Appendix B.

Proposition 2 states that the Ramsey tax is valid in all commonly used market models. Whatever the competition considered, it is always equal to a proportion r of the difference between the marginal utility parameter ai and the marginal cost ci . The simplicity of this formula is remarkable, allowing us to derive several identities. We can evaluate not only the impact of the Ramsey tax on market outcomes, but also as we shall see below, its consequences on welfare. Proposition 2 also points out that the Ramsey tax always leads in all models to the same proportional output reduction of

(1 − r )

and price rise of r ( ai − pi* ) . Finally, Proposition 2 implies a tax revenue equal

to Rt* = t′ x*t = r (1 − r )( a − c )′ x* . A quick computation yields that it is maximized when r = 0.5 . This is a generalization of Ramsey’s result. Indeed, recall that in Ramsey’s case (competitive market) the tax revenue maximization is equivalent to the social welfare maximization. In our case, the tax revenue maximization is not only that subject to a Ramsey tax, but also that subject to any unit tax.

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Let us now study the welfare effects of the Ramsey tax. From our framework presented in

Section

2,

we

can

CS = u ( x ) − p′ x = 0.5 x′ B x

write ,

π = ( p − c )′ x = ( a − c )′ x − x′ B x

the

pre-tax

consumer

pre-tax

total

pre-tax the

the ,

SW = CS + π = ( a − c )′ x − 0.5 x′ B x ,

the and

surplus

(CS)

as

profit

(π)

as

social

welfare

(SW)

as

after-tax

social

welfare

as

SWt* = CSt* + π t* + Rt* . By using the results from Proposition 2, we find: Proposition 3: Given a Ramsey tax t = r ( a − c ) with 0 < r ≤ 0.5 , the profit, consumer surplus and social welfare are: π i*, t = (1 − r ) π i* , CSt* = (1 − r ) CS * , and 2

2

SWt* = (1 − r 2 ) SW * − r (1 − r ) π * . Proof: see Appendix C.

Simple and uniform relationships between pre-tax and after-tax expressions can be extended to the welfare measures. While profits and consumer surplus always fall to

(1 − r )

2

of their pre-tax levels in all four markets, the relative social loss depends on the

nature of competition. Recall that in a competitive market, π * = 0 , the Ramsey tax leads to the loss of (1 − r 2 ) of the pre-tax social welfare. As market power increases, π * is not nil anymore and becomes higher, losses go up. Indeed, recall that in monopoly

π * = 2CS * , therefore SWt* = (1 − r )(1 + r 3) SW * .

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Let us now turn to the amount of tax revenue and its incidence on consumers and producers. Given our notations, the consumer burden and the producer burden can be written respectively as CB = ( p*t − p* )′ x*t and PB = ( p* − p*t + t )′ x*t .

Proposition 4: When a Ramsey tax is imposed, the tax incidence CB PB equals 2CS * π * . When the tax revenue is maximized, the consumer burden and producer burden reach their maximum of 2 CSt* and π t* respectively. Proof: see Appendix D. This proposition tells us that the maximum tax revenue is equal to 2 CSt* + π t* . Although the expression of the Ramsey tax is identical in all four markets, the corresponding revenues are different. Indeed, given t = r ( a − c ) , the corresponding tax revenue could be ranked according to the nature of the competition, but only if outputs can. Unfortunately, this is not the case. Although, the equilibrium quantities are the highest in a competitive market, they cannot always be comparable in imperfect competition. For instance, Bertrand and Cournot outputs cannot be compared with monopoly ones in a model with a mixture of substitute and complement goods [see Amir and Jin (2001)]. Nevertheless, revenue comparison for a given Ramsey tax leads to an interesting result.

Proposition follow:

5:

Given

any

2 RtM = RtPC > RtB > RtC

Ramsey .

In

tax

t = r (a − c) ,

addition,

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if

the

goods

tax are

revenues substitutes

( ∂xi ∂ p j ≥ 0 ), RtB > RtM , for complements ( ∂ pi ∂x j ≥ 0 ), RtM > RtC . Proof: see Appendix E.

Proposition 5 implies that for the same Ramsey tax, the more competitive a market is, the higher the outputs are, and so is the tax revenue.

Up to now, we have demonstrated that the Ramsey tax yields a number of simple relations for all four markets. Our next work is to check whether this tax is still efficient for any given tax revenue in imperfect competition. If not, we will need to ask what this tax maximizes then. To facilitate the understanding of our forthcoming results, let us consider two unit taxes imposed sequentially. At a given time, the government decides to levy a unit tax t and receives a tax revenue Rt = t′ H ( a − c − t ) . After that, if the government wants to implement an additional unit tax τ , it will receive a new revenue τ′ H ( a − c − t − τ ) . The maximum amount of revenue that this new tax brings up is

defined as “the potential tax revenue” left over by the current tax rate t. It is easy to show that τ * = 0.5 ( a − c − t ) , so Rτ* t = 0.25 ( a − c − t )′ H ( a − c − t ) .

Proposition 6: The Ramsey tax t = r ( a − c ) with 0 < r ≤ 0.5 , only maximizes the social welfare in a competitive market and in monopoly. Given any level of current tax rate , the

Ramsey

tax

always

maximizes

the

R ( t ) = 0.25 (1 r − 1) Rt .

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potential

tax

revenue

which

is

Proof: see Appendix F.

Proposition 6 implies the possibility of non efficiency of the Ramsey tax in imperfect competition. In the case of a competitive market, as profits are always zero, the potential revenue is equal to 2 CSt* , which is twice of the social welfare, therefore the Ramsey tax is efficient. Note that we also obtain SWt* = 0.5 (1 r − 1) Rt . In monopoly, π t* = 2CSt* , the potential tax revenue is 4 3 of the social welfare. Maximizing the potential tax revenue will again automatically make the tax efficient as in the competitive case. However, as Proposition 6 states, the Ramsey tax is not efficient in Cournot and Bertrand oligopolies. But it maximizes an alternative objective for the government whatever the market structure considered: its potential tax revenue, for a current level of taxation.

4. The Efficient Tax Rate Since Ramsey’s proportional tax is no longer efficient in imperfect competition, a natural question arises: what is the efficient tax in all markets? If, as argued by Ramsey (1927), the government wants to minimize the damage to social welfare for any given tax revenue, its tax rate should maximize a linear function of its tax revenue and the social welfare: Max Rt + γ SWt , where γ ≥ 0 , Rt = t′ x t and t

SWt = CSt + π t + Rt = ( a − c )′ x t − 0.5 x′t B x t . The expression of the tax rate derived from this maximization problem is not as simple

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as that of the Ramsey tax, especially in Cournot and Bertrand oligopolies. Let us emphasize that in this section our goal is to identify a uniform formula for the efficient tax applicable in all markets, and the corresponding pre-and after-tax relations for the market outcomes.

Proposition 7: In all four markets the efficient tax is t * = ( I − S )′ ( a − c ) where

S = (1 + γ )( 2I + γ B H )

−1

.

The

outputs

and

prices

are

x*t = S′ x*

and

p*t = ( I − S )′ a + S′ p* . Proof: Appendix G.

The efficient tax expressed in Proposition 7 cannot be called “Ramsey tax” as it does not always reduce all output proportionally. Nevertheless, it remains the “efficient Ramsey tax” in competitive and monopoly markets. In the former case, H = B −1 , S =

t* =

1+ γ I , so 2+γ

1 1+ γ 2−γ I , so t * = ( a − c ) . In the latter, H = 0.5 B −1 , S = ( a − c ) . In 2+γ 2 + 0.5γ 4+γ

Cournot and Bertrand competitions, as H = ( B + D ) and H = ( B + Λ −1 ) , S becomes −1

more

complicated

and

−1 (1 + γ ) ⎡⎣⎢ 2I + γ B ( B + Λ −1 ) ⎤⎦⎥

equal

to

−1

(1 + γ ) ⎡⎣ 2I + γ B ( B + D )

−1 −1

⎤ ⎦

and

−1

respectively.

Although we still have uniform formulae for the tax and the market outcomes in all four

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markets, the precise relationship of the tax rate and that of between the market outcomes are all different due to different values of S, in contrast to Proposition 2. Given the complicated formula for the efficient tax, one would ask how to find its exact value, especially in asymmetric Cournot and Bertrand markets, where the matrix B may be too complex. This problem can be solved by an adjustment process through market responses. We are able to show that there exists a uniform updating rule which always leads to the efficient tax in all four markets.

Proposition 8: If γ < 2 , the adjustment process of t k = 0.5 ⎡⎣a − (1 − γ ) c − γ p k −1 ⎤⎦ always converges to the efficient tax rate t* stated in Proposition 7 in all four markets. Proof: see Appendix H. If the government starts with a fixed γ (the marginal revenue increase for a unit of social loss), it will eventually end up with a certain amount of tax revenue causing the least social damage.

5. Concluding Comments This paper has assessed the scope of validity of the well-known Ramsey’s result in terms of efficiency in simple frameworks with imperfect competition and shows that it is wider than one could expect. First, we have demonstrated that the Ramsey tax implies simple and uniform output and price relationships between their pre- and after-tax levels. This part of work also

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established relations regarding profit, consumer surplus, social welfare, agents’ burdens and tax revenue. Second, we have shown that the efficiency property of the Ramsey’s proportional tax has been lost in imperfect competition except for the monopoly case. The Ramsey tax maximizes an alternative objective function: the potential tax revenue in all four markets. Third, we have given the expression of the efficient tax for each of the four markets. Although this expression is not identical in all markets, there exists a unique formula in all markets. Finally, we provide a simple adjustment process which always converges towards this tax in the four markets. The linear demand and cost structure are obviously a limitation of our model. However, it may be a compromise worthwhile to make in order to find tractable solutions in imperfect competition. It would be interesting to further investigate to what extent our results can be generalized to non-linear cases.

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Appendices Appendix A: We first consider the equilibrium output without tax, x* . In a competitive market, p = c . As p = a − B x , we obtain c = a − B x . Therefore, x* = B −1 ( a − c ) . In monopoly,

the

firm

chooses

its

output

to

maximize

its

profit

π = ( p − c )′ x = ( a − c − B x )′ x . Its first-order condition is a − c − 2B x = 0 , leading to x* = 0.5 B −1 ( a − c ) . In Cournot oligopoly, the first-order condition is p − c − D x = 0 . As

p = a − B x , we have a − c = ( B + D ) x , so x* = ( B + D )

−1

(a − c)

. In Bertrand

oligopoly, the first-order condition is x − Λ ( p − c ) = 0 . As p = a − B x , we have

Λ −1x − ( a − c − B x ) = 0 , so x* = ( B + Λ −1 ) . Since the equilibrium output vectors have −1

all the same structure, we can write them as x* = H ( a − c ) , where H is a symmetric and positive matrix, which is equal to B −1 , 0.5 B −1 , ( B + D )

−1

and ( B + Λ −1 ) in a −1

competitive market, monopoly, Cournot and Bertrand oligopolies, respectively. Given the unit tax t, firms’ marginal costs become c + t. So we can write x*t = H ( a − c − t ) .

Appendix B: x*t = H ( a − c − t ) = (1 − r ) H ( a − c ) = (1 − r ) x* .

p*t = a − B x*t = a − (1 − r ) B x* = r a + (1 − r ) ( a − B x* ) = r a + (1 − r ) p* . Since the equilibrium output can be written as x*t = H ( a − c − t ) , the tax revenue

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Rt = t′ x*t = t′ H ( a − c − t ) . From

∂Rt = H ( a − c − 2 t ) = 0 , we find t* = 0.5 ( a − c ) . ∂t

∂ 2 Rt = −2H is negative definite, as H is always positive definite in the four ∂t 2

Note that

markets. Hence the second-order condition holds.

Appendix C: As x*t = (1 − r ) x* , CSt* = 0.5 x*t ′B x*t = 0.5 (1 − r ) x*′B x* = (1 − r ) CS * . 2

2

π i*,t = ( pi*,t − ci − ti ) xi*,t = ⎡⎣ r ai + (1 − r ) pi* − ci − r ( ai − ci ) ⎤⎦ xi*,t = (1 − r ) ( pi* − ci ) xi*,t i

2 = (1 − r ) π i* . Recall Rt* = t*′x*t = r (1 − r )( a − c )′ x* , SWt* = CSt* + π t* + Rt* 2 = (1 − r ) ( CS * + π * ) + r (1 − r )( a − c )′ x* .

But

( a − c )′ x* = ( a − B x* − c )′ x* + x*′B x* = π * + 2CS *

.

Hence

SWt* = (1 − r ) ( CS * + π * ) + r (1 − r ) ( 2CS * + π * ) = (1 − r 2 ) SW * − r (1 − r ) π * . 2

Appendix D: CB = ( p*t − p* )′ x*t = r ( a − p* )′ x*t = r x*′ B x*t = r (1 − r ) x*′ B x* = 2r (1 − r ) CS * ′ PB = ( p* − p*t + t )′ x*t = ⎡⎣ r ( p* − a ) + r ( a − c ) ⎤⎦ x*t = r ( p* − c )′ x*t = r (1 − r ) ( p* − c )′ x* = r (1 − r ) π * . Therefore CB PB = 2CS * π * .

Moreover, with x*t = B −1 ( a − p*t ) , CB = ( p*t − p* )′ B −1 ( a − p*t ) . Simple derivations yield

∂CB ∂p*t = B −1 ( a − 2p*t + p* ) = 0

when

p*t = 0.5 ( a + p* ) .

Furthermore,

∂ 2CB ∂p*t 2 = −2 B −1 is negative definite. The consumer burden is maximized with r = 0.5 . Its value is 0.5 CS * = 2CSt* .

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The

producer

burden

is

PB = ( p* − p*t + t )′ x*t

.

As

p*t − p* = t′ B H

PB = t′ ( I − B H )′ x*t = t′ ( I − H B ) H ( a − c − t ) = t′ ( H − H B H )( a − c − t )

derivations

yield

.

∂PB ∂t = ( H − H B H ) ( a − c − 2t* ) = 0

,

Simple and

∂ 2 PB ∂t 2 = −2 ( H − H B H ) = −2H ( I − B H ) . The latter is positive definite if H −1 − B

is positive definite which is true for all markets. So the producer burden is maximized. With r = 0.5 , its value is 0.25 π * = π t* . Appendix

E:

The

tax

revenue

is

always

equal

to

Rt = t′ x*t = t′ B −1 ( a − p*t ) = r ( a − c )′ B −1 (1 − r ) ( a − p* ) = r (1 − r )( a − c )′ B −1 ( a − p* ) = r (1 − r )( a − c )′ B −1 ( a − p* ) = r (1 − r )( a − c )′ x* . Hence, we don’t need to consider r when we compare tax revenues, outputs’ comparison is sufficient. (1) Prove 2 RtM = RtPC . In a competitive market, x* PC = B −1 ( a − c ) and in monopoly, x* M = 0.5 B −1 ( a − c ) . So 2x* M = x* PC and hence 2 RtM = RtPC . −1 (2) Prove RtPC > RtB . It holds if ( a − c )′ ⎡B −1 − ( B + Λ −1 ) ⎤ ( a − c ) > 0 . ⎢⎣ ⎦⎥

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

−1

( a − c )′ B −1 ⎣⎡( B + Λ −1 ) − B −1 ⎦⎤ ( B + Λ −1 ) ( a − c ) = x′PC Λ −1x M > 0, as Λ −1 is positive definite. −1

−1 −1 (3) Prove RtB > RtC . It holds if ( a − c )′ ⎡( B + Λ −1 ) − ( B + D ) ⎤ ( a − c ) > 0 , that holds if ⎢⎣ ⎦⎥

( a − c )′ ⎡⎢⎣( B + Λ −1 ) ( a − c )′ ( B + Λ −1 )

−1

−1

−1 −1 −1 − ( B + D ) ⎤⎥ ( a − c ) = ( a − c )′ ( B + Λ −1 ) ⎡I − ( B + Λ −1 ) ( B + D ) ⎤ ( a − c ) = ⎣ ⎦ ⎦

⎡⎣B + D − B + Λ −1 ⎤⎦ ( B + D )

−1

( a − c ) = x′B ⎡⎣ D − Λ −1 ⎤⎦ xC

D − Λ −1 is positive matrix if bii > 1 β ii for every i. This is true as B is positive definite [see Amir and Jin (2001)]. (4) Prove RtB > RtM . Let us first compare the values of x B and x M .

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−1 −1 x B − x M = ⎡⎢( B + Λ −1 ) − 0.5 B −1 ⎤⎥ ( a − c ) = ( B + Λ −1 ) ⎡⎣I − 0.5 ( B + Λ −1 ) B −1 ⎤⎦ ( a − c ) . A few ⎣ ⎦

computations yield x B − x M = ( B + Λ −1 ) ⎡⎣ B − Λ −1 ⎤⎦ x M using x M = 0.5B −1 ( a − c ) . −1

Recall first that bii − 1 β ii > 0 . If all goods are substitute, ∂xi ∂ p j ≥ 0 for i ≠ j , so

β ij ≤ 0 which implies bij ≥ 0 . Recall that bii − 1 β ii > 0 , therefore x B > x M which implies RtB > RtM . (5)

Prove

RtC < RtM

:

We

first

write

their

difference

−1 ⎛ a − c ⎞′ ⎡ −1 ⎛ a − c ⎞ as RtC − RtM = ⎜ ⎟ ⎣( B + D ) − 0.5B ⎦⎤ ⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ −1 ⎛ a − c ⎞′ −1 =⎜ ⎟ ( B + D ) ⎡⎣ 2I − ( B + D ) B ⎤⎦ ( a − c ) ⎝ 8 ⎠

B −1 −1 ⎛ a − c ⎞′ RtC − RtM = ⎜ (a − c) ⎟ ( B + D ) [ B − D] 2 ⎝ 4 ⎠

.

.

A

Recall

few

computations

xC = ( B + D )

−1

(a − c)

yield

and

x M = 0.5B −1 ( a − c ) , so RtC − RtM becomes: C M 0.25 xC′ [ B − D] x M . If all goods are complements, bij ≤ 0 , and therefore Rt < Rt .

Appendix F: Given t, a new tax τ generates a revenue Rτ = τ′ H ( a − c − t − τ ) . The maximum value of this tax revenue is equal to 0.25 ( a − c − t )′ H ( a − c − t ) . Now we can

easily

show

that

t = r (a − c)

maximizes

the

expression

L:

L = t′ H ( a − c − t ) + 0.5 (1 − 2r )( a − c − t )′ H ( a − c − t ) (1 − r ) . When t = r ( a − c ) , one can

verify

∂ L ∂ t = H ⎡⎣a − c − 2t − (1 − 2r )( a − c − 2t ) (1 − r ) ⎤⎦ = 0

and

∂ 2 L ∂ t 2 = − H (1 − r ) is negative definite. So t maximizes the potential tax revenue.

Appendix G: Define the objective function L = t′ x t + γ ⎡( a − c )′ x t − 0.5 x t′ B x t ⎤ . As ⎢⎣ ⎥⎦

19

xt = H ( a − c − t )

derivations

,

L = ⎡⎣ t + γ ( a − c ) − 0.5γ B H ( a − c − t ) ⎤⎦′ H ( a − c − t ) .

Simple

∂ L ∂ t = H ⎡⎣a − c − 2t − γ ( a − c ) + γ B H ( a − c − t ) ⎤⎦

and

yield

∂ 2 L ∂ t 2 = −2 H − γ H B H which is negative definite. So L is maximized when ∂ L ∂ t = 0 , i.e. when ( 2 I + γ B H ) t = ⎡⎣(1 − γ ) I + γ B H ⎤⎦′ ( a − c ) . So we solve the efficient

tax

−1 t * = ( 2I + γ BH ) ⎡⎣(1 − γ ) I + BH ⎤⎦′ ( a − c ) = ( I − S )′ ( a − c )

where

S = (1 + γ )( 2I + γ B H ) . −1

x*t = H ( a − c − t * ) = HS′ ( a − c ) = (1 + γ ) ( 2H −1 + γ B )

−1

(a − c)

= (1 + γ )( 2I + γ HB ) H ( a − c ) = S′x* . −1

p*t = a − Bx*t = a − B S′ x* = a − (1 + γ ) B ( 2I + γ HB ) x* = a − (1 + γ ) ( 2B −1 + γ H ) x* −1

−1

−1 = a − (1 + γ )( 2I + γ BH ) Bx* = a − S′ ( a − p* ) = ( I − S )′ a + S′p* .

Appendix H: First note that a − c − 2t * − γ ( a − c ) + γ B H ( a − c − t* ) = 0 can be rewritten as: a − c − 2t * − γ ( a − c − B xT* ) = a − c − 2t* − γ ( pT* − c ) = 0 , so we can write the efficient tax t* as 0.5 ⎡⎣a − c − γ ( p*T − c ) ⎤⎦ . Hence we have t k − t * = 0.5 γ ( p*T − p k ) . But p k = a − B H ( a − c − t k −1 ) , so p k − p* = B H ( t k −1 − t* ) . Combining these two equations of tax and price differences, we obtain t k − t* = 0.5γ B H ( t k −1 − t* ) . By using our positive definite matrix B −1 , we can construct a non-negative series wk = ( t k − t* )′ B −1 ( t k − t* ) = 0.25γ 2 ( t k −1 − t* )′ HBH ( t k −1 − t * ) . t k converges to t* , if wk

converges

to

zero.

Since

wk +1 − wk = 0.25γ 2 ( t k − t* )′ HBH ( t k − t * ) − ( t k − t* )′ B −1 ( t k − t* ) .

20

we

know

= ( t k − t* )′ H ( 0.25γ 2 B − H −1B −1H −1 ) H ( t k − t* ) , the series of wk decreases for t k ≠ t* if the matrix K = 0.25γ 2 B − H −1B −1H −1 is negative definite. In a competitive market, H = B −1 , K = ( 0.25γ 2 − 1) B , which is negative definite for

γ < 2 . In monopoly, H = 0.5 B −1 , K = ( 0.25γ 2 − 4 ) B , which is negative definite for γ < 4 . In Cournot competition, H = ( B + D ) , K = ( 0.25γ 2 − 1) B − 2D − DB −1D . In −1

Bertrand competition, H = ( B + Λ −1 ) , K = ( 0.25γ 2 − 1) B − 2Λ −1 − Λ −1B −1Λ −1 . They −1

are both negative definite if γ < 2 . Hence, wk must have a limit, i.e. lim ( wk +1 − wk ) = 0 . k →∞

As K is negative definite, wk +1 − wk = ( t k − t * )′ K ( t k − t* ) = 0 only if t k = t* , i.e. lim {t k } = t* . k →∞

21

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