Journal of Public Economics 40 (1989) 359-367. North-Holland

COMMODITY TAXATION AND IMPERFECT COMPETITION

A Note on the Effects of Entry

Timothy BESLEY*

Woodrow Wilson School, Princeton University, Princeton, NJ 08544-1013, USA

Received August 1988, revised version received May 1989

This paper looks at the effects of allowing the number of firms to vary in Seade’s model of oligopoly and taxation. We show that the normative and positive consequences of a specific commodity tax are affected by entry in significant ways.

1. Introduction

The analysis of tax incidence in models of imperfect competition is a fairly recent undertaking. Two papers which appear in this journal are Seade (forthcoming) and Stern (1987). The objective of this note is to highlight the effects of entry into an imperfectly competitive industry when tax incidence is at issue. We do this in a model which directly extends that of Seade in the interest of making the consequences of entry as clear as possible. While Stern considers the possibility of entry, his model and its results are not directly comparable with those of Seade and hence how critical the role of entry might be is difficult to ascertain.

In the international trade literature the effects of entry have been shown to affect results quite markedly [see, for example, Horstmann and Markusen (1986)]. We shall also find this here. When the number of firms is endogenous, it is essential to consider the effects of taxation upon output per tirm and upon the number of firms in the market. We show that each of these may rise or fall in the face of a tax but that aggregate output will always fall. Normative consequences of taxation are also affected. Most striking in this respect is the fact that without entry, a small tax on an oligopoly is always welfare reducing, whereas with entry, it can be welfare increasing.

The analysis presented here complements that of Seade (forthcoming). The

*I am grateful to Kotaro Suzumura and John Vickers for discussions and to two anonymous referees for comments. This note is based on part of my Oxford University M.Phil. thesis (1985). All errors are my own.

0047-2727/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

360 T. Besley, Commodity taxarion and imperfect competition

results when entry is allowed are best thought of as being the effects of taxation in the long run, and those of Seade as being a short-run analysis. Whilst our model is similar to Seade’s,’ we make two simplifications to consider the effects of entry. First, we consider a model of Cournot oligopoly, while Seade considers a model with conjectural variations. Extension of the model presented here in this direction would be reasonably straightforward. Second, we focus on the symmetric equilibrium. Seade has considered some consequences of displacing non-symmetric equilibria.

This paper is structured as follows. In section 2 we consider the model and in section 3 we introduce taxation. Section 4 draws out some welfare implications while section 5 concludes.

2. The model

Consider an industry with n firms (2 sn5 co) producing a single homo- geneous product. We shall assume that there is Cournot competition. Let the inverse demand function be given by:

where p is price and qj is the output of firm j. We shall use Q ( -cjqj) to denote total output. We shall assume that the inverse demand function satisfies:

Assumption 1. p( ): R+ +R + is twice continuously differentiable and satisfies

p’(Q) <O for all Q such that p(Q) >O, where a prime denotes differentiation.

Below we shall introduce some further assumptions to ensure that the symmetric Cournot equilibrium is stable. The profit of firm i is:

where c( .) is the cost function which is assumed to be identical for each firm and to satisfy:

Assumption 2. c( .): R+ +R + is increasing and twice continuously differenti-

able with c(O) > 0.

‘The model of Suzumura and Kiyono (1987) is similar to that used here, except that they adopt a conjectural variation.

T. Be&y, Commodity taxation and imperfect competition 361

The assumption that c(0) is strictly positive says that there is a fixed cost faced by each firm. In analyzing the welfare properties of taxes we shall make the following stronger assumption:

Assumption 2’. c(qi) is of the form aqi + b.

In this instance we can interpret a and b as the marginal cost and fixed cost of producing an output qi, respectively. Firm i chooses its output to maximize profits given by (2). The first- and second-order conditions are:

and

n~=p( ‘) +p~( .)qi-C’( ‘)=O (3)

7&= 2pi( .) +p’,‘( .)qi-c”( .) <o, (4)

respectively. Since we shall focus entirely only on symmetric equilibria, (3) and (4) may be written, dropping the subscripts and superscripts, as:

7cq=p+p’(‘)q-c’(‘)=0

and

7cqq = 2p’ + p”q - c” < 0.

Defining [after Seade (1980a)]:

(3’)

(4’)

then (4’) may be written as:

$ {E+n+nk} CO. (4”)

Since p’( .) ~0, E + n + nk>O is necessary and sufficient for the second-order condition to hold.

We shall treat the number of firms as a continuous variable [a standard procedure in models such as this; see, for example, assume that firms enter the industry until profits are Cournot equilibrium level of output denoted by q*, number of firms must satisfy is therefore

x=p(nq*)q*-c(q*)=O.

Seade (1980a)] and will zero. Given a symmetric the condition which the

(5)

362 T. Besky, Commodity taxation and imperjkt competition

Hence the equilibrium value of n denoted as n* is determined simultaneously with q* by (5) and (3’).2 We assume throughout that an equilibrium exists. Furthermore, we impose stability conditions. In the present model, these are:

and

p’-c “<O

p”q + p’ < - t, [p’ - c”].

(64

(6b)

They are related to the stability conditions for fixed n models derived by Seade ( 1980b)3 and are equivalent, when the second-order condition holds, to the matrix in (8) below having a negative trace and a positive determinant (the RouthhHurwicz stability conditions).

3. The effects of taxation

Imagine a specific tax4 of f per unit, resulting in a profit for firm i of

ni=qip Cqj -c(qi)-tqi. ( 1 i (7)

It will be useful to write the inverse demand function, denoting competitors’ outputs by 7, as:

P=P(q+(n-1)7). (1’)

Totally differentiating (5) and (3’) once (2) is replaced by (7), and making use of (1’) and the envelope theorem, we have:

Solving (8), while making use of (7), we obtain:

(9)

‘There is nothing guaranteeing that n * is unique, see Vickers (1989). We shall, however, assume that it is. This would be so if all entry costs were sunk costs.

‘These conditions can also be found in Suzumura and Kiyono (1987). 4An earlier draft of this paper considered both ad valorem and specific taxes, which have

different effects under imperfect competition.

T. Besley, Commodity taxation and imperfect competition 363

and hence

dq sgn - = sgn E, dt

i.e. output per firm rises or falls when a tax is imposed, depending upon whether the inverse demand function is concave or convex. In Seade (forthcoming) it is shown that output per firm always decreases when a tax is imposed and since the number of firms is fixed, total industry output must also be lower. When the number of firms is allowed to vary, we need to know how n changes as well. Making use of (8) yields:

(10)

implying that the number of firms may rise or fall when a tax is imposed. Hence,

sgn$= -sgn{E+k+l}.

As with output per firm, there may be stable equilibria at which the number of firms increases with a tax. Even if n and q move in opposite directions, then total output must fall. This can be checked by considering

(11)

which is negative at stable equilibria since (6a) implies k>O. We turn next to a change in fixed costs. This is of interest since taxes or

subsidies to the set-up costs of firms are observed in practice. Hereafter, we shall adopt the cost function specification given in Assumption 2’. Proceeding analogously with the analysis of specific taxation, we have:

E+n ii- ““-‘I I p’q E+2n ’ (l-9

364 T. Besley, Commodity taxation and imperfect competition

which is ambiguous in sign. When E +n is positive,5 then output per firm increases. This is because the sign of E+n determines whether a firm’s marginal revenue rises or falls when the output of other firms in the industry changes. A rise in fixed costs reduces the number of other firms and acts just like a contraction in the output of the rest of the industry. When E +n>O, marginal revenue rises and an increase in output is required to restore the equality of marginal cost and revenue. Turning to the number of firms, there is no ambiguity:

drz n ~~ = ~_. db p’q2

(13)

which is negative after making use of (6b). This is unsurprising; a rise in the fixed cost means that a given revenue can profitably sustain the operation of fewer firms. It can also be shown that aggregate output decreases when fixed costs are increased.

4. Shifting of taxes into prices

It is also of interest to see how taxes are shifted into prices. Seade (forthcoming) showed that a necessary condition for a specific tax to be shifted by more than 100 percent when marginal costs were constant was that E < - 1.6 An analogous condition for the present model can be obtained by using the fact that

dp dQ dt =‘I dt ’

which implies, after using (1 l), that

dp ~ $1, as Eg$O. dt

(14)

If the demand function is concave, then the tax is shifted by less than 100 percent into the price, while if it is convex, it is shifted by more than 100 percent. Hence, in comparison with Seade’s case, over shifting is more likely in the free entry case. This seems intuitively reasonable; when both output and the number of firms adjust, then there is likely to be a larger shift in the price than when one of these variables (the number of firms) is held fixed.

%uzumura and Kiyono (1987) ~SSWX that E+n >O in most of their analysis. Indeed, they show that when this condition holds, then there is excessive entry into an oligopoly when firms use a Cournot pricing rule. The sign of E + n is however uncertain a priori. See also Mankiw and Whinston (1986).

‘In fact. if Assumption 2’ holds, then E< - 1 is both necessary and sufficient for over shifting.

T. Besley, Commodity taxation and imperfect competition 365

5. Some welfare implications

We turn next to the implications of taxation for economic welfare. Our analysis is limited by the fact that we shall consider only the partial equilibrium welfare measure of market surplus. Using this, we show that, contrary to the case analysed by Seade (forthcoming), welfare may rise when a tax is imposed when entry is taken into consideration. Assuming that tax revenue is redistributed back to consumers in a lump-sum fashion then total surplus (i.e. consumers’ surplus plus profits), denoted only as function of the tax rate, is given by:

act, W(t)= j {p(s)-afds-n(t)b,

0 (15)

where we have continued to make Assumption 2’ (which implies that k= 1). Differentiating (15) with respect to the tax rate, we have:

g!=(p-a) !!! -h$ (16)

Since in Seade’s case n is fixed, we can infer immediately that a small tax reduces welfare when aggregate output falls. Hence, in Seade’s model welfare is always reduced by a tax increase. Once the number of firms is allowed to vary there may however be a trade-off between effects upon the number of firms and upon output per firm. From Suzumura and Kiyono (1987) we know that a small reduction in the number of firms is beneficial when E + n >O. Generally speaking, the number of firms affects welfare since it determines how many times society incurs the fixed cost of setting up firms. On the other hand, since output is affected and price exceeds marginal cost, there is a second effect upon welfare. Hence, a tax which lowers the number of firms may make markets less competitive inducing a higher price cost margin. Making use of the zero profit condition in (16) yields:

dW dq dQ ~~ =n(p-a-t)dr+f dt dt

(17)

which shows that the welfare change depends upon the change in aggregate output und the change in output per firm. Evaluating (17) at the zero tax position, we have:

dW dq dt

--=n(p-a) 27’ (18)

366 T. Eesley, Commodity taxation and imperlect competition

and the change in output per firm is all that matters for welfare purposes. Putting together (18) with (9) and (14), we have:

Proposition 1. If E>(resp. <)O, then

(i) output per firm falls (resp. rises) with an increase in a specific tax on output;

(ii) prices rise by more (resp. less) than 100 percent in the face of a tax increase; and

(iii) welfare is reduced (resp. increased) by introducing a small specific tax on output.

If the tax were not initially zero, then the welfare analysis would be complicated by having to consider the effect of tax changes on tax revenues.

Consider next introducing a tax on set-up costs, denoted by r, which is used to finance a subsidy to output. We shall show that this strategy is unambiguously welfare improving. The government budget constraint is denoted by:

tn+tQ=O. (19)

Differentiating this with respect to r and t and evaluating at t =t=O gives the result that balanced budget strategies must satisfy:

ds

dt ” (20)

after remembering that Q/n=q. Differentiating the expression for welfare with respect to t and using (20) to keep the budget balanced and remember- ing that profits are zero, yields:

\F=n(p-a) {“;: -qz}=n(p-a);{$%]. (21)

after using (9), (12) and (20). Hence,

Proposition 2. A small tax on set-up costs used to finance a subsidy to output raises economic welfare.

This result shows that with two instruments at its disposal, the govern- ment has enough control to effect a welfare improvement. This result is in the spirit of these found in Konishi, Okuno-Fujiwara and Suzumura (1988).

T. Besley, Commodity taxation and imperfect competition 367

The reader is referred to this paper for a more extensive analysis of balanced budget welfare improvements in an oligopolistic setting.

5. Conclusion

This paper uses a simple model to place the effects of entry on tax incidence in sharp relief. We have demonstrated that entry under imperfect competition has consequences (positive and normative) for commodity taxation. We have also recognized that taxes and subsidies which affect the set-up costs of firms into an industry may be an important element of the fiscal armory under imperfect competition. Taking the view that tax incidence with entry represents a longer term effect, then this paper suggests that finding the appropriate time frame may be important in evaluating the effects of taxation in an imperfectly competitive setting.

References

Besley, T.J. and K. Suzumura, 1988, Taxation and welfare in an oligopoly with strategic commitment, typescript (All Souls College, Oxford).

Horstmann, I. and J.R. Markusen, 1986, Up your average cost curve: Inefficient entry and the new protectionism, Journal of International Economics 20, 225-247.

Konishi, H., M. Okuno-Fujiwara and K. Suzumura, 1988, Welfare improving tax-subsidy schemes in an oligopolistic setting, typescript.

Mankiw, N.G. and M.D. Whinston, 1986, Free entry and social inefficiency, Rand Journal of Economics 17, 48-58.

Seade, J.E., 1980a, On the effects of entry, Econometrica 48, 479-489. Seade, J.E., 1980b, The stability of Cournot revisited, Journal of Economic Theory 23, 15-27. Seade, J.E., Prolitable cost increases and the shifting of taxes in oligopoly, University of

Warwick economic research papers no. 260, Journal of Public Economics (forthcoming). Stern, N.H., 1987, The effects of taxation, price control and government contracts in oligopoly

and monopolistic competition, Journal of Public Economics 32, 1333158. Suzumura, K. and K. Kiyono, 1987, Entry barriers and economic welfare, Review of Economic

Studies 54, 157-167. Vickers, J.S., 1989, The nature of costs and the number of firms at equilibrium, International

Journal of Industrial Organization (forthcoming).