- Email: [email protected]
Ramsey tax rules for economies with imperfect competition
Journal
of Public
RAMSEY
Economics
38 (1989) 95115.
TAX RULES
FOR ECONOMIES COMPETITION Gareth
University Received
September
North-Holland
WITH
IMPERFECT
D. MYLES*
of Warwick,
Coventry
1987, revised version
CV4 7AL. UK received
November
1988
This paper derives optimal commodity tax rules for a general equilibrium model with imperfect competition. To achieve this, functions are constructed that describe the effects of taxation upon the equilibrium prices and profit levels of imperfectly competitive industries. The use of these functions allows existing tax rules to be generalised to accommodate imperfect competition.
1. Introduction This paper extends the theory of optimal commodity taxation to a general equilibrium model with imperfect competition. Important features of the model are non-linear responses of prices to tax changes, the connections between industries via their demand functions and the effects of taxation upon profits. Tax rules are constructed for a single-consumer ‘efficiency’ model and for a many-consumer model that incorporates considerations of equity. Industrial conduct is shown to be an important determinant of relative tax rates. Previous analyses of optimal taxation have been within the Arrow-Debreu paradigm of perfect competition, an avenue of research exemplified by Diamond and Mirrlees (1971). However, many issues in taxation, such as duties on whisky, petrol and cigarettes, are concerned with industries that are far from the competitive ideal. Although Myles (1987b) makes some progress in analysing taxation for a specific partial equilibrium model, no general equilibrium analysis of taxation and imperfect competition has yet been conducted. This paper is a first step in that direction. When designing optimal tax systems there are two aspects that are of critical importance in distinguishing imperfect from perfect competition: industries that are imperfectly competitive price above marginal costs so taxes may be over- or under-shifted and their behaviour is determined *I wish to thank invaluable comments W7-2727/89/$3.50
Terence Gorman, and criticisms. 0
Norman
Ireland
1989, Elsevier Science Publishers
and
an anonymous
B.V. (North-Holland)
referee
for their
96
G.D. Myles, Tax rules for economies with imperfect competition
conditionally upon other industries’ prices. Seade (1985) Stern (1987) and Myles (1987b) have analysed these features from a partial equilibrium viewpoint; here the insights of those papers are embedded within a general equilibrium model and this is employed to illustrate how existing tax rules are modified to accommodate the presence of imperfect competition. The paper is organised as follows: section 2 describes the model and the analytical method employed, the required analysis of imperfectly competitive industries is provided in section 3, and section 4 presents the one-consumer tax rule; this is generalised to many-consumers in the appendix. The implications of this rule for tax-setting and the importance of industrial conduct in determining relative taxes are discussed in section 5. Conclusions are given in section 6. 2. Description of the model The model consists of households, firms and a government. Households supply labour, demand goods and receive profits; they are described by their indirect utility functions and shareholdings in firms. Each firm belongs to a single industry; there is no joint production. Industries are either perfectly or imperfectly competitive; all firms comprising a perfectly competitive industry produce with the same constant returns to scale production technology. The role of the government is to choose the level of commodity tax rates to maximise a concave function of individual utilities subject to attaining a given level of revenue. To simplify the analysis I assume that labour, which is untaxed, is the only input into production and use it as numeraire throughout the paper; hence the wage rate remains constant at w. In conjunction with the assumption that competitive firms produce with constant returns to scale, this provides a set of effectively fixed pre-tax prices for these industries; their post-tax prices are then linear functions of tax rates. It also removes a number of difficult, and complicating, issues relating to the pricing of intermediate goods produced by imperfectly competitive firms. A tentative treatment of the model without this restriction can be found in Myles (1987a). In respect of the normalisation employed by Diamond and Mirrlees (1971), it is important to note that both the receipt of profits by consumers [Munk (1978)] and the existence of imperfect competition are individually sufficient to invalidate the separate normalisation of producer and consumer prices. Therefore, although using labour as numeraire does not affect the real equilibrium of the model [Cripps and Myles (1988)], the choice of labour as an untaxed good is not a trivial one. Three points can be made with respect to this choice. First, labour is the only good in the model supplied by households and consequently does play a different role to the taxed commodities. Secondly, many existing tax systems treat labour as distinct from other commodities and the
G.D. Myles, TAX rules for economies with imperfect competition
91
model reflects this fact. Thirdly, a tax upon labour can easily be incorporated into the model if desired and a rule derived for determining its optimal value. To analyse the optimal tax problem, the effects of tax rates upon prices and profits must be determined. The assumptions placed above upon the production processes of competitive industries imply that their equilibrium profits are zero and that their post-tax prices are the fixed pre-tax prices plus tax. Without the addition of imperfect competition the model is therefore equivalent to that of Atkinson and Stiglitz (1972). For more general production technologies it would be possible to follow Diamond and Mirrlees (1971) and control the competitive sector using the producer prices that support the equilibrium. The government choice variables would then be the consumer prices for goods produced competitively and the tax rates of those produced by imperfectly competitive firms; see also the discussion following eq. (12). In contrast, imperfectly competitive industries require rather more analysis. Consider first the determination of equilibrium prices. Each firm in the industry makes an output/price decision contingent upon the demand function they face, their perception of rivals’ behaviour and upon their cost structure, which includes the commodity tax on output. However, in a general equilibrium model, demand is contingent upon all prices in the economy. Consequently the equilibrium industry price is affected by the tax system in two ways. First, a variation in the tax rate upon a good represents a change in the cost structure of each firm producing it. This cost change will modify the profit-maximising decisions of these firms which, in turn, affect the equilibrium price of the good. This is the ‘direct’ effect of taxation and is present in both perfect and imperfect competition. The size of this effect is fixed for perfect competition but, in contrast, is dependent upon market structure and conduct when competition is imperfect; see in particular Stern (1987). Secondly, there are ‘induced’, or ‘indirect’ effects: taxes on other industries cause changes in their equilibrium prices, these price changes affect demand, and hence equilibrium price, for the industry in question. This effect is not present in perfect competition when producer prices, as here, are tixed. Seade (1985) and Stern (1987) have previously analysed the direct effects of taxation for a variety of market structures. Myles (1987b) has derived equations describing induced effects for a model of Cournot oligopoly. The approach taken was to differentiate the first-order conditions for profit maximisation and then solve the resulting equations for the derivative of equilibrium price with respect to either the tax rate or for prices that enter the demand function. In contrast, the approach used below is to solve directly for the functional dependence of each imperfectly competitive industry’s price upon the tax rate it faces and the prices of other goods, the derivatives of these functions are formally equivalent to those derived in previous papers, and to use these functions as the components of a general
98
G.D. Myles, Tax rules for economies
with imperfect competition
equilibrium model. The general equilibrium effects of taxation are then found by perturbing the complete system. The advantage of this approach is that it provides an organising idea that can be applied to many forms of imperfect competition and allows the structure of optimal tax rules to be analysed without requiring a precise specification of the nature of imperfect competition; only at the stage of application do the functions need to be specified. It also permits the analysis of taxation to be conducted within a general equilibrium model of imperfect competition. To this point I have concentrated on prices, all that has been said above applies equally to the profits of firms in imperfectly competitive industries. In an equivalent manner, there are direct and induced effects of taxation upon profits and functions will be constructed that capture these effects.
3.
Imperfect competition
and the effects of taxation
A central feature of the analysis below is the use of functions that govern the price of each good produced under conditions of imperfect competition, and the profits of each firm producing it, which have as their arguments the control variables of the government. The construction of these functions will be described for quantity-setting industries with fixed numbers of firms in this section. Quantity-setting with free-entry and price-setting industries are analysed in the appendix. At this stage, the precise interaction between firms does not need to be stated; the general equations derived encompass alternative forms of conjectural assumptions. Assuming that there are N goods available in the economy, these are partitioned into two subsets, P and I, with cardinality K and N-K, respectively. P contains all goods produced under constant returns to scale by perfectly competitive firms; the goods in P are indexed i= 1,. . . , k,. . . , K. I contains goods produced by imperfectly competitive industries; these are indexed i=K+l,..., r,..., N. As labour is the numeraire and the only input into production, the pre-tax price, pk, of a typical good k E P is predetermined at ckw, where w is the fixed wage rate and ck is the labour input per unit of k. Writing the tax rate on good k as t,, the post-tax price qk=pk+ t,, note that a&/& = 1. For each good rel that is produced by a quantity-setting industry with a fixed number of n, firms, n,L 1, and with each firm, j, in this industry earning profits rr!, it is assumed that the industry faces a demand function:
x,=x,
ql,...,qk,...,qK,qK+l,...,q,,...,qN,n-l+
2 dw j=l
> 3
(1)
G.D. Myles,
99
Tax rules for economies with imperfect competition
where X, is aggregate demand and n
-r=i=g+l jgl4 i#r
i.e. aggregate profit less that of the industry producing good r. The demand function has partial inverse:
q,=q,
qt ,*..,
qk ,...,
qK,qK+l,...,
x,3 . ..Y qNYxr+
2
n!;w
j=l
(
9
(2)
)
with X,=&i xi, xi the output of good I by the jth firm in the industry. Each firm chooses its optimal output level, xp, to maximise profits taking other industries prices as given: x!*=argmax{nj=xjq,-Cj(x:‘;w)-&xi},
j=l,...,n.
(3)
The output choices arising from this maximisation will be a function of the prices entering (2), the tax rate t,, profit income generated by all other industries K,, and the wage rate w; the form of the function will be partly determined by the oligopolistic interaction. Sufficient conditions that guarantee the existence of such maxima are derived in Cripps and Myles (1988); these essentially require the income derivative 8X,/&r to be bounded below 1 for large n. Assuming optimal choices to exist, they can be written: xp=arj(q1,...,qk,...,qK,qK+1,...,tr,...,qN,X-,;W), and aggregating over the
x;=
5 d’(.)=o’(q,
n,
j=l,
. . . . n,
(4)
firms: )...,
qk )...,
qK,qK+l,...,
t I,...)
qN,Lr;W).
(5)
j=l
Substituting (5) into the partial inverse demand function characterises the equilibrium prices as a function of other prices, the tax rate on good I, profit income and the wage rate: q*=f’(41,...,qk,...,qK,qK+1
,..‘, tI,..., q‘V,rL;w).
(6)
If (4) and (6) are substituted into the definition of firm j’s profit level, equilibrium profit can be expressed as: ~TI!=gj’(ql,...,qk,...,qK,qK+1
,..‘, t I,..., %v,n - ,; w).
(7)
The derivatives of the functions f’ and gj’ are equivalent to the partial equilibrium effects of taxation analysed by Seade (1985), Stern (1987) and Myles (1987b) for a number of market structures. In particular, it is noteworthy that af’/at,= f”: is >O and may be > 1, which represents
100
G.D. Myles,
Tax rules for economies with imperfect competition
over-shifting of taxes, when a condition on the convexity of the demand function is satisfied and that 3gP/atr-gJ;I can be positive, see Seade (1985). Both f: and gc, s#r, can be of either sign, the important determining factor being the sign of the second cross-derivative of inverse demand, a2q,/dX,dq, [Myles (1987b)]. The complete model of the imperfectly competitive sector consists of N-K equations of the form of (6) describing the determination of each imperfectly competitive price and crzK+ 1 n, functions determining each firm’s profit level (for free-entry industries where profits are delinitionally zero the g functions are not required, see appendix). Hence for given consumer prices for competitive goods and tax rates upon imperfectly competitive goods, the equilibrium in the imperfectly competitive sector is the simultaneous solution to eqs. (8) below:
qK+1
=fK+’
ql,...,qK,qK+lr
qK+2=fK+’
7(y =
q1,...,qK,tK+1,qK+2,...,qN,
g”
ql,...,qK,qK+l,...rtN,
t K+2r...,qN,
N
“,
2 i=K+2
c j=l
$;w
2 i=K+l i#K+2
f j=l
dw
, )
, )
(8)
To determine optimal tax policy it is necessary to calculate the effect that a change in policy will have upon the equilibrium. With constant returns in the competitive sector, the parameters under control of the government can be taken to be either the tax rates, t,,.. . , t,, or, recalling the process for price formation in the competitive sector, the prices ql,. . . ,qK together with the tax rates tK+ 1,. , t,. In what follows I choose to use the latter. There are two approaches that can be taken at this point: the implicit function theorem can be appealed to and (8) solved to give qK + 1,. . . , ~“~7~ in terms of the control variables or, as it is only the value of the derivatives of endogenous variables with respect to the control variables that are important
101
G.D. Myles, Tax rules for economies with imperfect competition
in the sequel, the system can be differentiated the latter and differentiating (8):
[*I
rr_
=
and solved.
Proceeding
with
(9)
LB]
or Adl=Bdt+Cdq, where A is of the form: I K+l I K+2 ..
-
with 1 +H,+~ dimensional price and profit derivatives f
K+l
0
1 . .
g;~++/.K+
1
...
...
...
...
diagonal
f0
0 ... ...
...
...
...
...
...
...
...
p++2z. K + 2
0
. ..
...
...
. .
0
Kf3
f K+3
0 .ft
...
Assuming A to be non-singular, this can be solved to give unique the endogenous variables in terms of the controls:
changes
dl=A-‘Bdt+A-‘Cdq, and
hence
a
and
...
K-t2 K+2
0
the main
0
0
K+l
Sk+“:
identity matrices down off this diagonal. B is:
set
of
in
(11) derivatives
aqK+l/aq,-~~+l,...,aqK+,/atK+l=
102
G.D. Myles, Tax rules for economies with imperfect competition
@K+l
K+ 1,. . . , a+K+ r/aqI = G?i+Kr+r,. . . , &P*N/at, E S2kNsN which captures the general equilibrium repercussions of any tax change. The derivatives @: and L?{ will be termed the direct effects of taxation and @s and Sz:, s# r, the indirect or induced effect. @F can also be given the interpretation of the degree of forward-shifting of the tax taking into account equilibriating adjustments; this is considered further in section 5. Alternatively, applying the implicit function theorem to (8):
It
“N.
N=
Q”N.
N
(4l,...,qK,tK+l,...,tN;W),
with the derivatives of these functions given by (11). Two special cases of the mode1 will be analysed in section 5. These are intended to illuminate the structure of the model, particularly the relationship of the f and @ functions, and to relate this analysis to previous work on taxation and imperfect competition. Before proceeding, it is worth noting the connections between this mode1 and Munk’s (1978) work on taxation with perfect competition and decreasing returns to scale. Munk chooses to use consumer prices as the control variables with producer prices, which are those necessary to decentralise the resulting equilibrium, functionally dependent upon the controls. The optima1 taxes are then the difference between the two. Furthermore, profit income is also a function of consumer prices. Differences between the models become apparent upon closer examination: with imperfect competition the taxes upon goods produced in the oligopolistic industries must be used as controls rather than consumer prices because the imperfect competition leads to functional relationships between consumer prices [i.e. eqs. (S)] which are absent in the Munk mode1 and, due to the decision problem of firms [eq. (3)], only a subset of prices in R, + are sustainable as imperfectly competitive equilibria. Furthermore, the mode1 presented in this paper does not allow for intermediate goods, primarily because there are a number of unresolved issues in modelling imperfect competition with intermediate goods [Hart (1985)]. Note that the assumption of constant returns to scale in the competitive sector in the present paper can be relaxed by following Munk’s derivation; this can be seen to introduce further profit effects into the tax rules.
4. A single-consumer
tax rule
This section derives a one-person tax rule, the derivation that of Atkinson and Stiglitz (1980) in order to highlight imperfect competition has upon the optimal tax rule.
follows closely the effect that
G.D. Myles,
103
Tax rules for economies with imperfect competition
The single consumer supplies labour to the firms, purchases their output and receives their profits as income. It is assumed that government policy is aiming to maximise his welfare, represented by the indirect utility function: (13) subject
to raising
revenue
to purchase
a quantity
with value R:
of labour
(14)
This maximisation
problem
max ql*....qX,fK+l,....t.
can be written:
~=Vq,,...,q,,rGw)
+,u
;
2
tiXi+
tiXi-R
i=K+l
i=l
1,
(15)
with i=K+l,...,N,
qi=~i(ql,...,qk,...rqx,tK+l,...,ti,...,tN;W),
and 7(i:=SZji(q1,...,qk,...,qK,tK+1,...,ti,...,tN;W),
j=l For a typical written:
good ke P, the first-order
condition
,..., n,,i=K+l,..., for this problem
N. may be
aqat,=av/aqk+ 5 av/aq,+;+av/an 5 f n$ s=K+l j=l s=K+l r +p
N
N
C ti aXi/aqk+ 1
1
N
1
X,+
i=l
i=l
tidXi/aq,'
+i$l s=i+l j$ltiaxila~.W 1=O. There condition
are two features for a perfectly
of (16) that competitive
@Sk
s=K+l
(16)
distinguish it from the equivalent model. First, (16) includes terms
G.D. My/es, Tax rulesfor economies with imperfect competition
104
involving @Sk,these describe the adjustment of equilibrium prices in the imperfectly competitive industries to the change in qk and affect both utility and the tax revenue received by the government. Secondly, profits are affected by the change in qk, the effect on each firm measured by Q;j; this has both welfare and revenue effects. Using Roy’s identity and the Slutsky equation, (16) can be written:
(17) where 8 = 1-(a/p) -
21 ti
dX,/&r,
i=
rk=a
f X3@“,--cr s=K-tl
-p$ f i=l
s=K+l
5 i Q-pi: 2 tiaxi/aq,. G; s=K+l
tidXi/d7c~
i=l
j=l
f j-
s=K+l
S$. 1
Ski is the substitution term and IX is the consumer’s marginal utility of income. Without the term (l/p)fk, (17) would be the standard Ramsey tax rule, so in a sense (l/p)rk represents the modification required to incorporate imperfect competition. Tk captures the induced effects of changing the tax on k upon profits and prices; if the induced effects were all zero, so too would be Tk, and the standard rule would apply to k. Treating the left-hand side of (17) to be approximately the reduction in compensated demand for k caused by the tax system and taking each component of Tk in turn, the reduction in demand will tend to be smaller if the tax on k: (1) increases the price of goods produced by imperfectly competitive industries (@i >O); (2) reduces profits (C$ cc 0), or (3) the induced effects lower tax revenue (tidXi/dqs. @;-CO, ti axi/anmf
From the above, (17) indicates that in the presence of imperfect competition an assessment of the effect of introducing the tax system must include the revenue and utility cost of induced price and profit changes. Neither of these effects would be present if all industries were perfectly competitive.
G.D. Myles, Tax rulesfor economies with imperfect competition
105
Proceeding in a similar manner, the first-order condition for the choice of tax rate for a typical good I E I is: N
aLlat,= av/aq,.q+
1 s=X+l
N
avpq,. q+
1
“S
avanC 0;’
s=K+l
j=l
Sfl
+p
2 5
X,+
[ +
t
i=l
ti
f
s=K+1
if1
tiaXJaq,‘@f
s=K+l
avfatt
f
j-l
1
=o.
52:
(18)
After employing Roy’s identity and the Slutsky equation, (18) becomes: ii1 riSli= -erx,+(ll~)rr,
r=K+l,...,N,
(19)
where @=(I/@;)-(+)-
f
i=l
tiaxilan
and l-l=(l/@;).
L7 (
-p
t i=l ifr
s=K+l Sfr
F s=K+l
s=K+l
tiaXi/a qs.@~--~
j=l
5 2 i+l s=K+l
tiaXi/&t
f
j=l
Qt
. >
In (19), (l/p)F again represents an adjustment of the standard Ramsey rule for the presence of induced price and profit changes. Its component terms may be interpreted as the utility and revenue effects of induced price and profit changes. The interaction of each of these with the reduction in compensated demand will be as for the competitive goods. Eq. (19) is distinguished from (17) by the dependence of 8’ upon r. This is due to the shifting term CJ:;8’ will equal 0 only if CD:=1, that is if tax shifting is 100 percent, otherwise the value of 8’ will be inversely related to @L. Hence, the greater the degree of over-shifting, the smaller will be the reduction in compensated demand. Section 5 provides examples that explore
106
G.D. My/es, Tax rules
for ecnnomies with impecfect competition
the consequences of variable degrees of tax-shifting for the choice of relative tax rates. To summarise this section, a Kamsey tax rule has been constructed for a single-consumer model with imperfect competition. For goods produced by perfectly competitive industries, this was equivalent to the standard Ramsey rule plus a term that adjusted for the induced price changes caused by the tax, the induced revenue changes and the direct and induced profit changes. For imperfectly competitive goods, the standard Ramsey rule was modified in two ways. First, 8 became dependent upon the good in question, its value being inversely related to the degree of forward-shifting of taxation. Secondly, the rule included an imperfect competition correction term.
5. Implications
for tax-setting
Two approaches are taken below to determine the implications of the above results for tax-setting. The first adopts the view of Mirrlees (1976) that the real property of the tax system is the extent to which it encourages, or discourages, the consumption of each good. This leads into an analysis of how the existence of imperfect competition modifies the usual rule for optimal discouragement. Secondly, two examples are analysed that provide insight into the tax rules by highlighting a number of features of general importance. Dividing (17) by X, gives:
i$l
fiSkiIXk=-BS(l/~)rklXk,
k=l,...,K,
(20)
where cr=, tiSki/Xk=dk is the ‘index of discouragement’ for good k. If Tk=O the optimum involves d,= -8, all k; the standard Ramsey rule. For two competitively produced goods i and j, whether disdj is dependent upon Ti/Xi 6 ri/Xj, thus if Xi = Xj, good i will be discouraged less than j if r’> rj. Consequently, the index of discouragement becomes dependent upon the interaction between the good in question and the imperfectly competitive sector. A good will be discouraged relatively less when an increase in its price leads to increases in the prices of imperfectly competitive goods, reductions in profits or to induced effects that lower tax revenue. Alternatively, at the optimum, the index of discouragement may be viewed as being equated to the sum of direct and indirect effects, captured by 8 and Tk, respectively. Similarly, from (19):
G.D. Myles,
jl
tiSrilXr=
Tax
rules for economies
-er+(1/P)fr/xr3
with imperfect
competition
107
r=K+l,...,N,
so that the same considerations apply as for (20) but with one important additional feature: both 8’ and r’ are dependent upon the degree of tax-shifting @:. To see the consequence of this, first take r’=O and write (21) as dr= -8’.
(22)
From (22) for two goods that are otherwise equivalent, the one with the greater value of @;, i.e. the good for which taxes are forward shifted most, will be discouraged less. As the example below illustrates, this is equivalent for simple models to a lower rate of tax upon this good. The mechanism behind this result is that raising a given level of revenue results in relatively higher consumer prices as the degree of tax-shifting increases. Consequently, those goods characterised by large values of @: are poor instruments for tax purposes and hence the lower index of discouragement. When r’ is non-zero, its value is inversely related to that of @F; this indicates that it is the values of the induced effects relative to @Lthat are of importance. This is about as far as the analysis of these tax rules is able to progress without specifying a precise mode1 of the economy but, as the arguments in the previous paragraphs illustrate, the rules are quite informative, even in their genera1 form. However, further insights into tax rules for the perfectly competitive model have been obtained by the study of special cases [Atkinson and Stiglitz (1972) Mirrlees (1975)] and it seems fruitful to proceed likewise. The structure of optima1 tax rules for perfectly competitive economies has led to a concentration upon consumer preferences as the major determinant of relative taxes, for example Deaton (1981). This focus has presumably developed owing to the passive role played by perfectly competitive firms: they simply shift the entire tax forward. In contrast, once imperfect competition enters the model there is scope for variations in firms’ behaviour; industries may vary as to the extent to which taxes are over- or under-shifted and in how responsive their price is to price variations elsewhere in the economy. Given these features of imperfect competition, the first example focuses on how relative taxes are affected by differences in industry conduct. To highlight the importance of conduct, assume that only two goods are produced with one of these, good 2, the output of an imperfectly competitive industry, that the government aims for a balanced budget (R=O) and that profits accrue to an actor outside the model. It should be noted that if both industries were competitive, the optimal tax rates under these conditions
108
G.D. Myles, Tax rules for economies
with imperfect competition
would both be zero: welfare could not be increased by any pair of taxes that satisfy the budget constraint. The deviations from zero taxes in the following solution can therefore be seen as a product of imperfect competition. For this example, the equilibrium price of the imperfectly competitive industry is given by q2
=
f2h
t2;
4
(23)
[cf. eqs. (S)]. The method of solution described in section 3 was to solve the system given by (8) to express the prices of goods produced under imperfect competition as functions of the prices of goods produced by perfectly competitive industries and the tax rates on the products of imperfectly competitive industries. From (23) it can be seen that this step is unnecessary for this example since the arguments of f’(.) are already those required. Hence, f2(41,
t,;
4
=
Q2(41,
t2;
4.
(24)
This equivalence occurs due to there being a single imperfectly competitive degree of industry. As noted following eq. (7), f’, IS ’ the partial equilibrium tax-shifting discussed by Seade (1985) and f: is the induced effect analysed by Myles (1987b). However, because of relation (24), these partial equilibrium effects are identical, for this example, to the general equilibrium effects @: and 0:. The necessary conditions for the choice of tax rates can be solved for t, to give: (25)
where
a=
@:w/~q,(~x,/aq,
--X,/Xl .axllaql)
As @i >O, if the goods are gross substitutes (dX,/dq, and aX,/aq, >O), then a ~0. To emphasise the importance of industrial behaviour, employ Roy’s identity to write (25) as:
The balanced budget requires one positive and one negative tax. From the tax on the imperfectly competitive industry, tZ, will be negative if
(26),
G.D. Myles, Tax rules for economies with imperfect
109
competition
x,(@:-l)-x,4:>0.
(27)
The conditions necessary for the inequality in (27) to hold can be stated or a loosely as ‘large @, negative @:‘, so that a high degree of over-shifting negative induced effect will lead to subsidisation. The reasoning behind this result is straightforward. If taxes are over-shifted the same will apply to any subsidy payment and, although the subsidy must be met by a tax on competitive firms, the final result of the policy may be a beneficial reduction of the general price level. This argument is reinforced if the imperfectly competitive industry also reduces its price in response to the tax on the competitive industry. A detailed analysis of a special case of this example, including explicit expressions for @: and @: for Cournot oligopoly with and without free-entry, is given in Myles (1987b). However, one simple example is worth noting: for a symmetrical n-firm quantity-setting oligopoly with industrial conduct represented by the conjectured response to output change 1,, 2 = a(xy= 1xi)/dxj, with 2 the same for all firms;
n aq2w2 : @ =(n+n) aq2j3~iGa2q2/ax:' where x is each firm’s equilibrium output. When competition is Bertrand, 1= 0 and @z = 1. As ;1 increases, with I. = 1 representing Cournot behaviour, @: will increase if aq2/aX2 > -nx d2q2/aX: and decrease otherwise. Correspondingly,
Q2
1
_pq,iaq,(aq,iax,
+ nx a2q2/ax:) - nx aq2m2. e2/ax2 ad (n+l)
aq2/a~ZZ2q2/ax:
pp--'
Thus, @i = 0 when 2 =0 and it may become positive or negative as 3, increases, depending upon the signs and values of the derivatives of demand. However, since @: and @i: are both zero for Bertrand competition it is possible to conclude that t, and t, will also be zero for Bertrand competition. Furthermore, substituting the expressions for @: and @: into (26) will provide a characterisation of t2 as 2 varies. The important point to note though is the manner in which relative taxes are determined by (27) with little explicit reference to consumer tastes. This is not to say preferences are unimportant, the terms @: and @: are dependent upon both the demand function, which reflects consumers tastes, and the conduct of the industry, so that relative taxes are determined jointly by tastes and conduct. However, in any application of these tax rules it is only aggregate demands and the values of @z and @: that need be utilised to find
G.D. Myles, Tax rules for economies with imperfect competition
110
the sign of the optimal taxes; it is in this sense that industrial behaviour becomes as important as tastes. The simplest extension of this example, making both industries imperfectly competitive, provides further insights. The profit-maximising prices are then given by
and q2 =f2(41? t2).
Solving
(29)
these:
@:=f:l(l -fX). The optimal
value of t,, again for a balanced
budget,
is given implicitly
by
where
To discuss the determinants of the sign of t,, I will take only the case when @i@: -@:@:>O and will assume that X,=X,. In these circumstances, sign {t2} = sign {(@: - @:) + (@: - @i)}, hence a sufficient condition for t, to be negative (which is equivalent to good 2 being taxed relatively less than good 1) is that @s> @: and ‘Pi >@:. These state that if the degree of tax-shifting on good 2 is greater than for good 1 and that the effect of a tax on good 2 upon the price of good 1 is greater than the indirect effect of a tax upon good 1, then good 2 should be subsidised. This argument obviously mirrors that connected with the index of discouragement above and illustrates the reasoning behind it. Finally, to relate this result to the partial
G.D. Myles, Tax rules
for economies
with imperfict competition
111
equilibrium results of Seade (1985) and Myles (1987b), @$ > @i is equivalent, from (30), to ft>f: and @:>@I to f:j:>f:f: so that knowledge of the partial equilibrium effects of taxation, the fys, is sufficient to establish the relative rates of taxation for the general equilibrium system.
6.
Conclusions
This paper has extended existing optimal tax rules to a general equilibrium model of imperfect competition. In making this extension, it has proved important to model the linkages that exist in the economy due to the decision-making structure of the imperfectly competitive industries. It was demonstrated that, for goods produced by competitive industries, the resulting optimal tax formulae could be presented in the form of the perfectly competitive rule plus an imperfect competition ‘correction’ term; this latter term capturing the indirect effects. The natural interpretation of this rule was modified likewise. For goods supplied under conditions of imperfect competition, the constant, 8, in the Ramsey rule became inversely related to the degree of tax-shifting. In summary, the tax rules derived illustrated the importance of: (a) differences between industries in the degree of forward-shifting of taxation; (b) effects of taxation upon profit income, and (c) the indirect effects of taxation due to the interdependences between industries. From these can be drawn the implication that it is not only consumer tastes that determine tax rates but also industrial structure and conduct. This point was illustrated by the examples of section 5. Finally, this paper has also provided a general method for approaching the analysis of imperfect competition. It is hoped that this framework should be of further value in the detailed analysis of specific issues in taxation.
Appendix In the main text equilibrium price and profit functions have been constructed for quantity-setting industries with fixed numbers of firms. This appendix constructs equivalent functions for quantity-setting with free entry and price-setting with differentiated products, A.1.
Free-entry
Each firm chooses
output
xi to maximise
n’;= xjq, - C,( x’;; w) - t,xJ;,
(A.11
G.D. Myles, Tax rules for economies with imperfect competition
112
where I have assumed that all firms face the same cost function so the optimal choice will be the same for all firms. Writing this common choice as x, and assuming it to be a continuous function of prices and the number n, of active firms, Xr*=~‘(q1,...,qk,...,qK,qK+I,...,tr,...,qN,7C-,,9;w). Entry
into the industry
is governed
n!=xJq,-C$x!;w)-t,x!=O, Substituting hence
(A.2) into
(A.2)
by the zero-profit all
this constraint
j=l,...,
constraint:
j ,..., n,.
provides
an implicit
(A.3) equation
for n,,
(A.4) which I assume can be solved for n,: (A.5) n’(.) can now be substituted of prices and tax rate:
into (A.2) to express
rI,...,
each output
x:=a’(q,,...,q,,...,q,,q,+,
,...,
XI*=a’(q1,...,qkr...,qK,qK+1
,...I t ,, . . ..qj+n_.;w).
as a function
qjV,n-‘, n’(.); w)
or
Finally, as inverse each firm’s output,
demand
is a function
qr=4,(41r...,4k,...,4K,4K+1,...,nx n’(‘) and or(.) can be substituted
of the number
, I,‘..,
qN,
% w),
64.6) of firms
and
of
(A.7)
into this to arrive at:
4,= f ‘(4 l,...,qk,...,qK,qx+l,...,t’,...,qN,~-’;W).
(A.8)
Eq. (A.8) characterises equilibrium price and, as profits are detinitionally zero, this is sufficient to describe the behaviour of the industry. A.2.
Price-setting
To treat price-setting I will consider each firm as an industry; its close neighbours will be identified by the conjectures that are incorporated into its maximisation. With this assumption, the firm faces a demand function: X,=X,(q,,...,q,
,...,qK,qK+1,...,q,,...,qN,n-,+n,;w),
(A.9)
and chooses q, to maximise: n, = x,q, - C,(X,; w) - r,X,, which will have the solution:
(A.lO)
113
G.D. Myles, Tax rules for economies with imperjtict competition
tI) . ..) qN,X_r;W).
qr=fl(q,,...,qk,...,qK,qK+l,...,
(A.ll)
This derivation is sufficiently general to incorporate many forms of market structure and conduct. Any oligopolistic interaction will be captured in the relation off’ to the primitives of demand and cost functions. Substituting f*(.) for qr in the definition of profit gives the equilibrium profit function:
?lr=~*(41,...,qk,...,4K,4K+1,...,t,, . . . . q?v,?L;w).
A.3.
Many-consumer
(A.12)
tax rules
Following the notation set out above, the components of the model are: (1) Social welfare. This is assumed to be a concave function of the indirect utilities of the H consumers
s.w.= w( v’(q,, .. .,qN, w,x1),.. ., vh(q,,...,qN, w,nh),.*., I/?q,, ...,qN, w,nH)), where h is a typical consumer and rch his profit consumer’s share of profits of the jth firm in industry
Xh,
$ 5 ~pT’~i i=K+l
and
t
income. 411’ is the r, so that
hth
@=l.
h=l
j=l
(2) Government revenue constraint. The government is attempting revenue to purchase a quantity of labour with value R. Hence,
to raise
(3) A set of functions a’(.), i=K $1,. . . , N, describing the evolution of prices and a set Qj’( .), j= 1,. . . , ni, i= K + 1,. . . , N, that determine each firm’s profits. Writing X, for x,,Xt/H and p” for the social marginal utility of income, dW/aV”.u”, the tax rule, for a typical good ke P, may be written:
Xk
=[
1-
k=l,...,
E(bhlH).(x;/x,) 1+ (1/PL)(~k/H8,)
h=l
k ,.,.,
K,
(A. 13)
G.D. Myles, Tax rules for economies
114
with imperfect competition
where
h=l
i=l
h=l
and bh is Diamond’s
s=K+
s=K+l
j=l
1 j=l
(1975) net social marginal
bh = fi”//t + 2
valuation
of income:
t; ?x;/&c”.
i=l
The tax rule given by (A.13) represents the standard many-consumer result with an additional term, (l/p)(zk/HXk), that corrects for the presence of imperfect competition. The correction term comprises the welfare and revenue value of the effect of the tax t, on prices and profits. Each component of rk will affect the left-hand side of (A.13) in the manner described for the equivalent term in Tk above. For each good produced by an imperfectly competitive industry, the tax rule is:
= -
(l/Q;) - 5
(bh/H) .(X:/x,)
h=l
r=K+l,...,
I ,...,
+( l/p)(z’/HXv), I
N,
(A.14)
where
h=l
Imperfect
competition
s=K+l
can be seen to have two effects upon
the structure
G.D. Myles, Tax rules for economies with imperfect competition
115
of (A.14). The degree of forward-shifting of the tax is accounted for by the appearance of @; in the term representing the standard many-consumer tax rule. The reduction in compensated demand becomes smaller the greater is the degree of overshifting. (A.14) also includes the imperfect competition correction term; this involves welfare and revenue effects of induced price changes and direct and induced effects upon profits. References Atkinson, Anthony B. and Joseph E. Stiglitz, 1972, The structure of indirect taxation and economic efftciencv, Journal of Public Economics 1. 97-l 19. Atkinson, Anthony B. and Joseph E. Stiglitz, 1980, Lectures on Public Economics (McGrawHill, New York). Cripps, Martin W. and Gareth D. Myles, 1988, General equilibrium and imperfect competition: Profit feedback effects and price normalisations, Warwick Economic Research Paper No. 295. Deaton, Angus S., 1981, Optimal taxes and the structure of preferences, Econometrica 49, 124551260. Diamond, Peter A., 1975, A many-person Ramsey tax rule, Journal of Public Economics 4, 335-342. Diamond, Peter A. and James A. Mirrlees, 1971, Optimal taxation and public production 1: Production efficiency and 2: Tax rules, American Economic Review 61, 8-27 and 261-278. Hart, Oliver, 1985, Imperfect competition in general equilibrium: An overview of recent work, in: Arrow, K.J. and S. Honkapohja, eds., Frontiers of Economics (Basil Blackwell, Oxford). Mirrlees, James A., 1975, Optimal commodity taxation in a two-class economy, Journal of Public Economics 4, 27-33. Mirrlees, James A., 1976, Optimal tax theory, Journal of Public Economics 6, 327-358. Munk, Knud J., 1978, Optimal taxation and pure profit, Scandinavian Journal of Economics 80, l-19. Myles, Gareth D., 1987a, Optimal commodity taxation with imperfect competition, Warwick Economic Research Paper, No. 280. Myles, Gareth D., 1987b. Tax design in the presence of imperfect competition: An example, Journal of Public Economics 34, 367-378. Seade, Jesus, 1985, Profitable cost increases, Warwick Economic Research Paper, No. 260. Stern, Nicholas H., 1987, The effects of taxation, price control and government contracts in oligopoly and monopolistic competition, Journal of Public Economics 32, 133-158.
of Public
RAMSEY
Economics
38 (1989) 95115.
TAX RULES
FOR ECONOMIES COMPETITION Gareth
University Received
September
North-Holland
WITH
IMPERFECT
D. MYLES*
of Warwick,
Coventry
1987, revised version
CV4 7AL. UK received
November
1988
This paper derives optimal commodity tax rules for a general equilibrium model with imperfect competition. To achieve this, functions are constructed that describe the effects of taxation upon the equilibrium prices and profit levels of imperfectly competitive industries. The use of these functions allows existing tax rules to be generalised to accommodate imperfect competition.
1. Introduction This paper extends the theory of optimal commodity taxation to a general equilibrium model with imperfect competition. Important features of the model are non-linear responses of prices to tax changes, the connections between industries via their demand functions and the effects of taxation upon profits. Tax rules are constructed for a single-consumer ‘efficiency’ model and for a many-consumer model that incorporates considerations of equity. Industrial conduct is shown to be an important determinant of relative tax rates. Previous analyses of optimal taxation have been within the Arrow-Debreu paradigm of perfect competition, an avenue of research exemplified by Diamond and Mirrlees (1971). However, many issues in taxation, such as duties on whisky, petrol and cigarettes, are concerned with industries that are far from the competitive ideal. Although Myles (1987b) makes some progress in analysing taxation for a specific partial equilibrium model, no general equilibrium analysis of taxation and imperfect competition has yet been conducted. This paper is a first step in that direction. When designing optimal tax systems there are two aspects that are of critical importance in distinguishing imperfect from perfect competition: industries that are imperfectly competitive price above marginal costs so taxes may be over- or under-shifted and their behaviour is determined *I wish to thank invaluable comments W7-2727/89/$3.50
Terence Gorman, and criticisms. 0
Norman
Ireland
1989, Elsevier Science Publishers
and
an anonymous
B.V. (North-Holland)
referee
for their
96
G.D. Myles, Tax rules for economies with imperfect competition
conditionally upon other industries’ prices. Seade (1985) Stern (1987) and Myles (1987b) have analysed these features from a partial equilibrium viewpoint; here the insights of those papers are embedded within a general equilibrium model and this is employed to illustrate how existing tax rules are modified to accommodate the presence of imperfect competition. The paper is organised as follows: section 2 describes the model and the analytical method employed, the required analysis of imperfectly competitive industries is provided in section 3, and section 4 presents the one-consumer tax rule; this is generalised to many-consumers in the appendix. The implications of this rule for tax-setting and the importance of industrial conduct in determining relative taxes are discussed in section 5. Conclusions are given in section 6. 2. Description of the model The model consists of households, firms and a government. Households supply labour, demand goods and receive profits; they are described by their indirect utility functions and shareholdings in firms. Each firm belongs to a single industry; there is no joint production. Industries are either perfectly or imperfectly competitive; all firms comprising a perfectly competitive industry produce with the same constant returns to scale production technology. The role of the government is to choose the level of commodity tax rates to maximise a concave function of individual utilities subject to attaining a given level of revenue. To simplify the analysis I assume that labour, which is untaxed, is the only input into production and use it as numeraire throughout the paper; hence the wage rate remains constant at w. In conjunction with the assumption that competitive firms produce with constant returns to scale, this provides a set of effectively fixed pre-tax prices for these industries; their post-tax prices are then linear functions of tax rates. It also removes a number of difficult, and complicating, issues relating to the pricing of intermediate goods produced by imperfectly competitive firms. A tentative treatment of the model without this restriction can be found in Myles (1987a). In respect of the normalisation employed by Diamond and Mirrlees (1971), it is important to note that both the receipt of profits by consumers [Munk (1978)] and the existence of imperfect competition are individually sufficient to invalidate the separate normalisation of producer and consumer prices. Therefore, although using labour as numeraire does not affect the real equilibrium of the model [Cripps and Myles (1988)], the choice of labour as an untaxed good is not a trivial one. Three points can be made with respect to this choice. First, labour is the only good in the model supplied by households and consequently does play a different role to the taxed commodities. Secondly, many existing tax systems treat labour as distinct from other commodities and the
G.D. Myles, TAX rules for economies with imperfect competition
91
model reflects this fact. Thirdly, a tax upon labour can easily be incorporated into the model if desired and a rule derived for determining its optimal value. To analyse the optimal tax problem, the effects of tax rates upon prices and profits must be determined. The assumptions placed above upon the production processes of competitive industries imply that their equilibrium profits are zero and that their post-tax prices are the fixed pre-tax prices plus tax. Without the addition of imperfect competition the model is therefore equivalent to that of Atkinson and Stiglitz (1972). For more general production technologies it would be possible to follow Diamond and Mirrlees (1971) and control the competitive sector using the producer prices that support the equilibrium. The government choice variables would then be the consumer prices for goods produced competitively and the tax rates of those produced by imperfectly competitive firms; see also the discussion following eq. (12). In contrast, imperfectly competitive industries require rather more analysis. Consider first the determination of equilibrium prices. Each firm in the industry makes an output/price decision contingent upon the demand function they face, their perception of rivals’ behaviour and upon their cost structure, which includes the commodity tax on output. However, in a general equilibrium model, demand is contingent upon all prices in the economy. Consequently the equilibrium industry price is affected by the tax system in two ways. First, a variation in the tax rate upon a good represents a change in the cost structure of each firm producing it. This cost change will modify the profit-maximising decisions of these firms which, in turn, affect the equilibrium price of the good. This is the ‘direct’ effect of taxation and is present in both perfect and imperfect competition. The size of this effect is fixed for perfect competition but, in contrast, is dependent upon market structure and conduct when competition is imperfect; see in particular Stern (1987). Secondly, there are ‘induced’, or ‘indirect’ effects: taxes on other industries cause changes in their equilibrium prices, these price changes affect demand, and hence equilibrium price, for the industry in question. This effect is not present in perfect competition when producer prices, as here, are tixed. Seade (1985) and Stern (1987) have previously analysed the direct effects of taxation for a variety of market structures. Myles (1987b) has derived equations describing induced effects for a model of Cournot oligopoly. The approach taken was to differentiate the first-order conditions for profit maximisation and then solve the resulting equations for the derivative of equilibrium price with respect to either the tax rate or for prices that enter the demand function. In contrast, the approach used below is to solve directly for the functional dependence of each imperfectly competitive industry’s price upon the tax rate it faces and the prices of other goods, the derivatives of these functions are formally equivalent to those derived in previous papers, and to use these functions as the components of a general
98
G.D. Myles, Tax rules for economies
with imperfect competition
equilibrium model. The general equilibrium effects of taxation are then found by perturbing the complete system. The advantage of this approach is that it provides an organising idea that can be applied to many forms of imperfect competition and allows the structure of optimal tax rules to be analysed without requiring a precise specification of the nature of imperfect competition; only at the stage of application do the functions need to be specified. It also permits the analysis of taxation to be conducted within a general equilibrium model of imperfect competition. To this point I have concentrated on prices, all that has been said above applies equally to the profits of firms in imperfectly competitive industries. In an equivalent manner, there are direct and induced effects of taxation upon profits and functions will be constructed that capture these effects.
3.
Imperfect competition
and the effects of taxation
A central feature of the analysis below is the use of functions that govern the price of each good produced under conditions of imperfect competition, and the profits of each firm producing it, which have as their arguments the control variables of the government. The construction of these functions will be described for quantity-setting industries with fixed numbers of firms in this section. Quantity-setting with free-entry and price-setting industries are analysed in the appendix. At this stage, the precise interaction between firms does not need to be stated; the general equations derived encompass alternative forms of conjectural assumptions. Assuming that there are N goods available in the economy, these are partitioned into two subsets, P and I, with cardinality K and N-K, respectively. P contains all goods produced under constant returns to scale by perfectly competitive firms; the goods in P are indexed i= 1,. . . , k,. . . , K. I contains goods produced by imperfectly competitive industries; these are indexed i=K+l,..., r,..., N. As labour is the numeraire and the only input into production, the pre-tax price, pk, of a typical good k E P is predetermined at ckw, where w is the fixed wage rate and ck is the labour input per unit of k. Writing the tax rate on good k as t,, the post-tax price qk=pk+ t,, note that a&/& = 1. For each good rel that is produced by a quantity-setting industry with a fixed number of n, firms, n,L 1, and with each firm, j, in this industry earning profits rr!, it is assumed that the industry faces a demand function:
x,=x,
ql,...,qk,...,qK,qK+l,...,q,,...,qN,n-l+
2 dw j=l
> 3
(1)
G.D. Myles,
99
Tax rules for economies with imperfect competition
where X, is aggregate demand and n
-r=i=g+l jgl4 i#r
i.e. aggregate profit less that of the industry producing good r. The demand function has partial inverse:
q,=q,
qt ,*..,
qk ,...,
qK,qK+l,...,
x,3 . ..Y qNYxr+
2
n!;w
j=l
(
9
(2)
)
with X,=&i xi, xi the output of good I by the jth firm in the industry. Each firm chooses its optimal output level, xp, to maximise profits taking other industries prices as given: x!*=argmax{nj=xjq,-Cj(x:‘;w)-&xi},
j=l,...,n.
(3)
The output choices arising from this maximisation will be a function of the prices entering (2), the tax rate t,, profit income generated by all other industries K,, and the wage rate w; the form of the function will be partly determined by the oligopolistic interaction. Sufficient conditions that guarantee the existence of such maxima are derived in Cripps and Myles (1988); these essentially require the income derivative 8X,/&r to be bounded below 1 for large n. Assuming optimal choices to exist, they can be written: xp=arj(q1,...,qk,...,qK,qK+1,...,tr,...,qN,X-,;W), and aggregating over the
x;=
5 d’(.)=o’(q,
n,
j=l,
. . . . n,
(4)
firms: )...,
qk )...,
qK,qK+l,...,
t I,...)
qN,Lr;W).
(5)
j=l
Substituting (5) into the partial inverse demand function characterises the equilibrium prices as a function of other prices, the tax rate on good I, profit income and the wage rate: q*=f’(41,...,qk,...,qK,qK+1
,..‘, tI,..., q‘V,rL;w).
(6)
If (4) and (6) are substituted into the definition of firm j’s profit level, equilibrium profit can be expressed as: ~TI!=gj’(ql,...,qk,...,qK,qK+1
,..‘, t I,..., %v,n - ,; w).
(7)
The derivatives of the functions f’ and gj’ are equivalent to the partial equilibrium effects of taxation analysed by Seade (1985), Stern (1987) and Myles (1987b) for a number of market structures. In particular, it is noteworthy that af’/at,= f”: is >O and may be > 1, which represents
100
G.D. Myles,
Tax rules for economies with imperfect competition
over-shifting of taxes, when a condition on the convexity of the demand function is satisfied and that 3gP/atr-gJ;I can be positive, see Seade (1985). Both f: and gc, s#r, can be of either sign, the important determining factor being the sign of the second cross-derivative of inverse demand, a2q,/dX,dq, [Myles (1987b)]. The complete model of the imperfectly competitive sector consists of N-K equations of the form of (6) describing the determination of each imperfectly competitive price and crzK+ 1 n, functions determining each firm’s profit level (for free-entry industries where profits are delinitionally zero the g functions are not required, see appendix). Hence for given consumer prices for competitive goods and tax rates upon imperfectly competitive goods, the equilibrium in the imperfectly competitive sector is the simultaneous solution to eqs. (8) below:
qK+1
=fK+’
ql,...,qK,qK+lr
qK+2=fK+’
7(y =
q1,...,qK,tK+1,qK+2,...,qN,
g”
ql,...,qK,qK+l,...rtN,
t K+2r...,qN,
N
“,
2 i=K+2
c j=l
$;w
2 i=K+l i#K+2
f j=l
dw
, )
, )
(8)
To determine optimal tax policy it is necessary to calculate the effect that a change in policy will have upon the equilibrium. With constant returns in the competitive sector, the parameters under control of the government can be taken to be either the tax rates, t,,.. . , t,, or, recalling the process for price formation in the competitive sector, the prices ql,. . . ,qK together with the tax rates tK+ 1,. , t,. In what follows I choose to use the latter. There are two approaches that can be taken at this point: the implicit function theorem can be appealed to and (8) solved to give qK + 1,. . . , ~“~7~ in terms of the control variables or, as it is only the value of the derivatives of endogenous variables with respect to the control variables that are important
101
G.D. Myles, Tax rules for economies with imperfect competition
in the sequel, the system can be differentiated the latter and differentiating (8):
[*I
rr_
=
and solved.
Proceeding
with
(9)
LB]
or Adl=Bdt+Cdq, where A is of the form: I K+l I K+2 ..
-
with 1 +H,+~ dimensional price and profit derivatives f
K+l
0
1 . .
g;~++/.K+
1
...
...
...
...
diagonal
f0
0 ... ...
...
...
...
...
...
...
...
p++2z. K + 2
0
. ..
...
...
. .
0
Kf3
f K+3
0 .ft
...
Assuming A to be non-singular, this can be solved to give unique the endogenous variables in terms of the controls:
changes
dl=A-‘Bdt+A-‘Cdq, and
hence
a
and
...
K-t2 K+2
0
the main
0
0
K+l
Sk+“:
identity matrices down off this diagonal. B is:
set
of
in
(11) derivatives
aqK+l/aq,-~~+l,...,aqK+,/atK+l=
102
G.D. Myles, Tax rules for economies with imperfect competition
@K+l
K+ 1,. . . , a+K+ r/aqI = G?i+Kr+r,. . . , &P*N/at, E S2kNsN which captures the general equilibrium repercussions of any tax change. The derivatives @: and L?{ will be termed the direct effects of taxation and @s and Sz:, s# r, the indirect or induced effect. @F can also be given the interpretation of the degree of forward-shifting of the tax taking into account equilibriating adjustments; this is considered further in section 5. Alternatively, applying the implicit function theorem to (8):
It
“N.
N=
Q”N.
N
(4l,...,qK,tK+l,...,tN;W),
with the derivatives of these functions given by (11). Two special cases of the mode1 will be analysed in section 5. These are intended to illuminate the structure of the model, particularly the relationship of the f and @ functions, and to relate this analysis to previous work on taxation and imperfect competition. Before proceeding, it is worth noting the connections between this mode1 and Munk’s (1978) work on taxation with perfect competition and decreasing returns to scale. Munk chooses to use consumer prices as the control variables with producer prices, which are those necessary to decentralise the resulting equilibrium, functionally dependent upon the controls. The optima1 taxes are then the difference between the two. Furthermore, profit income is also a function of consumer prices. Differences between the models become apparent upon closer examination: with imperfect competition the taxes upon goods produced in the oligopolistic industries must be used as controls rather than consumer prices because the imperfect competition leads to functional relationships between consumer prices [i.e. eqs. (S)] which are absent in the Munk mode1 and, due to the decision problem of firms [eq. (3)], only a subset of prices in R, + are sustainable as imperfectly competitive equilibria. Furthermore, the mode1 presented in this paper does not allow for intermediate goods, primarily because there are a number of unresolved issues in modelling imperfect competition with intermediate goods [Hart (1985)]. Note that the assumption of constant returns to scale in the competitive sector in the present paper can be relaxed by following Munk’s derivation; this can be seen to introduce further profit effects into the tax rules.
4. A single-consumer
tax rule
This section derives a one-person tax rule, the derivation that of Atkinson and Stiglitz (1980) in order to highlight imperfect competition has upon the optimal tax rule.
follows closely the effect that
G.D. Myles,
103
Tax rules for economies with imperfect competition
The single consumer supplies labour to the firms, purchases their output and receives their profits as income. It is assumed that government policy is aiming to maximise his welfare, represented by the indirect utility function: (13) subject
to raising
revenue
to purchase
a quantity
with value R:
of labour
(14)
This maximisation
problem
max ql*....qX,fK+l,....t.
can be written:
~=Vq,,...,q,,rGw)
+,u
;
2
tiXi+
tiXi-R
i=K+l
i=l
1,
(15)
with i=K+l,...,N,
qi=~i(ql,...,qk,...rqx,tK+l,...,ti,...,tN;W),
and 7(i:=SZji(q1,...,qk,...,qK,tK+1,...,ti,...,tN;W),
j=l For a typical written:
good ke P, the first-order
condition
,..., n,,i=K+l,..., for this problem
N. may be
aqat,=av/aqk+ 5 av/aq,+;+av/an 5 f n$ s=K+l j=l s=K+l r +p
N
N
C ti aXi/aqk+ 1
1
N
1
X,+
i=l
i=l
tidXi/aq,'
+i$l s=i+l j$ltiaxila~.W 1=O. There condition
are two features for a perfectly
of (16) that competitive
@Sk
s=K+l
(16)
distinguish it from the equivalent model. First, (16) includes terms
G.D. My/es, Tax rulesfor economies with imperfect competition
104
involving @Sk,these describe the adjustment of equilibrium prices in the imperfectly competitive industries to the change in qk and affect both utility and the tax revenue received by the government. Secondly, profits are affected by the change in qk, the effect on each firm measured by Q;j; this has both welfare and revenue effects. Using Roy’s identity and the Slutsky equation, (16) can be written:
(17) where 8 = 1-(a/p) -
21 ti
dX,/&r,
i=
rk=a
f X3@“,--cr s=K-tl
-p$ f i=l
s=K+l
5 i Q-pi: 2 tiaxi/aq,. G; s=K+l
tidXi/d7c~
i=l
j=l
f j-
s=K+l
S$. 1
Ski is the substitution term and IX is the consumer’s marginal utility of income. Without the term (l/p)fk, (17) would be the standard Ramsey tax rule, so in a sense (l/p)rk represents the modification required to incorporate imperfect competition. Tk captures the induced effects of changing the tax on k upon profits and prices; if the induced effects were all zero, so too would be Tk, and the standard rule would apply to k. Treating the left-hand side of (17) to be approximately the reduction in compensated demand for k caused by the tax system and taking each component of Tk in turn, the reduction in demand will tend to be smaller if the tax on k: (1) increases the price of goods produced by imperfectly competitive industries (@i >O); (2) reduces profits (C$ cc 0), or (3) the induced effects lower tax revenue (tidXi/dqs. @;-CO, ti axi/anmf
From the above, (17) indicates that in the presence of imperfect competition an assessment of the effect of introducing the tax system must include the revenue and utility cost of induced price and profit changes. Neither of these effects would be present if all industries were perfectly competitive.
G.D. Myles, Tax rulesfor economies with imperfect competition
105
Proceeding in a similar manner, the first-order condition for the choice of tax rate for a typical good I E I is: N
aLlat,= av/aq,.q+
1 s=X+l
N
avpq,. q+
1
“S
avanC 0;’
s=K+l
j=l
Sfl
+p
2 5
X,+
[ +
t
i=l
ti
f
s=K+1
if1
tiaXJaq,‘@f
s=K+l
avfatt
f
j-l
1
=o.
52:
(18)
After employing Roy’s identity and the Slutsky equation, (18) becomes: ii1 riSli= -erx,+(ll~)rr,
r=K+l,...,N,
(19)
where @=(I/@;)-(+)-
f
i=l
tiaxilan
and l-l=(l/@;).
L7 (
-p
t i=l ifr
s=K+l Sfr
F s=K+l
s=K+l
tiaXi/a qs.@~--~
j=l
5 2 i+l s=K+l
tiaXi/&t
f
j=l
Qt
. >
In (19), (l/p)F again represents an adjustment of the standard Ramsey rule for the presence of induced price and profit changes. Its component terms may be interpreted as the utility and revenue effects of induced price and profit changes. The interaction of each of these with the reduction in compensated demand will be as for the competitive goods. Eq. (19) is distinguished from (17) by the dependence of 8’ upon r. This is due to the shifting term CJ:;8’ will equal 0 only if CD:=1, that is if tax shifting is 100 percent, otherwise the value of 8’ will be inversely related to @L. Hence, the greater the degree of over-shifting, the smaller will be the reduction in compensated demand. Section 5 provides examples that explore
106
G.D. My/es, Tax rules
for ecnnomies with impecfect competition
the consequences of variable degrees of tax-shifting for the choice of relative tax rates. To summarise this section, a Kamsey tax rule has been constructed for a single-consumer model with imperfect competition. For goods produced by perfectly competitive industries, this was equivalent to the standard Ramsey rule plus a term that adjusted for the induced price changes caused by the tax, the induced revenue changes and the direct and induced profit changes. For imperfectly competitive goods, the standard Ramsey rule was modified in two ways. First, 8 became dependent upon the good in question, its value being inversely related to the degree of forward-shifting of taxation. Secondly, the rule included an imperfect competition correction term.
5. Implications
for tax-setting
Two approaches are taken below to determine the implications of the above results for tax-setting. The first adopts the view of Mirrlees (1976) that the real property of the tax system is the extent to which it encourages, or discourages, the consumption of each good. This leads into an analysis of how the existence of imperfect competition modifies the usual rule for optimal discouragement. Secondly, two examples are analysed that provide insight into the tax rules by highlighting a number of features of general importance. Dividing (17) by X, gives:
i$l
fiSkiIXk=-BS(l/~)rklXk,
k=l,...,K,
(20)
where cr=, tiSki/Xk=dk is the ‘index of discouragement’ for good k. If Tk=O the optimum involves d,= -8, all k; the standard Ramsey rule. For two competitively produced goods i and j, whether disdj is dependent upon Ti/Xi 6 ri/Xj, thus if Xi = Xj, good i will be discouraged less than j if r’> rj. Consequently, the index of discouragement becomes dependent upon the interaction between the good in question and the imperfectly competitive sector. A good will be discouraged relatively less when an increase in its price leads to increases in the prices of imperfectly competitive goods, reductions in profits or to induced effects that lower tax revenue. Alternatively, at the optimum, the index of discouragement may be viewed as being equated to the sum of direct and indirect effects, captured by 8 and Tk, respectively. Similarly, from (19):
G.D. Myles,
jl
tiSrilXr=
Tax
rules for economies
-er+(1/P)fr/xr3
with imperfect
competition
107
r=K+l,...,N,
so that the same considerations apply as for (20) but with one important additional feature: both 8’ and r’ are dependent upon the degree of tax-shifting @:. To see the consequence of this, first take r’=O and write (21) as dr= -8’.
(22)
From (22) for two goods that are otherwise equivalent, the one with the greater value of @;, i.e. the good for which taxes are forward shifted most, will be discouraged less. As the example below illustrates, this is equivalent for simple models to a lower rate of tax upon this good. The mechanism behind this result is that raising a given level of revenue results in relatively higher consumer prices as the degree of tax-shifting increases. Consequently, those goods characterised by large values of @: are poor instruments for tax purposes and hence the lower index of discouragement. When r’ is non-zero, its value is inversely related to that of @F; this indicates that it is the values of the induced effects relative to @Lthat are of importance. This is about as far as the analysis of these tax rules is able to progress without specifying a precise mode1 of the economy but, as the arguments in the previous paragraphs illustrate, the rules are quite informative, even in their genera1 form. However, further insights into tax rules for the perfectly competitive model have been obtained by the study of special cases [Atkinson and Stiglitz (1972) Mirrlees (1975)] and it seems fruitful to proceed likewise. The structure of optima1 tax rules for perfectly competitive economies has led to a concentration upon consumer preferences as the major determinant of relative taxes, for example Deaton (1981). This focus has presumably developed owing to the passive role played by perfectly competitive firms: they simply shift the entire tax forward. In contrast, once imperfect competition enters the model there is scope for variations in firms’ behaviour; industries may vary as to the extent to which taxes are over- or under-shifted and in how responsive their price is to price variations elsewhere in the economy. Given these features of imperfect competition, the first example focuses on how relative taxes are affected by differences in industry conduct. To highlight the importance of conduct, assume that only two goods are produced with one of these, good 2, the output of an imperfectly competitive industry, that the government aims for a balanced budget (R=O) and that profits accrue to an actor outside the model. It should be noted that if both industries were competitive, the optimal tax rates under these conditions
108
G.D. Myles, Tax rules for economies
with imperfect competition
would both be zero: welfare could not be increased by any pair of taxes that satisfy the budget constraint. The deviations from zero taxes in the following solution can therefore be seen as a product of imperfect competition. For this example, the equilibrium price of the imperfectly competitive industry is given by q2
=
f2h
t2;
4
(23)
[cf. eqs. (S)]. The method of solution described in section 3 was to solve the system given by (8) to express the prices of goods produced under imperfect competition as functions of the prices of goods produced by perfectly competitive industries and the tax rates on the products of imperfectly competitive industries. From (23) it can be seen that this step is unnecessary for this example since the arguments of f’(.) are already those required. Hence, f2(41,
t,;
4
=
Q2(41,
t2;
4.
(24)
This equivalence occurs due to there being a single imperfectly competitive degree of industry. As noted following eq. (7), f’, IS ’ the partial equilibrium tax-shifting discussed by Seade (1985) and f: is the induced effect analysed by Myles (1987b). However, because of relation (24), these partial equilibrium effects are identical, for this example, to the general equilibrium effects @: and 0:. The necessary conditions for the choice of tax rates can be solved for t, to give: (25)
where
a=
@:w/~q,(~x,/aq,
--X,/Xl .axllaql)
As @i >O, if the goods are gross substitutes (dX,/dq, and aX,/aq, >O), then a ~0. To emphasise the importance of industrial behaviour, employ Roy’s identity to write (25) as:
The balanced budget requires one positive and one negative tax. From the tax on the imperfectly competitive industry, tZ, will be negative if
(26),
G.D. Myles, Tax rules for economies with imperfect
109
competition
x,(@:-l)-x,4:>0.
(27)
The conditions necessary for the inequality in (27) to hold can be stated or a loosely as ‘large @, negative @:‘, so that a high degree of over-shifting negative induced effect will lead to subsidisation. The reasoning behind this result is straightforward. If taxes are over-shifted the same will apply to any subsidy payment and, although the subsidy must be met by a tax on competitive firms, the final result of the policy may be a beneficial reduction of the general price level. This argument is reinforced if the imperfectly competitive industry also reduces its price in response to the tax on the competitive industry. A detailed analysis of a special case of this example, including explicit expressions for @: and @: for Cournot oligopoly with and without free-entry, is given in Myles (1987b). However, one simple example is worth noting: for a symmetrical n-firm quantity-setting oligopoly with industrial conduct represented by the conjectured response to output change 1,, 2 = a(xy= 1xi)/dxj, with 2 the same for all firms;
n aq2w2 : @ =(n+n) aq2j3~iGa2q2/ax:' where x is each firm’s equilibrium output. When competition is Bertrand, 1= 0 and @z = 1. As ;1 increases, with I. = 1 representing Cournot behaviour, @: will increase if aq2/aX2 > -nx d2q2/aX: and decrease otherwise. Correspondingly,
Q2
1
_pq,iaq,(aq,iax,
+ nx a2q2/ax:) - nx aq2m2. e2/ax2 ad (n+l)
aq2/a~ZZ2q2/ax:
pp--'
Thus, @i = 0 when 2 =0 and it may become positive or negative as 3, increases, depending upon the signs and values of the derivatives of demand. However, since @: and @i: are both zero for Bertrand competition it is possible to conclude that t, and t, will also be zero for Bertrand competition. Furthermore, substituting the expressions for @: and @: into (26) will provide a characterisation of t2 as 2 varies. The important point to note though is the manner in which relative taxes are determined by (27) with little explicit reference to consumer tastes. This is not to say preferences are unimportant, the terms @: and @: are dependent upon both the demand function, which reflects consumers tastes, and the conduct of the industry, so that relative taxes are determined jointly by tastes and conduct. However, in any application of these tax rules it is only aggregate demands and the values of @z and @: that need be utilised to find
G.D. Myles, Tax rules for economies with imperfect competition
110
the sign of the optimal taxes; it is in this sense that industrial behaviour becomes as important as tastes. The simplest extension of this example, making both industries imperfectly competitive, provides further insights. The profit-maximising prices are then given by
and q2 =f2(41? t2).
Solving
(29)
these:
@:=f:l(l -fX). The optimal
value of t,, again for a balanced
budget,
is given implicitly
by
where
To discuss the determinants of the sign of t,, I will take only the case when @i@: -@:@:>O and will assume that X,=X,. In these circumstances, sign {t2} = sign {(@: - @:) + (@: - @i)}, hence a sufficient condition for t, to be negative (which is equivalent to good 2 being taxed relatively less than good 1) is that @s> @: and ‘Pi >@:. These state that if the degree of tax-shifting on good 2 is greater than for good 1 and that the effect of a tax on good 2 upon the price of good 1 is greater than the indirect effect of a tax upon good 1, then good 2 should be subsidised. This argument obviously mirrors that connected with the index of discouragement above and illustrates the reasoning behind it. Finally, to relate this result to the partial
G.D. Myles, Tax rules
for economies
with imperfict competition
111
equilibrium results of Seade (1985) and Myles (1987b), @$ > @i is equivalent, from (30), to ft>f: and @:>@I to f:j:>f:f: so that knowledge of the partial equilibrium effects of taxation, the fys, is sufficient to establish the relative rates of taxation for the general equilibrium system.
6.
Conclusions
This paper has extended existing optimal tax rules to a general equilibrium model of imperfect competition. In making this extension, it has proved important to model the linkages that exist in the economy due to the decision-making structure of the imperfectly competitive industries. It was demonstrated that, for goods produced by competitive industries, the resulting optimal tax formulae could be presented in the form of the perfectly competitive rule plus an imperfect competition ‘correction’ term; this latter term capturing the indirect effects. The natural interpretation of this rule was modified likewise. For goods supplied under conditions of imperfect competition, the constant, 8, in the Ramsey rule became inversely related to the degree of tax-shifting. In summary, the tax rules derived illustrated the importance of: (a) differences between industries in the degree of forward-shifting of taxation; (b) effects of taxation upon profit income, and (c) the indirect effects of taxation due to the interdependences between industries. From these can be drawn the implication that it is not only consumer tastes that determine tax rates but also industrial structure and conduct. This point was illustrated by the examples of section 5. Finally, this paper has also provided a general method for approaching the analysis of imperfect competition. It is hoped that this framework should be of further value in the detailed analysis of specific issues in taxation.
Appendix In the main text equilibrium price and profit functions have been constructed for quantity-setting industries with fixed numbers of firms. This appendix constructs equivalent functions for quantity-setting with free entry and price-setting with differentiated products, A.1.
Free-entry
Each firm chooses
output
xi to maximise
n’;= xjq, - C,( x’;; w) - t,xJ;,
(A.11
G.D. Myles, Tax rules for economies with imperfect competition
112
where I have assumed that all firms face the same cost function so the optimal choice will be the same for all firms. Writing this common choice as x, and assuming it to be a continuous function of prices and the number n, of active firms, Xr*=~‘(q1,...,qk,...,qK,qK+I,...,tr,...,qN,7C-,,9;w). Entry
into the industry
is governed
n!=xJq,-C$x!;w)-t,x!=O, Substituting hence
(A.2) into
(A.2)
by the zero-profit all
this constraint
j=l,...,
constraint:
j ,..., n,.
provides
an implicit
(A.3) equation
for n,,
(A.4) which I assume can be solved for n,: (A.5) n’(.) can now be substituted of prices and tax rate:
into (A.2) to express
rI,...,
each output
x:=a’(q,,...,q,,...,q,,q,+,
,...,
XI*=a’(q1,...,qkr...,qK,qK+1
,...I t ,, . . ..qj+n_.;w).
as a function
qjV,n-‘, n’(.); w)
or
Finally, as inverse each firm’s output,
demand
is a function
qr=4,(41r...,4k,...,4K,4K+1,...,nx n’(‘) and or(.) can be substituted
of the number
, I,‘..,
qN,
% w),
64.6) of firms
and
of
(A.7)
into this to arrive at:
4,= f ‘(4 l,...,qk,...,qK,qx+l,...,t’,...,qN,~-’;W).
(A.8)
Eq. (A.8) characterises equilibrium price and, as profits are detinitionally zero, this is sufficient to describe the behaviour of the industry. A.2.
Price-setting
To treat price-setting I will consider each firm as an industry; its close neighbours will be identified by the conjectures that are incorporated into its maximisation. With this assumption, the firm faces a demand function: X,=X,(q,,...,q,
,...,qK,qK+1,...,q,,...,qN,n-,+n,;w),
(A.9)
and chooses q, to maximise: n, = x,q, - C,(X,; w) - r,X,, which will have the solution:
(A.lO)
113
G.D. Myles, Tax rules for economies with imperjtict competition
tI) . ..) qN,X_r;W).
qr=fl(q,,...,qk,...,qK,qK+l,...,
(A.ll)
This derivation is sufficiently general to incorporate many forms of market structure and conduct. Any oligopolistic interaction will be captured in the relation off’ to the primitives of demand and cost functions. Substituting f*(.) for qr in the definition of profit gives the equilibrium profit function:
?lr=~*(41,...,qk,...,4K,4K+1,...,t,, . . . . q?v,?L;w).
A.3.
Many-consumer
(A.12)
tax rules
Following the notation set out above, the components of the model are: (1) Social welfare. This is assumed to be a concave function of the indirect utilities of the H consumers
s.w.= w( v’(q,, .. .,qN, w,x1),.. ., vh(q,,...,qN, w,nh),.*., I/?q,, ...,qN, w,nH)), where h is a typical consumer and rch his profit consumer’s share of profits of the jth firm in industry
Xh,
$ 5 ~pT’~i i=K+l
and
t
income. 411’ is the r, so that
hth
@=l.
h=l
j=l
(2) Government revenue constraint. The government is attempting revenue to purchase a quantity of labour with value R. Hence,
to raise
(3) A set of functions a’(.), i=K $1,. . . , N, describing the evolution of prices and a set Qj’( .), j= 1,. . . , ni, i= K + 1,. . . , N, that determine each firm’s profits. Writing X, for x,,Xt/H and p” for the social marginal utility of income, dW/aV”.u”, the tax rule, for a typical good ke P, may be written:
Xk
=[
1-
k=l,...,
E(bhlH).(x;/x,) 1+ (1/PL)(~k/H8,)
h=l
k ,.,.,
K,
(A. 13)
G.D. Myles, Tax rules for economies
114
with imperfect competition
where
h=l
i=l
h=l
and bh is Diamond’s
s=K+
s=K+l
j=l
1 j=l
(1975) net social marginal
bh = fi”//t + 2
valuation
of income:
t; ?x;/&c”.
i=l
The tax rule given by (A.13) represents the standard many-consumer result with an additional term, (l/p)(zk/HXk), that corrects for the presence of imperfect competition. The correction term comprises the welfare and revenue value of the effect of the tax t, on prices and profits. Each component of rk will affect the left-hand side of (A.13) in the manner described for the equivalent term in Tk above. For each good produced by an imperfectly competitive industry, the tax rule is:
= -
(l/Q;) - 5
(bh/H) .(X:/x,)
h=l
r=K+l,...,
I ,...,
+( l/p)(z’/HXv), I
N,
(A.14)
where
h=l
Imperfect
competition
s=K+l
can be seen to have two effects upon
the structure
G.D. Myles, Tax rules for economies with imperfect competition
115
of (A.14). The degree of forward-shifting of the tax is accounted for by the appearance of @; in the term representing the standard many-consumer tax rule. The reduction in compensated demand becomes smaller the greater is the degree of overshifting. (A.14) also includes the imperfect competition correction term; this involves welfare and revenue effects of induced price changes and direct and induced effects upon profits. References Atkinson, Anthony B. and Joseph E. Stiglitz, 1972, The structure of indirect taxation and economic efftciencv, Journal of Public Economics 1. 97-l 19. Atkinson, Anthony B. and Joseph E. Stiglitz, 1980, Lectures on Public Economics (McGrawHill, New York). Cripps, Martin W. and Gareth D. Myles, 1988, General equilibrium and imperfect competition: Profit feedback effects and price normalisations, Warwick Economic Research Paper No. 295. Deaton, Angus S., 1981, Optimal taxes and the structure of preferences, Econometrica 49, 124551260. Diamond, Peter A., 1975, A many-person Ramsey tax rule, Journal of Public Economics 4, 335-342. Diamond, Peter A. and James A. Mirrlees, 1971, Optimal taxation and public production 1: Production efficiency and 2: Tax rules, American Economic Review 61, 8-27 and 261-278. Hart, Oliver, 1985, Imperfect competition in general equilibrium: An overview of recent work, in: Arrow, K.J. and S. Honkapohja, eds., Frontiers of Economics (Basil Blackwell, Oxford). Mirrlees, James A., 1975, Optimal commodity taxation in a two-class economy, Journal of Public Economics 4, 27-33. Mirrlees, James A., 1976, Optimal tax theory, Journal of Public Economics 6, 327-358. Munk, Knud J., 1978, Optimal taxation and pure profit, Scandinavian Journal of Economics 80, l-19. Myles, Gareth D., 1987a, Optimal commodity taxation with imperfect competition, Warwick Economic Research Paper, No. 280. Myles, Gareth D., 1987b. Tax design in the presence of imperfect competition: An example, Journal of Public Economics 34, 367-378. Seade, Jesus, 1985, Profitable cost increases, Warwick Economic Research Paper, No. 260. Stern, Nicholas H., 1987, The effects of taxation, price control and government contracts in oligopoly and monopolistic competition, Journal of Public Economics 32, 133-158.