Public Choice 70: 225-237, 1991. © 1991 Kluwer Academic Publishers. Printed in the Netherlands.

Monopoly , rent-seeking, and second best theory*

R O G E R L A T H A M

F R A N C E S P E R I V A N C I C

Department o f Economics, York University, 4700 Keele Street, North York, Ontario, M3J 1P3

Submitted 13 June 1988; accepted 2 February 1990

Abstract. In the context of a simple general equilibrium model, in which there is a profit- maximizing monopolist, we show that in general the introduction of rent-seeking does not restore the first best pricing rule for the undistorted industry. This result is in direct contrast to that ob- tained when one assumes that the monopolist follows a full cost pricing rule with a constant mark- up ratio. Furthermore, it still holds even if the full cost pricing rule is profit-maximizing. We also investigate the conditions under which the first best pricing rule is reinstated for the undistorted industry.

1. Introduction

In a recent p a p e r A n a m and Katz (1988) (hereaf ter A&K) cons ider the impl ica-

t ions o f rent -seeking 1 for second best theory . More specif ical ly, in the context

o f a s imple genera l equ i l ib r ium mode l in which there is a monopo l i s t i c d is tor-

t ion, they show tha t the in t roduc t ion o f rent-seeking reinstates the first best

pr ic ing rule for the und i s to r t ed indust ry .

Fo l lowing Lipsey and Lancas te r (1956-1957) , however , A & K represent the

m o n o p o l i s t ' s decis ion in terms o f a full cost pr ic ing rule with a cons tan t m a r k -

up ra t io . 2 It is well known tha t , in general , such a rule nei ther implies nor is

impl ied by p ro f i t max imiza t ion . P ro f i t max imiza t ion , o f course, requires tha t

marg ina l revenue be equal to marg ina l cost .

In pa r t i a l equ i l ib r ium the full cost pr ic ing rule with a cons tan t m a r k - u p ra t io

is equiva lent to the marg ina l revenue equal to marg ina l cost cond i t ion if and

only if the m o n o p o l i s t ' s inverse d e m a n d func t ion is a rec tangula r hyperbo la . 3

In A & K ' s general equ i l ib r ium f r amework , it turns ou t tha t the rule is equiva-

lent to the cond i t i on if and only if the ut i l i ty func t ion no t only takes on a par t ic-

u lar f o rm with respect to the m o n o p o l i s t ' s ou tpu t but also s t rongly separa tes

it f rom the o ther ou tpu t s in the economy.

* The authors are grateful to Mahmudul Anam and Eliakim Katz for constructive comments on an earlier version and to Barry Smith for helpful conversations. However, the usual caveat applies.

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Consequent ly , it is o f interest to see whether A&K's result still holds if we

replace the full cost pricing rule with a constant m a r k - u p rat io with the margi- nal revenue equal to marginal cost condit ion. Tha t is the purpose o f this paper . We show that , in general , the in t roduct ion o f rent-seeking does not restore the

first best pricing rule for the undis tor ted industry. Section 2 contains the mode l and a descript ion of the A&K result for later

compar i son . Section 3 introduces the margina l revenue equal to marg ina l cost condi t ion and obtains a different result. Section 4 examines the circumstances in which the full cost pricing rule with a cons tant m a r k - u p ratio is equivalent

to the margina l revenue equal to margina l cost condit ion. Section 5 provides

a br ief s u m m a r y and conclusion.

2. The model

A&K, following Lipsey and Lancaster , assume that there are three single-

p roduc t industries producing outputs x, y, z. There is only one factor o f p roduct ion , 1 (say labor) , the to ta l supply o f which is fixed at i. The p roduc t ion

funct ions for each of the three outputs are given by

I x ly 1 z x - , y - , z - ,

a b c

where li, i = x, y, z, represents the a m o u n t of l abor employed in the ith indus- try, 1 x + ly + I z = i, and a, b, c are positive constants . Combin ing these as- sumpt ions yields the linear p roduc t ion possibil i ty front ier ,

ax + by + cz = i. (1)

The demand side is represented by the utility funct ion

u = u(x, y, z), (2)

which we assume to be strictly increasing in each o f its a rguments and strictly quasi-concave.

We find the first best o p t i m u m by maximizing the funct ion (2) subject to equat ion (1). Choosing x, y, z to maximize u(x, y, z) - k(ax + by + cz - i), where ~x is a Lagrange multiplier, yields the f i rs t -order condi t ions

u x - ka = 0, U y - ~,b = 0, u z - Xc = 0. (3)

Given the previous assumpt ions , these condit ions are not only necessary but

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also sufficient for the existence of a unique interior maximum. Eliminating X, we may express them as

u x a Uy b u x a

Uy b ' u z c ~ u z c (4)

It follows that the first best optimal pricing rules are

Px a py b Px a

py b Pz c Pz c (5)

where Pi, i = x, y, z, is the price of the ith output. Notice that, because the production possibility frontier is linear, we can infer these pricing rules directly.

Now A&K assume that an immovable monopolist produces x; that a com-

petitive industry produces z (designated as the numeraire commodity); that there exists a competitive wage rate, w; and that the optimal pricing rule for the industry producing y is to be determined. A&K postulate, for simplicity, that the monopolist producing x follows a full cost pricing rule with a constant

mark-up ratio; that is, Px = kaw, k > 1. Because price equals unit cost in the numeraire industry, that is, Pz = cw, we can express this rule as

Px a - k - - , k > l . (6)

Pz c

Then, using Px Ux - from the first-order conditions for utility maximization, Pz Uz

we can write equation (6) as

U x a - k - - , k > 1 . ( 7 )

u z c

We find the second best opt imum by maximizing the function (2) subject to

equations (7) and (1). We choose the outputs x, y, and z to maximize u(x, y, a] z) - o~ - k - ~(ax + by + c z - i), where c~ and ~ are Lagrange multi-

pliers. This process yields the first-order conditions:

Ux -- Ol OX -- ~ a = O; Uy -- Ol Oy - -

u ~ - ~ Oz - ~ c = O .

/ab = O;

(8)

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The previous assumptions do not guarantee that the second-order conditions for this problem, which depend inter alia upon third-order partial derivatives of the utility function, are satisfied. Consequently, although conditions (8) are necessary, they may not be sufficient.

Nevertheless, assuming that the second-order conditions are met, it follows from conditions (8) that the optimal pricing rule for the y-industry (in terms of the numeraire) is

O~ ff-z~-( ux \ / +

py = 0y \ u z / , (9)

0-~- + ~c

In general this pricing rule is different from the corresponding first best rule

in equations (5). But it will be the same if

0y = -~z = 0; ( lO)

U X that is, if the marginal rate of substitution of z for x, - - , is independent (at

U z

least locally) of both y and z. A&K introduce rent seeking by assuming the existence of a rent-seeking in-

dustry through which all of the monopoly rent is dissipated. This industry is taken to be in competitive equilibrium. It follows that the value of the labor, lr, employed in the rent-seeking industry must be equal to the value of the mo- nopoly rent (in terms of the numeraire); that is,

lrw a 1 - x. (11) Pz

Because price equals unit cost in the numeraire industry, that is, Pz -- cw, and because the monopolist follows the pricing rule in equation (6), that is,

Px a - - = k - - , k > 1, we can rewrite equation (11) as Pz c

1 r = (k-1)ax. (12)

Here it is important to notice that 1 r is independent of both y and z. Substituting equation (12) and the production functions into the constraint

that 1 x + ly + 1 z + 1 r = i yields

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kax + by + cz = i, (13)

which is the production possibility frontier modified to take into account both the monopolistic distortion and the associated rent-seeking. The new optimiza- tion problem is to maximize the function (2) subject to equation (13). Because the 'modif ied ' production possibility frontier (13) is linear, we can directly in- fer that the optimal pricing rules are

a py b Px a Px _ k - , - - = k - - . (14) Px b ' Pz c Pz c

Thus, the optimal pricing rule for the y-industry is the same as the correspond- ing first best rule in equations (5). Therefore, in these circumstances, the in- troduction of rent seeking removes the effect of the monopolistic distortion in the x-industry on the optimal pricing rule for the y-industry (as represented in equation (9)).

In summary, A&K show that

Proposition I: Given that the monopolist follows a full cost pricing rule with a constant mark-up ratio and that all o f the monopoly rent is dissipated through rent-seeking, the first best pricing rule is optimal for the undistorted industry.

3. The marginal revenue equal to marginal cost condition

In this section we assume that the monopolist , instead of following the rule in equation (6), chooses a level of x to maximize profit. The monopolis t ' s profit,

int msofthenumera e sg venby[ : a] Ux x. Because - f rom the c Pz Uz

first-order conditions for utility maximization, we can express this as

Ux(X y, z) a ] 7rX(x, y, z) = - x. (15)

L Uz(X, Y, z) c

The monopolist chooses x, given y and z, to maximize 7rX(x, y, z). In this con-

U x text the marginal rate of substitution of z for x, - - , plays the role of an inverse

U z

demand function (see Roberts and Sonnenschein, 1977). In the subsequent

analysis we assume that ~ (Uz) 2 u z U×x - u× Uz×

that this 'inverse demand function' is downward-sloping, and that

< O, SO

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[0 (u;) ux] x -~x + - - > 0, so that the monopolist 's marginal revenue is pos-

U z

itive. The first-order necessary condition for an interior optimum is

0 (U~z) u x a _ x 0x + 0, (16) U z C

an implicit function relating x, y, and z. This is the familiar marginal revenue

equal to marginal (average) cost condition. Because ~ < 0, it follows

ux Px a 0 [ a ( u ~ ) from condition (16) that - - > - - . Given that x +

Uz Pz c ~ ~-x 7

uf[ < 0, condition (16)becomes both necessary and sufficient for a unique ~zU

interior maximum. We find the second best optimum by maximizing the function (2) subject to

conditions (16) and (1). We choose the outputs x, y, and z to maximize u(x,

[ 0 U(~z) u x a ] y, z) - o~ x ~-x + . . . . . /3(ax + by + cz - i), where o~ and/3 U z C

are Lagrange multipliers. This yields the first-order conditions

~[ o(u~) ux] u x - c~ Ox x-~-x + - - - / 3 a = O,

U z

o [ o (u~) , ] U y - - O t ~ y X - ~ - X "-[- - - - / 3b = O,

U z

~ [ ~ (u~) ux] u x - ~ £ x3-Zx + - - U z

/3c = O.

(17)

Again, the previous assumptions are insufficient to guarantee that the second- order conditions for this problem are satisfied. Assuming that they are met, however, it follows from equations (17) that the optimal pricing rule for the y-industry (in terms of the numeraire) is

c ~ - - x + - - + ~ b P! = ay ~-x u z (18)

~z o~°[ o~°(u~) uxj x - = - + - + ~ c U z

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In general, this is different f rom both the corresponding first best rule in equa-

tions (5) and the second best rule in equat ion (9). It will be the same as that

in equat ions (5) if

o[ o(u ) .] o[ o(u ) "1 x + - - = x + - - = O; Oy -~x u z ~ z ~ u z

(19)

that is, if the monopol i s t ' s marginal revenue, x -~x + - - , is inde- U z

pendent (at least locally) o f bo th y and z. It will be the same as that in equat ion

(9) in those circumstances that we describe in Section 4.

We treat rent-seeking in the same way as in the previous section. In other

words, we assume that equat ion (11) holds. But now it follows f r o m condit ion

(16) t ha tPx a 0 (U~zz) - x - - . This condi t ion and Pz = cw permits us to Pz c Ox

rewrite equat ion (11) as

0(u ) lr = -cx2 OX

which is positive as a result o f the assumpt ion that

1 r depends on both y and z as well as on x. We can express equat ion (20) more

succinctly as

( 2 0 )

0 (U~XzX) < 0 . I n t h i s c a s e Ox

1 r = lr(x, y, z), (21)

01r 0 [ 0 (U~XzX) Ux] 01r 02 (UXzX) - - - - , - - , w h e r e 0 x - - c x 0x x ~ + - cx 2 and

u z 0y 0x0y

- cx 2 - - . Notice that - - is positive as a result o f the mono- Oz OxOz \ U z J Ox

01 r 01 r polist 's second-order condit ion, but that the signs o f - - and - - are indeter-

Oy Oz

minate.

Substituting equat ion (20) and the p roduc t ion funct ions into the constraint

that 1 x + ly + 1 z + 1 r = i yields

[ 0 a - ex 0x x + by + cz = i, (22)

a nonlinear p roduc t ion possibility frontier that takes into account both the

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monopolistic distortion (equation 16)) and the associated rent-seeking. Now the problem is to maximize the function (2) subject to condition (22). Choosing

x, y, and z to maximize u(x, y, z) - 0 a - cx Ox x + by + cz - i ,

where 0 is a Lagrange multiplier, yields the first-order conditions

0[ 0(u ) ,] Ux + x ×-O2x + - - U z

Uy h- qbCX 2 0x0y

Uz + 0CX2 0XOZ

0a = 0,

0b = 0, (23)

qSc = 0.

It follows from conditions (23) that the optimal pricing rule for the industry producing y (in terms of the numeraire) is

b - c x 2 02 (U~XzX) py OxOy

- . ( 2 4 )

c - cx 2 OxOz

Obviously, this is different from the corresponding first best rule in equations (5). Thus, given condition (16), rather than (7), the introduction of rent seeking does not resurrect the first best pricing rule for the y-industry.

Therefore,

Proposition II: Given that the monopolist follows a marginal revenue equal to marginal (average) cost pricing rule and that all o f the monopoly rent is dissi- pated through rent-seeking, the first best pricing rule is (in general) not optimal for the undistorted industry.

Clearly equation (24) will be the same as the corresponding first best rule in equations (5) if

OxOy - OxOz = O; (25)

that is, if the slope of the inverse demand function for the monopolist 's output,

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0 (u;) i nde endent,a leas ,oca,,y, ofbothya dz ,gx

Thus,

Proposition IlL" Given that the monopolist follows a marginal revenue equal to marginal (average) cost pricing rule, that all o f the monopoly rent is dissipat- ed through rent-seeking, and that the gradient o f the monopolist's inverse de- mand function is independent o f the other outputs in the economy, the first best pricing rule is optimal for the undistorted industry.

Notice that, using equation (21), we can equivalently express equation (24) as

01 r b + - -

py _ 0y. (26) Pz 01r

C + - - 0z

This restatement reveals that the effect of the monopolistic distortion in the x- industry on the optimal pricing rule for the y-industry (as represented in equa- tion (18)) has been replaced by the effect of the associated rent-seeking. It fol-

01 r 01 r lows that we can equivalently express condition (25) as - - 0. Namely,

0y 0z the first best pricing rule for the y-industry is optimal if the amount of labor employed in seeking the monopolist 's rent in the x-industry is independent of the amounts of y and z being produced.

4. E q u i v a l e n c e

Naturally, the question arises as to the circumstances in which the full cost pric- ing rule with a constant mark-up ratio of Section 2 is optimal, in the sense of being equivalent to the marginal revenue equal to marginal (average) cost con- dition of Section 3. Specifically, equation (7) is equivalent to equation (16) if and only if the utility function (2) satisfies

0 (U~z) ( k - l ) u x - - - O, k > 1, (27)

x 0x + k u z

a second-order partial differential equation. (We obtain equation (27) by

eliminating a between equations (7) and (16).) c

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Solving equation (27) we can show that

Theorem: The full cost pricing rule with a constant mark-up ratio (7) is optimal if and only if the utility function (2) takes on the form

1

I- 1 u = f kxk + h ( y , z ) , y ,

where fl, f2, hy, h z > O.

k > 1, (28)

Proof." See the appendix

Thus, conditions (7) and (16) are equivalent if and only if the utility function not only takes on a particular form with respect to x but also strongly separates it from y and z. If it is required that the monopolist 's full cost pricing rule with a constant mark-up ratio be optimal, then it is the analysis of Section 3, rather than that of Section 2, which is relevant. It is easily checked that the second proposition retains its validity when the utility function takes the form of the function (28).

Hence,

Proposition IV: Given that the monopolist follows an optimal full cost pricing rule with a constant mark-up ratio and that all o f the monopoly rent is dissi- pated through rent-seeking, the first best pricing rule is (in general) not optimal for the undistorted industry.

Before concluding, it is interesting to consider the special case in which the util-

ity function takes the form of the function (28) and, in addition, hzz and hzy are both equal to zero, that is, h(y, z) -- v(y) + tz, where t is a positive cons- tant. In this case we can show that condition (25) is satisfied, so the rule (24) is the same as the corresponding first best rule in equations (5). Thus, it appears that rent-seeking is effective in restoring the first best rule for the undistorted industry. In these circumstances, however, condition (19) (and condition (10))

is also satisfied, so that the second best rule (18) (and the second best rule (9)) is the same as the corresponding first best rule in equations (5). Therefore, as far as restoring the first best rule for the undistorted industry is concerned, the introduction of rent-seeking is superfluous. Consequently, for rent-seeking to have any significance in this regard, hzz and hzy cannot both be equal to zero. But then condition (25) is not met.

235

5. Conclusion

This paper shows that when we represent the monopolist's decision by a margi- nal revenue equal to marginal (average) cost condition, the introduction of rent-seeking does not (in general) restore the first best pricing rule for the un- distorted industry (proposition II). The effect of the monopolistic distortion is superseded by that of the associated rent-seeking. This result is in direct con- trast to that of A&K (proposition I), which is derived on the basis of the monopolist following a full cost pricing rule with a constant mark-up ratio.

We also investigate the conditions under which the introduction of rent- seeking does restore the first best pricing rule for the undistorted industry. Spe- cifically, we show that rent-seeking resuscitates the first being pricing rule if the gradient of the monopolist's inverse demand function is independent of the other outputs in the economy (proposition III). This result exposes the underly- ing partial equilibrium character of the A&K finding.

Finally, we examine the circumstances in which the full cost pricing rule with a constant mark-up ratio is profit-maximizing. This is the case if and only if the utility function takes on a particular form with respect to the monopolist's output and strongly separates it from the other outputs in the economy (theo- rem). Under these conditions rent-seeking remains ineffectual as far as restor- ing the first best pricing rule for the undistorted industry is concerned (proposi- tion IV).

Clearly, we have derived these results in the context of an extremely simple general equilibrium model. It seems reasonable to offer the conjecture, however, that similar results will hold, mutatis mutandis, in more complicated models, involving other kinds of imperfect competition and non-constant returns to scale.

Notes

1. Over the past decade the literature on rent-seeking has burgeoned. See the references contained in A&K (1988), especially the book edited by Buchanan, Tollison, and Tullock (1980).

2. To the best o f our knowledge, McManus (1959) was the first to criticize Lipsey and Lancaster in this respect.

3. Assuming constant average costs, as do A&K, the full cost pricing rule with a constant mark-up ratio k(> 1) is equivalent to the marginal revenue equal to marginal (average) cost condition if

~ dx Px . . . . . > 0. This is a first-order ordinary differen- and only if k, where r~ dp x x

i - k

(-U) tial equation and solving it yields Px = Ax , where A is an arbitrary positive constant.

2 3 6

R e f e r e n c e s

Anam, M. and Katz, E. (1988). Rent-seeking and second best economics. Public Choice 59:

215-224. Buchanan, J .M., Tollison, R.D. and Tullock, G. (Eds.) (1980). Towards a theory o f a rent-seeking

society. College Station, TX: Texas A&M University Press.

Lipsey, R.G. and Lancaster, K.J. (1956-1957). The general theory of the second best. Review of

Economic Studies 2 4 : 1 1 - 3 2 . McManus, M. (1959). Comments on the general theory of the second best. Review of Economic

Studies 26: 209-224. Roberts, J. and Sonnenschein, H. (1977). On the foundat ions of the theory of monopolistic com-

petition. Econometrica 45: 101 - 113.

A p p e n d i x

Proof of the theorem in Section 4.

Necessity u×(x, y, z)

Letting uz(x ' Y, z) - - - g(x, y, z), equation (27) becomes

0g ( k - 1) x0-~ + ~ g = 0 , k > 1.

The corresponding auxiliary equation is

( k - 1) dx dg

k x g

Integrating this equation yields

(k - 1)

g x k = [hzy ' z)]-l,

where h(y, z), and hence [hz(y, z)] -1, is an arbitrary function of the variables kept constant during

the integration. Therefore,

(k - 1)

x k u x - [h z (y ,z ) ] - l u z = 0

The corresponding auxiliary equation is

(1 - k)

x k dx = - h z (y, z)dz.

Integrating this equation yields

2 3 7

1

kxk + h(y, z) = m(y, u),

where m(y, u) is an arbitrary funct ion of the variables held constant during the integration. Taking

a partial inverse of this equat ion yields

1

u = f [kxk + h ( y , z ) , y ] , k > l ,

which is the funct ion (28) in the text. Notice that we assume that hy > 0, h z > 0, m u > 0, and

my < 0, so that u is strictly increasing in x, y, and z as required.

Sufficiency

Given the funct ion (28),

(1 - k )

k Ux X

u z hz

Therefore,

x - ~ - x +

( k - l ) u x ( l - k ) x -- X

k u z k h z

1 1 - - - 2 - - - 1 k ( k - l ) x k

+ - - k h z

= 0;

that is, equat ion (27) is satisfied. Q.E.D.