lobbying for protection by heterogeneous firms

European Journal of POLITICAL

European Journal of Political Economy ECONOMY Vol. 12 (1996) 19-32 ELSEVIER

Lobbying for protection by heterogeneous firms

Ngo Van Long a,b,*, Antoine Soubeyran b

a C1RANO and Department of Economics, McGill University, 855, Sherbrooke Street West, Montreal, Qud, Canada H3A 2T7

b GREQAM, Universit£ d'Aix-Marseille II, Ch~tteau La Farge, Route des Milles, 13290 Les Milles, France

Abstract

We show that total lobbying expenditure by domestic oligopolists is a function of the degree of heterogeneity of the industry, as well as of the demand condition. Sufficient conditions for big firms to contribute more (or less) than small firms are obtained. The effects of an increase in the degree of heterogeneity on lobbying expenditure are examined in two models of rent seeking.

JEL classification: A10; FI3; L13

Keywords: Lobbying; Tariff; Heterogeneity

1. Introduction

There is ample evidence that theories based on the paradigm of a benevolent government fail to explain the actual pattern of trade protection. (See the survey of the literature by Hil lman (1989, ch. 11).) Faced with the bewildering structure of trade barriers observed in any modern economy, economists have increasingly turned to the view that government 's economic policies such as taxes, tariffs and quotas are determined so as to equilibrate political markets rather than maximize some conception of social welfare (see Olson (1965), Peltzman (1976), Findlay

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20 N. V. Long, A. Soubeyran / European Journal of Political Economy 12 (1996) 19-32

and Wellisz (1982), Becker (1983) among others). Two groups of models have been developed to study how economic policies are politicized. The first group of models focuses on the economic calculus of policy makers as self-interested optimizing agents (see, for example, Hillman (1982), Brock and Magee (1978), Long and Vousden (1991)). The second group focuses on the economic calculus of interest groups (see Hillman and Riley (1989), Long and Vousden (1987), Cairns and Long (1991), and the survey by Nitzan (1994)). Grossman and Helpman (1994) view interest groups as principals and the government as their common agent.

For the sake of simplicity, most authors have modeled each interest group as a collection of identical agents. The price of this simplifying assumption is not negligible. In an insightful paper, Hillman (1991) convincingly argues that heterogeneity among firms and hence the distribution of market shares play a major role in political allocations of firms to influencing endogenous economic policies. The purpose of our paper is to provide some additional theoretical support for the idea that the degree of heterogeneity within a pressure group is an important determinant of the group's influence. Instead of postulating that the likelihood of success is a decreasing (or maybe increasing) function of the degree of heterogeneity of a pressure group, we approach this question with an open mind. We consider a simple model in which the dominant interest group consists of domestic firms in an oligopolistic industry facing competition from imports produced by foreign oligopolists. The domestic firms are heterogeneous in the sense that they have different unit costs. We use the variance of the distribution of unit costs within the group as a measure of the group's heterogeneity. The firms face a public good problem: they know that the tariff rate can be influenced by their aggregate lobbying expenditure. Two cases are considered. First, we study the non-cooperative Nash equilibrium in lobbying activities, in the spirit of Bergstrom et al. (1986), and Cornes and Sandler (1986). The results for this case are reported in Section 3. In Section 4, we turn to the case in which firms reach an agreement on how much each should contribute to a common lobbying budget. The theory of Nash cooperative bargaining is used here. Some concluding remarks are offered in Section 5. Our basic conclusions are that total lobbying expenditure depends on the degree of heterogeneity of the industry and on the curvature of the demand curve and that larger firms do not necessarily contribute more than smaller firms.

2. The basic model

We consider a market for a homogenous product in the home country. The demand function in the home country is

P=P(Q) (2.1)

N.V. Long, A. Soubeyran / European Journal of Political Economy 12 (1996) 19-32 21

with P'(Q) < 0). This market is supplied by m domestic finns and m* foreign firms. Let qi denote the output of firm i, to be sold in the home country.

re+m*

Q= ~ qi. (2.2) i = 1

The unit cost of firm i is denoted by Oi. Because of constant unit costs, the output decisions for the market in the home country are independent of the output decisions for the markets in other countries. The foreign firms are located in the foreign country. Their exports to the home country are subject to a specific tariff rate of t dollars per unit (the assumption that the tariff is specific, rather than ad valorem, is made to simplify notation; the results of our analysis remain basically unchanged for the case of an ad valorem tariff). The Cournot equilibrium is characterized by m + m* equations:

q , P ' ( Q ) + P ( Q ) = O i, i = 1 , 2 . . . . . m, (2.3)

q j P ' ( Q ) + P ( Q ) = O j + t , j = m + l . . . . . m+m*. (2.4)

The second-order conditions are

qkP"(Q)+2P ' (Q) < 0 , k = 1 , 2 . . . . . m+m*. (2.5)

Summing (2.3) and (2.4) over all firms, we obtain

QP'( Q) + (m + m* )P(Q) = Os + m* t (2.6)

where 19, is defined as

m+m*

O, = ~2 O h. (2.7) k = l

It follows from (2.6) that, given the number of firms, the industry's equilibrium output is a function of the sum of the unit costs and of the specific tariff rate times m*, the number of foreign finns, and is independent of the distribution of unit costs. Hence we can write

Q = Q( O~ + m* t). (2.8)

This result is a straightforward extension of a theorem of Bergstrom and Varian (1985). It is easy to see that the Cournot equilibrium output is a decreasing function of the tariff rate. From (2.6),

OQ m*

ot O P " + ( m + m * + l ) P " (2.9)

The denominator on the right-hand side of (2.9) is negative, due to the stability condition (see Dixit, 1986, condition 36-v). We are now ready to prove the following lemma:

22 N.V. Long, A. Soubeyran / European Journal of Political Economy 12 (1996) 19-32

Lemma 1. An increase in the tariff rate will increase the profit of all domestic f irms.

Proof From (2.3), for i = 1,2 . . . . . m,

P( Q) - 19, qi [ _ p , ]

Firms i's equilibrium profit is

(2.10)

[ P ( Q ) - 0,] 2 Hi = [ P - 19i]qi = [ _ p , ] = H i ( Q , 19i). (2.11)

It is a function of Q and ®i, where Q is given by (2.8). Differentiating (2.11) with respect to t:

at - OQ ]~ at (2.12)

an, [ (e- o,)P"- 2(P')2][e- o,] where

(2.13)

(2.14)

is negative, due to the second-order condition (2.5). It follows from (2.9), and (2.14) that all domestic firms gain from an increase in the tariff.

aQ [ _ p , ] 2

Substituting (2.3) into (2.13), we obtain

an, =qi[qiP" + 2P'] aQ

which (2.12)

We now introduce lobbying. We assume that the government of the home country will respond to the lobbying activities of the domestic oligopolists by raising the tariff rate. In principle, foreign firms could lobby, too; see Hillman (1989, ch. 8). However for simplicity we abstract from this consideration. Let x i denote the effective lobbying expenditure made by domestic firm i. The tariff is taken to be an increasing function of the sum of all the x,. Let

x = ~ x i. (2.15) i=1

Then, we postulate that

t = t ( x ) , t ' ( x ) > O , t " ( x ) < O . (2.16)

The signs of these derivatives reflect the usual assumptions that the marginal effectiveness of lobbying expenditure is positive but diminishing. There are

N. V. Long, A. Soubeyran / European Journal of Political Economy 12 (1996) 19-32 23

several ways of modeling the decision process of the domestic oligopolists. For simplicity, we will consider only two models: a non-cooperative lobbying model, and a cooperative lobbying model. In both models, we assume that domestic firms remain Cournot rivals when they make their output decisions, perhaps because of antitrust legislations.

3. Non-cooperative lobbying

The domestic firms' actions are taken in two stages. In stage I, domestic firm i spends x i dollars on lobbying. This is done simultaneously by all the domestic firms. The government then determines the tariff rate t. The government's decision rule t ( x ) , described by Eq. (2.16), is known by all firms. In stage II, the domestic firms, knowing the tariff rate announced by the government, decide how much to produce. They act as Cournot rivals, and therefore conditions (2.3) and (2.4) apply. Since we are interested only in equilibria that are sub-game perfect, we take it that, in stage I, all firms know that a Cournot equilibrium will obtain in stage II, and that this equilibrium will be parameterized by the specific tariff rate t, which they know they can influence by their effective lobbying expenditures in stage I. We assume that to achieve the effective lobbying level x i, the total cost to firm i is x i + l(xi), where I ( x i) represents the associated indirect cost. A possible interpretation is as follows: x i is the transfer payment made to the government officials, and I ( x i) is the cost of making such transfer payment less visible. We assume I ( x i) is strictly convex and increasing. In stage I, firm i must choose xi that maximizes its net profit N/:

N, = IIi[ Q( t),~gi] - x i - - I ( x i )

where I I i [ Q ( t ) , @ i] is its Cournot equilibrium profit in stage II, and where t stands for t(x~ + . . . +Xm). In maximizing its net profit, the firm takes into account the dependence of the tariff rate on its own lobbying expenditure x i. All domestic firms move simultaneously. The necessary condition for maximizing N~ is

r ( x ) - 1 _<0 (3 .2)

where the strict inequality holds only if the firm's optimal choice is a corner solution. Do bigger firms spend more on lobbying? The answer depends on the elasticity of the slope of the demand curve, evaluated at the Cournot equilibrium output curve. Define this elasticity as

R = - Q P " / P ' . (3.3)

(Notice that R has the sign of P".) We can now state our first proposition.


Proposition 1. Assume that I ( x ) is strictly convex. For any two domestic firms i and j, where @i < @j (i.e. firm i is the bigger firm), the following results hold:

(i) The bigger firm spends more on lobbying if R < 0. (ii) The smaller firm spends more on lobbying if R is a sufficiently great

positive number.

Proof Assume an interior solution for both x i and x;. Then (3.2) gives

OIIj/OQ I'( xi)

o~/OQ I'(xj)

Using

Since

(3.4)

(2.13), the left-hand side of (3.4) is the ratio

(P-Oi)(2p'2-P"(P-Oi)) ( P - Oj ) (2P ' 2 - P"( P - Oj)) = riJ" (3.5)

P - ®~ > P - O j, it follows that if P " < 0 then the ratio rij in (3.5) is greater than 1. This implies I ' (x i ) > I ' (x j ) , which is equivalent to x~ > xj due to the strict convexity of the function I. This proves (i). If P" is positive and is sufficiently great such that the ratio rij in (3.5) is smaller than 1, x i < xj. It should be noted, due to (2.5) and (2.10),

2P ' 2 - P " ( P - Oi) > O. (3.6)

This completes the proof of Proposition 1.

To understand why a smaller firm may spend more on lobbying than a bigger one, it is useful to bear in mind that if R is positive, an increase in the tariff rate will lead to a relative expansion of the smaller domestic firms. This result is stated and proved below.

Proposition 2. With an increased tariff, smaller domestic firms expand relatively to larger firms if and only if R is positive.

Proof From (2.3), for all i, k < m,

(Ok - Oi) qi - qk -- _ p,

It follows that

d(q i - -qk ) ( O k -- O i ) P " ( Q ) ( O Q / O t )

at [_p,]2

(3.7)

(3.8)

We now turn to the main question stated in the introduction: does an increase in the degree of heterogeneity of domestic firms lead to an increase in aggregate


lobbying expenditure? To simplify the algebra, let us consider the case where there are only two domestic firms, with unit cost O H and O e respectively (O n > OL). Corresponding to (3.2), we now have two first-order conditions:

OH H F ( X " ' X L ' O " ' O L ) = at t ' ( x ) - l - / ' ( x . ) = O , (3.9)

OHL , G(x n , x L , O n , O L ) = - - ~ t t ( x ) - l - I ' ( x L ) = 0 . (3.10)

The second-order conditions are:

OF F . - < 0 , (3.11)

Ox H

OG G L - < 0 . ( 3 . 1 2 )

Ox L

We show in the appendix that at the Nash equilibrium of the lobbying game, the following condition holds:

D = FHG L - FLG H > O. (3.13)

From (2.8), we know that if O H + {~L is a constant, then, for given t and foreign costs ®j ( j = m + 1 . . . . . m + m* ), the industry's equilibrium output is uniquely determined. In particular, if we increase O H by ~ > 0 and reduce O L by the same amount, there will be no change in industry output, if t stays constant. We interpret an increase in ~ as an increase in the degree of heterogeneity of the domestic finns. Let us now determine the impact of such an increase in heterogeneity of the total lobbying expenditure. Write

O H = O H + e,

O e = O L - e.

Differentiate (3.9) and (3.10) with respect to e:

F.( Oxn ] + OXL ] - OF --~-e ] FL(---~-e ]

GH( OXH t -'I-GL( ] ""~-ff ) = - - - -

The following proposition

OE '

OG

Oe

follows immediately from (3.16) and (3.17).

(3.14)

(3.15)

(3.161)

(3.17)

Proposition 3. An increase in the degree of heterogeneity of the domestic firms will increase the total lobbying expenditure if P" < 0 and I"( x) is non-increasing.

Proof See the appendix

26 N. V. Long, A. Soubeyran / European Journal o[ Political Economy 12 (1996) 19-32

Intuitively, if P" is negative, an increased tariff tends to benefit large firms relative to small firms. This bias is more pronounced, the higher the variance of the distribution of unit costs. This explains the result. It is important to note that the above assumptions ( P " < O, l " ( x ) non-increasing) are sufficient conditions: the result holds under weaker conditions, as can be seen from the proof of Proposition 3. Sufficient conditions for an increase in the degree of heterogeneity to decrease total lobbying expenditure can also be found from the Eq. (A.16) in the appendix. However, they are rather unwieldy. We now turn to the case of cooperative lobbying, where the necessary and sufficient conditions for increased lobbying are much sharper.

4. Cooperative lobbying

When firms cooperate in their lobbying activities, they collectively decide how much each must contribute. Since we are dealing with firms having different unit costs of production, it is clear that in general they do not contribute equal amounts. In what follows, we use the Nash cooperative bargaining approach to model the cooperative lobbying decision. We continue to assume that the firms remain Cournot rivals in the production stage. Consider the simple case where there are two domestic firms. If they fail to agree on how much each should contribute to the common lobbying budget, they will lobby separately, and the outcomes will be as described in the previous section. Their net profits in the non-cooperative lobbying cases will be denoted by d i ( i = 1,2); these are often called the disagreement payoffs. Suppose that, in the cooperative case, an amount x is transferred to the policy maker in order to secure a tariff rate t (x) . The total cost is then x + l(x), where l ( x ) is the associated indirect cost of lobbying. The two firms must contribute amounts yl and Y2 such that

Y, + Y2= x + l ( x ) . (4.1)

Firm i 's net revenue is the profit it obtains from selling its output as a Cournot rival, minus the amount y /

Ri = Hi - Yi. (4.2)

In Eq. (4.2), II i is the profit in a Cournot equilibrium. It is given by Eq. (2.11). According to the theory of Nash cooperative bargaining, the equilibrium values Yl and Y2 must maximize the Nash product • defined below:

l/f= (R 1 _ d l ) (R2 _d2 ) (4.3)

subject to

2

R 1 + R 2 = Y ' ~ I I i [ O i , Q ( O , . + m * t ( x ) ) ] - x - l ( x ) . (4.4) i--I


Clearly, to maximize (4.3), it is necessary that the right-hand side of (4.4) be maximized by choosing the value of the transfer, x. It turns out to be more convenient to use the inverse function,

x = x ( t ) . (4.5)

This function tells us the amount of transfer necessary to achieve a given tariff rate. Because it is obtained from (2.16), it has the following properties:

x ' ( t ) > O, x" ( t ) > 0. (4.6)

Generalizing the above arguments, if there are m domestic firms, their cooperative problem in the lobbying stage consists of finding a value of the tariff, t, that maximizes the sum of their net profit, B:

B = ~ IIi[ Oi, Q( O s + m* t)] - x ( t ) - l( x( t ) ) . (4.7) i=1

The first-order condition for the maximization problem is

OH i OQ [1 + l ' ( x ( t ) ) ] x ' ( t ) = 0 . (4.8) i=10Q Ot

This equation determines t, and hence the effective total lobbying expenditure x. Our central question is: does an increase in the degree of heterogeneity of the domestic firms lead to an increase in x? To answer this question, it is useful to re-write the first-order condition (4.8) in such a way that an index of heterogeneity appears on the left-hand side of that equation. For that purpose, the following lemma turns out to be useful:

Lemma 2. Let 0 M and V M denote respectively the mean unit cost of the domestic firms, and the variance (or dispersion) of the distribution of the unit costs within the group of domestic firms. Then the mean profit of the domestic firms in a Cournot equilibrium is given by

V M + ( P - OM) 2 H M = (4.9)

- P ' ( Q )

where

1 O M = - - ~ Oi, (4.10)

mi=l

1 m

vM = - E ( o i - o M ) 2. mi=l

(4.11)

Proof See the appendix.


The first-order condition (4.8) can now be rewritten as

OH m OQ m - - [1 + l ' ( x ( t ) ) ] x ' ( t ) = 0 . (4.12)

OQ at

On the other hand, it is clear from (4.9) that

OHM_ l [sR(VM+tZ~)_Zmlx~] (4.13) cgQ mtz M

where P'M is the mean equilibrium mark-up of the domestic firms,

ix M = P(Q) - O M , (4.14)

and s is the market share of the domestic firms,

1 m

s= -~ i~= lq i, (4.15)

while R is the elasticity of the slope of the demand curve, defined by Eq. (3.3). It is important to recall that an increase in the variance (or the degree of heterogeneity of the domestic firms), keeping the mean unit cost constant, does not change industry output and price. This was clear from Eq. (2.6) and the comments that follow it. Substituting (4.13) into (4.12), we can find the comparative static response of t (and hence x) with respect to an increase in the degree of heterogeneity of the domestic firms. Since industry price and output remain constant, there will be no change in R, s and ix M. The following result concludes this section:

Proposition 4. If the elasticity of the slope of the demand curue at the Cournot equilibrium is negative [ respectiuely, positiue], then an increase in the degree of heterogeneity of the domestic firms will increase [ respectiuely, decrease] the total lobbying expenditure for the tariff protection.

Proof Let +(V M, t) denote the left-hand side of the first-order condition (4.12). Then

OV~ - ~ ~t (4.16)

Since (Ocb/Ot)< 0 is the second-order condition, the sign of 0t/0V M is the same as the sign of 0qb/0V M, which, using (4.13), is

1 OQ - - s R (4.17) IzM Ot

where OQ/Ot is negative, by (2.9). This completes the proof.


The above proposition shows that the sign of R (or of P") plays a crucial role in determining the effect of increased heterogeneity. If R is positive, then an increase in the tariff will lead to the relative expansion of smaller domestic firms (i.e. those domestic firms with high unit costs), as proved in Proposition 2. As a result, the gain in profits for the domestic firms as a group is to some extent dissipated by a relative decline in efficiency. This dissipation is larger, the bigger is the gap between unit costs. Therefore, a greater dispersion of unit costs implies a lower marginal benefit of a given tariff increase. This explains why, when R is positive, a higher variance of unit costs results in a lower level of lobbying expenditure. A parallel argument applies to the case where R is negative.

5. Concluding remarks

We have shown that the degree of heterogeneity of a pressure group has important implications for its total lobbying expenditure and hence its degree of success. In the cooperative lobbying case we obtained a sharp result that an increase in heterogeneity will increase total lobbying expenditure if and only if, at the equilibrium, the elasticity of the slope of the demand curve is negative. In the non-cooperative lobbying case, a sufficient condition for a positive correlation between heterogeneity and lobbying expenditure was obtained. It is broadly in the line with the cooperative lobbying case. For simplicity, we have restricted attention to one kind of heterogeneity: differences in unit costs of production. The resources used in lobbying activities were assumed to have no effects on production costs. An alternative formulation would he to introduce into our oligopoly model the assumption of Hillman (1991) that each firm is endowed with a fixed stock of resources (entrepreneurial and managerial abilities) to be allocated between political and economic activities. Heterogeneity then appears in the form of differences in endowments of abilities. This approach adds another dimension to the analysis of rent-seeking under oligopolistic conditions.

Appendix A

A. 1. Proof of Proposition 3

We apply Cramer's rule to Eqs. (3.16) and (3.17). From (2.9), (2.12) and (2.13), "

OH. [( P - O ) n l P " - ZP'Z]( P - O)n)m *

Ot = [ Q p " + ( m + m * + l ) p ' ] p '2 > 0 , ( a .1 )


0171 L [( P - O L ) P " - 2p '2 ] ( p - O L ) m *

0, [ee"+Cm+m ÷ 1 ) e ' ] e '2 >o . (1.2)

Let A and B denote the right-hand side of (A. 1) and (A.2) respectively. Then the coefficients of the system (3.1 6)-(3.1 7) are

F n = A t " ( x ) - - I " ( X H ) ,

F L = A t " ( x ) ,

OF - - = t ' C x ) [ 2 p ' 2 - Z p " C P - O H ) ] C , OE

where

m* C = <0 ,

[QP" + ( m + m* + 1)P'] p,2

G. =Bt"(x), a L = B t " ( x ) - Z"(XL)

OG

Therefore

OG

Oe

Let

- - - - - t ' ( x ) [ 2 p ' 2 - - 2 p " ( p - - O L ) ] C .

OF

OE + 2t'( x ) C P " ( O H - OL).

(A.3) (A.4)

(A.5)

(A.6)

(A.7)

(A.8)

(A.9)

(A.10)

(A.14)

(A.15)

(A.16)

6~X H

&

C~X L

&

It follows that

a(x. ÷x~) &

1 -- ~ [ a ( l " ( x , ) - I"(XL) ) + bI" (xH) ] .

1 [( aA + aB + bA)t"C x ) - aI"( XL)],

D

1 D [Ca + b ) I " ( X H ) -- (aA + aB + b A ) t " ( x ) ] .

OF a - ae ' (A'11)

b = 2 t ' ( x ) C P " ( O H - OL). (A.12)

The system (3.16)-(3.17) becomes

(1.13) Bt"(x) B t " ( x ) - - I " ( x L ) l l d x L I = ( - a - b)de] '

It is easy to verify that the determinant D is positive. Applying Cramer's rule,

N.V. Long, A. Soubeyran/European Journal of Political Economy 12 (1996) 19-32 31

If P" < 0, then x H < x L by Proposi t ion 1. If I " ( x ) is non- increas ing, l " ( x . ) - ,, > p" _ I ( x e) _ 0. Also, < 0 implies that a > 0 and b > 0. Since D > 0 the r ight-hand

side of (A.16) is posit ive under the above assumptions.

A.2. Proof of Lemma 2

From (2.3),

1 qi = p _ [ P ( Q ) - Oi]. ( A . 1 7 )

Fi rm i ' s profit at the Cournot equi l ibr ium is

[ P ( Q ) - 0~] 2 H i = [ P ( Q ) - Oi]qi = - P ' ( A . 1 8 )

Therefore

- 1 ~ [ ( P - OM) + (OM _ Oi) ] 2 ( 1 . 1 9 ) m H M = ~_, Hi -- p, i = 1 i = 1

and, f inally

mHM = _ P--;1 [ m ( P - OM) 2 71- mVM] . ( A . 2 0 )

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