# Allocative inefficiency and measures of market power

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• Allocative Inefficiency and Measures of Market Power

JOHN P. FORMBY AND STEPHEN LAYSON*

Introduction

Lerner's classic paper on the measurement of monopoly power presents the well known in- dex that is now widely used as a conceptual measure of market power [Lerner, 1934]. The tack of marginal cost data often leads to the substitution of a profit rate for the Lerner in- dex in empirical studies. Both measures have appeal because it is thought that they are use- ful predictors of market performance [Scherer, 1980, p. 56]. Since the work of Bain, it is gen- erally recognized that there is not a one-to-one correspondence between monopoly power and market performances [Bain, 1959]. However, the exact relation between market power and performance is left somewhat vague. This paper investigates the precise relation of the Lerner index and profit rates to market performance as measured by allocative economic efficiency.

It is shown that changes in market power, as measured by a Lerner index or a profit rate, can easily move in the opposite direction of the static welfare losses from monopoly. This result is demonstrated using both linear and constant price elasticity demand functions. To keep the analysis simple, it is assumed throughout that the firm's production function exhibits constant returns to scale and that the input prices facing the monopolist are constant.

Linear Demand

Let demand, constant long-run marginal cost, and marginal revenue be given by:

P=a -bQ; (1)

MR = a - 2bQ; and (2)

MC = MCo . (3)

*University of Alabama and University of North Carolina-Greensboro, respectively.

Setting MR = MC and solving for monopoly output and price:

Qm = (a - MCo )/2b; (4)

e m = (a + MCo)/2. (5)

The efficient output level, found by equating P and MC is:

Q* = (a - MCo )/b. (6)

The welfare losses from monopoly are:

W = 1/2(P,n -MCo) (Q* - Qrn). (7)

Substituting for Pro, Q*, and Qm in (7) yields:

W= 1/2[(a-MCo)/2] [(a-MCo)/2b], (8)

and

W = (a - MCo )2/8b. (9)

The Lerner index of monopoly is:

L = (I0)

(Pro -MCo )/P,,, = (a -MCo )/(a + MCo ).

Because long run marginal cost and long run average cost are equal, in this case, the profit rate, expressed as a proportion of total revenue, is identical to the Lerner index:

rr/PQ = ((P -MC)Q) /PQ = L, (11 )

The relationship between the Lerner index and profits, expressed as a percentage of owners' equity, V, is given by:

rr/V= (P -MC) (Q/V) . (t2)

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Owners' equity is simply the value of the firm's capital stock. Letting Pk be the price per unit of the capital stock and K be the units of capi- tal,

V=PkK. (13)

Assuming units of the capital stock can be purchased at a constant price, and that the flow ....... of capital services is proportional to the stock of capital, the ratio Q/V is constant for a firm with a constant retnrns to scale (CRS) produc- tion function.' Note that a CRS production function was initially assumed in order to gen- erate the constant long run marginal and average ,'~" '~ costs. Treating Q/V as a constant, profits ex- pressed as a percentage of owners' equity, (12), is proportional to (P- MC).

Now, comparisons of (9), (10), and (11), re- veals that b, the absolute value of the slope of the demand curve, is inversely related to the welfare losses from monopoly but has no im- pact on the Lerner index or the profit rate, ex- pressed as a proportion of total revenue. From Equation 5, a change in b also leaves the monopoly price unaffected; hence, 7r/V in Equation 12 is also unaffected by a change in b.

The implications of (5), (9), (10), (11), and (12) are clear. A specific Lerner index or profit rate, expressed as a proportion of total revenue or owners' equity, is consistent with any mag- nitude of the welfare loss from monopoly. This is illustrated in Figure I for two demand curves

Price

D1 and D2 with common values of a, MCo, and ~,~ P but different values of b. Because a and MCo are unchanged, L is the same for both D~ and ~'~ D2 .~ Because Q/V can be treated as a constant, the constancy of P and MC also imply that

1 Holding input p~ices constant and assuming a CRS production function, the cost minimizing ratio of capital services to other inputs remains the same for all levels of output. It follows, in the long run, that output is proportional to the flow of eapital services.

2 Another way to show that the Lerner index of monopoly power is consistent with widely different amounts of welfare losses from monopoly is to rotate the MC curve in Figure I counter-clockwise through point 6". This decreases the welfare losses from monop- oly but leaves the Lerner index unchanged.

profit as a proportion of owners' equity, given by (12), is also the same on D1 and D2, but as can easily be seen in Figure I, the welfare losses, W1 and W2, differ widely.

Figure l

Ri\$in~ Welfare Losses WiEh a Constant Lerner Index

\\, \

\ \ "\ \ \, , \

\

l 'MR2 1 ,1~ ~ ,1~ ~ ~ .........

To show that the welfare losses may move in the opposite direction of the Lerner index and the profit rates, compare demand curves D1 and D2 in Figure II. Because the vertical inter- cept for D2 is lower than the vertical inter- cept of D1, the value of Pa is less than P1

Figure II

Welfare Losses and Lerner Indexes Move in the opposite Direction

• FORMBY AND LAYSON: ALLOCATIVE INEFFICIENCY 69

and the Lerner index and the profit rates are lower for D23 but the welfare losses from mo- nopoly are clearly larger, W2 > Wl.

As before, the profit rates expressed as a pro- portion of total sales and owners' equity are, respectively,

Constant Demand Elasticity

Results shnflar to the linear demand case can be derived for constant price elasticity demand functions. Because empirical econo- mists often use constant price elasticity func- tions to estimate demand, the demonstration that the Lerner index and the profit rates may move in the opposite direction of the welfare losses for shifts in constant price elasticity de- mand functions makes this point much stronger. Let the demand function be:

Q=oa -B ~ > 0,B > 1, (14)

where B is the price elasticity of demand. Changes in c~, the multiplicative constant, result in equi-proportional changes in quantity de- manded for a given price.

Average revenue is:

p = al/B Q -vB (15)

rr/PQ = L, and (19)

~r/V = (P -MC)(Q/V). (20)

It is clear from (17), (18), (19), and (20) that when demand is characterized by a constant price elasticity, changes in the multiplicative constant, o~, have no effect on Pro, L, ~r/PQ, or ./v.

The welfare loss from monopoly in the case of constant price elasticity is given by:

(21) W = fQ* oy~ Q-1/t~dQ-MCo(Q,-Qra),

Qm

where Q* and Qm are the efficient and monop- olyoutputs, respectively. The monopoly out- put can be shown to be:

Qm = a(MCo/(1 - 1/13)) -t~ , (22)

The first order condition for maximum monop- oly profit, for constant marginal cost is:

MR =P(1 - l /B)=MCo, (16)

and the efficient output is:

Q* = o&ICo -~. (23)

or Evaluating the integral in (2 t) yields,

Pm =MCo/(1 - l/B).

The Lerner index in this case is

(17) W=(a 1/t~'/(1 - 1/13))Q -l~ + '1~* (24)

-MCo(Q*- Qm) .

L=I /B . (18) Substituting for Q* and Qm in (24) gives:

3 To show that the Lerner index and the profit rate vary directly with a, differentiate (10) and (12) with respect to a:

0L - - : 2MCo/(a + MCo) a. > 0 i)a

1 ~Or/v)/~a=-(Q/V) > O.

2

W = a[(1/(1 - 1//3)) (25)

(MCo 1-fl _ (MCo/ (1 - 1/t3)) l-fl) - MCo (MCo -~ - (MCo/ (1 - 1/13))-~) ].

The terms within the brackets in (25) must sum to a positive number i.e., the welfare losses from monopoly are necessarily positive. For this reason, an increase in the multiplicative

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constant, a, in a constant elasticity demand function, increases the welfare loss from monopoly. As a increases (decreases), the wd- fare losses from monopoly increase (decrease) but the Lerner index and the profit rates re- main unchanged. A given Lerner index or profit rate is consistent with literally any amount of welfare losses from monopoly.

To illustrate that the Lerner index and the profit rates may move in the opposite direction of the welfare losses, for the case of constant price elasticity demand functions, let long run marginal cost be .5, let the ratio Q/V be con- stant at .4, and let the parameters of the demand functions in periods 1 and 2 be denoted by:

cq = 100 a2 = 200

/31 = 1.8 /32 = 2.

The profit maximizing values of P and Q in period 1 and 2 are:

P1 = 1.125 P2 = 1

Q1 = 80.90 Q2 = 200.

Solving for the Lerner indexes, the profit rates and the welfare losses in periods 1 and 2 gives:

L1 = .56 L2 = .50

7h /(el at ) = .56 7h /(P2 Q2 ) = .50

zq/v1 = .25 zr2/V2 = .20

W1 = 53.32 W2 = 100.

In this example, both the Lerner index and the profit rates are less in period 2 than period 1, yet the welfare losses from monopoly are larger in period 2.

Conclusion

This paper has shown that under conditions of both linear and constant price elasticity de- mand functions, market power as measured by the Lerner index and profit rates may easily move in the opposite direction of the welfare losses from monopoly. This means that changes in monopoly power, as measured by either Lerner indexes or profit rates, are not adequate to predict qualitative changes in allocative in- efficiency.

More (less) market power does not necessarily imply more (less) atlocative inefficiency whether one is comparing different industries at a point in time or a given industry at different points in time. For these reasons, interpretation of the relation between market power as measured by the Lerner index or profit rates and market performance must proceed with caution.

REFERENCES

A. P. Lerner, "The Concept of Monopoly and the Measurement of Monopoly Power," Review of Eco- nomic Studies, Vol. I, June 1934, pp. 157-75.

F. M. Scherer, Industrial Market Structure and

Economic Performance, 2rid ed., Chicago, Rand McNally, 1980.

J. S. Bain, Industrial Organization, New York, Wiley, 1959.