Monopoly vs. Competition in Light of Extraction Norms By Arkadi Koziashvili♣, Shmuel Nitzan♣ and Yossef Tobol♠
Abstract This note demonstrates that whether the market is competitive or monopolistic need not be the result of ideology, the relative political power of a certain producer or the nature (non-convexity) of the technology. The form of the market structure can be determined by a government that maximizes the extracted resources when it considers the two alternative market structures. The extracted profits hinge on the demand and cost characteristics and on the prevailing norms of extraction. Our claim is illustrated by assuming that both demand and supply are linear functions and that the government can extract a fixed rate of every competitive producer's profit or the resources expended by the producers when they contest on the monopoly profit.
Keywords: monopoly, competition, norms of extraction, JEL classification codes: D72; D78; O50 ♣
♠
Department of Economics, Bar Ilan University, Ramat Gan, 52900 Israel, Corresponding author, Tel: +972 3 531 8345, Fax: + 972 3 535 3180, e-mail:
[email protected]. School of Industrial Management, Jerusalem College of Technology (JTC). e-mail:
[email protected] .
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1. Introduction Suppose that social norms allow the use of political office for privileged distribution such that society has a political culture of extraction, Congleton, Hillman and Konrad (2008), Epstein and Nitzan (2007), Konrad (2008) or corruption, Ades and Di Tella (1999), Bliss and Di Tella (1997), Treisman (2000). Specifically, this culture is represented by two options available to the government. First, there exists a fixed profit share R the existing n competitive producers are required to give up in order to secure the government approval of their production activity. Second, the government can assign the monopoly status to one of the existing producers extracting the monopoly profits from the producers that take part in the contest on the award of the monopoly rent. The objective of this note is to demonstrate the determination of market structure by the government in the binary case of monopoly vs. competition. Assuming that the cost and demand functions are linear, we examine the effect of the market characteristics as well as of the prevailing norm of extraction or corruption on the government's choice between monopoly and competition.
2. Monopoly vs. competition Suppose that there are n identical producers in the industry. One option the government has is to award the exclusive right of production to one of these producers, the winner in the contest on the monopoly profit 1 . In such a case, we assume that the government is able to extract the monopoly profit A from the n riskneutral contestants. This assumption is plausible under certain conditions that are well known in the contest-theory literature. For a recent study of the class of contests that satisfy this assumption see Alcalde and Dahm (2009).2 The other possibility is to let these n identical producers get the profit of a competitive market, subject to the existing simple political culture that extracts a fixed rate R of the profit. In such a case the government receives RE, where E is the total profit of the n competitive 1
The possible technological advantage of a monopoly due, for example, to his ability to invest more in research and development is disregarded. In other words, we assume that his cost function is identical to that of the competitive producers. 2 Complete rent dissipation in a contest, where all monopoly profit is transferred to the government, has already been noted in the early studies of Krueger (1974) and Posner (1975) and in Tullock's (1980) rent-seeking contest, where the government applies the simple lottery contest success function and the number of firms competing on the monopoly profit is sufficiently large In the later case, the government actually extracts (
n −1 ) A . Although our complete dissipation assumption simplifies the n
illustration, it is inessential qualitatively.
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producers. The market structure determined by the government hinges on the relationship between A and RE. If R< A/E, the government prefers the monopoly market structure. If R > A/E, the n-producer competitive industry is the preferred market structure of the government. In the former case, monopoly is not due to non convexity of the technology (increasing returns to scale) or to the political power of some interest group. In the latter case, competition among the existing n producers is not the result of ideology or the benevolence of the government. In both cases the preferred market structure reflects the maximization of the extracted resources from the producers. Market structure is determined then by the prevailing norms of extraction (the ability to award the monopoly rent the fixed profit share R that can be extracted from the competitive producers) and the characteristics of the supply and demand for the produced good. To illustrate how demand, supply and the norms of extraction determine market structure, consider the stylized case of linear demand and supply functions.
3. Linear demand and supply
Suppose that the demand function p ( x ) is linear and the cost function c( x) of every producer is quadratic and there are no fixed costs. That is, the demand and marginal cost function are linear, p ( x) = δ − κ x, κ > 0 and c′( x ) = µ x, µ > 0 . Figure 1 presents the demand curve, marginal revenue curve, marginal cost curve of the monopoly and the supply curve of the n- producer industry. The monopoly output xm
is determined by the intersection of the monopoly marginal cost and marginal revenue curves. The competitive equilibrium output xe is determined by the intersection between the market demand and supply curves. The profit of a monopoly is represented by the trapeze A. The total profit of the n competitive producers E is given by the triangle E. Let Κ =
µ A and λ = . Κ is the relation between the profits of the κ E
monopoly and the competitive industry. λ , which represents the market characteristics, is the relation between the slopes in absolute terms of the marginal cost and the demand curves.
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δ
c′ = µ x
D
pm
p( x) = δ − κ x n
∑ c ' = ( µ / n) x
A
pe
i =1
E MR( x) 0
xm
xe
x
Figure 1: Monopoly vs. competition by n producers The following result clarifies how market structure is determined by the specific form of the demand and supply functions, represented by the parameter λ and by the political culture, represented by the parameter R. Notice that the parameter
δ is inconsequential in determining the selected form of market structure. Proposition 1: The government sets the monopoly market structure rather than the
competitive one, if R < K =
1 (λ + n) 2 . The government prefers the competitive n (λ + 2)λ
market structure, if the inequality is reversed.3 The proof of the proposition and its corollaries is relegated to the Appendix.
Notice that R > K is a sufficient condition for the superiority of the competitive market structure, regardless of the proportion of the monopoly profit A extracted by the government (see footnote 2). 3
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One implication of Proposition 1 is that K can be larger than or equal to 1. That is, the profit of the monopoly can exceed the profit of the competitive industry. In such a case the government certainly prefers the monopoly to the competitive market structure. Whether this situation arises depends on the parameters n and λ . Specifically, Corollary 1.1: The monopoly is always preferred to the competitive market structure
if λ
λ ), then an increase in n requires a decrease (an increase) in λ . In other words, A typical iso- Κ curve is negatively or (positively) sloped when n < λ (n > λ ). Figure 2 presents such
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an iso- Κ curve for Κ = 0.36. In this case, a government that can extract 36% of the profit made by a competitive producer would prefer the competitive system rather than the existence of a monopoly for any combination of n and λ represented by a point above the iso-0.36 curve.
λ
K = 36%
n
Figure 2: The iso- Κ curve, for Κ = 36%.
Suppose that the government does not take industry size as given, but rather selects the number of producers that yields the maximal profit E. The second part of Corollary 1.2 implies that the preferred industry size is n = n* = λ . In this case, Proposition 1 takes the following form: Corollary 1.3: If the government can set the optimal competitive industry size n*,
then it prefers the monopoly market structure rather than the competitive one, if
R 2 , that is, the slope of the marginal cost must be larger than two times of the absolute value of the slope of the demand function. Notice that in this case the first part of Corollary 1.2 is still valid. Technologies and consumers may vary across different regions or states as well as along time in the same place. This variability certainly affects the threshold Κ that determines the government decision. One might be interested in computing the average threshold in different geographical areas or along time, taking into account the variability in both n and λ . Clearly if the average K in one environment is smaller than the average K in another environment, then the probability that the government in the former environment prefers and set the competitive market structure is higher. The following corollary of Proposition 1 presents the average threshold K corresponding to n , assuming that λ is uniformly distributed over [ λmin , λmax ] . Corollary 1.4: If λ is uniformly distributed over [ λmin , λmax ] and λmax − λmin λmax ,
λmax (n − 2) 2 ⎡ (λmax + 2) ⎤ 1 n ln − then K = + ⎢ln ⎥. n 2λmax λmin 2nλmax ⎣ (λmin + 2) ⎦ Corollary 1.3 implies that for λmax = ∞ , K = 1/n. In this case, on average, the minimum extraction rate required for the government to prefer the competitive market structure exceeds 1/n. Assuming that λmin = 0.1, we obtain the following table that
λmax
3
5
7
10
15
∞
n=2
1.63
1.28
1.11
0.96
0.83
0.5
n=3
1.68
1.47
1.21
0.99
0.81
0.33
Table 1: K corresponding to n =2, 3, λmin = 0.1 and λmax =3, 5, 7, 10, 15 and ∞ .
presents the values of the average threshold K corresponding to n = 2 ,3 and λmax =3, 5, 7, 10 , 15 and ∞ . Notice that the minimum necessary average threshold of the extracted profit share that induces the government to prefer in higher probability the
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two-producer or the three-producer competitive system requires that the maximal relation between µ and κ is at least 10, λmax ≥ 10 . Otherwise, the profit of the competitive market structure is insufficient to make it probabilistically advantageous relative to the monopoly market structure.
4. Summary
We have demonstrated how the technology and market characteristics together with the prevailing extraction norms (the ability to extract the monopoly profit and the fixed profit share that can be extracted from the competitive producers) determine whether the market is monopolistic or competitive. The comparison between monopoly and competition with limited entry was based on the simplifying assumption of linear demand and supply functions. We derived the criterion used by the government to decide which of the two possible market structures is preferred (Proposition 1). This criterion depends on the slopes of the demand and marginal cost curves and on the existing number of producers. Proposition 1 was applied to compute the combinations of linear demand and marginal cost functions and number of producers that ensure the superiority of the monopoly (Corollary 1.1), to examine the effect of changes in demand, supply and the existing number of producers on the applied criterion (Corollary 1.2), to derive the criterion used by the government to decide which of the two possible market structures is preferred, when it can set the optimal industry size that yields the largest competitive profit (Corollary 1.3) and to compute the average level of the criterion in an environment where the parameters of the demand and supply functions are random variables (Corollary 1.4).
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Appendix: Proofs
Proposition 1:
In Figure 1, xm and x e are determined, respectively, by the intersection of the marginal revenue and marginal cost curves and the demand curve and aggregate marginal cost curves of the n producers. Hence, (A1)
δ − 2κ x = µ x ⇒ xm =
(A2)
δ −κ x =
µ n
x ⇒ xe =
δ µ + 2κ
µ n
δ +κ
,
.
We can now use xm and x e to compute A and E, the profits of the monopoly and the competitive industry. Notice that in Figure 1, A= (A+D) –D. xm2 1 Since ( A + D) = xm [ 2δ − xm (κ + µ ) ] and D = κ , 2 2 1 A= (A+D) –D = δ xm − xm2 (2κ + µ ) . 2
(A3)
Since E =
1 2µ xe , 2 n 2
(A4)
⎛ x ⎞ 2+λ x A = 2δ 2 m −⎜ m ⎟ . E xe ( µ / n) ⎝ xe ⎠ µ + 2κ
Notice that 2
(A5)
δ xm 2 e
x
=
δ
2
δ
2
µ + 2κ (
µ n
⎛µ ⎞ ⎜ +κ ⎟ n ⎠ . =⎝ ( µ + 2κ )
+ κ )2
Hence, by (A4), (A5), (A1) and (A2), we obtain that
µ
(A6)
2
µ
⎛ µ ⎞ ( + κ )( + κ ) ⎜ ( + κ ) ⎟ λ+2 A n =2 n −⎜ n = ⎟ µ E ( 2 ) / n + µ κ λ ⎜⎜ ⎟⎟ ( µ + 2κ ) n ⎝ ⎠
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λ
( + 1) 2 1 (λ + n) 2 = n n =K. {2 − 1} = (λ + 2)λ n (λ + 2)λ
We have thus obtained that the government sets the monopoly market structure rather 1 (λ + n) 2 than the competitive one, if and only if R < K = n (λ + 2)λ
and it prefers the
competitive market structure if the inequality is reversed.
Corollary 1.1:
By (A6), K > 1 ⇒ (λ + n) 2 > (λ + 2)λ n . That is,
λ 2 + 2λ n + n 2 > λ 2 n + 2λ n or λ 2 + n2 > λ 2n ⇒ λ 2
> ⎡⎣ n 2 − λ 2 ⎤⎦ 0 ⇔ n λ . = 2 ∂n n λ (λ + 2) <
