Unification of Oligopolistic Markets for a Homogeneous Good in the Presence of an Antitrust Commission

Abstract

A basic framework is provided to explain the integration process experienced by oligopolistic markets serving a homogeneous good in different countries. Over the past few decades, such processes have been observed, for instance, in some European markets – in particular, in the energy sector. The idiosyncratic element here is the introduction of an exogenously given antitrust commission that supervises competition in the market and has the authority to fine firms for anticompetitive behavior. We model the unification decision as a simple cooperative non-transferable utility game. We find that the creation of an antitrust commission plays a major role in providing the necessary incentives for market unification. In particular, the commission is able to induce unification of all markets via appropriate choice of antitrust policy. This stands in stark contrast to the benchmark scenario in which the antitrust commission is absent – here, market unification never occurs. We propose the Iberian Electricity Market (MIBEL) as a case study.

Appendices

Appendix: A

Proof of Proposition 1

In the segregated case, each firm in country i ∈ M obtains the profit

$$V(\{i\}) = {\Pi}^{NU}_{i} = \frac{\mu L_{i} \rho}{(N_{i}+1)^{2}}. $$

Let S be a coalition of J > 1 markets. To show that S cannot improve upon the segregated markets payoff vector x, we must show that the system of J inequalities defined by, for each j ∈ S, ${{\Pi }_{S}^{U}}> {\Pi }_{j}^{NU}$, cannot all be satisfied simultaneously. Indeed, they cannot all be satisfied even replacing the strict inequalities with weak inequalities. Suppose, to the contrary, that, for all j ∈ S, it is true that

$$ \frac{\mu \left({\sum}_{s\in S} L_{s}\right) \rho}{\left({\sum}_{s\in S}N_{s}+1\right)^{2}}\geq \frac{\mu L_{j} \rho}{(N_{j}+1)^{2}}. $$

(2)

Cancelling μ and ρ and rearranging, this can be written as

$$\left(\sum\limits_{s\in S}L_{s}\right)(N_{j}+1)^{2}\geq L_{j}\left(\sum\limits_{s\in S} N_{s}+1\right)^{2} $$

for all j ∈ S. Summing over j, this implies that

$$ \sum\limits_{s\in S} (N_{s}+1)^{2}\geq \left({\sum}_{s\in S}N_{s}+1\right)^{2}. $$

(3)

Expanding (3) leads to the condition

$$\sum\limits_{s\in S} {N_{s}^{2}}+2\sum\limits_{s\in S}N_{s}+J \geq \sum\limits_{s\in S} {N_{s}^{2}} +\sum\limits_{s\in S}\sum\limits_{j\in S\setminus\{s\}} N_{s}N_{j} +2\sum\limits_{s\in S} N_{s}+1, $$

whence

$$J-1\geq \sum\limits_{s\in S}\sum\limits_{j\in S\setminus\{s\}} N_{s}N_{j} \geq \sum\limits_{s\in S} (J-1)N_{s} \geq J(J-1). $$

This, however, implies that 1 ≥ J, contradicting our initial assumption that S is a coalition comprising at least two markets.

Hence, at least one country is strictly better off keeping its market segregated. Since the weak inequalities in Eq. 2 are a consequence of the strict inequalities, the same is true in the case of strict inequalities: for some j ∈ S, it must be that ${\Pi }_{j}^{NU}>{{\Pi }_{S}^{U}}$. It follows that, since S is an arbitrary coalition, there cannot be a core payoff vector in which there is any market unification at all – in any non-singleton coalition, one of the countries would deviate. □

Proof of Proposition 2

Let S be some coalition of J markets. Without loss of generality, write S = {1, … , J}. If it is the case that, for all j ∈ S, $V(S)_{j}={\Pi }^{\prime U}_{j}>{\Pi }^{\prime NU}_{j}=V(\{j\})$, then the coalition improves upon their payoffs under segregated markets. We can write this set of conditions as: for each j ∈ S,

$$\frac{\mu \left({\sum}_{s\in S} L_{s}\right) \rho}{\left({\sum}_{s\in S} N_{s}+1\right)^{2}}-F\cdot \left[{\sum}_{N_{1}+\cdots+N_{J}}^{\infty} \frac{\lambda^{x} e^{-\lambda}}{x!}\right]> \frac{\mu L_{j} \rho}{(N_{j}+1)^{2}}-F\cdot \left[{\sum}_{N_{j}}^{\infty} \frac{\lambda^{x} e^{-\lambda}}{x!}\right], $$

which, collecting terms multiplied by F and separating the sums, can be rewritten

$$ \frac{\mu \left({\sum}_{s\in S} L_{s}\right) \rho}{\left({\sum}_{s\in S} N_{s}+1\right)^{2}}+F\cdot \left[\sum\limits_{N_{j}}^{N_{1}+\cdots+N_{J}-1} \frac{\lambda^{x} e^{-\lambda}}{x!}\right]> \frac{\mu L_{j} \rho}{(N_{j}+1)^{2}}. $$

(4)

Since the term in brackets multiplying F is strictly positive, it follows that, if F is chosen large enough to ensure that the inequality holds for a given j ∈ S, then the firms in market j prefer to unify their market with the other markets in coalition S. In particular, if F is chosen large enough so that the above inequality holds for each j ∈ S, then the coalition can improve upon their payoffs under segregated markets.

Moreover, it is clear that the case of a fully unified market will be the unique element of the core if F and λ are chosen appropriately (if S ⊂ M, think of S as a single firm s, where $N_{s}={\sum }_{j\in S} N_{j}$ and $L_{s}={\sum }_{j\in S}L_{j}$, and then apply the above argument repeatedly to show that, for appropriate F and λ, no subset of the grand coalition will deviate). □

Appendix: B

We provide a brief discussion further to footnotes 9 and 20. In some situations it would be justified to make the alternative assumption that the probability of being fined is, rather than depending on the absolute number of firms in the market, strictly decreasing in the ratio of firms to consumers, N _i /L _i . That is, the more firms per citizen, the less likely an individual firm is to be fined. Denote the probability by Q(N _i /L _i ). To simplify notation, write K _i ≡N _i /L _i and $K\equiv \frac {L_{1}+L_{2}}{N_{1}+N_{2}}$. Then, firms 1 and 2 will want to unify their markets if and only if, for j = 1, 2, it is true that ${\Pi }^{U}-F\cdot Q(K)> {\Pi }^{NU}_{j}-F\cdot Q(K_{j}),$ or, equivalently,

$$ {\Pi}^{U}+F\cdot(Q(K_{j})-Q(K)) > {\Pi}^{NU}_{j} \quad \text{for ~} j=1,2. $$

(5)

It follows by basic algebra that, without loss of generality, K ₁ ≤ K ≤ K ₂, with strict inequalities whenever the ratios are not the same. If the ratios are the same, then the fines cancel out and we are back in the case without an antitrust commission and, hence, the two inequalities can not hold simultaneously by Result 1. Assume, then, the ratios are not the same, so that Q(K ₁) > Q(K) > Q(K ₂). The term multiplying F in Eq. 5 is positive for firm 1 and negative for firm 2. In particular, the following condition must be satisfied:

$$\frac{{\Pi}^{U}-{\Pi}^{NU}_{2}}{Q(K)-Q(K_{2})}> F > \frac{{\Pi}^{NU}_{1}-{\Pi}^{U}}{Q(K_{1})-Q(K)} . $$

A necessary (but not sufficient) condition for this to hold for some F is that firm 2 would benefit from unification whereas firm 1 would suffer, i.e., ${\Pi }^{U}> {\Pi }^{NU}_{2}$ and ${\Pi }^{NU}_{1}> {\Pi }^{U}$.

There are cases where this is satisfied. For instance, if L ₁ = 2, L ₂ = 1, N ₁ = N ₂ = 1, then the condition reduces to $\frac {1/12}{Q(K)-Q(K_{2})}> F > \frac {1/6}{Q(K_{1})-Q(K)}.$ Then if, say, Q(K ₁) = 0.9, Q(K) = 0.4 and Q(K ₂) = 0.2 (clearly, a probability function satisfying these properties exists), we find that any F satisfying 5/12 > F > 1/3 induces unification.

On the other hand, in some cases unification cannot be induced for any fine, regardless of the probability function. For example, the introduction of an antitrust commission does not lead to unification in any situation in which, in the absense of a commission, firms in neither market would be set to benefit from integration.

Thus, Result 2 extends partially to this case – an antitrust commission can exploit favorable conditions in the individual markets to induce integration. On the other hand, if these favorable conditions are lacking, integration cannot be induced regardless of the policy.