Cartel Harm with Input Substitutability: Theory and an Application

Transcription

1 Cartel Harm with Input Substitutability: Theory and an Application by: Daan in 't Veld This article examines the substitution effects of cartels. After a group of input producers has fixed a high price, its customers may change the relative demand of their inputs. To model this, the elasticity of substitution is used. Surprisingly, it is found that both with close substitutes and close complements, forming a cartel may not be profitable. Introduction A cartel is an agreement between firms to fix prices above competitive levels. While higher prices typically lead to a transfer in wealth from customers towards cartel members, the main concern for antitrust authorities is the accompanying loss in social welfare. For this reason, antitrust law is assigned to ex ante deter cartel agreements, and ex post punish the offenders and return the wealth to the rightful owner. Simple as this may sound, the analysis of cartels has turned out to be very complex. Unlike other deprivations (for example, a bank robbery, a cartel often provokes a cycle of reactions from many relevant agents in the economy; wealth is redistributed among all these agents. To model a number of vertical cartel effects, Han, Schinkel and Tuinstra (2008 use a fixed proportions production chain with an arbitrary number of layers. My purpose is to investigate horizontal or substitution effects, connected with firms outside the cartel, but within the same production layer. Covering the behavioural changes of the different agents, game theory is a useful tool to explain and describe cartel effects. In this article I will compare two equilibria attached to situations before and after the cartel. Surprisingly, it will be found that forming a cartel is not always profitable. This rather paradoxical proposition was for the first time set forth in a famous paper of Salant, Switzer and Reynolds (983 in the context of mergers. Intuitively, the possibility of a disadvantageous cartel arises because other firms in the same production layer react to the new strategy of the cartel members. Model of production and assumptions We will look at the three-layer industry presented in Figure. There are n downstream firms producing consumer goods, using two inputs from m and m 2 upstream suppliers. Downstream quantity and price is denoted by (Q, p; upstream market quantities and prices by, w and, respectively. The separation of the upstream firms is the major deviation from the mainstream cartel literature. I will explore the implications of price-fixing agreements by the m primary input producers under the existence of secondary input market. The industry will be subject to the following assumptions. First, the relation between the two inputs and the output is specified. We use a CES (constant elasticity of substitution production function: σ σ σ σ σ σ QZ (, Z2 = ( Z + Z, σ The CES production function is constructed in such a way that σ equals the elasticity of substitution, defined as Daan in 't Veld [ Z / Z ] w / w [ w / w ] Z / Z Daan in t Veld completed the mathematical economics track of the MSc in Econometrics (cum laude at the University of Amsterdam. This article is a summary of his master thesis written under the supervision of dr. Jan Tuinstra. Currently, Daan stays at the University St. Petersburg to fully master the Russian language and attempt to understand the economic situation in Russia. ( AENORM vol. 8 (67 June 200 9

2 Figure. The three-layer industry with two groups of input producers. individual profits (indicated with *, and the cartel state in which the m firms collude (with C. Game-theoretically, in the latter state there remains only one decision-making entity in the primary input market. Equilibrium analysis Given the demand curve p(q and prices w and w 2, every downstream firm i maximises individual profits π i : max π = max( p( Q mc( w, w q, i i (2 where mc(w are the constant marginal costs of the downstream firms, and can be derived from (. This is a quite simple optimization problem in one dimension, where only the produced quantities of the n firms other than i are unknown. Using the fact that all firms are identical, the first order condition for (2 provides the competitive Cournot equilibrium quantity: In words, the elasticity of substitution measures how the demand ratio of inputs Z reacts to a change in the price ratio w /w 2. The higher σ, the more elastic the input substitution, and, for example, the more the demand ratio Z will fall after a price increase w. More insight in the effect of σ can be obtained by looking at two limit cases. For σ 0 there are no substitution possibilities for firms, and the inputs, here called perfect complements, are used in equal proportions. Indeed, it is possible to derive that lim σ 0 Q,Z 2 = min(z,z. Alternatively, when σ approaches infinity, Z becomes very sensitive to the input price ratio, and every downstream firm will tend more and more towards the input with the lowest price. Thus inputs become perfect substitutes. It is easy to see that ( now becomes linear: Q,Z 2 = Z 2 + Z 2 2. The typical assumption is that firms compete in quantities, and that prices are determined by market forces. In other words, all three markets are Cournot oligopolies. Naturally, upstream production precedes downstream production; downstream production precedes consumption. We use backward induction for our sequential game with two stages. Therefore, we will start with the downstream market equilibrium for given input prices w and w 2. Then we will derive the optimal choices for the upstream firms that will lead to an equilibrium with certain (w, using the knowledge of the strategies of the downstream firms. Finally, there are two technical assumptions to simplify the analysis. Marginal costs in the input markets are set constant and equal to d and e, respectively. Next, consumer demand will be expressed by a linear demand curve p(q = a bq. As our point of interest in this industry is cartel effects, we compare the market outcomes in two states: the competitive state in which all firms are maximising ( w, w2 = ( a mc( w, w2. ( n+ b For the two upstream markets some complications arise. Consider the primary input market. While it is assumed that upstream firms recognise the optimal downstream strategy given by (3, they also have to reckon with the simultaneous production of the secondary input firms. When all m firms behave competitive, they are maximizing profits as follows: max π = max( w ( Z, Z d z, j j z j z j where w, Z is an inverse demand function that 2 determines the market-clearing price as a function of Z and Z 2. For the cartel, the counterpart problem of (4 is: max Π ( Z = max( w ( Z, Z d z. Z Z In both the competitive and cartel state, the optimization problem results in a first order condition of the form FOC = 0. This first order condition can be rewritten as an aggregate reaction function: R {Z FOC = 0} (3 (4 (5 (6 Note that this is a rather peculiar reaction function, because it describes the optimal reaction of the complete primary input market to the production of the secondary market. This notion of aggregate reaction functions was earlier used by Salant, Switzer and Reynolds (983. Due to the symmetry, we can collect all individual reactions together in one reaction. 0 AENORM vol. 8 (67 June 200

3 TNO Mijn fascinatie Doelgericht innoveren. Nieuwe producten, nieuwe diensten, nieuwe mogelijkheden creëren. Creatieve antwoorden vinden op de vragen die de samenleving stelt. Werken aan betere oplossingen: sneller, veiliger, slimmer, efficiënter. Dat is mijn fascinatie.

4 Figure 2. Aggregate reaction functions and the equilibrium for σ. Figure 3. The merger paradox for σ. The equilibria are found in the intersection points of R and R 2, where R belongs to firms either competing or colluding. In Figure 2, the reaction functions and the equilibrium are drawn in the {Z } plane. Clearly, in the intersection point no firm of any market can raise higher profits by changing his production level. Although it is difficult to derive the reaction functions explicitly for general σ, the two limit cases can be worked out. This will enable us to understand two possible situations in which forming a cartel is not profitable: the merger paradox and the reinforced race-to-the-bottom. The merger paradox We start with perfect substitutes (σ. Now downstream firms only choose the input with the lowest price, and have a linear production function Q = Z 2 + Z 2 2. This means that downstream marginal costs equal mc(w = 2min(w. Only when prices are equal, both inputs are produced. Combining these results with (3, it is not difficult to derive that under w = w 2, ( n+ b w( Z, Z2 = ( a ( Z + Z n With this relatively simple formula for w, the first order condition of (4 can be solved to the aggregate reaction of competing firms m n 2( a 2 d R ( Z = ( Z * 2 m + n+ b (7 and we see that the reaction function is linear and downward-sloping. In order to obtain the reaction function in the cartel state, it suffices now in (7 to replace m by. The equilibria in the two states for perfect substitutes are represented in Figure 3. By forming a cartel, the reaction function of the primary input firms shifts proportionally towards the vertical axis. The cartel profits C C are maximised by R (Z, and therefore Π ( R > Π ( R * (Z for every Z 2. However, in the cartel equilibrium, the secondary input market produces more, and therefore it is possible that the cartel makes fewer profits than the sum of the competitive primary firms. This result has become known as the merger paradox since the famous paper of Salant, Switzer and Reynolds (983. The merger paradox, as I will also call it in the present context of cartels, occurs as a special case in our model. Salant, Switzer and Reynolds gave a necessary condition for the advantageousness of the cartel if d=e. For us, it is enough to know that there exist some upperbound σ, given the other parameters in the model, where for σ>σ forming a cartel is not profitable because of the merger paradox. The reinforced race-to-the-bottom For close complements, as well as for close substitutes, it will turn out that there exist disadvantageous cartels. In the limit case of perfect complements, our model reproduces a puzzling result of Sonnenschein (968. He showed that firms producing perfect complements optimally choose their quantity just below the choice of the opponent, hereby creating a shortage of their own product, and an accompanying high price. This leads to a so-called race-to-the-bottom until Z = Z 2 = 0. The race-to-the-bottom holds even when there exist perfect substitutes for every complement, i.e. as in our situation where m, m 2 >, which was also shown by Dari- 2 AENORM vol. 8 (67 June 200

5 Figure 4. The reinforced race-to-the-bottom for σ=0.25. relatively well documented. The application underlined the relevance for taking substitution effects into account. References Dari-Mattiaci, G. and F. Parisi. Substituting complements. Journal of Competition Law and Economics 2.3 (2006: Han, M.A., M.P. Schinkel. and J. Tuinstra. The overcharge as measure for antitrust damages. Amsterdam center for law and economics working paper series Salant, S.W., S. Switzer. and R.J. Reynolds. Losses from horizontal merger: the effects of an exogenous change in industry structure on Cournot-Nash equilibrium. Quarterly Journal of Economics 48 (983: Mattiaci and Parisi (2006. This means that for perfect complements, the Cournot equilibrium constitutes to zero production in the entire industry. In accordance with Sonnenschein, a cartel is unable to resolve this situation. It may seem trivial that a cartel cannot succeed in raising profits in a paralysed industry. However, a cartel may also create such a situation itself. Any cartel will reduce the primary input production Z, in order to profit from the excess demand. This means that for σ close to 0, the cartel actually reinforces the race to the bottom, and therefore even fewer profits are made. Again the strong reaction of the secondary input producers is critical for the loss of profits. This reinforced race-to-the-bottom is illustrated in Figure 4. Even when in the cartel equilibrium both inputs are produced with a positive quantity, it is possible that the rigorous restriction of the production, initiated by the cartel formation, leads to fewer profits. This means that, depending on the remaining parameters, there exists a underbound σ, for which σ < σ the (reinforced race-tothe-bottom occurs. Sonnenschein, H. The dual of duopoly is complementary monopoly; or, two of Cournot s theories are one. Journal of Political Economy 76 (968: Conclusions We have seen that cartels may be disadvantageous if there is either a close substitute or close complement available. Given the remaining parameters, a cartel is profitable only if σ > σ > σ. The merger paradox and race-to-the-bottom, known for limit cases, were previously not connected and interpreted in terms of the σ-spectrum. Other conclusions in the thesis were related to the ex post estimation of cartel harm using the common overcharge measure, and the possible compensation to the secondary input producers. Additionaly, I applied the model to the lysine conspiracy of the 990 s. This cartel between four global producers of feed supplements is one of the most harmful ever discovered, and, consequently, AENORM vol. 8 (67 June 200 3