Effect of changes in efficiency

To understand how cost asymmetries affect the cartel’s pricing I perform some comparative statics. When the outside option is the Bertrand equilib-rium the results coincide with those by Harrington (1991).

Proposition 11. Cost reductions for the efficient firm leads to lower cartel prices.

When it is most profitable to deviate to the Bertrand equilibrium there is no valuable outside option. The effect on price from a change in costs is therefore only determined by how the cost change affects the firms’ optimal

prices. A reduction of the efficient firms’ costs lowers the efficient firms’

monopoly prices, but does not affect the inefficient firm’s monopoly price. A cost reduction therefore reduces the cartel price.

When the most most profitable option is to deviate to a subcartel there are also profits in the punishment phase. Lower costs increase the value of the outside option and from proposition 9 we know that a more valuable outside option reduces price. Both effects hence pull in the same direction and prices fall faster as a result of cost reductions by the efficient firms when they would deviate to a subcartel. Thus, irrespective of the deviation strategy, ^∂P

∂cE > 0 holds.

Whenδ ∈ (δ, δE) the cartel cannot agree on the Nash bargaining outcome and therefore agrees onP_E^m. A reduction ofcEreducesP_E^m. Hence the effect of cost changes is the same as above.

Proposition 12. Cost reductions for the inefficient firm can either reduce or increase the cartel price.

When the best outside option is to deviate to the Bertrand equilibrium, reducingcIhas the same effect on price as a reduction incE, i.e. a reduction in costs reduces the monopoly price - in this case for the inefficient firm.

Lower costs therefore reduces the cartel price. When the best option is to deviate to a subcartel the relationship is more complicated since changes in costs, besides changing the optimal price, also influences the value of the outside option. The outside option for the efficient firms is ^(cI−cE₂^)D(cI) and reducingcI reduces the value of the outside option which gives the efficient firms less leverage and pushes the cartel price upwards. The two effects work in opposite directions. The effect of the cost reduction on prices hence depends on which of these two effects are strongest.

When the cost differences are small the price increasing effect of the outside option is small and prices fall with cost reductions from the inefficient firm

∂P

∂cI > 0. But for sufficiently large cost differentials, the price will increase as the inefficient firm becomes more efficient because the efficient firm looses bargaining leverage, i.e. ^∂P

∂cI < 0. Berg (2011) defines the critical cost dif-ference that turns the relation negative in a two player game using a linear demand function. Without further assumptions on the concavity of demand, it is not possible to determine the effect when the best outside option is to deviate to a subcartel.

In the constrained bargaining situation, when the firms cannot agree on the Nash bargaining outcome, the cartel price is set atP_E^m. In this case changes in the efficient firm’s costs do not affect the cartel price.

Proposition 13. Cost reductions only affect the optimal market share when the most profitable deviation strategy is a subcartel.

When it is most profitable to deviate to the Bertrand equilibrium, the optimal market share is given by

s^bert=1

3 (16)

Hence all firms will receive the same market share in the cartel, independent of the firm level of efficiency since there are no profits for the efficient firms in the punishment phase. Consequently there is no valuable outside option that confers bargaining leverage that enables the efficient firms to require a higher market share despite being more efficient.

When it is most profitable to deviate to a subcartel the optimal market share is given by

s^sub=1

3+ (cI− cE)D (cI)

6 (P − cE)D (P ) (17)

When the members are symmetric all firms get the same market share. But, this is not the case when there are cost asymmetries. In similarity to Har-rington (1991) I conjecture that^dssub

dcE < 0 and ^d_dc^ssub_I > 0 because an efficient firm should get a higher market when its outside option improves.¹⁹ The direct effect ofcE ons, holding P constant, is indeed negative. But as seen in Proposition 11, the cartel price is affected bycE where ^∂P

∂cE > 0 and from footnote (18) ^∂s

∂P > 0. There is hence a countervailing effect through the prices where ^∂ssub

∂cE > 0. Although I cannot generally determine the total effect, numerical simulations using linear demand function²⁰ confirms that the overall effect is negative in that case.

HoldingP fixed the partial effect ^∂_∂c^ssub_I > 0.²¹ But from proposition 12 cI

also affectsP and the effect depends on the size of the cost difference. Total

19I have only been able to verify this numerically for a linear demand function.

20Demand function: D (P ) = a − P .

21 ∂s

∂cI=^(cI−c_{6(P −c}^{E)D(cI)+D(cI)}

E)D(P ) > 0 since cI< p^mE.

differentiation gives ^dssub

dcI > 0 if _∂c^∂P_I > 0 and this is the case for small cost differences. The logic is that increased costs for the inefficient firm gives a higher price. Due to higher prices the efficient firms needs more compensation with market share to stay in the cartel. The finding which is consistent with that of Bae (1987), is confirmed by numerical simulation in the case of linear demand.