In this section, the analysis is extended to allow for free entry. Let be the fixed entry cost per firm. The main questions that shall be answered are whether the result of Proposition 1 continues to hold under free entry conditions, and whether a regulator can use a subsidy to entry to improve welfare.
0 F ≥
As a benchmark, consider first a situation with zero entry costs. To maximize welfare, the social planner offers a continuum of product qualities (the interval between the lowest and the highest preferred quality) at marginal cost, and each consumer type chooses her preferred quality level q t( ). Using (1), (7), (13), and (18), we obtain for the utility of type t :
This is the maximum welfare that can be achieved in this market in the absence of entry costs.
This formula can be used to compute a simple estimate for an upper bound to welfare improvements by any type of regulation. Note, that, under free entry conditions, the market equilibrium with zero entry costs coincides with the planner’s solution: an infinite number of firms enter, and welfare is as in (36).
In the presence of entry costs, the optimal number of firms is finite. (36), then, always overestimates the scope for welfare improvements by regulation. When 3 firms are in the market, using (33) and (36), we obtain as an estimate of the scope for welfare improvements:
*
∞− = percent. This illustrates that the market solution with 3 firms leaves rather
little scope for welfare improvements even when compared to the first best solution with an infinite number of product versions where entry costs are neglected.
When entry costs are taken into account, the scope for welfare improvements is smaller. The following Proposition extends the result of Proposition 1, taking into account entry costs:
Proposition 2: When three firms are in the market, the upper bound to the percentage welfare
increase by any regulation that leaves the number of competitors unchanged equals 1.6 percent.
Proof:
When entry costs are taken into account, the formula for welfare (19) becomes:
W = + Π −S nF (37)
, where Π is industry profit before entry costs.
Suppose, entry occurs in stage 0, before firms coordinate on their positions in the quality dimension. Therefore, entry occurs as long as the following condition holds:
( )n nF 0
Π − ≥ (38)
To obtain an upper limit to potential welfare improvements, we want to compute R as in n (34), but now the maximization is also over , and we have to subtract the total entry cost from all expressions for W . In the numerator, entry costs drop out, while the denominator decreases in .
profit, then, equals zero, so welfare equals consumer surplus. For n= , this is given by: 3
* 2
The difference between the results of Propositions 1 and 2 can be interpreted as follows.
When fixed entry costs are taken into account, profits are partially or fully dissipated, so total welfare is reduced, while the difference in welfare with / without regulation stays the same.
This yields a slightly higher estimate of the scope for welfare improvements measured in relative terms. However, the basic result, that the scope for welfare improvements is small when 3 firms are in the market, remains valid.
It has been shown that, when 3 or more firms are in the market, there is little scope for welfare improvements, so a laissez-faire policy seems generally appropriate. The following results, thus, focus on the duopoly case where (depending on the parameter values) substantial welfare improvements may be possible. In particular, I will analyze whether a subsidy to entry may be welfare improving, which seems plausible as welfare is almost maximized for
3
n= . Note, that a subsidy to entry has (compared to other policies) the advantage that the regulator does not need to influence the firms’ choices of qualities and prices directly.
Proposition 3: When two firms are in the market, there is an interval of fixed entry costs where entry is welfare improving but does not occur under market conditions. A subsidy to entry is, thus, welfare improving.
F
Proof:
Under free entry conditions, at most two firms enter the market if Π(3) 3− F<0. Using (15), this yields the condition:
51 F 4096
> α (40)
Starting from a situation with two firms, entry by a third firm leads to the following change in welfare (using (32) and (33)):
* *
3 2
797 12288
W W F F
− − = α − (41)
Therefore, for 51 797
4096 F 12288
α < < α , the entrant stays out, although entry would be desirable from a welfare perspective.
The result of Proposition 3 stems from the fact that there is insufficient entry under market conditions. This result is in contrast to standard oligopoly theory (e.g. Cournot competition), where prices in excess of competitive prices usually induce more entry than optimal, due to an inefficient replication of entry costs. In the present model, there is a substantial increase in consumer surplus at the transition from n= to 2 n= because over-differentiation almost 3 vanishes, while profits decrease sharply.130 This leads to insufficient entry.
Note, that, if a subsidy is used to trigger entry, it must be at least as high as 51 F 4096
− α to make the entrant’s profit non-negative. A subsidy should not be granted if 797
12288
F > α . If the actual level of the fixed entry cost is unknown to the regulator, the optimal policy is to F
130 Note, that, unlike in the present model where a regime change occurs at the transition from n=2 to n=3, in Cournot markets, the transition to competitive pricing as the number of firms increases is a gradual one.
offer a subsidy of 797 51 161
12288α −4096α =3072α to a potential entrant, since entry will, then, occur if and only if it is socially desirable.
The result of Proposition 3 raises the question where to look for an optimal policy. In the following, I want to answer the question whether a subsidy to entry is superior to a hypothetical optimal policy that leaves the number of competitors unchanged (starting from a situation with two firms in the market). The result is summarized in Proposition 4.
Proposition 4: If two firms are in the market, a subsidy to entry is inferior to an optimal policy that leaves the number of competitors unchanged.
Proof:
A subsidy to entry is superior to an optimal policy that leaves the number of firms unchanged if W3*− >F W2sp, where W2sp is welfare in the planner’s case for n= . Using (32) and (33), 2 this yields the condition:
29 12288 F< α
(42)
However, by (40), when there are two competitors in a market, we must infer that entry costs are at least as high as 51
4096α . Therefore, condition (42) is violated.
Proposition 4 implies that a policy that installs optimal prices and qualities in the duopoly case, without triggering entry, is always superior to a subsidy to entry. Nevertheless, there are arguments in favor of an entry subsidy. If welfare is to be increased by means of taxation, minimum quality standards, or other policies that aim at establishing quality choices that are close to the optimum, then the regulator must have sufficient knowledge about the demand and cost structure in the market, which enables him to compute the optimum. A subsidy to entry, on the other hand, does not require as much information. The welfare enhancing effect of this policy relies strictly on competitive forces, and as long as the subsidy is not too high, it will always be welfare improving.