To compare welfare in a market equilibrium with the optimum (the social planner’s case) for a varying number of firms, I will first derive an expression that allows to compute welfare for a given set of quality levels and mark-up prices.
I assume consumer surplus ( ) and industry profit (S Π ) enter welfare in an identical way:
W = + Π S (19)
Consumer surplus is computed by summing over the different products and integrating over consumer types, taking into account consumers’ product choices:
1 1
Using (1), (6), and (7), and evaluating the integral, this can be written as:
(
Industry profit is simply the sum over the firms’ profits. Using (4) and (6), we obtain: n
Using (21) and (22) in (19), we obtain the following expression for welfare:
( )
Note, that prices affect welfare only via market shares (that are determined by the ’s). This is because prices are simply transfers from consumers to firm owners and, thus, have no direct effect upon welfare. Market shares and quality levels, however, affect welfare directly.
Inefficiencies in vertically differentiated markets can, thus, arise due to distorted quality levels, or due to distortions in market shares, which implies that some consumers do not purchase from the same firm where they would buy in the planner’s case (or both).
ti
125
The Social Planner’s case:
The planner’s problem is to maximize welfare by choosing prices and qualities optimally.
Since welfare depends upon prices only via the ’s, the problem can be decomposed into two steps: 1. determine the vector
ti
1 1
( ,..., , ,...,q q tn tn−1) that maximizes (23) for a given n , and 2.
choose mark-up prices such that (24) is fulfilled ri ∀ ∈i {1,...,n−1}. Note, that in step 2, the regulator has one variable more than necessary to fulfill the n− conditions, because there 1 are mark-up prices to choose. Intuitively, suppose the planner chooses for product 1 some price greater than marginal cost ( ). This does not cause an inefficiency if the planner chooses the remaining prices and all quality levels such that the optimal allocation is reached.
Now all mark-up prices are greater than zero. Therefore, if the planner values consumer surplus only marginally more than profits, the optimal choice will always be , and, not surprisingly, all other mark-up prices will optimally be zero as well. However, rather than imposing this, this should be the result of the planner’s optimization problem, because the
n
125 Another potential source of inefficiency arises in uncovered market situations when too many consumers (as compared to the planner’s case) do not make a purchase. As pointed out in Section 4.2, this chapter focuses on covered market outcomes, so this possibility does not arise.
planner may set prices different from marginal costs to correct for other inefficiencies.
Therefore, we are only free to choose one mark-up price.126 I, thus, assume r1=0. Using (13) in (23), the first step of the planner’s problem reads:
( )
The FOC can be written as follows:
ti =α(qi+1+qi)−β , i=1,...,n− 1 (26) 0 2= β−4αqi+ +ti ti−1 , i=1,...,n (27) Plugging the ’s from (26) into (27), we obtain the following relation: ti
1 1
(28) illustrates that, in the optimum, there are equal distances between all quality levels.
To solve the system of linear equations (26) and (27) for an arbitrary number of firms , the easiest approach is to guess (part of) the solution, and then to verify that the candidate solution fulfills the equations. Since quality levels are equally spaced, an obvious guess is that all market shares are identical in the planner’s solution. I, thus, assume:
n
, 1,..., 1
i
t i i n
=n = − (29)
Plugging this into (27), we obtain for the planner’s quality choices:
1/ 2 , 1,...,
All that remains to be done is to use (29) and (30) in (26) to show that the candidate solution is the correct solution to step 1 of the planner’s problem (this is straight-forward).
Step 2 of the planner’s problem is to choose mark-up prices that fulfill (24). Solving (24) for , we obtain (using (29) and (30)):
All mark-up prices are, thus, identical, which, combined with the assumption , yields , . The planner, thus, sells all product versions at marginal cost.
1 0
r =
i 0
r = i=1,...,n
Using the above results, welfare in a market equilibrium can be compared with the respective
126 Furthermore, this mark-up price must be sufficiently low to yield market coverage.
planner’s outcome for an arbitrary number of firms. For the purpose of this chapter, it suffices
Note, that the welfare reductions in the market equilibrium relative to the planner’s case result for different reasons. For , both firms’ market shares are equal to 1/2 in the planner’s case and in the market equilibrium. The lower welfare in the market equilibrium is, thus, due to the fact that qualities are too dispersed (over-differentiation). For , qualities in the market equilibrium are (slightly) too dispersed, and in addition, market shares are skewed
2 n=
3 n=
127, which implies that there is a misallocation of consumers over the firms (some consumers purchase from a different supplier than in the planner’s case).
Note, that all expressions for welfare in (32) and (33) are of the same form:
2
4
const
W V β β
α
= + + + . This permits quantitative predictions about potential welfare
improvements by regulation. Consider the following measure for the maximum percentage welfare increase that a regulator can achieve by setting prices and qualities optimally (for a given number of firms ): n
R is, thus, an upper bound to the percentage welfare increase a regulator can achieve, which n
means that we can not find parameter values that yield a higher relative welfare increase. It is a measure of the scope for welfare improvements that can be achieved by regulation.
Computing the ratio drop out in the numerator, and in the denominator, they appear with positive signs. This implies that the ratio strictly decreases in V and β . Therefore, to compute an upper bound to the percentage welfare increase, simply set V = =β 0.128 The parameter α, then, drops out.
127 The high- and the low-quality provider have a market share of about 27%, and the medium firm of 46%.
Mark-up prices are also not identical (see (15)).
128 A parameter restriction of the form β β≥ > or 0 V V≥ >0 is sometimes necessary to assure non-negativity of the lowest equilibrium quality and/or market coverage. These restrictions are for simplicity neglected in the
Evaluating (34) for the duopoly case, we obtain an upper bound to the percentage welfare increase of 200 percent. This implies that, depending on the actual parameter values, there can be a substantial scope for welfare improving policy interventions in the duopoly case. Results are, however, markedly different when the number of competitors is larger. The main result of this Section is summarized by the following Proposition:
Proposition 1: When three firms are in the market, there is an upper bound to the percentage welfare increase by regulation of only 0.66 percent.
Proof:
Using (33) and (34), we obtain for V = =β 0: 3 35 /108 989 / 3072
0.0066 989 / 3072
R = − ≅ .
According to Proposition 1, when there are three firms in the market, the scope for welfare improving policies is small.129 It should be pointed out that for any plausible parameterization of the model (β >0), the actual scope for welfare improvements is even smaller, as the upper bound has been computed for extreme parameter values.
The reason for behind the result is that the equilibrium qualities when three (or more) firms are in the market are close to the optimum, as can easily be confirmed by comparing (15) and (30) (for ). The over-differentiation result that drives the inefficiencies in the duopoly case, thus, almost disappears at the transition from
3 n=
2
n= to n= or more firms. Furthermore, 3 even if the specific value for the upper bound to potential welfare improvements that was computed above relies on the specific modeling choices made, the main result, namely that the scope for welfare improvements decreases dramatically at the transition from n= to 2 firms, holds more generally and does not depend on the specific modeling choices made (such as the linear-quadratic specification of the function ).
3 n=
(.) f
As a consequence, we can conclude that a laissez-faire policy seems appropriate whenever 3 or more firms are in the market. This is especially true if the regulator faces some uncertainty
0
computation of the upper bound to potential welfare increases. However, in both cases, (β > or V >0), incorporating the restriction into the computation would only lead to a reduction in the upper bound, thus strengthening the main results of the analysis further.
129 This holds a fortifiori for an even larger number of firms since prices and quality choices are even closer to the planner’s solution.
about the true parameter values of the model (as is always the case in reality). A policy that is intended to increase welfare may, then, even turn out to be harmful.