Shaked Sutton (1982) - Relaxing Price Competition Through Product Differentiation

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• Shaked, A. and J. Sutton (1982), "Relaxing price competition through product differentiation", Review of Economic Studies 49, 3-13. pdf

@article{shaked1982relaxing,
  title={Relaxing price competition through product differentiation},
  author={Shaked, A. and Sutton, J.},
  journal={The Review of Economic Studies},
  pages={3--13},
  year={1982},
  publisher={JSTOR}
}

Abstract

The notion of a Perfect Equilibrium in a multi-stage game is used to characterize industry equilibrium under Monopolistic Competition, where products are differentiated by quality. Central to the problem of providing adequate foundations for the analysis of monopolistic competition, is the problem of describing market equilibria in which firms choose both the specification of their respective products, and their prices. The present paper is concerned with a very particular model of such a market equilibrium. In this equilibrium, exactly two potential entrants will choose to enter the industry; they will choose to produce differentiated products; and both will make positive profits.

The Model

The game has three stages:

1. Entry Decisions by [math]n\,[/math] firms
2. Entrants choose product quality [math]q \in [q',q'']\,[/math]
3. Price Competition

The set up is as follows:

• Unit mass of consumers with type [math]t \in [t', t'']\,[/math] where [math]t'' = t'+1\,[/math]
• Unit demand. A consumer who purchases from firm [math]i\,[/math] gets [math]u(t) = t\cdot q_i - p_i\,[/math]
• [math]mc=0\,[/math]
• Cost of entry [math]\epsilon\,[/math]

Assumptions:

1. [math]t''\gt 2t'\,[/math]: There is enough heterogeneity to prevent one firm from capturing the whole market
2. For two firms with qualities [math]q_1 \lt q_2: t'q_1 \gt \frac{(t''-2t')\Delta q_2}{3}\,[/math], where [math]\Delta q_2 = q_2=q_1\,[/math]. This prevents unserved customers.
3. [math]t'' \lt 4t'\,[/math] there is not excessive heterogeneity. This limits the number of firms that can survive.

Stages and Possibilities

 Entry              Quality        Prices and Profits
 ----------------------------------------------------
 
 2 firms            Same           [math]p=0,\; \pi=0\,[/math]
 2 firms            Different      [math]0\lt p_1\lt p_2,\; 0\lt \pi_1\lt \pi_2\,[/math]
 3 or more firms    Same           [math]p=0,\; \pi=0\,[/math]
 3 or more firms    Different      [math]0\lt p_{n-1}\lt p_n,\; 0\lt \pi_1\lt \pi_2, \pi_{i\ne1,2} = 0\,[/math]
 3 or more firms    Top 2 Same     [math]0=p_{n-1}=p_n,\; \pi = 0\,[/math]
 

There are two steps to answering which are equilibria:

1. Lemma 1 - with unequal qualities on the top two products make money
2. Proposition 2 - in any NE of the quality choice subgame, profits are zero

From the above it can be shown that more than two entrants is not an equilbrium with [math]\epsilon \gt 0\,[/math]. Furthermore, they will differentiate their products (as shown by proposition 1 and lemma 2). So only the bold row is an equilibrium.

Price competition and Lemma 1

Suppose there are [math]n\,[/math] differentiated firms with qualities [math]q_1,\ldots,q_n\,[/math]. Only the top two will have positive profits.

A consumer indifferent between consuming goods [math]k\,[/math] and [math]k-1\,[/math] has equal utilities:

[math]t_k\cdot q_{k-1} - p_{k-1} = t_k\cdot q_{k} - p_{k} \implies t_k = \frac{p_k-p_{k-1}}{q_k-q_{k-1}} = \frac{\Delta p_k}{\Delta q_k}\,[/math]

Lemma 1

The objective function for firm [math]k\,[/math] is:

[math]\pi_k = (t_{k+1}-t_k)\cdot p_k = (\frac{\Delta p_{k+1}}{\Delta q_{k+1}} - \frac{\Delta p_{k}}{\Delta q_{k}})\cdot p_k\,[/math]

Likewise for firm [math]n\,[/math]:

[math]\pi_n = (t''-t_n)\cdot p_n = (\frac{t'' - \Delta p_{n}}{\Delta q_{n}})\cdot p_n\,[/math]

The respective first order conditions wrt [math]p_n\,[/math] give (subbing back in [math]t\,[/math]'s):

[math]-\frac{p_k}{\Delta q_{k+1}} - \frac{p_k}{\Delta q_{k}} +t_{k+1} - t_{k} = t_{k+1} - \frac{p_k}{\Delta q_{k+1}} - \frac{p_{k-1}}{\Delta q_{k}} - 2t_k = 0\,[/math]

[math]-\frac{p_n}{\Delta q_n} + t'' -t_n = t''-2t_n -\frac{p_{n-1}}{\Delta q_n} = 0\,[/math]

From the first FOC above (for [math]k\,[/math]), if firms have positive profits and [math]p\gt 0\,[/math] then it must be the case that [math]t_{k-1} \gt 2t_k\,[/math] and the case that [math]t'' \gt 2t_n\,[/math].

We have therefore shown that [math]t'' \gt 2t_n \gt 4t_{n-1}\,[/math], but for [math]n\ge 3 t''\gt 4t'\,[/math] which violates assumption 3. The customer that is indiffent between buying from firms 1 and 2 is poorer than the poorest customer, so only firms 1 and 2 get positive market shares.

A limitation on the heterogeneity limits the number of firms in equilibrium even with free entry, but active firms make positive profits. This is in contrast to spatial models (like the circular city) where as fixed costs go to 0 the number of firms goes to infinity and price approaches (true) marginal cost (i.e. transport costs go to zero).

Proposition 2

With more than 2 entrants, at least two qualities are [math]q''\,[/math] and [math]\pi = 0\,[/math] for all firms.

The explanation is as follows:

1. At least one quality is [math]q''\,[/math] (easily proved by contradiction, as a firm could choose [math]q''\,[/math] and suddenly make non-zero profits)
2. At least a second firm will pick [math]q''\,[/math] too, as only the top two qualities make money, picking anything else would yield zero profits.
3. Bertrand competition results at [math]q''\,[/math], and [math]q''\,[/math] makes zero profits.

Proposition 3

Under assumptions 1-3, for any [math]\epsilon\,[/math] exactly two firms enter and choose unequal qualities to make positive profits.

Proposition 1 and Lemma 2

Solving by backwards induction.

In the third stage take qualities [math]q' \le q_1 \le q_2 \le q''\,[/math] as given.

Firm 2 maximizes:

[math]\max_{p_2} (t'' - t_2)\cdot p_2 = (t'' - \frac{\Delta p_2}{\Delta q_2})\cdot p_2\,[/math]

[math]\therefore p_2 = \frac{t'' \Delta q_2 +p_1}{2}\,[/math]

Likewise firm 1 maximizes:

[math]\max_{p_1} (t_2 - t')\cdot p_1 = (\frac{\Delta p_2}{\Delta q_2} - t')\cdot p_2\,[/math]

[math]\therefore p_1 = \frac{p_2 - t' \Delta q_2}{2}\,[/math]

Solving together gives:

[math]p_2 = \frac{(2t'' - t')\Delta q_2}{3} \,[/math]

[math]p_1 = \frac{(t'' - 2t')\Delta q_2}{3} \,[/math]

By assumption 1, [math]p_1 \gt 0\,[/math]. By assumption 2 there are no unserved customers.

Given these prices:

[math]t_2 = \frac{(p_2 - p_1)}{\Delta q_2} = \frac{t''+t'}{3}\,[/math]

[math]D_1 = t_2 - t_' = \frac{t''-2t'}{3}\,[/math]

[math]D_2 = t_'' - t_2 = \frac{2t''-t'}{3}\,[/math]

[math]\pi_1 = D_1 p_1 \gt 0\,[/math]

[math]\pi_2 = D_2 p_2 \gt \pi_2\,[/math]

Now firms choose qualities!

Clearly both would like to pick the high quality, unless it has already been picked, in which case they want to pick the lowest possible quality. Quality choice is therefore a coordination game with lopsided payoffs. Presumably perturbation to timing will allow one to pick first.

Conclusion

Vertical differentiation is not isomorphic to horizontal differentiation. In horizontal differentiation, competitors with two equal prices have local market power. Here lower quality firms are driven out, and only high quality firms exploit some market power. However, quality and price competition with many firms will be too fierce, so even under free entry conditions we will get a finite number of firms, quality differentiation and positive profits.