Salop (1979) - Monopolistic Competition With Outside Goods

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Reference(s)

• Salop (1979), "Monopolistic competition with outside goods", Bell Journal of Economics 10, 141-156. pdf

@article{salop1979monopolistic,
  title={Monopolistic competition with outside goods},
  author={Salop, S.C.},
  journal={The Bell Journal of Economics},
  pages={141--156},
  year={1979},
  publisher={JSTOR}
}

Abstract

The Chamberlinian monopolistically competitive equilibrium has been explored and extended in a number of recent papers. These analyses have paid only cursory attention to the existence of an industry outside the Chamberlinian group. In this article I analyze a model of spatial competition in which a second commodity is explicitly treated. In this two-industry economy, a zero-profit equilibrium with symmetrically located firms may exhibit rather strange properties. First, demand curves are kinked, although firms make "Nash" conjectures. If equilibrium lies at the kink, the effects of parameter changes are perverse. In the short run, prices are rigid in the face of small cost changes. In the long run, increases in costs lower equilibrium prices. Increases in market size raise prices. The welfare properties are also perverse at a kinked equilibrium.

Summary

The model exhibits kinked demand curves, but then so can the Hotelling model. However, here it is possible to study symmetric equilibria and so equilibrium brand differentiation.

The Model

The model is set up as follows:

• A unit mass of customers is uniformly located around a circumference 1 circle
• [math]mc=c\,[/math]
• Fixed cost of entry = [math]f\,[/math]
• Unit demand for customers with valuation [math]v\,[/math]
• [math]n\,[/math] firms
• linear transport costs [math]t\,[/math]
• Two stage game:
  • Decide whether to pay [math]f\,[/math] and enter
  • Compete in prices

The solution concept is backwards induction, with symmetric equilibria. Note that nothing gaurantees that demand is well behaved.

A distance [math]x\,[/math] from firm [math]i\,[/math], the percieved total cost is:

[math]p_i + xt\,[/math]

From firm [math]j\,[/math]:

[math]p_j+ (\frac{1}{n} -x ) t\,[/math]

Solving for the inverse demand function

A rival is located [math]{1}{n}\,[/math] away and charges [math]p_j\,[/math]. A customer [math]k\,[/math] is indifferent between buying from [math]j\,[/math] and not buying at all:

[math]v - p_j - (\frac{1}{n} -k)t = 0\,[/math]

All customers closer than [math]k\,[/math] are 'captive'.

Charging [math]p_i\,[/math] only gets customers exactly at [math]i\,[/math]'s location - lowering price towards [math]c\,[/math] (note that prices don't get there even under perfect competition because of transportation costs) gets more customers. A customer at [math]x\,[/math] is indifferent between buying and not buying from [math]i\,[/math]:

[math]v-p_i-tx = 0 \quad \therefore x = \frac{v-p_i}{t}\,[/math]

Total demand is twice this:

[math]q_i^m = 2 \cdot \frac{v-p_i}{t} \quad \therefore p_i^m = v - \frac{t}{2} \cdot q_i\,[/math]

Note the slope of the inverse demand function is [math]- \frac{t}{2}\,[/math].

Lowering price to [math]p_i = v-tk\,[/math] gets the whole captive market and starts competiting for [math]j\,[/math]'s customers. Further decreases steal customers.

An indifferent consumer at [math]x \gt k\,[/math] gives:

[math]\underbrace{p_i + tx}_{\mbox{buy from i}} = \underbrace{p_j + t(\frac{1}{n} -x)}_{\mbox{buy from j}}\,[/math]

[math]\therefore x = \frac{p_j - p_i + \frac{t}{n}}{2t}\,[/math]

but demand is two sided, so [math]q_i^c = 2x\,[/math].

Therefore under perfect competition: [math]p_i^c = tp_j +\frac{t}{n} - t q_i^c\,[/math]

Which has a slope of [math]-t\,[/math], therefore the demand curve is kinked where competition meets monopoly at [math]p_i = v - tk\,[/math] (monopoly to the left of the kink, competition to the right, naturally).

Finding the equilibrium

The equilibrium is where the isoprofit hyperbola is tandent to the demand curve. The isoprofit hyperbola is:

[math]d \pi = dq_i(p_i-c) + q_i dp_i\,[/math]

(according to the notes, but [math]\frac{d\pi}{dq_i} = \frac{dq_i}{dq_i} (p - c) + q_i \frac{dp_i}{dq_i}\,[/math]).

The slope of this is [math]\frac{dp_i}{dq_i}\,[/math], which equals:

[math]\frac{dp_i}{dq_q} = -\frac{p_i - c}{q_i}\,[/math]

Therefore:

• If [math]c\,[/math] is low and [math]t\,[/math] is high, the isoprofit curves are steep relative to demand, and there is more likely to be a competitive equilibrium
• There is a range of values for which equilibrium lies in the kink and the comparative statics are perverse
  • For instance a small increase in [math]c\,[/math] has no short run effect - but with free entry and zero profits, it must result in loses and so exits. This lengthens the distance to competitors and increases the ran of the monopolistic part of the demand function (move the kink right), which leads to lower prices.

Solving the competitive equilibrium by backward induction

In the 2nd stage demand per firm is:

[math]q_i^c = \frac{p_j - p_i + \frac{t}{n}}{t}\,[/math]

[math]\pi_i^c = (p-c)\cdot \left (\frac{p_j - p_i + \frac{t}{n}}{t}\right) -f\,[/math]

FOC assuming symmetry gives:

[math]p_* = \frac{t}{n} + c\,[/math]

[math]q^* = \frac{1}{n}\,[/math]

In the first stage, firms enter if they'll make (weakly) positive profits:

[math](p^* - c)\cdot q^* - f = 0\,[/math]

[math]( \frac{t}{n} + c -c) \cdot \frac{1}{n} = f\,[/math]

And the number of entrants must satisfy a zero profit condition:

[math]n^* = \sqrt{\frac{t}{f}}\,[/math]

Therefore equilibrium prices are:

[math]p_* = \frac{t}{n^*} + c = p_* = \frac{t}{\sqrt{\frac{t}{f}}} + c = c + \sqrt{tf}\,[/math]

Welfare comparison

Is [math]n^*\,[/math] the right number of firms? To maximize welfare:

[math]n \cdot \underbrace{\frac{1}{n}(v-c)}_{\mbox{Surplus from a firm}} - \underbrace{\frac{t}{4n}}_{\mbox{Transport Costs}} - \underbrace{nf}_{\mbox{Entry}} = v - c -\frac{t}{4n} - nf\,[/math]

Taking the FOC wrt [math]n\,[/math] gives:

[math]n = \sqrt{\frac{t}{4f}} = \frac{1}{2}\cdot n^*\,[/math]

Therefore the market generates twices as much entry as is efficient. Thus there is too much variety, because each entrant captures part of the market that would have been served by others. Private returns are higher than entry costs, but much of the private return comes from "business stealing" and does not contribute to welfare.