SUELO URBANIZABLE

To examine competitive nonlinear quantity-based pricing, we use the fol- lowing model.13 Two …rms, noted 1 and 2, compete by setting two-part tari¤s of the form Ti(q) = mi + piq, where mi is a …xed fee and pi is a variable fee. As described above, such two-part tari¤s are used in a large variety of context. For instance, in telephony, the …xed fee can be viewed as a subscription fee for telephone services and the variable fee as the price per minute of communication. As in the previous model, we assume that the …rms are located at the two endpoints of the unit interval (…rm 1 at 0 and …rm 2 at 1). They sell competing brands (produced at a constant marginal cost of c) to a unit mass of consumers. Consumers are uniformly distributed

on [0; 1] and are supposed to be one-stop shoppers (i.e., they buy from at most one …rm).i Instead of supposing as before that consumers have unit demands, we assume now that they can consume any quantity from the …rm they decide to patronize. Let u (q) denote a consumer’s gross utility (excluding transport costs and payments to the …rm) if she buys a quantity q. So, when a consumer faces a variable fee p, the quantity consumed q is such that u0_{(q) = p. Inverting the latter expression, we obtain the demand} function q (p), which we assume to be decreasing in p. All consumers of …rm i buy the same quantity and their net surplus (still excluding transport costs) is wi(pi; mi) = v (pi) mi, where v (pi) is given by

v (pi) = max

q fu (q) piqg with v 0_(p

i) = q.

Consumers enjoy a …xed surplus r from consumption and incur a linear transportation cost, , per unit of distance. So, a consumer located at x 2 [0; 1] derives net utility v1 = r x + w1(p1; m1) when buying from …rm 1 and net utility v2 = r (1 x) + w2(p2; m2) when buying from …rm 2. We assume that r is large enough, so that all consumers choose to participate in the market with the relevant range of tari¤s. Then, the indi¤erent consumer is such that

r x + w^ 1 = r (1 x) + w^ 2, ^x (w1; w2) = ₂1 ( + w1 w2) . Let us now analyse the pricing game between the two …rms. Competition in variable and …xed fees (piand mi) can equivalently be viewed as competi- tion in variable fee and net surpluses (pi and wi). Then, as mi= v (pi) wi, …rm 1 maximizes

max p1;w1 1

= ₂1 ( + w1 w2) [(p1 c) q (p1) + v (p1) w1] ,

where the …rst term is the mass of consumers buying from …rm 1 and the second term is the per consumer pro…t. The …rst-order condition for pro…t- maximization with respect to p1 is

@ 1

@p1 = 0 , q (p1) + q 0_(p

1) (p1 c) + v0(p1) = 0:

i

This assumption is quite realistic in markets where competition with two-part tari¤s prevails: customers purchase only one …rm’s product or service so as to save …xed fees.

Since v0_{(p1) = q (p1), the condition boils down to q}0_{(p1) (p1 c) = 0. As} q0(p1) < 0 and applying the same argument to …rm 2, we have that

p₁ = p₂ = c.

Next, the …rst-order condition with respect to w1 is

@ 1

@w1 = 0 , [(p1 c) q (p1) + v (p1) w1] ( + w1 w2) = 0:

Using the previous result (p₁ = c) and invoking the symmetry of the model (which implies that w1 = w2 at the equilibrium), we can rewrite the latter condition as v (p1) w1 = 0. Recalling that mi = v (pi) wi and using the same argument for …rm 2, we obtain that

m₁ = m₂ = .

At the equilibrium, each …rm makes a pro…t equal to ₁ = ₂ = =2. For future reference, we note the equilibrium industry pro…t under nonlinear tari¤s as N L= .

We record the following result.

Lesson 10.7 If two …rms compete over the Hotelling line with two-part tar- i¤ s and if all consumers are served over the relevant range of tari¤ s, then the unique symmetric equilibrium outcome is such that a consumer who buys quantity q makes payment T (q) = +cq (where is the consumer’s transport cost and c is the …rms’ marginal cost of production). Because of marginal cost pricing, the equilibrium involves e¢ cient consumption.

In this equilibrium, the variable fee is equal to the …rms’marginal cost. A general condition for this result is that the demand of the marginal con- sumer be equal to the average demand. This condition is ful…lled under the particular utility functions we use here (because, as we have seen, all consumers purchase the same amount whatever their location).14

A corollary of the previous result is that competition with two-part tari¤ s improves welfare compared to competition with linear tari¤ s. Indeed, two- part tari¤s induce marginal-cost pricing, whereas linear prices drive …rms to set positive price-cost margins since there is no other way to generate pro…t. It is not clear, however, how this increase in welfare is split between

the …rms and the consumers. This issue is a bit technical to analyze. The reader may want to skip what follows and go immediately to Lesson 10.8.

We introduce some additional pieces of notation. Let i(pi) (pi c) q (pi)

denote per consumer pro…t under linear price pi, and for a candidate equilibrium

pricep, de…ne

s (k) v (c + k (p c)) ,

where k is a scalar. Hence, s (0) = v (c) and s (1) = v (p). Noting sk(k) the

derivative ofs (k) with respect tok, we havesk(k) = (p c) v0(c + k (p c)).

Recalling thatv0(pi) = q (pi), we have then the following equality:

(c + k (p c)) = k (p c) q (c + k (p c)) = ksk(k) .

In particular, (p) = sk(1).

Suppose …rmj chooses pj = p. Forpto be a symmetric equilibrium price, it

must be thatpi= pmaximizes …rmi’s pro…t, which writes as

i = ₂1 ( + v (pi) v (p)) i(pi) :

Lettingpi = c + k (p c), we can use the notation we just introduced to perform

the analysis in terms of the scalarkrather than the pricep. Firmi’s pro…t rewrites as

i= ₂1 ( + s (k) s (1)) ( ksk(k)) ;

and it is necessary thatk = 1maximizes this expression for pto be a symmetric equilibrium. The …rst-order condition gives

k (sk(k))2 ( + s (k) s (1)) (sk(k) + kskk(k)) = 0.

If we evaluate the condition atk = 1, we have

(sk(1))2 (sk(1) + skk(1)) = 0 , sk(1) = skk(1) + 1(sk(1))2.

Recalling that (p) = sk(1), we can express industry pro…t at the symmetric

equilibrium with linear prices in two di¤erent ways:

L = sk(1) and (10.5)

L = skk(1) + 1(sk(1))2. (10.6)

Now, as we assumed thatq0(pi) < 0, we have that skk(1) > 0. So, using (10.6),

we have that L> 1(sk(1))2. But, using (10.5), we also have that 1 (sk(1))2 =

1 2

L. Combining these two results, we …nd that

Hence, we conclude that pro…ts are higher when nonlinear pricing is employed rather than linear pricing.

Let us now turn to consumers. At some price p = c + k (p^ c), welfare is the sum of consumer surplus (v (^p)) and of industry pro…t ( (^p)). Using the previous equivalences, we can rewrite welfare as a function of the scalark: W (k) s (k) ksk(k). Under two-part tari¤s,p = c^ (i.e.,k = 0) and soWN L= W (0);

under linear pricing,p = p^ (i.e.,k = 1) and soWL= W (1). Suppose that total

welfare is concave in linear prices and thus is concave ink. This implies in particular thatW (k)lies below its tangent atk = 1. It also follows that the welfare di¤erence between two-part tari¤s and linear pricing, W = WN L WL= W (0) W (1),

satis…es

W W0(1) = skk(1) .

Now, using (10.5) and (10.6), we know thatskk(1) = L 1 2L= L( L).

Furthermore, we have just shown that L< , which implies that L= < 1and

thus thatskk(1) = ( L= ) ( L) < L. Combining the previous results,

we obtain that

WN L WL< N L L, CSN L WN L N L < CSL WL L;

meaning that consumers in aggregate are worse o¤ when nonlinear tari¤s are used. To understand the previous results, let us compare how prices are …xed in the two cases. With nonlinear pricing, lowering the …xed fee has two opposite e¤ects: (i) it induces a loss of pro…t on existing consumers, and (ii) it attracts pro…table consumers from the other …rm. The optimal …xed fee balances these two forces. By contrast, with linear pricing, lowering price has the same two e¤ects, but also an additional positive e¤ect: with elastic demands, a lower price expands demand from each type of consumer and thus entails a gain in average pro…t per consumer. As a result, there is more incentive to lower prices with linear pricing, which explains why consumers do better. Yet, as welfare is unambiguously higher under two-part tari¤ because the variable fee is equal to marginal cost, it must be that industry pro…ts are larger too.

We summarize the comparison between linear and nonlinear tari¤s as follows.

Lesson 10.8 Competition based on two-part tari¤ s rather than on linear tari¤ s increases industry pro…ts and welfare. Yet, if welfare is concave in linear prices, the increase in industry pro…ts is larger than the increase in welfare, which implies that consumers are harmed by nonlinear pricing.

Intertemporal price

discrimination

In many markets …rm o¤er the same product in di¤erent periods and con- sumers buy only one item over the whole time horizon. This description of consumer behavior …ts in particularly well for durable goods such as cars, washing machines (and other household appliances), computers and software (which does not wear out at all unless obsolescence is arti…cially imposed). It also …ts for particular items which can be ordered in advance such as a holiday package, a plane ticket and a concert ticket. While our insight de- rived in the …rst two sections of this chapter only applies to durable goods, we then obtain results that also apply to ticket sales.

In the case of durable goods, consumers derive the bene…t from the purchase of the good over a number of periods. Also, consumers can decide on the timing of their purchase. An example is furniture consumers may want to replace. They may buy the desired piece immediately and replace the old piece; or they keep their old piece for some more time and thus postpone the purchase of the desired piece. Suppose a …rm sells a product over a number of periods and that both …rm and consumers have discount factor . If the …rm sells the product in the …rst period only at a uniform price, it makes the standard monopoly pro…t (see Chapter 2). If the …rm can sell the product over several periods, we may think that this opens up the possibility of price discrimination, which is bene…cial for the …rm. Whether this conjecture is correct depends on a number of circumstances, as we will explore in this section.

An important issue is whether a …rm can commit to future prices and if the answer is negative, what kind of prices consumers expect. Clearly, even if a …rm preannounces future prices, we must ask whether the …rm has an incentive to deviate at some later point. Therefore, as a starting point, we consider the situation in which the …rm lacks any commitment power and sets the period t-price in that period, not earlier.

11.1

Durable good monopoly without commitment

A monopolist who sells a product at various instances in time may be thought to be in a better position to extract surplus from consumers than a monopolist who only sells once, because, as already pointed out above, selling at various points in time opens up the possibility of intertemporal price discrimination. As we will see, we can give some logic to this rea- soning, only to subsequently question its applicability to many markets of consumer goods. To this end, we will contrast a model with a small number of consumers (here: two) and a model with a large number of consumers (here: a continuum)