Chapter 7: Product DifferentiationA1. Firms meet only once in the market.Relax A2. Products are differentiated.A3. No capacity constraints.

Timing:1. firms choose simultaneously their location in the productspace,

2. given the location, price competition.

• Spatial-differentiation model– Linear city (Hotelling, 1929)– Circular city (Salop, 1979)

• Vertical differentiation model– Gabszwicz and Thisse (1979, 1980);– Shaked and Sutton (1982, 1983)

• Monopolistic competition (Chamberlin, 1933)• Advertising and Informational product differentiation(Grossman and Shapiro, 1984)

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1 Spatial Competition1.1 The linear city (Hotelling, 1929)• Linear city of length 1.• Duopoly with same physical good.• Consumers are distributed uniformly along the city,N = 1

• Quadratic transportation costs t per unit of length.• They consume either 0 or 1 unit of the good.• If locations are given, what is the NE in price?Price CompetitionMaximal differentiation• 2 shops are located at the 2 ends of the city, shop 1 is atx = 0 and of shop 2 is at x = 1. c unit cost

• p1 and p2 are the prices charged by the 2 shops.• Price of going to shop 1 for a consumer at x is p1 + tx2.• Price of going to shop 2 for a cons. at x p2 + t(1− x)2.• The utility of a consumer located at x is

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U =

s− p1 − tx2 if he buys from shop 1

s− p2 − t(1− x)2 if he buys from shop 20 otherwise

• Assumption: prices are not too high (2 firms serve themarket)

• Demands areD1(p1, p2) =

p2 − p1 + t2t

D2(p1, p2) =p1 − p2 + t

2t• and profit

Πi(pi, pj) = (pi − c)pj − pi + t2t

• So each firm maximizes its profit and the FOC givespi =

pj + t− c2

for each firm i

• Prices are strategic complements:

∂2Πi(pi, pj)

∂pi∂pj> 0.

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• The Nash equilibrium in price isp∗i = p∗j = c + t

• The equilibrium profits areΠ1 = Π2 =

t

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Minimal differentiation• 2 shops are located at the same location xo.• p1 and p2 are the prices charged by the 2 shops.• Price of going to shop 1 for a consumer at x isp1 + t(xo − x)2.

• Price of going to shop 2 for a consumer at x isp2 + t(xo − x)2.

• The consumers compare prices.... Bertrand competition• Nash equilibrium in prices is

p∗i = p∗j = c

• and the equilibrium profits areΠ1 = Π2 = 0

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Different locations• 2 shops are located at x = a and of shop 2 is at x = 1−bwhere 1− a− b ≥ 0.

• If a = b = 0: maximal differentiation• If a + b = 1: minimal differentiation• p1 and p2 are the prices charged by the 2 shops.• Price of going to shop 1 for a consumer at x isp1 + t(x− a)2.

• Price of going to shop 2 for a consumer at x isp2 + t(1− b− x)2.

• Thus there exists an indifferent consumer located at exp1 + t(ex− a)2 = p2 + t(1− b− ex)2⇒ ex = p2 − p1

2(1− a− b) +1− b + a

2• and thus the demand for each firm is

D1(p1, p2) = a +1− b− a

2+

p2 − p12(1− a− b)

D2(p1, p2) = b +1− b− a

2+

p1 − p22(1− a− b)

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• The Nash equilibrium in price isp∗1(a, b) = c + t(1− a− b)(1 + a− b

3)

p∗2(a, b) = c + t(1− a− b)(1 + b− a3)

• Profits are

Π1(a, b) = [p∗1(a, b)− c]D1(a, p∗1(a, b), p∗2(a, b))

Π2(a, b) = [p∗2(a, b)− c]D2(b, p∗1(a, b), p∗2(a, b))Product ChoiceTiming:

1. firms choose their location simultaneously2. given the location, they simultaneously choose prices

• Firm 1 chooses a that maximizes Π1(a, b)⇒ a(b)

• Firm 2 chooses b that maximizes Π2(a, b)⇒ b(a)

• and then (a∗, b∗).• What is the optimal choice of location?

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dΠ1(a, b)

da=

∂Π1(a, b)

∂p1

∂p1∂a

+∂Π1(a, b)

∂a+

∂Π1(a, b)

∂p2

∂p2∂a

where∂Π1(a, b)

∂p1

∂p1∂a

= 0 (due to envelope theorem)

• We can rewrite

dΠ1(a, b)

da= [p∗1(a, b)− c](

∂D1(.)

∂a+

∂D2(.)

∂p2

∂p2∂a)

where∂D1(.)

∂a=

3− 5a− b6(1− a− b) Demand Effect (DE)

and∂D2(.)

∂p2

∂p2∂a

=−2 + a

3(1− a− b) < 0 Strategic Effect (SE)

Thus,dΠ1(a, b)

da= [p∗1(a, b)− c](

−1− 3a− b6(1− a− b)) < 0

• As a decreases, Π1(a, b) increases.Result The Nash Equilibrium is such that there ismaximal differentiation, i.e. (a∗ = 0, b∗ = 0)

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• 2 effects may work in opposite direction– SE<0 (always): price competition pushes firms tolocate as far as possible.

– DE can be >0 if a ≤ 1/2, to increasemarket share,given prices, pushes firms toward the center.

– But overall SE>DE, and they locate at the twoextremes.

1.1.1 Social planner

• Minimizes the average transportation costs.• Thus locates firms at

as =1

4, bs =

1

4Result Maximal differentiation yields too much productdifferentiation compared to what is socially optimal.

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1.2 The circular city (Salop, 1979)• circular city• large number of identical potential firms• Free entry condition• consumers are located uniformly on a circle of perimeterequal to 1

• Density of unitary around the circle.• Each consumer has a unit demand• unit transportation cost• gross surplus s.• f fixed cost of entry• marginal cost is c• Firm i’s profit is

Πi =

((pi − c)Di − f if entry0 otherwise

• How many firms enter the market? (entry decision)

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Timing: two-stage game1. Potential entrants simultaneously choose whether or notto enter (n). They are automatically located equidistantfrom one another on the circle.

2. Price competition given these locations.

Price Competition• Equilibrium is such that all firms charge the same price.• Firm i has only 2 real competitors, on the left and right.• Firm i charges pi.• Consumer indifferent is located at x ∈ (0, 1/n) from i

pi + tx = p+ t(1

n− x)

• Thus demand for i isDi(pi, p) = 2x =

p + tn − pit

• Firm i maximizes its profit(pi − c)Di(pi, p)− f

• Because of symmetry pi = p, and the FOC givesp = c +

t

n

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How many firms?Because of free entry condition

(pi − c)1n− f = 0⇒ t

n2− f = 0

which givesn∗ = (

t

f)12

and the price isp∗ = c + (tf)

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• Remark: p− c > 0 but profit =0....• If f increases, n decreases, and p− c increases.• If t increases, n increases, and p− c increases.• If f → 0, n→∞ and p→ c (competitive market).

• Average transportation cost is2n

Z 12n

0

xtdx =t

4n• From a social viewpoint

Minn[nf + 2n

Z 12n

0

xtdx]

⇒ ns =1

2n∗ < n∗

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Result The market generates too many firms.

• Firms have too much an incentive to enter: incentive isstealing the business of other firms.

• Natural extensions:– location choice– sequential entry– brand proliferation

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1.3 Maximal or minimal differentiation• Spatial or vertical differentiation models make importantprediction about business strategies

Firms want to differentiate to soften price compe-titionIn some case: maximal product differentiation.

• Opposition to maximal differentiation– Be where the demand is (near the center of linearcity)

– Positive externalities between firms (many firms maylocate near a source of raw materials for instance)

– Absence of price competition (prices of ticket airlinebefore deregulation)

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2 Vertical differentiation• Gabszwicz and Thisse (1979, 1980); Shaked and Sutton(1982, 1983)

Timing: two-stage game1. Simultaneously choice of quality.2. Price competition given these qualities.

• Duopoly• Each consumer consumes 0 or 1 unit of a good.• N = 1 consumers.• A consumer has the following preferences:u =

(θs− p if he buys the good of quality s at price p0 if he does not buy

where θ > 0 is a taste parameter.• θ is uniformly distributed between θ and θ = θ + 1;density f(θ) = 1; cumulative distribution F (θ) ∈[0,∞)

• F (θ): fraction of consumers with a taste parameter < θ.• 2 qualities s2 > s1

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• Quality differential: ∆s = s2 − s1• Unit cost of production: c• Assumptions

A1. θ > 2θ (insure demand for the two qualities)A2. c + θ−2θ

3 ∆s < θs1 (insure that p∗1

s1< θ)

• Firms choose p1 and p2

Price competition (given qualities)• There exists an indifferent consumer: eθ = p2−p1

s2−s1 .– A consumer with θ ≥ eθ buys the quality 2 (θ ≥

p2−p1s2−s1 ). The proportion of consumers who will buygood of quality 2 is F (θ)− F (eθ).

– A consumer with θ < eθ and θ ≥ θ > p1s1buys low

quality 1. So the proportion of consumers who willbuy good of quality 1 is F (p2−p1s2−s1)− F (θ).

– if θ < θ no purchase.• Then demands are

D1(p1, p2) =p2 − p1∆s

− θ

D2(p1, p2) = θ − p2 − p1∆s

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• Each firm maximizes its profitMaxpi(pi − c)Di(pi, pj)

• The reaction functions areR1(p2) = p1 =

1

2[p2 + c− θ∆s]

R2(p1) = p2 =1

2[p1 + c + θ∆s]

• The Nash equilibrium isp∗1 = c +

θ − 2θ3

∆s (A2.)

p∗2 = c +2θ − θ

3∆s > p∗1

• Demands areD∗1 =

θ − 2θ3

(A1.)

D∗2 =2θ − θ

3• and profits

Π1(s1, s2) =(θ − 2θ)2

9∆s

Π2(s1, s2) =(2θ − θ)2

9∆s

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• High quality firm charges a higher price than low qualityfirm.

• High quality firm makes more profit.Choice of quality• Simultaneous choice of quality.• Quality is costless.• si ∈ [s, s], where s and s satisfy A2.• Each firm chooses si that maximizes its profit Πi(si, sj)• s1 = s2 cannot be an equilibrium, as they can do betterif s1 6= s2 (profit increases)

• If s1 < s2, as ∂Πi(.)/∂∆s > 0 both firms make moreprofit if more differentiation.

• Firm 1 reduces its quality towards s, firm 2 increases itsquality towards s.

• 2 Nash equilibrium in quality: {s∗1 = s, s∗2 = s} and{s∗2 = s, s∗1 = s}

Result The equilibrium is such that there is maximaldifferentiation.

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⇒Firms try to relax the price competition through productdifferentiation.

• If sequential entry – the first chooses s and the secondchooses s. (unique NE)

• But then race to be first...

• Even if quality is costless to produce, the low qualityfirm gains from reducing its quality to the minimum(because it softens price competition).

• Difference with location model:– if A1. does not hold anymore: only one firm makesprofit in the market.

– A “low” low quality cannot compete with a highquality,

– A “high” low quality trigger tough price competition.

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3 Monopolistic competition• Chamberlin (1933)• Dixit and Stiglitz (1977), Spence (1976)• Monopolistic competition corresponds to the followingindustry configuration:– each firm faces a downward sloping demand,– each firm makes no profit (free entry condition),– the price of one firm does not affect the demand ofany other firm

• No strategic aspect;• Too many of too few products?

• First idea (Chamberlin (1933)): too few• In fact not true (Dixit and Stiglitz (1977), Spence(1976))

• 2 effects work in opposite direction:– non appropriability of social surplus (too fewproducts)

– business stealing (too many products)

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4 Advertising and Informationalproduct differentiation• Effect of ad on consumer demand and product differ.• Ad conveys information on existence and price.• Information issue can be solved, at some cost, throughadvertising (search good).

• Monopolistic competition (Butters, 1977)• Oligopoly (Grossman and Shapiro, 1984)• Socially too much or too little advertising?• Firms are differentiated along two dimensions:– information,– location.

• What is the effect of advertising on the elasticity ofindividual demands and on the appropriability andbusiness stealing effects?

• Linear-city model• 2 firms locates at the two extremes• consumers are distributed uniformly

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• s gross surplus• t transportation cost• The only way to reach consumers is to send adsrandomly.

• Advertising: information about the product, and price.– If a consumer receives no ads, he does not buy.– If he receives 1 ad he buys from the firm.– If he receives 2 ads he chooses the closest firm.

• Fraction of consumers who receive an ad from i is φi,i = 1, 2

• Consumers located along the segment have equalchances of receiving a given ad.

• Cost of reaching fraction φi is A(φi) = aφ2i /2• Firms choose p1 and p2• Firms compete for the “common demand”• Demand for 1 is

D1 = φ1[(1− φ2) + φ2p2 − p1 + t

2t]

– A fraction 1− φ2 does not receive ad from 2– A fraction φ2 receives at least one ad from 2

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• Elasticity of demand at price p1 = p2 = p andφ1 = φ2 = φ is

ε1 = − p1D1

∂D1∂p1

=φp

(2− φ)t

– it is increasing with φ, so with ad.• Consider that firms simultaneously choose prices andlevels of ads.

• Firm 1Maxp1,φ1

(p1 − c)φ1[(1− φ2) + φ2p2 − p1 + t

2t]−A(φ1)

• FOC arep1 =

p2 + c + t

2+1− φ2φ2

t

φ1 =1

a(p1 − c)[(1− φ2) + φ2

p2 − p1 + t2t

]

• Symmetric gamep∗1 = p∗2 = p

∗

φ∗1 = φ∗2 = φ∗

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• Assume a ≥ t/2p∗ = c + (2at)

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φ∗ =2

1 + (2at )12

Π1 = Π2 =2a

(1 + (2at )12)2

• p∗ > c + t (price under full information)– the price increases with t, and with a.

• The lower the advertising cost, and the higher thehorizontal differentiation, the more the firms advertise.

• Profits are– increasing with t– increasing with a because of 2 effects. If a increases,∗ DE: it induces profit to decrease,∗ SE: it decreases ad, and thus increases informa-tional PD. Firm raises the price.

• The market level of ad can be greater or smaller than thesocially optimal level of ad.– non appropriability of SS (low incentive to ad)– business stealing (excessive advertising)

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