Econ 705: Lecture 3 1

Static Oligopoly Games

The Symmetric Linear Cournot Model

Suppose n ≥ 1 identical firms produce a homogeneous good and possess constant returns technologywith average/marginal costs of c. inverse Market demand is linear and given by

p = a − Q.

Suppose that each firm i simultaneously decides how much output, qi ∈ R+, to put on the market.Also, assume that all output is sold at the market clearing price. Then given a conjecture aboutq−i, firm i’s problem is

maxqi∈R+

πi(qi, q−i) ≡

a −

n∑

j=1

qj

qi − cqi.

This yields the linear system of first-order conditions

∂πi

∂qi= a −

∑

j 6=i

qj

− 2qi − c = 0. i = 1, . . . , n.

Solving this for qi yields the system of best-response functions

q∗i (q−i) =12

a − c −

∑

j 6=i

qj

, i = 1, . . . , n.

At a (Cournot) Nash equilibrium, each firm’s conjecture about its rivals’ output levels is correct(i.e., qj = q∗j for all j). With this fact in hand, the resulting system of n symmetric linear equationscan easily be solved because q∗i = q∗j

1 gives the Cournot-Nash equilibrium of this game:

q∗i = (a − c − (n − 1)q∗i )/2⇔ (n + 1)q∗i = a − c

⇔ q∗i = (a − c)/(n + 1),

for i = 1, . . . , n. The equilibrium industry output and price are, thus,

Q∗ = nq∗i =n

n + 1(a − c)

andp∗ = c +

a − c

n + 1.

Equilibrium profit per firm is

π∗i =

(a − c

n + 1

)2

.

Observe that this yields the monopoly outcome for n = 1, and the competitive outcome in the limitas n tends to infinity.

1This only works because the model is symmetric. Generally, we must actually solve the system by brute-force(linear algebra) methods.

Econ 705: Lecture 3 2

Example 1 (Symmetric Cournot Duopoly). Let n = 2 (duopoly), a = 80, and c = 20. Figure1 below illustrates the best-response or “reaction” functions of the two firms. Figure 2 shows theaggregate demand and the equilibrium quantities and prices under monopoly, Cournot, and perfectcompetition.

30

30

60

60

q∗2 = 20

q∗1 = 20 q1

q2

q∗1(q2)

q∗2(q1)

Figure 1: Cournot reaction functions

80

80

c = 20

6040

40

30

50

Q

P

Competitive outcome

Cournot outcome

Monopoly outcome

Figure 2: Aggregate demand and equilibriumunder monopoly and Cournot competition

Econ 705: Lecture 3 3

Symmetric Bertrand Competition with Homogeneous Goods

In the case of monopoly, choosing quantity is equivalent to choosing price. Suppose, however, thatn ≥ 2, and that firms strategically choose prices rather than quantities (i.e., each firm i = 1, . . . , nposts a price pi ∈ R). Also suppose that consumers buy from the firm posting the lowest price andthat firms are never capacity constrained.

Proposition 1 (The Bertrand Paradox). There exists at least one pure-strategy Nash equilib-rium of the symmetric, homogeneous-goods Bertrand-pricing game. Moreover, in any such equi-librium at least two firms (call them 1 and 2) set the competitive price p∗1 = p∗2 = c. Aggregateequilibrium sales are Q∗ = a − c (the competitive quantity). No firm sets a price lower than c, andfirms setting higher prices make no sales. Hence all firms earn zero profit.

Proof. Note that the prescribed behavior is an equilibrium because all firms make zero profit andnone of them can do strictly better by unilaterally deviating (prices higher than c result in no salesand prices lower than c result in losses). We now show that no other behavior is an equilibrium.(Looking for a fixed point of the aggregate best-response correspondence is problematic in thissetting because a firm’s best response is not defined whenever the lowest price posted by its rivalsexceeds c.)

Case 1: Suppose that a single firm (say 1) sets the lowest price p > c, then it captures the entiremarket and earns profit of (p − c)(a − p). This cannot be an equilibrium because firm 2 canincrease its profit from zero to (p− ε− c)(a−p+ ε) > (p− c)(a−p)− ε(a− c) > 0 by choosinga price p − ε ∈ (c, p).

Case 2: Suppose that m ≥ 2 firms set the lowest price of p > c. To see that this cannot be anequilibrium, note that the least profit earned by any of the firms charging p (say firm 2) isno more than (p− c)(a− p)/m. However, by under-cutting p by ε > 0, 2 would receive profitin excess of (p − c)(a − p) − ε(a − c) which is strictly greater than (p − c)(a − p)/m for

ε <m − 1

m(a − c)(p − c)(a − p).

Case 3: Suppose that firm 1 sets p1 = c and the next lowest price set is p2 > c. This cannot be anequilibrium because 1 could raise its profit from (c−c)(a−c) = 0 to (p2−ε−c)(a−p2 +ε) > 0by setting p1 = p2 − ε ∈ (c, p2).

Case 4: Suppose that one or more firms set the lowest price p < c. Then at least one of thesefirms makes positive sales and negative profit. This firm could do better by raising its priceto c and earning zero.

If firms choose prices and produce to meet demand, then n = 1 still yields the monopolyoutcome, but n ≥ 2 results in the perfectly competitive outcome, Q∗ = a − c and p∗ = c. This isknown as the Bertrand paradox or the bang-bang property of Bertrand equilibrium.

The critical assumptions underlying the Bertrand paradox are:

1. Firms produce homogeneous (not differentiated) goods.

2. Firms have CRS technology and (in particular) no capacity constraints (otherwise a pure-strategy Nash equilibrium does not exist).

3. Consumers know all prices and can shift their purchases to the low-price firms without cost.

Econ 705: Lecture 3 4

Strategic Complements, Strategic Substitutes, and ComparativeStatics

For notational ease, denote the partial derivative of a function F : RN → R with respect to its kthargument by ∂kF .

Definition 1 (Strategic Complements and Substitutes). Consider a game in which the strat-egy sets of all players are connected subsets of the real line and in which best-response correspon-dences are differentiable functions. Player i regards player j’s strategy, sj, as a strategic comple-ment if ∂jBRi ≥ 0, and he regards sj as a strategic substitute if ∂jBRi ≤ 0.

Strategic complements and strategic substitutes correspond respectively to upward-slopingand downward-sloping best-response functions. Suppose ui(s) is twice continuously differentiable.Player i’s best-response function is defined implicitly by the first-order condition

∂iui(BRi(s−i), s−i) ≡ 0.

Differentiating this w.r.t. sj and assuming ∂iiui < 0 yields

∂jBRi(s−i) = −∂ijui

∂iiui.

Hence, a necessary and sufficient condition for player i to regard sj as a strategic complement is∂ijui > 0. If this is true for all i and j ∈ N , then the payoff functions are supermodular in thestrategies, and the game is itself said to be supermodular.

Likewise, a necessary and sufficient condition for player i to regard sj as a strategic substituteis ∂ijui < 0. If this is true for all i and j ∈ N , then the payoff functions are submodular in thestrategies, and the game is itself said to be submodular.

Consider a Nash equilibrium s∗ = (s∗1, . . . , s∗n). Comparative statics evaluated as s∗ will haveintuitive signs iff the matrix of second partials

∂11u1 ∂12u1 · · · ∂1nu1

∂21u2 ∂22u2 · · · ∂2nu2...

.... . .

...∂n1un ∂n2un · · · ∂nnun

is negative definite. When n = 2, a sufficient condition for this is that the equilibrium is stable.

Definition 2 (Strategic Stability). Suppose n = 2. A Nash equilibrium s∗ is said to be stableif ∣∣BR′

1(s∗2)BR′

2(s∗1)

∣∣ < 1

or equivalently|∂12u1∂21u2| < ∂11u1∂22u2

In words, S∗ is stable if player 1’s best-response function is steeper in absolute value than player2’s at s∗ when considering the functions in (s1, s2) space. (Observe that the slope of 1’s best-responsein (s1, s2) space is 1

BR′1(s2) .) Note that a non-degenerate mixed-strategy Nash equilibrium in a 2×2

matrix game is always unstable because player 1’s best-response function is horizontal and player2’s is vertical at the equilibrium.

Econ 705: Lecture 3 5

Comparative static analysis also differs depending on whether strategies are strategic comple-ments or strategic substitutes. Specifically, suppose that payoff functions are u1(s1, s2; c1) andu2(s1, s2; c2) where ci is a parameter. Consider a stable equilibrium s∗(c1, c2) defined by the systemof first-order conditions

[∂1u1(s∗1(c1, c2), s∗2(c1, c2), c1)∂2u2(s∗1(c1, c2), s∗2(c1, c2), c2)

]=

[00

]

Differentiating this system w.r.t. c1 yields[∂11u1 ∂12u1

∂21u2 ∂22u2

] [∂1s

∗1(c1, c2)

∂1s∗2(c1, c2)

]=

[−∂13u1

0

]

Applying Cramer’s Rule yields

∂1s∗1(c1, c2) =

−∂13u1∂22u2

∂11u1∂22u2 − ∂12u1∂21u2

and∂1s

∗2(c1, c2) =

∂13u1∂21u2

∂11u1∂22u2 − ∂12u1∂21u2.

Stability of the equilibrium implies that the denominator on the right side of these expressions ispositive. Moreover, ∂22u2 < 0 (because s∗2 maximizes u2). Hence, a rise in c1 causes player 1 tolower its equilibrium strategy s∗1 if ∂13u1 < 0 and causes it to raise s∗1 if ∂13u1 > 0. This makessense because ∂13u1 is the impact of raising c1 on player 1’s marginal utility. Player 2’s equilibriumresponse to a rise in c1 depends both on the sign of ∂13u1 and on the sign of ∂21u2. In particular,s∗2 rises with c1 if and only if (i) ∂13u1 > 0 and ∂21u2 > 0 (strategic complements) or (ii) ∂13u1 < 0and ∂21u2 < 0 (strategic substitutes). In case (i), ∂13u1 > 0 means that a rise in c1 causes 1’sbest-response function to shift up. Moreover ∂21u2 > 0 implies that 2’s best-response function isupward sloping so that the shift in 1’s best-response function leads to higher values of both s∗1 ands∗2. In case (ii), 1’s best-response function shifts down when c1 rises. However, 2’s best-responsefunction is downward sloping so that the shift down in 1’s best-response function leads to a lowervalue for s∗1 but a higher value for s∗2. These effects are illustrated further in the following example.

Example 2 (Differentiated Duopolists). Consider duopolists who produce differentiated goodswith CRS technology. In particular, the demand for firm i’s product is

qi = 1 − pi + (1/2)pj . (1)

Begin by supposing that the firms both simultaneously set prices. The problem facing firm i is

maxpi∈R+

πi(p1, p2; ci) ≡ (1 − pi + (1/2)pj)(pi − ci).

The best-response function is

p∗i (pj) = (1/2)(1 + ci) + (1/4)pj , i = 1, 2, i 6= j.

Hence, firm i’s best-response function is linear with a vertical intercept of (1 + ci)/2 and slopeof 1/4. This is a setting of strategic complements since i’s optimal price is increasing in pj (i.e.,best-response functions slope up). Note also that the unique Nash equilibrium in this setting isstable because dp∗i /dpj = 1/4; in other words, 2’s best-response function has a slope of 1/4 and 1’a slope of 4 (in (p1, p2) space). Finally, observe that ∂i,3πi = 1 > 0 so that a rise in ci causes i’sbest-response function to shift up.

Econ 705: Lecture 3 6

p2

p1

12 (1 + c2)

12(1 + c1) p∗1

p∗2p∗2(p1)

p∗1(p2)

Figure 3: Strategic complements

Direct substitution yields

pi = (1/2)(1 + ci) + (1/4)((1/2)(1 + cj) + (1/4)pi),

orp∗i = (2/3) + (8/15)ci + (2/15)cj , i = 1, 2, i 6= j.

Hence, as indicated by the above analysis, i’s equilibrium price, p∗i , is increasing in j’s unit cost,cj. The idea is that a rise in cj causes firm j to raise its price (i.e., its best-response function shiftsup); since firm i’s best-response function slopes up, the new equilibrium involves higher prices forboth firms. Hence, if cj rises, then firm i also raises its price. In other words, i competes lessaggressively in equilibrium when j’s marginal cost rises.

Now, consider precisely the same model but suppose instead that both firms simultaneously setquantities. Inverting the demand system (1) gives

pi(qi, qj) = 2 − (4/3)qi − (2/3)qj , i = 1, 2, i 6= j. (2)

Hence, payoffs are

πi(q1, q2; ci) ≡ (2 − (4/3)qi − (2/3)qj)qi − ciqi, i = 1, 2, i 6= j.

Maximizing this w.r.t. qi, holding qj fixed, yields the best-response function:

q∗i (qj) = (3/4) − (3/8)ci − (1/4)qj , i = 1, 2, i 6= j.

Again, the best-response functions are linear, but in this case, the setting is one of strategic sub-stitutes (i.e., the best-response functions slope down). Once again, the unique Nash equilibrium isstable because the slope of 2’s best-response function is −1/4 and the slope of 1’s is −4 (in (q1, q2)space). Note also, that i’s best-response is decreasing in ci since ∂i3πi = −1.

Direct substitution yields

qi = (3/4) − (3/8)ci − (1/4)((3/4) − (3/8)cj − (1/4)qi),

orq∗i = (3/5) − (2/5)ci + (1/10)cj , i = 1, 2, i 6= j.

Econ 705: Lecture 3 7

q2

q1

34 − 3

8c2

34 − 3

8c1

q∗2(q1)

q∗1(q2)

q∗1

q∗2

Figure 4: Strategic substitutes

Here, an increase in cj leads to a decrease in q∗j and an increase in q∗i . In other words, i competesmore (rather than less) aggressively in response to a rise in cj when the firms compete in quantities.

Spatial Competition

Hotelling (1929) argued that firms (also religious sects, and political platforms) will tend to clusterat the center of the spectrum of consumer tastes. As an example of this phenomenon, supposethat consumers are uniformly distributed along the unit interval, which is intended to represent thepopulation distribution in a city or the spectrum of consumer tastes for some product characteristic.Two firms compete in this market by choosing their locations: firm 1’s location (distance from theorigin) is a, and firm 2’s location is b with 0 ≤ a ≤ b ≤ 1 w.l.o.g.. Suppose that consumers facea linear transportation cost t per unit of distance from the firm at which they buy. Also, supposethat the prices of the firms are exogenously fixed at p1 = p2 = p > c, where c is the constantmarginal cost of production. Assume that each consumer values one unit of the good at v andadditional units at zero (the unit-demand assumption), where v − t − p > 0 (i.e., the market iscovered). Suppose that total demand is split equally between the firms if a = b (firms locate atthe same spot). We seek an equilibrium in locations. (As with undifferentiated price competition,best-response correspondences often do not exist in this setting due to payoff discontinuities.)

Lemma 1 (No Differentiation). In any Nash equilibrium, the firms choose the same location,a∗ = b∗.

Proof. By way of contradiction, suppose a∗ < b∗. In this case, 2 would increase its demand bymoving closer to firm 1, choosing b ∈ (a∗, b∗), since it keeps all of the customers to its right, andcaptures some of the middle customers.

Lemma 2 (The Median Customer Theorem). In any Nash equilibrium the firms locate at thecenter, a∗ = b∗ = 1/2.

Proof. By way of contradiction, suppose (w.l.o.g.) that a∗ = b∗ < 1/2. In this case, the firms eachserve half the market, but if 2 moves an infinitesimal distance to the right, its demand jumps tob∗ − ε > 1/2.

Proposition 2 (Equilibrium Location). The unique Nash equilibrium is for both firms to locateat the center, a∗ = b∗ = 1/2.

Econ 705: Lecture 3 8

Proof. By Lemmas 1 and 2, no other location choices constitute an equilibrium. To see thata∗ = b∗ = 1/2 is a Nash equilibrium, note that firm 2 gets demand of 1/2 in this case and wouldget strictly less demand by choosing any other location.

Let a and b be general locations of firms 1 and 2. Then aggregate transportation cost iscomposed of four pieces:

1. the consumers to the left of firm 1∫ a

0t(a − x) dx =

ta2

2,

2. the consumers between the firms that are closer to firm 1∫ a+b

2

at(x − a) dx =

t (b − a)2

8,

3. the consumers between the firms that are closer to firm 2∫ b

a+b2

t(b − x) dx =t (b − a)2

8,

4. the consumers to the right of firm 2

∫ 1

bt(x − b) dx =

t (1 − b)2

2.

Hence, aggregate transportation cost is

TC(a, b) ≡ t(0.5a2 + 0.5(1 − b)2 + 0.25(b − a)2

).

At the Nash equilibrium locations, a∗ = b∗ = 12 , aggregate transportation cost is TC = t

4 . To findthe locations that minimize transportation cost, we derive the first-order conditions

∂TC

∂a=

t(3a − b)2

= 0

and∂TC∂b =

t(3b − a − 2)2

= 0.

Solving these yields a∗∗ = 14 and b∗∗ = 3

4 . This yields aggregate transportation cost of TC = t8 , or

half of the Nash equilibrium level. This makes sense. There might as well be only a single firm ifthey both locate at 1

2 ! The spectrum of consumer tastes is better served with some differentiation.Note that at the locations a∗∗ = 1

4 and b∗∗ = 34 , the firms are no worse off as compared with

the Nash equilibrium locations since they each still serve half the market. Of course, some of theconsumers in the middle of the city (spectrum) are worse off if the firms locate at 1/4 and 3/4.

Finally, observe in passing that no pure strategy equilibrium exists when there are three firmsthat simultaneously choose locations.