PQ-Nash Duopoly:A Computational Characterization

J. Huston McCullochOhio State University

18th Int’l. Conf. on Computing in Economics and FinancePrague, 2012

PDF with color figures at www.econ.ohio-state.edu/jhm/papers/PQNash.pdf

Cournot vs Bertrand Duopoly

Cournot (1838): Each chooses P and Q, taking other’s Q as given

Bertrand (1883): Each chooses P and Q, taking other’s P as given

Cournot vs Bertrand Duopoly

Cournot (1838): Each chooses P and Q, taking other’s Q as given

Bertrand (1883): Each chooses P and Q, taking other’s P as given

Neither takes other’s full (P, Q) strategy as given: Both half right, half wrong!

Duopoly Model:

2 firms produce Q1, Q2 of identical good

Increasing Marginal (Variable) Cost schedules MC1(Q1), MC2(Q2)Decreasing Demand function D(P)

Atomistic, price-taking consumersOutput perishable at end of period, but freely disposableProducers can’t price discriminateFull information – each knows D(P), other’s MC“Parallel Rationing”:

If Pi < Pj , Firm i sells out first to highest-demand consumers, Firm j gets residual demand. (as if curb market with free entry efficiently allocates total Q to consumers)

Simple illustrative straight-line model:

MC1(Q1) = Q1/a, MC2(Q2) = Q2/(1-a), a (0, .5],

so a is share of smaller firm in horizontally summed MC

D(P) = 1 + e – e P,

so P = 1, Q = 1 is quasi-competitive (P-taking) equilibrium

and e > 0 is absolute price elasticity of D at P = 1, Q = 1.

Eg a = 1/3, e = 1:

With 2-plant monopoly (+) and quasi-competitive () equilibria.

Some well-known Duopoly models:

But none is a 1-stage Nash equilibrium

Cournot (1838) popularly said to be Q-Nash

eg by Singh and Vives (1984)

Firm 1’s decision, taking Q2 given

But in fact firms can and do set P as well as Q. Firm 1 would undercut P2 and raise Q1, so not simple Nash.

Cournot is, however, outcome of 2-stage Q-then-P Nash game – Kreps and Scheinkman (1983).

Bertrand (1883) popularly thought to be P-Nash

Firm 1’s decision, taking P2 as given

But in fact, firms can and do set Q as well as P. Firm 1 would reduce Q and thereby raise P1 above P2, so Bertrand not simple Nash

Is there a 1-stage PQ Nash equilibrium?

No PQ-Nash equilibrium in pure strategies:

Can find P2, Q2 so 3 intersections of MC1 and MR1: Middle one clears market

No PQ-Nash equilibrium in pure strategies:

Can find P2, Q2 so 3 intersections of MC1 and MR1: Middle one clears market, but is local profit minimum. Left and right intersections are maxima but don’t clear market.

Is there an equilibrium in mixed strategies?

Nash (1950):

Every finite bimatrix game has at least 1 equilibrium in pure or mixed strategies.

Computing in Economics and Finance:

One equilibrium of discrete approximation to bimatrix game may be found numerically using algorithm of Mangasarian & Stone (1964).

(Matlab thanks to Bapi Chatterjee 2008)

With n strategies each, takes about n3 flops:n iterations, n2 flops each

But with h x h grid of (P, Q) strategies, each has n = h2 strategies

So takes about h6 flops!

h = 11: several secondsh = 15: several minutesh = 21: several hours

With mixed strategies, combined output may exceed or fall short of D.

Let qi = amount sold by firm i, Qi = amount produced, as above.

If Pi < Pj, “parallel rationing:” qi = min(Qi, D(Pi)), qj = min(Qj, max(0, D(Pj) – qi)).

In rare event Pi = Pj and Q1 + Q2 > D(P), assume sales proportional to production: qi = min(Qi, D(P) Qi / (Qi + Qj)).

Then i = Pi qi – Ci(Qi).

Solution generally is high “regular” price with low Q, plus string of low “sale” prices with higher Q’s:

2-step 15x15 grid for each with ½ step offset in P to reduce ties.Probabilities proportional to area of symbols, = ¼ in legend.

D moreelastic

D less elastic

Firm 1verysmall

Firms equal in size

Same story with other parameter values:

Note that max P is always less than Cournot, while min P always > Bertrand.But sometimes unsold Q. Which more efficient?

Robt. Gertner (unpublished 1986 MIT dissertation!)

• Finds FOC for PQ-Nash problem!• Support for Pi

is [Pmin, Pmax], same for both firms• Qi conditional on Pi is unique• Mass point at Pmax, continuous distribution below• If both charge Pmax market clears; else, overproduction• Gives differential eq’n for density, but “intractable”• Gives no solution, even for linear or const. elast. schedules• Only treats symmetric case MC1(Q) = MC2(Q)

Now at Chicago Booth School My nomination for next Nobel Prize!

Valid simultaneous move Nash Duopoly models are now: • Gertner PQ 1-stage accounts for random sales!• Cournot-Kreps-Scheinkman Q-then-P 2-stage

generalized by Wu, Zhu & Sun IJIO 2012• Production-to-Order P-Qmax 1-stage (next slide)

Levitan and Shubik (1978) anticipate Gertner with zero MC and costly disposal (or storage)

Kreps and Sheinkman (1983) anticipate Gertner with vertical MC in 2nd stage game à la Edgeworth (1925) but KS requires parallel rationing – Davidson & Deneckere (1986)

Singh and Vives (1984) – firms artificially constrained to either pick Pi and meet all D at that Pi or pick Qi and sell at D-1(Q1+Q2)

Production-to-Order (PTO) PQ Duopoly

Each firm simultaneously sets Pi and Qimax

Then passively takes orders qi up to Qimax

Produces Qi = qi, so never overproductionCould be underproduction in mixed equilibrium

If Pi < Pj, qi = min(Qi

max, D(Pi)) qj = min(Qj

max, max(D(Pj)-qi, 0))

If Pi = Pj, assume (less compellingly) qi = min(Qi

max, D(P) Qimax / (Qi

max + Qjmax)

Then i = Pi qi – Ci(qi).

Computational example of PTO Duopoly with a = 1/3

Primary strategy of smaller firm now lowest price charged by either, not highest price as in Gertner PQ Duopoly.(There are also pure NE with P = 1, but these are dominated.)

PTO Duopoly with a = ½:

Multiple pure Nash Equilibria arise, with P1 = P2 = 1 up to P1 = P2 = 1.065 (with e = 1)

High price equilibrium is “Superlative Nash Eq” ie the optimal one for both firms, so would be chosen

I conjecture this pattern is valid in general: Mixed strategies if a < ½ with smaller asymmetrically favoring lowest price Converges on pure strategies as a ½ Lower avg. P than Gertner

Unsolved issues: 1. Solve Gertner differential eq’n for simple cases:

Linear MC, affine D Qi|Pi affine?Constant elasticity D with Cobb-Douglas or CES MVC (w/ 2nd factor fixed)Or at least quick, accurate algorithm

2. FOC for asymmetric MCsThen solve for simple cases

3. Find FOC for PTO duopolyThen solve for simple cases

4. Solve PQ-Stackelberg modelApproximates Gertner?

Application: (McC 1993) Costs vs benefits of antitrust with increasing returns to scale, sunk factor, duopoly alternative

Q: What is a duopolist’s favorite music genre?

Q: What is a duopolist’s favorite music genre?

A: Doo-wop!

Stackelberg ModelsPseudo-Nash arguments invalid

Q-Stackelberg – 3-stage sequential game valid“Leader” chooses QL, then“Follower” chooses QF, then P’s determined by Kreps-Scheinkman game

P-Stackelberg (McC 1993)Leader chooses PL, QL, while Follower chooses PF, QF as a fn of PL only. inconsistent perception of Leader’s strategy space

PQ-Stackelberg (I haven’t seen this proposed or implemented)Leader chooses PL, QL, then Follower chooses PF, QF as a fn of both PL, QL.However, Leader can induce Follower to be P- or Q-taker -- High QL and/or PL – F acts as P-taker (as in Fig. 4) Low QL and/or PL – F acts as Q-taker (as in Fig. 6) At boundary, F is indifferent (as in Fig. 7), so easy to find. So outcome conceivably Q- (or P-) Stackelberg!?

But might induced zone changes alter outcome? Does Leader always choose same zone?

Do both firms agree on who the “Leader” should be?? If so, how?