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Event DrivenDynamic Systems

Bujor PVLOIU

2010

University "Politehnica" of BucharestFaculty of Engineering in Foreign LanguagesMaster Studies in Business Administration



2Event Driven Dynamic Systems

Dynamic SystemsDynamic Systems

Discrete Event ExamplesDiscrete Event Examples Untimed DESUntimed DES

Markov ChainsMarkov Chains

Petri NetsPetri Nets Business Process ManagementBusiness Process Management

MaxMax--Plus AlgebraPlus Algebra

Game TheoryGame Theory -- NashNash Game TheoryGame Theory -- OligopolyOligopoly




Measures of MarketMeasures of Market

StructureStructure HerfindahlHerfindahl--Hirschman Index:Hirschman Index:

summed squares of all firm marketsummed squares of all firm market

shares in the industryshares in the industrySSii = market share (in percent) of the= market share (in percent) of thei i thth firmfirm

HHI =HHI = (S(Sii ))22

R ange: 0 (perfect competition) toR ange: 0 (perfect competition) to10,000 (monopoly)10,000 (monopoly)




Measured Industry Concentrationin Manufacturing

Second and third columns: Percentage in value added in theindustry

Last column: Herfindahl index from the 50 largest firms

Data based on the 2002 Economic Census.




Capacity + Bertrand = CournotCapacity + Bertrand = Cournot

Bertrand (1883) criticized Cournot¶s modelBertrand (1883) criticized Cournot¶s model(1838) on the grounds that firms compete by(1838) on the grounds that firms compete bysetting prices and not by setting quantities.setting prices and not by setting quantities.

Kreps and Scheinkman (1983) defendedKreps and Scheinkman (1983) defendedCournot¶s model. They developed a twoCournot¶s model. They developed a two--stagestagegame with capacitiesgame with capacities

and proved that capacities in a Nash equilibriumand proved that capacities in a Nash equilibriumare determined by Cournot¶s model.are determined by Cournot¶s model.

p1

p2

k 1

k 2

1

2




Cournot duopoly (simultaneousCournot duopoly (simultaneousquantity competition)quantity competition)

Stackelberg duopoly (sequentialStackelberg duopoly (sequential

quantity competition)quantity competition)

x2x1

x1

x2

1

2

1

2




Homogeneous DuopolyHomogeneous Duopoly Two firms (i=1,2) produce a homogenous

good.

Outputs: q1 and q2,

Q= q1+q2

Marginal costs:

c1 and c2

Inverse demand function:p(q)=a-bq Q

P

a

a/bq

p

0




Homogeneous DuopolyHomogeneous Duopoly

There are no fixed costs and the totalcost of producing the quantity qi is

cciiqqii

Profit function of firm 1:

11=p(Q)q=p(Q)q11--cc11qq11=(=(a-bQ)qq11--cc11qq11

=(=(a-b(q1+q2))qq11--cc11qq11===((=((a-c1)-b(q1+q2))qq11




CournotCournot--Nash equilibriumNash equilibrium

Profit functions:Profit functions:

11((qq11,q,q22),), 22((qq11,q,q22))

R eaction Functions:

R eaction Functions:qq11

R R (q(q22)=argmax)=argmaxq1q1 11((qq11,q,q22),),

qq22R R (q(q11)=argmax)=argmaxq2q2 22((qq11,q,q22),),

Nash equilibrium: (Nash equilibrium: (qq11

CC,q,q22

CC))

qq11R R (q(q22

CC)=q)=q11CC

qq22R R (q(q11

CC)=q)=q22CC




Cournot equilibriumCournot equilibrium(accommodation)(accommodation)

Profit function of firm 1Profit function of firm 1

11(q(q11,q,q22 )=(()=((aa--cc11)-b(q1+q2))qq11

R eaction function of firm 1R eaction function of firm 1

dd11(q(q11,q,q22 )/dq)/dq11=0=0

--2b2bq1+ ((+ ((aa--cc11)-bq2)=0

qq11R R (q(q22)=()=(aa--cc11)/2b)/2b-q2 /2 Analogous:Analogous:

qq22R R (q(q11)=()=(aa--cc22)/2b)/2b-q1 /2




Cournot equilibriumCournot equilibrium(accommodation)(accommodation)

Nash equilibriumNash equilibrium

qq11R R (q(q22

CC)=q)=q11CC ,, qq22

R R (q(q11CC)=q)=q22

CC

((aa--cc11)/2b)/2b-q2C /2= q1C

((aa--cc22)/2b)/2b-q1C /2=q2

C

««

q1C= ((aa--2c2c11+c+c22)/3b)/3b

q2C= ((aa--2c2c22+c+c11)/3b)/3b




b )c c (a

b

)c c (a

)c c (a p

C

C

C

9

2

9

23

2

12

2

2

21

1

21

!

!

!




Monopoly GameMonopoly Game

A firm produces a homogenous good.

Outputs: qM, Q= qM

Marginal costs: cM

Inverse demand function:p(q)=a-bq

There are no fixed costs and the total cost

of producing the quantity qM is ccMM

p(qM)=a-bqM




Monopoly GameMonopoly Game Profit of the firm:

=p(Q)q=p(Q)qMM--ccMMqqMM=(=(a-bqM)qqMM--ccMMqqMM

=(=(a-bqM-ccMM)qqMM

qM1=0; qM2= (a-ccMM)/)/b;

qM= (a-ccMM)/2)/2b

== (a-ccMM))22 /4 /4b

qM1 qM2qM

M




Graphical RepresentationGraphical Representation

Q1

Cournot-NashequilibriumC

1 qq

R

1qqR

Q

M q

C q

C q1M q1




Numerical ExampleNumerical Example

a = 130, b = 1, c1 = c2 = 10

q1C= ((aa--2c2c11+c+c22)/3b=40)/3b=40

1C

= ((aa--2c2c11+c+c22))22

/9b=1600 /9b=1600ppCC= (= (a+ca+c11+c+c22)/3=50)/3=50

q1M

= (a-cc1M1M

)/2)/2b=60

1M1M== (a-cc1M1M))22 /4 /4b=3600

pp1M1M= (= (a+ca+c11)/2=70)/2=70




Oligopoly versus MonopolyOligopoly versus Monopoly

Companies, when engaged in competition,end up producing less than what they wouldproduce otherwise.

However, the total quantity to the market q1+q2 would be higher than what a

monopolist would produce and sell, qM . As the number of firms in the market

increases, each firm¶s profitability

decreases. The firms begin to have smaller market

shares and the end-customers benefit moreand more from the tough competition.




Cartel GameCartel Game

If both produces half of quantity, the monopolyIf both produces half of quantity, the monopolyquantity, the profit is 3600/2=quantity, the profit is 3600/2=18001800, more than, more thanthe Cournot profit =the Cournot profit = 16001600. (Pareto domination,. (Pareto domination,like in the prisoner dilemma).like in the prisoner dilemma).

This can lead to tacit collusion to raise the pricesto the jointly optimal level and it is called cartel.

A cartel ³is a combination of producers of anyproduct joined together to control its productionsale, and price, so as to obtain a monopoly andrestrict competition in any particular industry orcommodity´




ExerciseExercise

Find the equilibrium in a CournotFind the equilibrium in a Cournotcompetition.competition.

Suppose that the demand function isSuppose that the demand function isgiven by:given by:

p(Q)=25p(Q)=25 -- QQ

and the costs per unit by:and the costs per unit by:cc11=3 and c=3 and c22=2.=2.




Bertrand gameBertrand game

The firms set prices rather than quantitiesas in Cournot¶s model.

Assume that two firms produce identicalgoods which are perfect substitutes fromthe consumers¶ perspective, i.e.,consumers buy from the producer whocharges the lowest price.

If the firms charge the same price, they

will split the market evenly. There are no fixed costs of production and

the marginal costs are constant at c1=c2=c.




Bertrand GameBertrand Game

Demand/quantity depends on price

qiB=qi(pi,p j), with i,j in {1,2}

Profit function of firm i for price piis:

iiBB=p=piiqi

B - ccii qiB=(p(pii--c)c)qi(pi,p j)

Where the demand is given by:Where the demand is given by:

q(pi), pi<p j

qi(pi, p j)= q(pi)/2, pi=p j

0, pi>p j±°

±¯®




Bertrand GameBertrand Game

This time the strategies available toeach firm are the different prices itmight charge, rather than thequantities it might produce.

The Bertrand paradox states that theunique equilibrium has the firms

charge the competitive price:p1

B=p2B=c

Why?




The Bertrand game suggests that whenthe firms compete on prices (and the costsare symmetric) we obtain a perfectly

competitive market even in a duopolysituation.

However, in real life, most customers donot only choose based on price, but also

based on other factors such as quality andconvenience of location.




Bertrand gameBertrand game

The firms set prices rather than quantitiesas in Cournot¶s model.

Assume that two firms produce identicalgoods which are perfect substitutes fromthe consumers¶ perspective, i.e.,consumers buy from the producer whocharges the lowest price.

If the firms charge the same price, they

will split the market evenly. There are no fixed costs of production and

the marginal costs are constant at c.




Stackelberg gameStackelberg game

In many business situations, firmschoose their actions sequentiallyrather than simultaneously.

To be more precise, one firm choosesits action after it observes anotherfirm¶s action and responds to it.

Let it be Firm 1.




Stackelberg gameStackelberg game

Firm 1 moves first and chooses herquantity to sell q1. We call Firm 1 the ³Stackelberg leader.´

Firm 2 observes q1 before choosinghis own output level q2. We call firm2 the ³Stackelberg follower.´

The total market demand isQ=q1+q1

Both firms seek to maximize theirprofits.




This game is known as Stackelberg game,and it is a dynamic game of complete andperfect information.

The game has two players and twostages: In stage 1, player 1¶s action set is [0,1), as

player 1 can produce any non-negativequantity, whereas player 2¶s only available

action is ³do nothing.´ In stage 2, player 2¶s action set is [0,1), and

player 1¶s only available action is ³do nothing.´




The Firm 2 will respond to theThe Firm 2 will respond to theactions of the Firm 1 like inactions of the Firm 1 like in CournotCournot

q2

R

= ((aa--cc22)/2b)/2b-q1

S

/2The profit of Firm 1 is

(backpropagating):

11 (q(q11,q,q22 )=(()=((aa--cc11)-b(q1+q2

R

))qq1111 (q(q11,q,q22 )=(()=((aa--cc11)-b(q1+ ((aa--cc22)/2b)/2b-

q1 /2))qq11




Prices, Quantities, ProfitsPrices, Quantities, Profits




Total potential surplus in themarket

Total consumersurplus is consumerswillingness to pay thatis not captured by thefirm. It is positive if the firm sells belowconsumers¶ willingness to pay.Deadweight loss isdemand that is notsatisfied by the firmand is positive if thefirm sells below themaximum marketdemand.The Total potentialsurplus is the totalpiece of the market




Simple Double Auction GameSimple Double Auction Game

Single buyerSingle buyer

Single sellerSingle seller

Seller valuation cSeller valuation c Buyer valuation p, p>cBuyer valuation p, p>c

Goods are traded through doubleGoods are traded through double

auction, using bids bc and bpauction, using bids bc and bp If bc>bp, no trade; otherwiseIf bc>bp, no trade; otherwise

p=(bc+bp)/2p=(bc+bp)/2





Complete information game, p and cComplete information game, p and cknownknown

Play the game for p=80 and c=50Play the game for p=80 and c=50 Play repeatedly the gamePlay repeatedly the game

Incomplete information game. YouIncomplete information game. Youknow just your own valuation andknow just your own valuation andthe result from the auctioneerthe result from the auctioneer





HOMEWOR KHOMEWOR K

Try to make an Excel/Matlab/C«.Try to make an Excel/Matlab/C«.Program that will act like anProgram that will act like anincomplete information agent (sellerincomplete information agent (selleror buyer)or buyer)

Discuss the strategies (comment theDiscuss the strategies (comment theactions)actions)

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