# duopoly - DUOPOLY STATIC COMPETITION HOMOGENEOUS GOODS Lets...

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D UOPOLY S TATIC C OMPETITION : H OMOGENEOUS G OODS Lets find the Nash equilibrium of a game where firms choose prices. The game has 2 players, who simultaneously chose prices, under the following assumptions. Assumptions: i. Homogenous goods ii. static competition (i.e., it is a one-shot game) iii. firms choose prices iv. marginal cost=c and no FC (which implies CRT). The first step to study this game is to figure out how does the demand function, faced by each firm, look like. Denote by D 1 (P 1 ,P 2 ) the quantity demanded from firm 1 when they charge price P 1 and their rival, firm 2 charges P 2 . How does D 1 (P 1 ,P 2 ) look like? These 2 firms are selling homogeneous or identical goods. Homogenous products are indistinguishable from each other by the consumer. Buyers simply choose the cheapest brand (an example are IBM clones, vegetables) since they are perceived identical they do not care which one they purchase. To simplify the model assume the market demand is: Q(P)=10- P. Then: D 1 (P 1 ,P 2 ) = { If firm 1 offers the product cheaper than firm 2, firm 1 sells to all buyers willing to buy at that price. If both firms charge the same price, they split sales. If firm 1 charges a higher price than firm 2 then no customer buys form firms 1 (notice, this makes sense since they offer identical goods). What about profits? Profits are the markup of each firm, ( P 1 – c ), times the quantity sold by each firm, D 1 (P 1 ,P 2 ), namely the quantity firm 1 sells when it charges price P 1 and firm 2 charges P 2 . π (P 1 ,P 2 ) = ( P 1 – c ) D 1 (P 1 ,P 2 ) Observe the strategic interdependence among firms by looking at their profit functions. The optimal price for firm1 depends on the price charged by firm 2. Find the Nash equilibrium. To find an equilibrium just assume a pair prices P 1 and P 2 are a candidate for equilibrium and try to rule out all cases which one of the players would have an incentive to deviate. 10 - P 1 if P1<P2 (10 - P 1 ) / 2 if P1=P2 0 if P2<P1

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Case 1: p i <c (namely, either price is lower than marginal cost). In such case, one of the firms would sell D(p i )=10-p i units and make a loss of (c-p i )\$ per unit. Can this be an equilibrium behavior? To rule it out find a profitable deviation. For example, the firm making losses would increase the price to avoid selling at a loss. Case 2: p 1 >p 2 >c. Firm 1 gets no profits, but they could get positive profits if they charged p 2 (or less), since there is a profitable deviation no pair of prices such p 1 >p 2 >c can be part of an equilibrium. Case 3: p 1 =p 2 >c. Both sell (10-p 1 )/2. However, by charging one cent less, they lose \$(10-p)/200 but gain the other half of the market: \$p 1 (10-p 1 )/2 hence such pair of prices cannot form an equilibrium. Case 4: p
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duopoly - DUOPOLY STATIC COMPETITION HOMOGENEOUS GOODS Lets...

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