Nash Equilibrium and Duopoly Theory


 Dominic Haynes
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1 ash Equilibrium Economics A ection otes GI: David Albouy ash Equilibrium and Duopoly Theory Considerthecasewherethecasewith =firms, indexed by i =,. Most of what we consider here is generalizable for larger (general oligopoly) but working with firms makes things much easier. Let firm s profit depend its own action a ("action" is defined very broadly here) as well as firm s action, so we can write π (a,a ), and similarly for firm π (a,a ). The general ideas here would also be applicable if we used utility, e.g. U (a,a ),ratherthanprofits.. Definition A set of actions a,a constitutes a ash equilibrium iff π a,a π a,a for all a,and π a,a π a,a for all a In other words a set of actions is a ash equilibrium if each firm cannot do better for itself playing its ash equilibrium action given other firms play their ash equilibrium action.. olving for ash Equilibria To find the ash equilibrium, we should consider each firm s profit maximization problem where each firm takes each other s action as given parametrically, but which are resolved simultaneously: max π (a,a ) and max π (a,a ) a a. Oftentimes finding a ash involves checking all the possible combinations (a,a ) and asking yourself "is this a ash equilibrium?" ometimes it is possible to eliminate dominated actions iteratively (see a book on game theory) to narrow the cases that need to be checked. However, assuming profit functions are continuously differentiable, concave, and a and a are both positive, we can take first order conditions. The separate first order conditions for each firm is just π a,a π a = and,a = (ash FOC) a a which is a system of equations in unknowns a,a, and so usually a little algebra will yield the solution. Additionally, the second order conditions imply that the profit functions should be concave in each firm s own action, namely π a,a π a = and,a = (ash OC) a The second order conditions say that marginal profits should slope downwards with respect to the firm s own action..3 Best Response Functions By the implicit function theorem the FOC for firm alone defines what it will play given a, i.e. firm s best response function (or reaction curve) a = r (a ). By solving the FOC for firm for a in terms of a we can find the best response function as the FOC a π (r (a ),a ) a =
2 holds for all a (not just the ash). Geometrically the bestresponsecutvescanbeseenasfinding a point in the (a,a ) where the firm s isoprofit curve, which has slope π / a π / a is tangent to the horizontal line defined by a, which has slope zero as firm takes firm s action as fixed as if it were a constraint. This can be interpreted as firm solving for its maximum profit subjectto the constraint that a is a fixed number. A similar best response function r (a ) can be defined for firm, where π (r (a ),a ) = a The best response curve for firm lies where the isoprofit curveforfirm is vertical as it is tangent to the line for a fixed value of a. A ash equilibrium can be seen as where each action is a best response to the other firm s action a = r a and a = r a This is where the best response curves cross in a graph with a on one axis and a on the other. ubstituting in one best response function into the other gives a = r r a which states that firm s action a is its best response to firm s best response to firm s action. Thus, the ash equilibrium concept produces a consistent set of responses. Firm cannot do better given that firmisusingitsbestresponsetofirm.. trategic Complements and ubstitutes It is useful to know how one firm will react if the other firm changes its action. Cases where a greater action dr by elicits more of a response by, i.e. da >, identifies a situation where a and a are called strategic complements. The alternate case where dr da <, iswherea and a are called strategic substitutes. Recall that the equation π(r(a),a) a =holds for all a and therefore we can differentiate this expression with respect to a to get d π (r (a ),a ) = π dr da a a + π = da a a and so solving for the slope of the best response curve µ dr π π = da a a a The sign of this expression depends on the sign of the second derivatives of the profit function. The sign of π is negative because of (ash OC), i.e. the marginal profit function is downward sloping in a a. Therefore we can conclude that the sign of dr da will have the same sign as the cross partial derivative π a a, i.e. µ µ dr π sign = sign da a a o if firm s action a increases the marginal profit offirm s own action π a,thenfirm will increase its action a. The following example should help make this clearer. The isoprofit curveisdefined by the equation π (a,a )=k where k is a constant. Treating a as a function of a and differentiating totally π (a,a (a )) = k with respect to a and solving for the implicit derivative gives da = π / a da π / a
3 Example Cournot Competition In this case firms compete in quantities q and q (which are a and a above). Take the case where inverse demand is given by p = q q and costs are zero. o π (q,q )=( q q ) q and π (q,q )=( q q ) q. The first order conditions for firms and are π / q = q q =and π / q = q q =. olving for the reaction functions we get r (q )= q / and r (q )= q /. Below are two graphs which give a graphical derivation of the best response function for firm. The first gives the profit functions and marginal profit functions for firm givenfirm produces zero q =, the cartel quantity q C =3(see below), or the ash quantity q =. As q increases the marginal profit functionforfirm shifts down, as π q q = <. This shifts firm s optimal response q to the left as marginal profit is downward sloping, i.e. as π =, implyingthat q dr /da = / so quantities are strategic substitutes in Cournot competition. The second graph shows how the best response curve of firm is given by the points where the isoprofit curvesoffirm are perfectly horizontal, i.e. they are tangent to the constraint that a is a fixed number. ote that profits are higher on isoprofit curves which are lower and that the monopoly case corresponds to where the isoprofit curveis tangent to the q axis where q =. profit 3 q 3  q q Profit andmarginalprofit offirm Deriving the Best Response Function The graphical derivation of the best response curve for firm is similar. The two curves intersect at Cournotash equilibrium quantities q = q =,asshownbelowontheleft,andeachfirm makes a profit of π (, ) = π (, ) =. ow say that firm were to suddenly experience higher costs c =3rather than zero. A little work shows that this would shift firm s reaction curve inwards to r (a )=9/ r /, causing firm s output to fall to, and because quantities are strategic substitutes, firm s output would rise to 7. q q Best Response Curves of Both Firms Example Hotelling Competition Assume that firms sell goods in a partly differentiated market with their demands decreasing in their own price but increasing in their competitors price,withspecific functional This specific modelwouldbejustified by assuming the two competitors are located on opposite ends of a road with a length 3
4 forms given by q = p + p and q = p + p. Assuming costs are zero, profits for each firm as afunctionofpricesisthengivenbyπ (p,p )=( p + p ) p and π (p,p )=( p + p ) p. The FOC for firm is π / p = p + p =which yields the best response function r (p )=+p /. imilarly the best response function for firm is given by r (p ) = + p /. The two best response functions are increasing in each others prices dr /dp = dr /dp =/, exhibiting that prices are strategic complements in the Hotelling model. The best response curves intersect at the equilibrium prices p = p = as shown below, leading to profits of π (, ) = π (, ) =. p Hotelling Best Responses p Joint Profit Maximization Because the firms profits depend directly on actions they do not control, the firms could jointly raise their profits by coordinating their actions.. First Order Conditions ay the firms were to collude to maximize the sum of their profits, solving the following problem In this case there are two first order conditions max π (a,a )+π (a,a ) a,a π (a a,a )+ π (a a,a )= π (a a,a )+ π (a a,a )= (Joint FOC). Externalities The latter derivative in each of these FOC, i.e. π / a and π / a represents the externality that each firm has on the other through its action. If the action of firm increases the profits of firm, π / a >, then firm s action is said to have a positive externality on firm. If instead the action of firm decreases the profits of firm, π / a <, then firm sactionissaidtohaveanegative externality of kilometers with consumers uniformly distributed along the road of one per kilometer, each demanding one unit of the good, and incurring a cost of cents per kilometer in travel costs (both ways). In this case where all consumers buy, then the amount sold by firm will be determined by the consumer at distance q who will be indifferent between butying at firm and firm. Thus q is determined by the equation p + q = p + q which yields the demand function for q and q = q.
5 of firm. In the ash equilibrium, firms ignore the externalities (look at the ash FOC) yielding a lower joint profit maximization then would be obtained than if they were taken into account. In the case where firms exert positive externalities on each other they would do better by increasing their level of a, while if they exert negative externalities on each other, they would collectively do better by decreasing their level of a. 3.3 Tangency ote that the (Joint FOC) can be rewritten and combined to show that π / a = π / a = π / a = π / a π / a π / a π / a π / a which implies that the isoprofit curvesaretangentatthejointprofit maximum. The isoprofit curvesare not tangent, but perpendicular, at the ash equilibrium because where the best response curves cross firm s isoprofit curve is horizontal while firm s isoprofit curve is vertical. This means that there are points on the graph inside both isoprofit curves (below and to the left) where both firms could be better off. However without coordination, these superior points are not an equilibrium as they are off the firms best response curves. If firm chooses its best response it can increase its profit although its competitor firm will lose more profits than firm will gain. Example 3 Cournot Competition Just differentiating the profit functions we can see that if a firm increases its output then it exerts negative externalities on the other as π / q = q < and π / q = q < ay both firms coordinate to maximize the sum of their profits, which is given by π (q,q )+π (q,q )=( q q ) q +( q q ) q =( q q )(q + q ) =(q + q ) (q + q ) The FOC in this case will be identical for both q and q (q + q )= Therefore we can conclude that q + q =which is the monopoly quantity q M. Howeverthisdoesnotsay how production should be split between the two firms (not surprising since marginal costs are always the same, zero, at both firms). A typical assumption is to assume that the firms act as a cartel and split production (and profits) equally, so that each firm produces q C = qc =3,andearnsprofits π (3, 3) = π (3, 3) =. The graph below shows a pair of isoprofit curves for each firms. At the cartel solution (3, 3) two of the curves are tangent; at the ash equilibrium (, ) two of the curves are perpendicular. q 3 3 q However firm s best response to q =3is not to produce 3 but to produce r (3) = 9/. This gives a higher profit tofirm of π (9/, 3) = /,againof9/, althoughfirm would then earn a profit of π (9/, 3) = 7/, incurring a loss of 9/ which is greater than firm s gain. Firm would not cooperate with firm if it believed that firm would cheat on the cartel arrangement. 3 Of course we are leaving out the effects on consumers. In some cases (like Cournot competition) the joint profit maximizing solution makes firms off but hurts consumers more than it helps firms, as illustrated in the dead weight loss created.
6 Example Hotelling Competition The joint profits of the two firms here are given by π (p,p )+π (p,p )=( p + p ) p +( p + p ) p =(p + p ) p +p p p =(p + p ) (p p ) In this case profitscanbemadeinfinite by choosing equal prices for both p = p which makes the second term zero, and then setting the prices arbitrarily high to increase the first term without bound. Of course this is not a realistic conclusion, it comes from the fact that we used a very simple model of Hotelling competition. 3 equential Equilibrium Thepreviousanalysisassumedthatbothfirms moved simultaneously. As we will show below, things change considerably if one firm moves before the other in a sequence of two rounds. Assume now that firm moves before firm, and that it is able to anticipate firm s reaction. These types of situations are often known as tackelberg Games. 3. Backward Induction The optimization technique used to solve for how the firms will behave is known as backward induction. This means that we will first solve how firm will act in the second round in response to how firm acts in the first round. econd we move back in time to solve for how firm will act in the first round, using our solution for the second round to model firm s anticipation of how firm will react. The technique of backward induction is a fundamental technique in determining how to solve for optimal strategies in a number of game theoretic situations, from tictoe and chess, to more serious situations. 3. Round Two Firm s problem is identical to that found in the ash equilibrium. profit overa, producing a best response function r (a ). It takes a as given and maximizes its 3.3 Round One max a π (a,a ) \The difference lies in firm s problem as it no longer takes a as fixed number but as a best response to what it does, a = r (a ). Thus instead of taking a as a fixed number as its constraint (a horizontal line on the (a,a ) graph) it takes a = r (a ) as its constraint (which is the best response curve on the graph). ubstituting in the constraint into firm s objective function we have max π (a,r (a )) a Differentiating totally with respect to a yields the first order condition π a a,r a π + a a,r a dr = (equential FOC) da otice that the second term in this FOC does not appear in the (ash FOC). This equation can be solved for a provided we know the reaction curve of firm (recall dr /da = π / a π / a a ), The FOC in this case are given by (p p )=and + (p p )=which together imply p p =and p p = which is impossible. The FOC do not help in finding a maximum because no maximum existes (infiniti is not a real maximum). The problem with this model lies in our assumption that all of the people along the road in the Hotelling model buy regardless of price  Hotelling s original model is in fact much richer and allows people to not buy at all if the price is too high. a
7 and a can be found just by plugging in a into firm s reaction function, i.e. a = r a. the FOC we have dr = π / a da π / a which means that the reaction curve must be tangent to firm s isoprofit curve. Rearranging 3. equential versus imultaneous Actions An interesting question is whether firm will perform more or less of a than it does when it moves simultaneously with firm. This depends ultimately on the sign of the second term in (equential FOC) as it implies π a a,r a π = a a,r a dr a da Recall in the simultaneous ash solution π a =at a. Because π is concave, we know π / a is decreasing and so a will be greater than α if π / a in the above equation is negative. Conversely a will be less than α if π / a in the equation above is positive. On the righthand side of the equation we can see that the π / a is related to two terms we have already encountered. The first term includes π / a which is related to whether firm s action has a positive or negative externality. The second term involves dr /da whose sign depends on whether a and a are strategic substitutes or complements. Becausetherearefourdifferent cases we can make a little chart to separate the various cases (work through them to see if you understand). Positive Externality π trategic Complement π a a > a a > >a a >a egative Externality a π a < <a a <a trategic ubstitute π a a < a <a a >a a >a a <a In the chart we were also able to determine how firm s action would be different: when a and a are strategic complements a will be higher if a is higher, while if a and a are strategic substitutes, a will be lower if a is higher (and viceversa). Example Cournot Competition The reaction curve for firm is r (q )= q /. firm s profit function gives Plugging this into π (q,r (q )) = ( q +q /) q =( q /) q Taking the FOC we get q = q =and q = r () = 3. Profits are given by π (, 3) = and π (, 3) = 9 so here firm benefits from a "first mover advantage" as it gets more profits than firm. otice that q => =q and q =3< =q as q and q are strategic substitutes and exert negative externalities on the other firm, corresponding to the case in the lower righthand corner of the table above. The tangency of the isoprofit curveoffirm and the best response curve of firm mentioned above is shown It is just a coincidence that the tackelberg quantity is the same as the monopoly quantity q = qm. 7
8 in the graph below. q equential Cournot Equilibrium q Example Hotelling Competition Pluggin in the reaction function of firm, r (p )=+p / into firm s profit functionweget π (p,r (p )) = ( p ++p /) p =( p /) p TakingFOCthistimeweget p = p = and p = r () =. Profits are given by π (, ) = and π (, ) =. With Hotelling competition firm gets higher profits as there is a "second mover advantage" unlike the Cournot model. Both firm and firm s prices are higher than in the simultaneous ash equilibrium case as higher prices exert positive externalities on competitors and as prices are strategic complements (the upper righthand side of the table above). Below is a graph showing the tangency condition  note that profits are higher above the isoprofit curve (with higher p ) than below it. p 3 3 equential Hotelling Equilibrium p
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