4. CONTINUOUS VARIABLES AND ECONOMIC APPLICATIONS

Transcription

1 STATIC GAMES 4. CONTINUOUS VARIABLES AND ECONOMIC APPLICATIONS Universidad Carlos III de Madrid

2 CONTINUOUS VARIABLES In many games, ure strategies that layers can choose are not only, 3 or any other finite number, sometimes there are infinitely many. Consider two oligoolistic firms must decide how much to sell. The strategy sace is the interval [0,q i ], where q i is the maximum caacity of Firm i. A mixed strategy is another examle of continuous variable We will solve these games by finding the best-resonse function or reaction function of each layer to the ossible actions of his rivals.

3 CALCULATION OF THE BEST RESPONSE FUNCTION n Two cases: q Differentiable ayoff function (e.g., quantity cometition roblems, voluntary contributions to ublic goods.) Best resonse function arises from the first order condition of maximization (if the objective function is concave.) For non-concave functions, we need to consider higher order conditions. q Linear or non-differentiable ayoff function (e.g.: allocation and distribution roblems.)

4 CONTINUOUS VARIABLES n In most economic alications, layers have continuous strategies. n We will study: q Cometition in Quantities or Price. q Public Good Contribution.

5 Quantity cometition (Cournot) Simultaneous cometition in quantities Homogeneous roduct There is a market clearing rice Direct demand: Q = D() ; D () < 0 Inverse demand: P = (Q) Costs: C (q ),...,C n (q n ) Objective: max i = max (Q) q i C i (q i ) q i >0

6 Cournot duooly n A good is roduced by two firms: y. They roduce quantities q and q, resectively. Each firm chooses its quantity without knowing the rival s decision. n The market rice is P(Q)=a-Q, where a is a constant and Q=q +q. n The cost for Firm i when it roduces q i is C i (q i )=c i q i.

7 The game with identical costs The normal form game is: Players: {Firm, Firm } Strategies: S =[0, + ), S =[0, + ) Payoffs: u (q, q ) = q (a-(q +q )-c) u (q, q ) = q (a-(q +q )-c)

8 n A Nash equilibrium The Equilibrium q A air of quantities (q *, q *) such that q * is Firm s best rely against q *, and q * is Firm s best rely against q * This means that q * solves Max u (q, q *) = q (a-(q +q *)-c) subject to 0 q + q * solves Max u (q *, q ) = q (a-(q *+q )-c) subject to 0 q +

9 Solution Finding the NE Firm s bestrely Max u (q *, q ) = q (a-(q +q )-c) s.t 0 q + FOC: a - q q c = 0 q = max{0, (a q c)/} Firm s bestrely q = max{0, (a q c)/}

10 Solution Assume q i * a-c, (later on we will check that thiscondition holds) The air (q *, q *) is a NE if q * = (a q * c)/ q * = (a q * c)/ Solving the system we have q * = q * = (a c)/3 (smallerthan a-c, as assumed)

11 Solution n Usingthe best rely funcions: Firm BR: BR (q ) = (a q c)/ if q < a c = 0, otherwise Firm s BR BR (q ) = (a q c)/ if q < a c = 0, otherwise a c q Nash equilibrium (a c)/ (a c)/ a c q

12 Cournot duooly: Asymmetric costs Linear demand: =a-q, with Q = q + q Constant marginal cost : c i < a, i=, q Max (a-q i -q j -c i ) q i FOC: Firm i s best rely function i=, R R R i (q j ) = q i = max{0,(a-q j -c i )/} q Notice that R i decreases with q j : quantities are strategic substitutes.

13 Solving the system for q y q, and assuming non negative solutions we get q* i = (a- c i + c j )/3 Firm i s roduction is decreasing on it s own costs and increasing on the rival s. The effect of own costs are greater than that of the rival s. More efficient firms have greater market shares. Still, the market is inefficient.

14 General results in the Cournot model. Price above marginal cost. There are incentives to reduce roduction if cometitors roduce at marginal cost.. Price inferior to monoolistic Price. Incentives to increase roduction if cometitors roduce the monooly quantity. 3. If the number of firms increases, the rice decreases. The rice goes to the marginal cost as the number of firms goes to infinity.

15 Price cometition. Bertrand model The model. Homogeneous good.. Simultaneous cometition in rices. 3. Consumers buy from the firm offering the lowest Price. 4. No caacity restrictions, either firm can roduce any quantity at marginal cost c. ì0 q i = íd( i )/ î D( i ) if i > j if i = j if i < j

16 Firm i s demand function given P j Notice the discontinuity i j D D( j )/ D( j ) q i

17 Bertrand game The Normal form Players: {Firm, Firm } Strategies: S =[0, + ), S =[0, + ) Payoff functions: ï î ï í ì > = - - < - - = ï î ï í ì > = - - < - - = if 0 if ) / )( ( if ) )( ( ), ( if 0 if ) / )( ( if ) )( ( ), ( a c a c u a c a c u

18 The reaction funtion $ & ( ) = % & '& $ & ( ) = % & '& c ε monooly c ε monooly si c if monooly > c if > monoolio si c if monooly > c if > monooly Notice that if j c the best rely for i is any rice suerior or equal to j. For simlicity we consider that, when indifferent among different rices, and if one of them is c, then the firm chooses =c.

19 () < c, < 0 Þ Increase the rice: = 0 () = > c Þ = - e Bertrand model (3) > > c, = 0 Þ = - e, > 0 (4) > = c, = = 0 Þ = - e, > 0 Unique Nash Equilibrium: = = c, = = 0 Þ Perfect cometition with firms!

20 n Bertrand s aradox: Bertrand s aradox q Two firms is enough to find erfect cometition. n Bertrand s aradox arises because of the discontinuity of the demand, which imlies the discontinuity of rofits. q The firm selling at the lowest rice serves the whole market. n How to escae from the Bertrand s aradox? q Diferentiated roducts. q Caatity choice and, then, rice.

21 Price cometition with differentiated roducts n Two firms roduce differentiated roducts. Both choose rices to maximize rofits. n Let be the rice of Firm and the rice of Firm. Given these rices, Firms demands are given by q = a - + b q = a - + b. n Both firms have constant marginal costs equal to c.

22 Reaction functions n Firm rofits maximization roblem is max ( c)(a + b ) f.o.c. : a + b + c = 0 BR ( ) = a + c + b n We do the same thing for Firm. n The NE is the calculated solving the system formed by the two reaction functions (best rely) taking into account that rices must be higher than c. n See that as rival s rice increase, the best resonse is to also increase own rice: rices are strategic comlements.

23 Voluntary contributions to ublic goods n n n Two consumers, and, must decide how much to contribute to a ublic good. Denote by c and c the resective contributions. Consumer has a net wealth of w whereas consumer has w. The total quantity of ublic goods is the sum of the contributions. Payoffs: u (c, c ) = (c + c )(w c ) u (c, c ) = (c + c )(w c )

24 Comuting the NE n Solve consumers maximization roblems: Max u (c, c ) = (c + c )(w c ) s.t. 0 c w FOC: c = w c Similarly for Consumer : FOC: c = w c

25 Comuting the NE The FOC constitute consumers best rely functions. Solving the system we find: c = w w 3 c = w w 3 One can show that the solution is not otimal.