Consider two firms who are engaged in a Research and Development (R&D) "con-test". Both firms simultaneously expend resources to try to win the contest (whichmay mean developing a superior product or developing a product before the com-petitor). If the two firms expend bị and b2, respectively, on R&D, the probabilitythat firm 1 wins the contest isif b1 = b2 =0BE otherwiseP1(b1, b2) =where r is some exogenous constant, r E (0, 0). If firm 1 wins the contest, it willsubsequently earn revenue of 1 (not including the cost of R&D, b1). If firm 1loses the contest, it will earn zero revenue, and thus lose bj in total. Hence, firml's expected profit is n'(b1,b2) = p1(b1,b2) - b1. Everything is symmetric forfirm 2.i. How does p1(b1,0) depend on b1? Is it an equilibrium for both firms tospend nothing on R&D (b1 = b2 = 0)? Prove and explain your answer. Forwhich values of r is n' (b,b2) concave in b1 when b2 > 0?ii. Consider the possibility of a symmetric pure-strategy equilibrium. Assum-ing such an equilibrium exists, derive it and prove it is unique. Is the equi-librium efficient? Under what conditions onr do a symmetric pure-strategyequilibrium exists? Prove and explain your answers.iii. Assume r = 1 and that firm 2 observes firm 1's expenditures before it hasto make its own investment. Derive the subgame perfect Nash equilibrium.How does it differ from the equilibrium in the simultaneous-move game? Isthere a first-mover advantage in this game? Does firm 2 prefer the sequentialgame or the simultaneous game?iv. Assume r = 2. Does firm 2 have a unique best response to all bi E (0, 00)?Is there an equilibrium of the sequential game in which firm 1 invests morethan in the simultaneous-move game? If firm 1 invests less than insimultaneous game, does firm 2 invest more or less than firm 1? Describean equilibrium of the sequantial game and compare it to the equilibriumof the simultanous game. Does firm 1 prefer the sequential game or thesimultaneous game? How about firm 2? Prove and explain your answers.

When the equilibrium state of the perfect competition is being considered, it is not applicable in…

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equilibrium

ENGR.ECONOMIC ANALYSIS

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ISBN:9780190931919

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Chapter1: Making Economics Decisions

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Question

Consider two firms who are engaged in a Research and Development (R&D) "con-
test". Both firms simultaneously expend resources to try to win the contest (which
may mean developing a superior product or developing a product before the com-
petitor). If the two firms expend bị and b2, respectively, on R&D, the probability
that firm 1 wins the contest is
if b1 = b2 =0
BE otherwise
P1(b1, b2) =
where r is some exogenous constant, r E (0, 0). If firm 1 wins the contest, it will
subsequently earn revenue of 1 (not including the cost of R&D, b1). If firm 1
loses the contest, it will earn zero revenue, and thus lose bj in total. Hence, firm
l's expected profit is n'(b1,b2) = p1(b1,b2) - b1. Everything is symmetric for
firm 2.
i. How does p1(b1,0) depend on b1? Is it an equilibrium for both firms to
spend nothing on R&D (b1 = b2 = 0)? Prove and explain your answer. For
which values of r is n' (b,b2) concave in b1 when b2 > 0?
ii. Consider the possibility of a symmetric pure-strategy equilibrium. Assum-
ing such an equilibrium exists, derive it and prove it is unique. Is the equi-
librium efficient? Under what conditions onr do a symmetric pure-strategy
equilibrium exists? Prove and explain your answers.
iii. Assume r = 1 and that firm 2 observes firm 1's expenditures before it has
to make its own investment. Derive the subgame perfect Nash equilibrium.
How does it differ from the equilibrium in the simultaneous-move game? Is
there a first-mover advantage in this game? Does firm 2 prefer the sequential
game or the simultaneous game?
iv. Assume r = 2. Does firm 2 have a unique best response to all bi E (0, 00)?
Is there an equilibrium of the sequential game in which firm 1 invests more
than in the simultaneous-move game? If firm 1 invests less than in
simultaneous game, does firm 2 invest more or less than firm 1? Describe
an equilibrium of the sequantial game and compare it to the equilibrium
of the simultanous game. Does firm 1 prefer the sequential game or the
simultaneous game? How about firm 2? Prove and explain your answers.

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