1 PATENT RACING-THE GAME THEORETIC APPrOaCh Assume two firms a and b who want to decide whether they should attempt to produce a new product with marginal cost c. The demand for the good is P=a-bQ and as for the r&d effort, the costs of sett a research lab are K and the probability that the lab will successfully the product is p If both firms successfully develop the product they will be a Cournot duopoly 4g, PC=93,9f=92=3), while if only one of the two develops the new product she will be rewarded with monopoly profits (M=(a2,PM=盟,QM=2B) The expected profit net of setup costs if only A establishes an R&d lab while b does not is E(B I R&DA=0, R&DB>0)=E(A I R&DA >0, R&DB=O) 1+(1-p)0-K=pM-K If both establish R&D labs, then E(TB I R&Da>0, R&Db >0)=E(TA R&da>0, R&zDB>0)= =p(1-p 4+n2(a 96--K=pMa p-K 9 No R&d r&d No R&d0,0 I R&d pM-K, 0 pM(1-p)-K,pM(1-p)-K 1.1 Possible Nash equilibrium outcomes 1. Neither firm establishes an R&D lab if pM (no R&D, no R&D)(if pM>K)and if 2. Only one firm establishes an R&d division in equilibrium b(R&D, no R&d)>(r&D, R&D) 0>pM(1-8)-K→K>pM(1-8) ining a and b gives M>K>PM(1-P) 3. Both firms will R&D in Nash equilibrium if pM(1-6p)> K. In this case b prefers to r&d irrespective of whether A r&zDs or not and the same applies for A. Hence the(r&D, R&D) is a dominant strategies equilibrium 1
1.2 Complications Note that(r&D, R&D) may be a Nash equilibrium, but it will not necessarily be a Pareto optimum from the firms' perspective if the sum of profits under this equilibrium is less than the sum of profits under a(r&D, no R&D)or(no R&D, R&D)combination. This will be the case if 2pM(1-。0)-2KpM(1 nd combining this with the fact that(R&D, R&D)is a Nash equilibrium oM(1-P)>K>pM(1 10 Hence the above condition gives us the case where both firms doing R&D is a Nash equilibrium which is not Pareto optimal from the firms' point of view, in the sense that firms would be jointly better off with only one of the two firms having an R&D Ia 3 Consumer surplus In the case of monopoly consumer surplus is equal to (a-)(2M phile in the Cournot case it is equal to (a-"2)(号) CSc=2 4M 8M The expected social surplus ignoring the R&d costs if only one firm, say a established an R&d lab is equal to E(TA R&DA>0, R&DB=0)+E(TB R&DA>0, R&DB=0)+PCS M+0+p-==pM The expected social surplus ignoring the r&d costs if both firms set up an R&D lab are E(TA R&DA>0, R&DB >0)+E(TB R&DA>0, R&DB>0) CSc +2p(1-p)SM
=2(4-)M+p29+p2g+201-)2=pM3-92) The second lab is socially desirable only if M(3 )-K-K>5M-K→K0.75(since then pM(1.5-9)K>pM(.5-111 for p>0.75 If both firms doing R&D is Pareto optimal from the firms'point of because KpM(1-p)it follows that double R&d will also be optimum from the societys point of view, but the reverse does not always necessarily hold since we may have that pM(15-4p)>K>pM(1-g