The surprisingly hilarious show \"BattleBots\" showed remote control robots in m

Question

The surprisingly hilarious show "BattleBots" showed remote control robots in mortal combat. (It used to be on Comedy Central, is still available at battlebots.com, and was resurrected in the 2015 summer on ABC.) The show reminded me of a more general phenomenon. Inventors typically do not capture the full benefits of their creations (e.g., a new robotic design inspires an engineer to approach a design differently, etc.). Suppose that the market for killer robots shares this characteristic: Society benefits whenever another robot is provided. This benefit, however, increases as the number of provided robots increases, as the number of potential inspirations build upon each other. Consumers' willingness to pay for these robots is P = 50 - 3Q, while the marginal cost of building them is constant at MC = 20. (Prices and costs are in $1000s, while quantities are unsealed.) In this particular case, the marginal benefit that is external to buyers and sellers is EMB = Q: the first robot provides $1000 in external benefits, the second $2000, etc. Find the efficient quantity of killer robots

Explanation / Answer

Private demand curve: P=50-3Q

Externality = Q

Social demand curve = Private demand curve + Externality = 50-3Q+Q = 50-2Q

Thus, Social Benefit = d(Social demand curve)/dQ = 50-4Q

SMC = 20

Thus, at the efficient point, SMC = SB

That is,

50-4Q = 20

Q = 7.5

Thus, the efficient quantity of robots = 7.5

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