Question

Problem 3: A Consumption Externality (34p) Consider two drivers, indexed by A and B Driver A chooses a driving speed, denoted 4, and similarly, driver B chooses a driving speed, denoted 2. Each driver likes more speed, but more speed also increases the risk that they have an accident. The risk of an accident is p + 3. If an accident occurs, each suffer a loss of c. The expected) utility of driver i 44 = log (24)-(24+2) and similarly, the utility for driver B is P = log (2")-0(24+2B) a) What speed will each driver choose in a Nash equilibrium? b) What is the socially optimal speed of each driver? c) Suppose that each driver is fined an amount t if an accident occurs (adding to the cost c), what level of t induces each driver to choose the socially optimal speed?

Answer 1

OXB 8 a) 4A = log xA - CCXA+ XB) so at&am: auA ) t cao so IXA* = 10 & for B, Quß, a to -C =0, xp*=lle) Abe 6.) at 8 ocially optimal level, Max UA tuB o us = log xA + log x B - 26 CXA+XB) 80 Burn > xth - 20 =0, 24% - to-dc=0 80 fxr Etme | XBS - c) UA' = log xA - (C++) (XA + XB) - to then at Eqm a UA! = 1.- (ett) =0 then XA = X AS = + + gf Tt C (8imilarly for B) To so t= c) Ane 2XA X Fts aXA2 X