Consider a health insurance market. The market has many insurers so that each insurance company offers insurance at the fair insurance premium. Each consumer has utility function, U(X) = \\j X and has an initial wealth of $200. Consumers have access to a (magical) gym that reduces their probability of needing to go to the doctor from 90% to 10%. A gym membership costs $6. A trip to the doctor costs $120. Suppose insurance companies charge the fair insurance premium which assumes all consumers go to the gym. This fair insurance premium is $ 12 v and the insurance company makes a prot of $ v per consumer. Suppose insurance companies continue to charge the fair insurance premium that assumes all consumers go to the gym, but now charges a copay of $15 for a doctor's visit. The insurance company now makes a profit of $ v per consumer. By implementing a copay, the insurance company v the v