Question 1 (50 pts) Consider a rational addiction model with flow utility u(e₂, h. y) where e is consumption of the additive good, h is health, and y is consumption of the non-addictive good. The flow utility is increasing in all three inputs. The present value of lifetime income is A, the discount factor is 3, and the prices of the addictive and non-addictive goods are pe and 1, respectively. Health depends on prior consumption of the addictive good as follows: h=h(₁-2) a. (10 pts) What does the flow utility need to satisfy for the tolerance property to be satisfied? Explain your answer. b. (10 pts) What does the flow utility need to satisfy for the reinforcement property to be satisfied? Assume tolerance is satisfied. Explain your answer./nc. (8 pts) Write the setup of the maximization problem. d. (5 pts) Write the Lagrangian of the problem./ne. (17 pts) Compute the first order condition with respect to the addictive good at t. Assume tolerance is satisfied. Explain your FOC.

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