Home Assignment 8 Due November 7th, 23:00, online submission only Study Guide Watch how Nobel laureate Robert Shiller provides a narrative for investment (here from 01:24 to 28:11). Also read section “Optimality” from chapter 5 in Williamson. Home Assignment 1. [40 points] Recall consumption-leisure model from chapters 4-5: there are n people in the economy, and each person is endowed with 1 unit of time. Everyone can rest l amount of time and produce the rest of the time, 1 − l. Production increases with work effort, c = z (1 − l)α , where 0 < α < 1, z is an exogenous productivity, and all output is consumed. The preferences are described by log c + log l. (a) Find the “first best” allocation in this economy. (b) How the amount of leisure in this economy depends on productivity z? Explain intuitively. 2. [60 points] Recall the prison economy from HA2: there are n prisoners, each is given 1 cigarette and 1 serving of ramen. Half of them really like to smoke and their preferences are described by 2 log cs + log rs . The other half generally prefers ramen to smoking, and their total utility is log cn + 2 log rn . (a) Find competitive equilibrium (that is all prices and quantities) in the exchange economy, where all prisoners can trade with each other. (b) Now suppose that one prisoner takes all the power, and thus she can decide what to do with the total endowment of n cigarettes and n ramen. Luckily, she is benevolent, that is she maximizes 1 1 n (2 log cs + log rs ) + n (log cn + 2 log rn ) . 2 2 Decide what is her “technological” constraint, that is what are the limits on how she can allocate goods across prisoners. Remember that now there are no prices and prisoners can 1 not exchange goods with each other. Find the “first best” allocation, that is the allocation that this powerful prisoner chooses.1 1 Recall that if you need to maximize one function over 2 variables, max F (x, y), the necessary conditions for optimum x,y are just ∂F (x, y) /∂x = 0 and ∂F (x, y) /∂y = 0. 2
