q1 as attached

1. (34 points) All individuals in the economy earn hourly wage in and have the follow- ing Cobb-Douglass utility function over consumption and hours worked each day: 1 2 u(c,h) = lnc+ ln(24 h) 3 3 Each individual earns labor income wh if she works h hours, but also generates environmental damage by working. This damage makes_everyone in the economy poorer. Specically, if everyone in the economy works h hours, each individual's budget constraint is reduced by D x )1. Since each individual is a small part of the economy, she does not consider the damage she causes when choosing her own h. Rather, she takes the hours worked by everyone else, h, as given and solves max 1lnc+ Eln(24 h) Si. c S wh DE. (1,}: 3 3 (a) (4 points) First assume working causes no environmental damage: D = 0. What is each individual's privately optimal hours worked 12*? Holw does this compare to the socially optimal hours worked hSP? (b) (6 points) Now suppose working does cause environmental damage: D > (J. i. Taking hours worked by everyone else 71 as given, what is each individual's privately optimal hours worked If? This may be a function of D, h, and w. ii. Briey comment on how hours worked differs from part (a). How is your answer related to the income and substitution effects? For the rest of the problem, assume that m = $50 and D = $15. (c) (6 points) Find the hours each individual works in equilibrium. Remember that each individual's hours worked must be optimal given how much everyone else is working, so if = h. Denote the equilibrium hours worked by h\". hSP hSP (d) (6 points) Find the socially optimal hours worked . How does compare to the privately optimal hours h\" from part (c)? How does it compare to the social optimum from part (a)? (e) (6 points) An environmental advocacy group reads your analysis and claims there must be a mistake. Since working causes environmental damage, they argue, it must be true that the socially optimal hours worked is lower than it would be if there were no externality. Why is their reasoning incorrect? (f) (6 points) After you correct the advocacy group's misunderstanding, they pro- pose a Pigouvian Tax of 1' = D on hours worked. This way, they argue, each worker will internalize the external damage they cause by working. The revenue will be used for unrelated government expenditures that do not affect workers\