Macroeconomics 411

Consider H households, with household h. consisting of E1 members. There is a single consump- tion good in this economy. Individuals also care about leisure. thus their per-period utility is u\". [ct-1h. lm). where i denotes the individual and h denotes the household. This utility function is differentiable in both arguments and satisfies all usual conditions. Individuals discount the future at rate ,3 and main'rnize expected utility. Assume that there exist a countable set of payoff relevant states of nature 3: in period t. and we denote a generic state by s! [3; E 3;}. As usual we denote a history by s' = [s]. _... 3;}. The probability of history 5* is 1t[s*]l. The sources of individual income are wage income and non labor inconle. Non-labor income in history at! is 34th {3;} and the hourly wage rate as mm {st Ll. Note that both inconle and wages only depend on the current state 3;. Thus. total wage income of individual 1' in household in. and state 5:; will be the wage rate my, {3,} times the number of units of time worked. Each individual has a total time endowment ofTiIJ. per period. Finally. there is a transfer schedule amongst households. Let 1']. {3} denote the {net} transfer received by household 1!. when state 3 is realised. Household Level Analysis \"re begin by assuming that the risk sharing unit is the household. Assume a unitarian household model in which allocations are decided as a result of an eificient social-planner-like decision rule with weights Fe!- on individual utility fmetions. {1} 1W'rite down the program that a household h solves when deciding consumption and labor allocations for its members. {2} Characterize the solution to the allocation problem. Please be explicit on which variables I:"=f1 and ii=h depend. Provide a precise intuition why the solution depends on those variables (and why not on some others}. [3} 1'J'i'rith your answer to the previous question in mind: what do you think or!" the usual risk sharing regressions? Why might a signicant effect of individual income in the consumption regression not be informative about the absence of risk-sharing? [4} Now suppose that consumption and leisure are separable in individuals'I prefer- ences. Formally, suppose that ui'h {4:11} = vi'h {c} + gi'h {I}. Which variables determine individual leisure and consumption now?I {5} How could this allocation be decentralised, [assuming that each agent is free to decide how much to wrk}? {6} Does individual labor supply ofagent i in household in. depend on wages and incomes of individuals in the household? 1i.i'ai'hy or why not? {'7} Suppose onlyr for this question that the utility function was CARA only in consump- tion [i.e.,ui=h{e} = e_\"=h'] and you would like to identify the risk aversion of agents in a household. Could you identifyr the parameter cm. using the usual risk-sharing regression? 1Why or why not? {3} Express the indirect utility function of household h, wk, inlplicitly. What are the arguments of the indirect utility function? 1illage Level Analysis 'We now assume that the village is the risk sharing unit. {Ell} Set up the planning problem for the village and prove that this problem can be solved by the determination ofstate contingent transfers ml to Inaximise the weighted sum of household indirect utilities. [1D] Characterize the allocation. How would the allocation rule differ in two different states, 3 and .'..3 that satisfy the following property EELI 21:1 y'J' =Ef=l Ei=l 34' '[h {3'}