• Suppose there are two differently productive individuals i ∈ {h, l} in an economy. They are identical up to their work productivity. Both have a quasi-linear production function in leisure and consumption. Leisure is the fraction of a day not worked. Denote the fraction of a day worked as L and consumption as c. Then both individual’s preferences are given by the utility function: u(Li, ci) = (1 − Li)1/2 + ci, Denote the two individuals’ productivity by wh and wl, respectively. Normalize the price for consumption to unity such that the individual i’s consumption is given by ci = wi*Li. 1. Find the fraction of a day an optimizing individual with productivity wi chooses to work. (Note that L cannot be negative!) 2. Find the indirect utility an individual with wi receives. (Hint: indirect utility is the maximised utility as a function of wi) From now on assume that wl ≤ 1/2 and wh > 1/2. 3. Who has got the higher indirect utility. Why? 4. A social planner, who can choose the work levels of both the individuals and also redistribute units of consumption between the two individuals. Suppose further that he want to implement a utilitarian social welfare function. Which allocation (Lh, Ll, ch, cl) would he choose. (Hint: First set up your Social Welfare Function, then the budget constraint and maximize). Does he care how consumption is distributed across individuals? Why or why not? 5. Is this allocation Pareto efficient? 6. Now suppose that he would like to implement a Rawlsian Social Welfare Function. Which allocation would he implement? (Hint: A Rawlsian Social Planner does not want to leave any welfare on the table when choosing working times but then likes to redistribute via a transfer of the consumption good). 7. Is this allocation Pareto efficient? 8. How does the optimal transfer from the high to the low productivity worker depend on wh. Draw a graph. 9. Now suppose that only tool is a tax with tax rate τ , which is levied on income and the revenue is redistributed to the other individual. Without calculating anything yet, explain if you can implement the Utilitarian optimum from above. What about the Rawlsian optimum you just calculated? How do your findings relate to the second fundamental theorem of welfare economics? 10. Calculate how much an individual with productivity wi works. How does the tax rate influence the working decision? Relate this finding to the notion of Equity-Efficiency Trade-Off. 11. Calculate the optimal Rawlsian tax rate for wh = 1. How much utility does the less productive individual gain from redistribution? How much does the more productive lose? What is the change in the sum of utilities? 12. Repeat the calculations for wh = 2 and compare your results from above. Comment.