Tax Incidence and Optimal Taxation

Questions

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. Suppose you are the 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 you want to implement a utilitarian social welfare function. Which allocation (lh, ll
   , ch, cl) would you
   choose. (Hint: First set up your Social Welfare Function, then the budget
   constraint and maximize). Do you care how consumption is distributed
   across individuals? Why or why not?
5. Is this allocation Pareto efficient?
6. Now suppose that you would like to implement a Rawlsian Social Welfare
   Function. Which allocation would you 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).
   1
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 your 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?

1. 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.
2. 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?
3. Repeat the calculations for wh = 2 and compare to your results from
   above. Comment.
   Hints

1. This means that there is a corner solution for low productivity workers (i.e.
   for low w the marginal utility of increasing work from l = 0 is negative),
   where they do not work at all. When w increases beyond a certain point
   it becomes optimal to work.
2. Up to the critical w, where the worker starts to work the maximized utility
   is given by plugging l
   ∗ = 0 into the utility function. When l
   ∗ becomes
   positive then you have to substitute the labour supply into the utility
   function.
3. Follows straight from the point above.
4. The hint is already in the question.
5. Can I change any of the choice variables to make one of the two better off
   without making the other worse off?
6. Rawls maximizes the utlity of the poorest. First observe that I can freely
   distribute utility through the transfer of consumption (as utility is linear in
   consumption). This implies that the planner wants to make sure that the
   pie is as large as possible, such that he then can redistribute. To see what
   maximizes the pie consider the following. If an individual works less than
   optimal for herself, then the planner could ask her to work more, which
   would generate more additional consumption than would be needed to
   2
   compensate the worker for the loss in utility from working more. Suppose,
   a worker works more than individually optimal. Then the planner could
   make this worker better off by asking him to work less. The additional
   utility increase would be larger than the loss in consumption utility. So
   the social planner could take some consumption away and redistribute it
   without making him worse off. Hence, the social planner does not want
   to distort working hours but then would redistribute consumption such
   that the poorest is best off, which means that both have the same indirect
   utility after redistribution.
7. Should follow directly from what I wrote above.
8. This should be quite mechanical.
9. I leave this one to you . . .
10. . . . and this one
11. also this one
12. and also the last one.