Suppose the current tax system is such that the government takes some fixed percentage t of any labor income that you make.
A. Some in Congress have proposed the following alternative type of tax system known as the negative income tax: You get a certain guaranteed income x even if you do not work at all. Then, for any income you earn in the labor market, the government takes a certain percentage k in taxes. In order to finance the guaranteed income x, the tax rate on labor income in this alternative system has to be higher than the tax rate under the current system (i.e. t < k).
(a) On a graph with leisure on the horizontal axis and consumption on the vertical, illustrate what your budget constraint under the current tax system looks like—and indicate what the intercepts and slopes are assuming a leisure endowment of E and before tax wage w.
(b) On a similar graph, illustrate what your budget constraint looks like under the alternative system. (c) You hear me say: “You know what—after looking at the details of the tax proposal, I can honestly say I don’t care whether we keep the current system or switch to the proposed one.” Without knowing what kind of goods leisure and consumption are for me, can you tell whether I would work more or less under the negative income tax? Explain.
(d) What would your tastes have to look like in order for you to be equally happy under the two systems while also working exactly the same number of hours in each case?
(e) True or False: The less substitutable consumption and leisure are, the less policymakers have to worry about changes in people’s willingness to work as we switch from one system to the other.
B. Consider your weekly decision of how much to work, and suppose that you have 60 hours of available time to split between leisure and work. Suppose further that your tastes over consumption and leisure can be captured by the utility function u(c,ℓ) = cℓ and that your market wage is w = 20 per hour.
(a) Write down the budget constraint under the two different tax policies described above; i.e. write down the first budget constraint as a function of c, ℓ and t and the second as a function of c, ℓ, k and x.
(b) Derive the optimal choice under the current tax system(as a function of t .) In the absence of anything else changing, do changes in wage taxes cause you to change how much you work? Can you relate your answer (intuitively) to wealth and substitution effects?
(c) Now derive your optimal leisure choice under a negative income tax (as a function of k and x). How is your work decision now affected by an increase in k or an increase in x?
(d) Suppose that t = 0.2. Using your utility function to measure happiness, what utility level do you attain under the current tax system?
(e) Now the government wants to set k = 0.3. Suppose you are the pivotal voter—if you approve of the switch to the negative income tax, then it will pass. What is the minimum level of guaranteed income x that the negative income tax proposal would have to include in order to win your support?
(f) How much less will you work if this negative income tax is implemented (assuming x is the minimum necessary to get your support)?