Roving bandit vs. stationary bandit

Assume that there are two periods, 0 and 1. The first-period output from the economy is 1, an autocrat can tax it with a tax rate, 0t1.

1. Denote the tax revenue asc0. How much is it in terms oft? How much of the output is left after tax, i.e., how much is 1c0in terms oft?
2. Assume that the second-period output from the economy is 2(1c0), which is increasing in how much of the first-period output is left after tax. Assume that the autocrat will tax all of the second-period output. Denote the second-period tax revenue asc1. How much is it in terms oft?
3. Now assume that the autocrat can choose the first-period tax rate,t, to maximize the net present value of tax revenues from the two periods, with a discount factor, 01. In which case the autocrat is more patient and has a longer horizon,= 0, or= 0.5, or= 1?
4. What is the net present value of tax revenues from the two periods in terms oftand?
5. What is the optimal tax rate for the autocrat, in terms of?
6. Consider the case in which= 0.49 and the case in which= 0.51. In which case
7. will the autocrat have a higher optimal tax rate?
8. If the less the output in the first period is left after-tax, the more output will be produced in the second period, what is the optimal tax rate in the first period?
9. What can you conclude from this exercise?