Question

1 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, 0 ≤ t ≤ 1.

a. Denote the tax revenue as c0. How much is it in terms of t? How much of the output is left after tax, i.e., how much is 1 − c0 in terms of t?

b. Assume that the second-period output from the economy is 2(1 − c0), 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 as c1. How much is it in terms of t?

c. 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, 0 ≤ β ≤ 1. In which case the autocrat is more patient and has a longer horizon, β = 0, or β = 0.5, or β = 1?

d. What is the net present value of tax revenues from the two periods in terms of t and β?

e. What is the optimal tax rate for the autocrat, in terms of β?

f. Consider the case in which β = 0.49 and the case in which β = 0.51. In which case will the autocrat have a higher optimal tax rate?

g. 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?

h. What can you conclude from this exercise?

Answer 1

So, in this problem there are two periods “1” and “2”. Let’s assume that “Y0” and “Y1” be the output of period “1” and “2” respectively. So, we have given that “Y0=1” and “t” be the tax rate of period “0”. So, the “tax revenue” of period “1” is given by, “C0 = t*Y0 = t”, where “t” is between “(0, 1)”. So, “C0 = t”.

b).

Now, the 2^nd periods output from the economy is “Y1 = 2*(1-C0)”, is increasing in “1^st period’s after tax output”. Now, since the autocrat will tax all the 2^nd period output, => the 2^nd period tax revenue is given by, “C1 = Y1 =2*(1 – C0)”.

=> C1 = 2*(1 – C0) = 2*(1 – t), => C1 = 2*(1 – t).

c).

Let’s assume that “β” be the discount factor, so the “net present value of tax revenue” from these 2 periods is given below.

=> T = C0 + β*C1 = t + β*2*(1-t) = t + 2β – 2β*t, => T = t + 2β – 2β*t, be the “net present value of tax revenue”.

Now, if “β=0”, => T = t, => as the tax rate of the period 1 increases, => the overall tax revenue will also increase, => optimum “t” should be “t=1”.

If “β=1”, => T = 2 - t, => as the tax rate of the period 1 increases, => the overall tax revenue decrease, => optimum “t” should be “t=0”.

Now, if “β=0.5”, => T = 1, => “T” is totally independent of “t”, => as the tax rate of the period 1 increases, => the overall tax revenue will remain same, => optimum “t” can be anything.

So, here “β=1”, => autocrat is more patient and has a longer horizon.

d).

So, the present value of tax revenues from the two periods in terms of “t” and “β” is given below.

=> T = C0 + β*C1 = t + β*2*(1-t) = t + 2β – 2β*t, => T = t + 2β – 2β*t.

e)

So, we can see that “T” depends on “t” and “β”, so can reduce the above equation in the following forms.

=> T = 2β + (1 - 2β)*t, so, if “(1-2β) > 0, => β < 1/2”, => as “t” increases, => “T” increase, => the optimum “t” is “1”. If “(1-2β) < 0, => β > 1/2”, => as “t” increases, => “T” decrease, => the optimum “t” is “0”. Now, if “(1-2β) = 0, => β = 1/2”, => as “t” increases, => “T” remain same, => the optimum “t” is any value between “0” to “1”.