Taxation of Inheritances

Consider a representative individual from generation t that lives for one period t and whose

utility function is given by

U(Ct , lt , bt) = ln(Ct) + ln(L L lt) +

ln(bt)

2

Where Ct is her consumption level, L is her maximum number of units of labor available, lt

represents her lifetime labor supplied and bt represents the net-of-tax bequests left to the her

only child who will live in period t + 1. The individual living in period t faces the following

budget constraint:

bt = (wlt Ct + btt 1)R

In other words, the bequest she leaves for her child is equal to her lifetime labor income, minus

her total consumption plus the initial wealth she had at the start of her life1 , all that multiplied

by R which is a rate that tells us by how much she is able to augment her savings throughout

her life2.

(a) Set up and solve the utility maximization problem for an individual from the fifirst gener

ation to show that the optimal consumption, labor and net-of-tax bequests for the fifirst

generation are C1 = 2

5 (Lw + b0), l1 = 3

5L L 2

5

b

0

w

and b1 = R

5 (Lw + b0), respectively.3

1Which is nothing but the bequest she got from her parent.

2Another way to read this problem is thinking of each t as a period of L years, where each individual can

choose the number of years she works (lt), the total consumption for the L years (Ct) and the bequest left at the

L year bt. Here w could be interpreted as the annual wage rate and R would be equivalent to (1 + r)L, where r

represents an approximation of the annual interest rate.

3For this part and for the remaining of this math problem assume that all the solutions are interior, that is,

there is no need to check for corner solutions.

56(b) Consider now an individual from the second generation whose initial wealth is given by

the bequest left by the individual from generation 1. Find the optimal C2, l2, and b2 as

functions of the parameters w and R and the initial wealth of the fifirst generation b0 [Hint:

You don't have to solve the maximization problem again].

7Suppose now that the government decides to introduce an estate tax of 40% on all the bequests.

The tax is unanticipated and is introduced at the end of period one, after the before-tax be

quest of the fifirst generation has been determined. The tax applies to the bequests of the fifirst

generation and will also be applied to the subsequent generations.

(c) Find the the optimal consumption level, labor supply and the net-of-tax bequests left by

individuals in generation 1 and generation 2 under this scenario.

(d) Compare your answers in (c) to the optimal values in (a) and (b), Did anything change?

What is the intuition behind these changes?

8In the same setting as before, suppose now that 100 individuals are born each period. Again,

each individual has only one child. All individuals have a maximum of L = 80 units of labor

available. In period 1 there are three types of individuals: There is 1 super wealthy individual

(SW) whose initial wealth b0,SW is equal to $6, 000 and 9 wealthy (W) individuals with b0,SW =

$600. Everyone else has an initial wealth b0,E = $40. We'll also assume that the super wealthy

individual and her offffspring have constant wage rate wSW = 400 and "return" rate RSW = 10,

while the wealthy and her offffspring have wW = 50 and "return" rate RW = 4. Finally, the rest

of the people and their offffspring face wE = 10 and RE = 1.5.4

(e) For the fifirst generation, calculate the share of the total initial wealth that is inherited by

the top 1%, the top 10% and the share that is held by the bottom 90% at the beginning of

the fifirst generation. How far are this numbers from the ones found by Saez and Zucman

'16 for the US in 2016 (Covered in lecture)?

(f) Using the results you found in (a), calculate the labor supply and lifetime labor income

for each type of individual in the fifirst generation.

4The "return" rates of 10, 4 and 1.5 are associated with "annual" interest rates of approximately 3%, 1.8%

and 0.5%, respectively.

9(g) From the previous results, calculate the share of the lifetime labor income that was earned

by the top 1%, the top 10% and the share that was earned by the bottom 90%. How far

are this numbers from the ones found by Saez and Zucman '19 and Piketty, Saez, and

Zucman '18 for the US by 2017 (Covered in lecture)?

(h) Using the results you found in (a), calculate the bequest that each type of individual from

the fifirst generation would leave to her child in a world without taxation. Using these

results, compute the share of the total initial wealth for the second generation that is

inherited by the top 1%, the top 10% and the share that is held by the bottom 90% at

the beginning of the period 2.

10(i) As in (c), suppose that the government introduces a estate tax of 40% on the bequests

left by the super wealthy of the fifirst generation. Assume also that the revenue collected

from that policy is distributed in equal parts among the 100 individuals at the start of

period 2. Compute the new share of the total initial wealth for the second generation that

is held by the top 1%, the top 10% and the share that is held by the bottom 90%

(j) Compare the results from (h) and (i) to the results in (e). From the evidence seen in class,

which situation (between (h) and (i) ) Americans seem to prefer?