1.Taxation of Inheritances

Consider a representative individual from generation tthat lives for one period tand whose utility function is given by

U) = ln(Ct) + ln(L - l_t)+ln(bt) /2

Where C_tis her consumption level, L is her maximum number of units of labor available, l_t represents her lifetime labor supplied and btrepresents 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+ bt_\)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 life, all that multiplied by Rwhich is a rate that tells us by how much she is able to augment her savings throughout her life.

(a)Set up and solve the utility maximization problem for an individual from the first generation to show that the optimal consumption, labor and net-of-tax bequests for the first generation are Ci = ²2 (Lw+ bo), li = |L |b⁰and bi = R(Lw+ bo), respectively.

(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, I2, and b₂as functions of the parameters wand Rand the initial wealth of the first generation bo [Hint: You don't have to solve the maximization problem again].

Suppose 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 bequest of the first generation has been determined. The tax applies to the bequests of the first 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?

In 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 bo,swis equal to $6, 000 and 9 wealthy (W) individuals with bo,sw= $600. Everyone else has an initial wealth by= $40. We'll also assume that the super wealthy individual and her offspring have constant wage rate wsw= 400 and "return" rate Rsw= 10, while the wealthy and her offspring have ww = 50 and "return" rate Rw = 4. Finally, the rest of the people and their offspring face We= 10 and Re= 1.5.

(e)For the first 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 first 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 first generation.

(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 first 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.

(i)As in (c), suppose that the government introduces a estate tax of 40% on the bequests left by the super wealthy of the first 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?