Uncertainty and Risk - DARP

Microeconomics CHAPTER 8. UNCERTAINTY AND RISK
Exercise 8.10 A person has an objective function Eu(y) where u is an increasing,
strictly concave, twice-di¤erentiable function, and y is the monetary value
of his …nal wealth after tax. He has an initial stock of assets K which he may
keep either in the form of bonds, where they earn a return at a stochastic rate
r, or in the form of cash where they earn a return of zero. Assume that Er > 0
and that Prfr < 0g > 0.
1. If he invests an amount in bonds (0 < < K) and is taxed at rate t
on his income, write down the expression for his disposable …nal wealth y,
assuming full loss o¤set of the tax.
2. Find the …rst-order condition which determines his optimal bond portfolio
.
3. Examine the way in which a small increase in t will a¤ect .
4. What would be the e¤ect of basing the tax on the person’s wealth rather
than income?
Outline Answer:
1. Suppose the person puts an amount in bonds leaving the remaining
K of assets in cash. Then, given that the rate of return on cash is
zero and on bonds is the stochastic variable r, income is
[K ] 0 + r = r
If the tax rate is t then, given that full loss o¤set implies that losses and
gains are treated symmetrically, disposable income is
and (disposable) …nal wealth is
[1 t] r
x = [K ] + + [1 t] r
[cash] [value of bonds] [income]
= K + [1 t] r: (8.16)
Note that x is a stochastic variable and could be greater or less than initial
wealth K.
2. The individual’s optimisation problem is to choose to maximise Eu(x).
Using (8.16) the FOC for an interior solution is
which implies
E (u x (x) [1 t] r) = 0;
E (u x (x)r) = 0: (8.17)
Solving this determines = (t; K), the optimal bond purchases that
depends on the tax rate and initial wealth as well as the distribution of
returns and risk aversion.
cFrank Cowell 2006 126