Uncertainty and Risk - DARP
Uncertainty and Risk - DARP 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
Microeconomics 3. Take the FOC (8.17). Substituting for x from (8.16) and di¤erentiating with respect to t we get E u xx (x) r + @ @t [1 t] r r = 0; so that E u xx (x)r 2 + @ @t + @ @t [1 [1 t] = 0 @ @t = t] = 0: 1 t : An increase in the tax rate increases the demand for bonds. 4. Final wealth is initial wealth plus income. If the tax is on wealth then disposable …nal wealth is x = [1 t] K + [1 t] r (8.18) instead of (8.16). Clearly the FOC (8.17) remains essentially unaltered (the new tax just reduces total wealth). Di¤erentiating the FOC with x de…ned by (8.18) we now …nd This implies E u xx (x) KE (u xx (x)r) + E K r + @ @t [1 u xx (x)r 2 t] r r = 0; + @ @t [1 t] = 0: K E (u xx(x)r) E (u xx (x)r 2 ) + @ [1 t] = 0: @t @ @t [1 @ @t t] = + K E (u xx(x)r) E (u xx (x)r 2 ) : = 1 t + K 1 t E (u xx (x)r) E (u xx (x)r 2 ) : The …rst term on the right-hand side is positive; as for the second term, the denominator is negative and the numerator is positive, given DARA. So the impact of tax on bond-holding is now ambiguous. cFrank Cowell 2006 127
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Microeconomics<br />
3. Take the FOC (8.17). Substituting for x from (8.16) <strong>and</strong> di¤erentiating<br />
with respect to t we get<br />
<br />
<br />
E u xx (x) r + @<br />
@t [1 t] r r = 0;<br />
so that<br />
<br />
E<br />
u xx (x)r 2<br />
+ @<br />
@t<br />
+ @<br />
@t [1<br />
[1 t] = 0<br />
@ <br />
@t<br />
=<br />
<br />
t] = 0:<br />
<br />
1 t :<br />
An increase in the tax rate increases the dem<strong>and</strong> for bonds.<br />
4. Final wealth is initial wealth plus income. If the tax is on wealth then<br />
disposable …nal wealth is<br />
x = [1 t] K + [1 t] r (8.18)<br />
instead of (8.16). Clearly the FOC (8.17) remains essentially unaltered<br />
(the new tax just reduces total wealth). Di¤erentiating the FOC with x<br />
de…ned by (8.18) we now …nd<br />
This implies<br />
E<br />
<br />
u xx (x)<br />
KE (u xx (x)r) + E<br />
K r + @<br />
@t [1<br />
<br />
u xx (x)r 2<br />
<br />
t] r r = 0;<br />
+ @<br />
@t [1<br />
<br />
t] = 0:<br />
<br />
K E (u xx(x)r)<br />
E (u xx (x)r 2 ) + @ [1 t] = 0:<br />
@t<br />
@ <br />
@t [1<br />
@ <br />
@t<br />
t] = + K E (u xx(x)r)<br />
E (u xx (x)r 2 ) :<br />
= <br />
1 t + K<br />
1 t<br />
E (u xx (x)r)<br />
E (u xx (x)r 2 ) :<br />
The …rst term on the right-h<strong>and</strong> side is positive; as for the second term,<br />
the denominator is negative <strong>and</strong> the numerator is positive, given DARA.<br />
So the impact of tax on bond-holding is now ambiguous.<br />
cFrank Cowell 2006 127
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