STOCHASTIC

EFFECTS OF TAXATION ON RISK-TAKING 277
w0
FIQUSE IVB
Budget Constraints: No Loss-Offset
with r=0.
w(e,)
S[-U"{e(l~v)-r(l-t)}eaW0-V'e]dF(e)
o
which is unambiguously positive. Hence, we have:
Proposition 4(b). The demand for the risky asset is always less with
no loss-offset or partial loss-offset than with full loss-offset.
We shall now attempt to find some more precise conditions under
which risk-taking unambiguously increases or decreases. For simplicity
we limit ourselves to the case where r = 0. We can write the
first order conditions for expected utility maximization as
(12) SU'e(l-t)dF(ff)+fu'edF(0)=O
y o
so the sign of da/dt is that of
(13) k-U"aem-t)W0-U'e]dF(8)
= f[- U " iW - Wo) -l]uWS):
r u U J
We thus obtain:
Proposition 4(c). If r=0 the imposition of an income tax with no
loss-offset decreases the demand for the risky asset if relative risk
aversion is less than or equal to unity. 1
4. The Bernoulli utility function, V = lnW, has constant relative risk
EFFECTS OF TAXES ON RISK TAKING