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Unformatted text preview: ility maximizer who is offered fair insurance
optimally chooses to be fully insured. Investment on a Risky Asset
initial wealth w
two states, “good” or “bad”
risky assets returns rg > 0 or rb < 0, depending on the state
If consumer invests x in risky asset he gets
(
w C xrg W w/ prob.
0
wD
w C xrb W w/ prob. 1 so his expected utility is EU.x/ D u.w C xrg / C .1 /u.w C xrb / Marginal utility of x :
EU 0.x/ D rg u0.w C xrg / C .1 /rb u0.w C xrb / Investment on a Risky Asset
Consider second derivative:
2
EU 00.x/ D rg u00.w C xrg / C .1 2
/rb u00.w C xrb /; which is negative if consumer is risk averse. Look at marginal utility at x D 0:
EU 0.0/ D u0.w/ . rg C .1 /rb /
„
ƒ‚
…
expected return A risk averse consumer does not invest in a risky asset with negative
expected return; A risk averse consumer always invests a positive amount in a risky asset
with positive expected return. Taxing a Risky Asset
after tax returns: .1 t /rg in the good state and .1 t /rb in the bad state The FOC determines his optimal investment level:
EU 0.x/ D 0;
that is, .1 t /rg u0.w C x.1 t /rg / C .1 /.1 t /rb u0.w C x.1 t /rb / D 0: Does imposing a tax encourage or discourage investment in the risky
asset? Taxing a Risky Asset
The answer, surprisingly, is it encourages investment in the risky asset!
Let x be the optimal investment level when t D 0, and x the optimal
O
investment under t > 0.
Then we must have:
xD
O x
1 t And the proof comes from the FOC:
.1 t /rg u0.w C x.1
O t /rg / C .1 /.1 t /rb u0.w C x.1
O t /rb / D 0; see blackboard. The tax reduces the expected return but it also reduces the risk,
because in the bad state the tax is actually a subsidy. By increasing investment, consumer can get the same consumption
pattern as before, offsetting the tax....
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This note was uploaded on 03/01/2012 for the course ECON 121 taught by Professor Samuelson during the Spring '09 term at Yale.
 Spring '09
 SAMUELSON
 Microeconomics

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