Investment on a risky asset initial wealth w two

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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.

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