Suppose a consumer has wealth w and is considering investing some amount,x,in ri

Question

Suppose a consumer has wealth w and is considering investing some amount,x,in risky asset.This asset can earn a return of Rg(which is positive return and therefore a good outcome or Rb (a negative return that ultimately reduces initial wealth per dollar invested in the asset).Thus wealth in the good and bad states will be:

Wg=(w-x)+x(1+Rg)=w+wRg where Rg>0

Wb=(w-x)+x(1+Rb)=w+wRb where Rb<0

the probability of the good state is and the probability of the bad state is 1-.

a)Find the general expression for the consumer's expected utility maximization problem for investing x dollars in the risky asst.Let u(w) represent the general utility function.Characterize the solution to this problem by finding the first order condition and check to see if (or when ) the second order condition ( or will be)satisfied.

b)Let the individual's utility function be given by u(w)=ln w .Using this,solve the consumer's expected utility maximization problem.(Assume that the amount invested is positive)

c)suppose the individual has to pay a (positive)tax,t,on the return of the asset,whether it is positive or negative.Thus the after tax returns will be (1-t)Rg and (1-t)Rb.Wealth in each state is no:

Wg=w+x(1-t)Rg where Rg>0

Wb=w+x(1-t)Rb where Rb<0

Given the specific utility function in (2b),find the optimal amount of investment when t>0.

d)What is the relationship between investment when the tax rate is zero versus when it is strictly positive?Does your make sense?Explain.

Explanation / Answer

For part A of this wuestion, it tells us to use first order condition. This is another wording for take the derivative. We take the derivative of the following two functions with respect to w Wg=(w-x)+x(1+Rg)=w+wRg where Rg>0 First rewrite and simplify to make the derivative easier Wg=w-x+x+xRg=w+wRg where Rg>0 FOC Wg=1-0+0+0=1+Rg Wg=Rg and we know Rg is >0 Wb=(w-x)+x(1+Rb)=w+wRb where Rb

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