1. EU=  U(W+x* r g ) + (1-  )U(W+x* r b ) Last week saw consumer with wealth W, chose to invest an amount x* when returns were r g in the good state with probability  and r b in the bad state with probability 1- .

2. EU=  U(W+x(1-t)r g ) + (1-  )U(W+x (1-t)r b ) What is the effect of taxation on the amount we choose to invest in the risky investment? Now only get (1-t)r g in the good state with probability  and (1-t)r b in the bad state with probability 1- .

3. EU=  U(W+x(1-t)r g ) + (1-  )U(W+x (1-t)r b ) What are the FOC’s?

4. EU=  U(W+x(1-t)r g ) + (1-  )U(W+x (1-t)r b ) What are the FOC’s?

5. EU=  U(W+x(1-t)r g ) + (1-  )U(W+x (1-t)r b ) What are the FOC’s? Like last case except that there is no guarantee that x with taxes is the same as x* Why? Payoff in good and bad states different.

6. EU=  U(W+x(1-t)r g ) + (1-  )U(W+x (1-t)r b ) What are the FOC’s? Like last case except that there is no guarantee that x with taxes is the same as x* Why? Payoff in good and bad states different. W+x[1-t]r s

7. What is the relationship between x* and Usually think, if return on asset goes down, want less of it. That is, might think x*> To see why this is wrong set Then MRS between the two states is:

8. To see this suppose set MRS between the two states is:

9. To see this suppose set MRS between the two states is: but since 0<t<1, (1-t)<1

10. Why does investment rise? When the good state occurs t is a tax. However, if losses can be offset against tax then t is a subsidy when r b occurs rgrg (1-t)r g (1-t)r b rbrb So t reduces spread of returns and therefore risk Only way can recreate original spread is to invest more (i.e. x up)

11. Mean-Variance Analysis

12. Suppose W = £100 and bet £50 on flip of a coin EU=0.5U(50) + 0.5 U(100) Probability 0.5 £50 £150 Outcomes

13. More outcomes => More complexity Probability 1/6 £10£60 Outcomes E.g W = £30, Bet £30 on throw of dice Prizes 1=£10, 2= £20, 3=£30, 4=£40, 5=£50,6=£60, each with probability 1/6 £30£40£50£20

14. More outcomes => More complexity Probability 1/6 £10£60 Outcomes EU= 1/6 U(10)+ 1/6 U(20)+ 1/6 U(30)+ 1/6 U(40)+ 1/6 U(50)+ 1/6 U(60) £30£40£50£20

15. More complex still: Probability of returns on investment follows a normal distribution Probability Returns With EU need to consider every possible outcome and probability Too complex, need something simpler

16. Probability of returns on investment follows a normal distribution Probability Returns If we can use some representative information that would be simpler.

17. Probability of returns on investment follows a normal distribution Probability Returns Called Mean-Variance Analysis U=U( ,  2 )

18. Like returns to be high Like Risk to be low U=U( ,  2 ) U=U( ,  ) (measured by Variance  2 or Standard Deviation,  ) Risk Return U0U0 U2U2

19. The Mean-Variance approach says that consumers preferences can be captured by using two summary statistics of a distribution : Mean Variance

20. Mean  =  1 w 1 +  2 w 2 +  3 w 3 +  4 w 4 + … …….+  s w s +... Or in other words

21. Mean

22. Similarly Variance is  2 (w 2 -  ) 2 + ………...  3 (w 3 -  ) 2 +  4 (w 4 -  ) 2 + … …….+  s (w s -  ) 2 +... Or in other words   =  1 (w 1 -  ) 2

23. Variance

24. We like return- measured by the Mean We dislike risk - measured by the Variance

25. Suppose we have two assets, one risk-free and one risky, e.g. Stock The return on the risk-free asset is r f and its variance is  f 2 =0 The return on the risky asset is r s with probability s, but on average it is r m and its variance is  m 2

26. If we had a portfolio composed of x of the risky asset and (1-x) of the risk-free asset, what would its properties be?

27. Return

28. Variance

30. Results Return Variance: