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Last week saw consumer with wealth W, chose to invest an amount x* when returns were rg in the good state with probability p and rb in the bad state with probability 1-p. • EU=pU(W+x* rg ) + (1-p)U(W+x* rb) What is the effect of taxation on the amount we choose to invest in the risky investment? Now only get (1-t)rg in the good state with probability p and (1-t)rb in the bad state with probability 1-p. • EU=pU(W+x(1-t)rg) + (1-p)U(W+x (1-t)rb) • EU=pU(W+x(1-t)rg) + (1-p)U(W+x (1-t)rb) What are the FOC’s? U g U b (1 t )rb (1 t) rg (1 p ) p x x • EU=pU(W+x(1-t)rg) + (1-p)U(W+x (1-t)rb) What are the FOC’s? U g U b p (1 t )rg (1 p ) (1 t )rb 0 x x U g U b p rg (1 p ) rb 0 x x • EU=pU(W+x(1-t)rg) + (1-p)U(W+x (1-t)rb) What are the FOC’s? U g U b p rg (1 p ) rb 0 x x 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. • EU=pU(W+x(1-t)rg) + (1-p)U(W+x (1-t)rb) What are the FOC’s? U g ^ U b p [(W x(1 t )rg ] rg (1 p ) [(W x(1 t )rb ] rb 0 x x ^ 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]rs ^ • What is the relationship between x* and x Usually think, if return on asset goes down, want less of it. That is, might ^ think x*> x To see why this is wrong set U g Then MRS between the two states is: x* x 1 t ^ ^ [W x(1 t )rg ] (1 p )rb x ^ U b prg [W x(1 t )rb ] x x* x 1 t ^ To see this suppose set U g MRS between the two states is: ^ [W x(1 t )rg ] x ^ U b [W x(1 t )rb ] x x* x 1 t ^ To see this suppose set MRS between the two states is: but since 0<t<1, (1-t)<1 x* x x* 1 t ^ So U g x* [W (1 t )rg ] x 1 t U b x* [W (1 t )rb ] x 1 t U g [W x * rg ] (1 p )rb x U b p r g [W x * rb ] x 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 rb occurs rb (1-t)rb (1-t)rg rg •So t reduces spread of returns and therefore risk •Only way can recreate original spread is to invest more (i.e. x up) Mean-Variance Analysis Suppose W = £100 and bet £50 on flip of a coin Probability 0.5 Outcomes £50 £150 • EU=0.5U(50) + 0.5 U(100) More outcomes => More complexity • 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 Probability 1/6 Outcomes £10 £20 £30 £40 £50 £60 More outcomes => More complexity • 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) Probability 1/6 Outcomes £10 £20 £30 £40 £50 £60 More complex still: Probability of returns on investment follows a normal distribution • With EU need to consider every possible outcome and probability • Too complex, need something simpler Probability Returns Probability of returns on investment follows a normal distribution • If we can use some representative information that would be simpler. Probability Returns Probability of returns on investment follows a normal distribution Called Mean-Variance Analysis U=U(m,s2) Probability Returns Like returns to be high Like Risk to be low U=U(m,s2) Return U=U(,) U2 U0 Risk (measured by Variance s2 or Standard Deviation, s) The Mean-Variance approach says that consumers preferences can be captured by using two summary statistics of a distribution : • Mean • Variance Mean m=p1w1+ p2w2+ p3w3+ p4w4+ … …….+psws+ ... Or in other words S m ps ws s1 Mean S m ps ws s1 Similarly Variance is s2=p1(w1-m)2 p2 (w2-m)2 + ………...p3 (w3-m)2 + p4 (w4-m)2 + … …….+ps (ws-m)2 + ... Or in other words S s 2 p s (ws m)2 s1 Variance S 2 2 s p s (ws m) s1 We like return- measured by the Mean We dislike risk - measured by the Variance 2 U U (m ,s ) U U(, ) Suppose we have two assets, one risk-free and one risky, e.g. Stock The return on the risk-free asset is rf and its variance is sf2=0 The return on the risky asset is rs with probability s , but on average it is rm and its variance is sm2 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? Return S rx p s( xrs (1 x)r f ) s1 S rx x p s( rs) (1 x)r f s1 rx xrm (1 x)r f Variance S s 2x p s ( xrs (1 x)rf rx )2 s1 S s 2x p s ( xrs (1 x)r f xrm (1 x)rf )2 s1 S s 2x p s ( xrs (1 x)r f xrm (1 x)r f )2 s1 S 2 p s ( xrs xrm)2 sx s1 S 2 x2 p s (rs rm)2 sx s1 2s 2 s2 x x m Results Return rx xrm (1 x)r f Variance: 2 s 2x x2sm