First best cR We measure welfare using long-run self-O utility cR The first best solution does not depend on degree of time-inconsistency maxc cu(ci+u(c,) StC+C2/R=1 FOC R
We measure welfare using long-run self-0 utility The first best solution does not depend on degree of time-inconsistency First Best R u c u c FOC c c R u c u c f b f b c c = + = + '( ) '( ) : s.t. / 1 max ( ) ( ) 2 1 1 2 1 , 2 1 2
Diagram of proof fb fb l(c1)+(c2) R
Diagram of Proof R 1 c2 c1 ( ) ( ) 1 2 u c +u c ( , ) 1 2 fb fb c c
Autarky cR Investors cannot commit, and liquidation has no cost, so they will liquidate some of the investment for consumption at date 1 based on their date 1 preference regardless what they believe at date 0 maX Gl,c u(C)+ Bulc Stc+C/r=1 FO、t(C b at fb at BR<R→c
Investors cannot commit, and liquidation has no cost, so they will liquidate some of the investment for consumption at date 1 based on their date 1 preference regardless what they believe at date 0 Autarky a t f b a t f b a t a t c c βR R c c c c u c u c FOC c c R u c βu c 1 1 2 2 2 1 1 2 , 1 2 , '( ) '( ) : s.t. / 1 max ( ) ( ) 1 2 = + = +
Diagram of proof (c1)+B(c2) l(c1)+l(c2)
Diagram of Proof R 1 c2 c1 ( ) ( ) 1 2 u c +u c ( , ) 1 2 at at c c ( ) ( ) 1 2 ( , ) u c + βu c 1 2 fb fb c c
Ineffective market CR In the autarky case, if we allow for trading at date 1, that is, an investor can trade his date 2 consumption from his investment for date 1 consumption, investors will have the same consumptions as in autarky case CR Proof: The price of date 2 consumption, p, must be 1/R, otherwise either(1, 0)or(O,R)will dominate all other points on the budget line and it cannot be equilibrium maX CI,C2,C1, C2 u(C1+ Bu(c2) C,+ pc,=co + pco S t 0+c0/R=1
In the autarky case, if we allow for trading at date 1, that is, an investor can trade his date 2 consumption from his investment for date 1 consumption, investors will have the same consumptions as in autarky case Proof: The price of date 2 consumption, p, must be 1/R, otherwise either (1,0) or (0,R) will dominate all other points on the budget line and it cannot be equilibrium Ineffective Market / 1 s.t. max ( ) ( ) 0 2 0 1 0 2 0 1 2 1 , , , 1 2 1 2 0 2 0 1 + = + = + + c c R c pc c pc u c βu c c c c c