Introduction to choice over time

Transcription

1 Microeconomic Theory -- Choice over ime Inroducion o choice over ime Individual choice Income and subsiuion effecs 7 Walrasian equilibrium ineres rae 9 pages John Riley Ocober 9, 08

2 Microeconomic Theory -- Choice over ime Inroducion o choice over ime Two periods: Uiliy U( c, c ) is sricly increasing Ineres rae on savings: r Income sequence { y }. Consumpion sequence { c } Financial capial sequence { K }. 3 Period budge consrains: ( r)( K y c ) K () ( r)( K y c ) K () * 3 John Riley Ocober 9, 08

3 Microeconomic Theory -3- Choice over ime G. Inroducion o choice over ime Two periods: Uiliy U( c, c ) is sricly increasing Ineres rae on savings: y Income sequence { y }. Consumpion sequence { c } Financial capial sequence { K }. 3 Period budge consrains: ( r)( K y c ) K () ( r)( K y c ) K () 3 Consumer saves Consumer s problem: Choose sequences { c}, { K } o solve Max { U( c, c ) ( r)( K y c ) K, ( r)( K y c ) K } 3 John Riley Ocober 9, 08

4 Microeconomic Theory -4- Choice over ime Consumer s problem: Choose sequences { c}, { K } o solve Max { U( c, c ) ( r)( K y c ) K, ( r)( K y c ) K } 3 Sep : Try o reduce he problem o a simpler consrain problem (Divide he firs equaion by r and he second by K y c K r ( r) ( ) K y c K 3 r ( r) To maximize uiliy K3 0. Then add ( ) and ( ) y c K ( y ) ( c ) 0 r r ( ) John Riley Ocober 9, 08

5 Microeconomic Theory -5- Choice over ime y c K ( y ) ( c ) 0 r r Noe ha he higher he ineres rae, he Slope = + r seeper is he life-ime budge line. The relaive price of fuure consumpion has fallen. Consumer saves Group Exercise : U( c) lnc lnc r 0., { y } {00,7}. Solve for { c } Group Exercise : U( c) lnc lnc lnc r 0., { y } {00,7,7}. Solve for { c } 3 John Riley Ocober 9, 08

6 Microeconomic Theory -6- Choice over ime Skech of he answer o group exercise Sep : how ha he life-ime budge consrain is 7 7 c ( ) c ( ) c 3 y ( ) y ( ) y r ( r) r ( r). (.) Sep : Obain he FOC Noe ha we can hink of price of he period commodiy. So he prices are 3 The FOC are U U U p c p c p c 3 3 r as he price of he period commodiy. Similarly p3 is he ( r) ( p, p, p ) (,, ) (,, ) r ( r)..44 Sep 3: Solve using he FOC and budge consrain Using hese condiions you should be able o show ha c 70. John Riley Ocober 9, 08

7 Microeconomic Theory -7- Choice over ime Comparison wih he sandard budge consrain c y c K y r r p x p x p p Slope = + r p p x x p p Thus he lifeime budge consrain is jus like he one period budge consrain. p p r is he marke value of fuure goods relaive o curren goods Consumer saves We call his he presen value of fuure goods. The life-ime budge consrain is PV{ c } K PV{ y } John Riley Ocober 9, 08

8 Microeconomic Theory -8- Choice over ime Decomposiion ino subsiuion and income effecs of an increase in he ineres rae Subsiuion Effec The relaive price of fuure goods / ( r) is lower so firs period consumpion falls and second period consumpion rises. ** Consumer saves John Riley Ocober 9, 08

9 Microeconomic Theory -9- Choice over ime Decomposiion ino subsiuion and income effecs of an increase in he ineres rae Subsiuion Effec The relaive price of fuure goods / ( r) is lower so firs period consumpion falls and second period consumpion rises. Income effec Since he consumer is a saver, a higher ineres rae Makes him beer off. Thus income has o be reduced o keep her on he same level se. Assuming ha goods are normal, giving he income back leads o higher consumpion in boh periods. Consumer saves In paricular, firs period consumpion rises. * John Riley Ocober 9, 08

10 Microeconomic Theory -0- Choice over ime Decomposiion ino subsiuion and income effecs Subsiuion Effec The relaive price of fuure goods / ( r) is lower so firs period consumpion falls and second period consumpion rises. Income effec Since he consumer is a saver, a higher ineres rae Makes him beer off. Thus income has o be reduced o keep her on he same level se. Assuming ha goods are normal, giving he income back leads o higher consumpion in boh periods. Consumer saves In paricular, firs period consumpion rises. Toal effec The wo effecs are off-seing for firs period consumpion and hence for firs period saving. Daa analyics ypically show small ineres rae effecs on personal saving. John Riley Ocober 9, 08

11 Microeconomic Theory -- Choice over ime Class Exercise: Are he effecs reinforcing for borrowers? If he ineres rae rise he new life-ime budge line (he dashed line hrough ( y, y ) in he figure) is seeper. The consumer is worse off. If he consumer is subsidized o ge her back o he original level se, he consumes he bundle ĉ depiced. Thus he subsiuion effec is o move Around he level se reducing c and increasing c. Exercise: Discuss he income effec when he subsidy is removed Consumer borrows Hence explain why (assuming normal goods) he wo effecs are reinforcing. Boh lower firs period consumpion hence lower borrowing. John Riley Ocober 9, 08

12 Microeconomic Theory -- Choice over ime Walrasian Equilibrium wih wo consumers, Alex and Bev Example : Exchange economy wih wo periods A B { y } (4,6) { y } (,0), No iniial financial capial K 0 U ( c, c ) u( c ) u( c ), h A, B, where h h h h h Period goods canno be sored. u() c c / John Riley Ocober 9, 08

13 Microeconomic Theory -3- Choice over ime Wha is he marginal rae of subsiuion a he 45 line? U U uc ( ) MRS ( c, c) / c c u ( c ). line Therefore U U u ( a) MRS ( a, a) / c c u () a is he discoun facor for fuure consumpion The smaller is he discoun facor he smaller is he value of fuure consumpion. Thus he consumer is willing o give up more period consumpion for an addiional uni of curren consumpion John Riley Ocober 9, 08

14 Microeconomic Theory -4- Choice over ime Walrasian Equilibrium wih wo consumers, Alex and Bev Class example: Exchange economy wih wo periods A B { } {4,6} { } {,0}, h h h h / h / U ( c, c ) ( c ) ( c ), h A, B. Period goods canno be sored. For simpliciy assume (a) Are here poenial gains from exchange? (b) Solve for he equilibrium ineres rae. HINT: Can we solve a simpler problem firs? (c) How does he oucome change if period goods can be coslessly sored and consumed in period? (d) If wha is he new equilibrium ineres rae? John Riley Ocober 9, 08

15 Microeconomic Theory -5- Choice over ime (a) Gains from exchange John Riley Ocober 9, 08

16 Microeconomic Theory -6- Choice over ime (b) Walrasian equilibrium ineres rae A B { y } {4,6}, { y } {,0}, U ( c, c ) u( c ) u( c ) ( c ) ( c ), h A, B. h h h h h h / h / Hin: Wha is special abou his model? John Riley Ocober 9, 08

17 Microeconomic Theory -7- Choice over ime Represenaive agen U MRS ( c, c ) c u ( c ) c c ( ) MRS y y / U / u ( c ) c c c / (, ) ( ) y y / line ( y, y) (5,36), y / 6 MRS ( y, y) ( ) y 5 John Riley Ocober 9, 08

18 Microeconomic Theory -8- Choice over ime Equilibrium rading (c) How do we add sorage o he model? Hin: Wha is he echnology? John Riley Ocober 9, 08

19 Microeconomic Theory -9- Choice over ime Group problem Example : Exchange economy wih wo periods A B { y } {6,4} { y } {0,}, Assume and ha goods canno be sored (a) Are here poenial gains from exchange? line (b) Solve for he equilibrium ineres rae. (c) How does he oucome change if period goods can be sored a no cos and consumed in period? John Riley Ocober 9, 08

20 Microeconomic Theory -0- Choice over ime Cosless sorage A very simple example of consan reurns o scale. Each uni sored o day yields one uni omorrow. The se of feasible plans is Y {( z, q) q z} line slope = - John Riley Ocober 9, 08