MACROECONOMIC THEORY T. J. KEHOE ECON 85 FALL 7 ANSWERS TO MIDTERM EXAMINATION. (a) Wh an Arrow-Debreu markes sruure fuures markes for goods are open n perod. Consumers rade fuures onras among hemselves. An Arrow-Debreu equlbrum s sequene of pres p, p, p, and onsumpon levels,,, ;,,, suh ha Gven p, p, p,, onsumer,,, hooses,,, o solve s.. p max log pw. w w,,,. (b) Wh sequenal marke markes sruure, here are markes for goods and bonds open every perod. Consumers rade goods and bonds among hemselves. A sequenal markes equlbrum s sequenes of neres raes r,, onsumpon levels,,, ;,,,, and asse holdngs b, b, b,... ; b, b, b,... suh ha Gven r,, he onsumer,,, hooses,,, ; b, b, b,... o solve b Here oherwse bnd n equlbrum. max log s.. b w b B, b w r b,,, b. B, where B s hosen large enough, rules ou Ponz shemes bu does no w w,,, b b,,,
() Proposon : Suppose ha p, p, p, ;,,, ;,,, s an Arrow- Debreu equlbrum. Then r, ;,,, ;,,, ; b, b, b,... ; b, b, b,... s a sequenal markes equlbrum where p r p b w ( ) b w r b,,,. Proposon : Suppose ha r, ;,,, ;,,, ; b, b, b,... ; b, b, b,... s a sequenal markes equlbrum. Then p, p, p, ;,,, ;,,, s an Arrow-Debreu equlbrum where p p,,,. ( r ) (d) Usng he wo onsumers frs order ondons we an wre In every perod, Sne hs mples ha ondon o wre whh mples ha Consequenly, p,.. s onsan, we an normalze p and use he frs order p p,.
whh mples ha p pw, ( ) p p. ( ). (We an even work ou and, alhough he queson does no requre hs and would be a wase of preous me o do so durng he exam. Chek:..) ( ) To alulae he sequenal markes equlbrum, we jus use he formulas from proposon n par. For example, p r. p Noe ha, n,
b. b b Consequenly, onsumer lends /( ) n even perods, and onsumer pays bak ( r) b n odd perods. Tha s, ( b, b, b, b4,...),,,,... ( b, b, b, b4,...),,,,.... (e) A sequenal markes equlbrum s sequenes of neres raes r,, onsumpon levels,,, ;,,,, asse holdngs b, b, b,... ; b, b, b,..., and sorage levels x, x, x,...; x, x, x,...,suh ha Gven r,, he onsumer,,, hooses,,, ; b, b, b,... ; x, x, x,... o solve b Here oherwse bnd n equlbrum. log max s.. b x w b x w r b x,,, b B,, x. B, where B s hosen large enough, rules ou Ponz shemes bu does no x x w w x x w w x x,,, b,,, b The sorage ehnology wll no be used n he equlbrum beause savng n he good, whh has a gross reurn of, s domnaed by savng n asses, whh has a gross reurn of r. To see hs, ake he frs-order ondon of he onsumer s problem wh respe o b, ( r ) 4
and he frs order ondon wh respe o x :, f x. Sne n he equlbrum of he model whou sorage, r /, whh mples ha x. ( ), r. (a) An Arrow-Debreu equlbrum s a sequene of pres p, p, and an alloaon,(, ),(, ), suh ha Gven p, onsumer hooses ĉ o solve max s. p pw m. Gven p, p, onsumer,,,, hooses (, ) o solve s.. max log p p pw p w,. w w,,,. (b) Wh sequenal marke markes sruure, here are markes for goods and asses open every perod. The onsumers n generaons and rade goods and asses among hemselves. A sequenal markes equlbrum s a sequene of neres raes r,, an alloaon,(, ),(, ),, and asse holdngs b, b, suh ha Consumer hooses ĉ o solve max s. w m. 5
Gven r, onsumer,,,, hooses (, ) and b o solve max log s.. b w ( w r ) b,. w w,,,. b m, b ( r ) m,,,. () Sne here s no fa money, here s only one good per perod, here s only one onsumer ype n eah generaon, and onsumers lve for only wo perods, he equlbrum alloaon s auarky: ĉ w (, ) (, ) w w The frs order ondons from he onsumers problems n he Arrow-Debreu equlbrum are p, f. p Sne w, he seond ondon holds wh equaly. Consequenly, p p p p w p w. p Normalzng p, we oban ( /) p w. Smlarly, he frs order ondons from he onsumers problems n he sequenal markes equlbrum, mply ha r w or r / w. Sne he equlbrum alloaon s auarky, b. (d) An alloaon,(, ),(, ), s feasble f 6
w w,,,. An alloaon s Pareo effen f s feasble and here exss no oher alloaon,(, ),(, ), ha s also feasble and sasfes wh a leas one nequaly sr. log log,,,..., If ( w, w) (,), he equlbrum alloaon s Pareo effen. We prove hs by onradon. If he equlbrum alloaon,(, ),(, ), s no Pareo effen, hen here s anoher feasble alloaon ha,(, ),(, ), s Pareo superor o,(, ),(, ), If log log, hen p p pw p w. Oherwse, (, ) would no solve he maxmzaon problem of generaon. Smlarly, mples p pw. Suppose ha bu ha Then le and. Then bu log log p p pw p w. pw p w p p p log log p p pw p w. One agan, hs would mply ha (, ) would no solve he maxmzaon problem of generaon, whh s mpossble. Consequenly, p p pw p w. 7
Smlarly, mples p pw. Therefore p pw p p pw p w,,,..., wh a leas one nequaly sr. Addng hese nequales up, we oban I s here ha onverge. p ( w /). p ( ) ( p ww), where w / p w w w w w w plays s role n ensurng ha hese seres w w ( ) ( /) ( ) 4 Mulplyng he feasbly ondon n perod by p and addng up yelds p ( ) ( p w w) 4, whh s a onradon. (e) A sequenal markes equlbrum s a sequene of neres raes r,, an alloaon,,(, ),(, ),(, ),(, ), and asse holdngs b, b, b, b suh Consumer hooses,,, o solve max u ( ) s. w m. Gven r, onsumer,,,,,, hooses (, ) and b o solve max u(, ) s.. b w w ( r ) b,. w w w w,,,. b b m m b b ( r ) ( m m ),,,. 8
. (a) An Arrow-Debreu equlbrum s sequenes of pres of goods p, p, p,, wages w, w,, renal raes r,, onsumpon levels,, ;,,, and apal soks k, k, ; k, k, suh ha Gven p, p, p,, w, w,, and r, onsumer hooses,, o solve max log s.. p w ( p ( ) r ) k. (Here we have he onsumers sell her nal apal o frms and have frms make apal aumulaon desons. If we have onsumers make apal aumulaon desons, hen onsumers hoose k,, and he budge onsran s k p ( ) k k w rk.) r p( k k ) ( ) p( k k ),,, w p k k,,, ( ) ( ) ( ( ) ), f k,,, p r p (A good answer would explan ha hese are he prof maxmzaon ondons for onsan reurns. Noe ha, f we have onsumers make apal aumulaon desons, hen he zero prof ondon on aumulang apal s a frs order ondon for uly maxmzaon and does no need o be nluded as a separae equlbrum ondon.) k k ( )( k k ) ( k k) ( ) ( k k),,, (b) A sequenal markes equlbrum s sequenes of wages w, w,..., renal raes k k b b r, r,, neres raes r,, onsumpon levels,, ;,,, apal soks k, k, ; k, k, and bond holdngs b, b, ; b, b,, suh ha k k b b Gven w, w,...,,, onsumer,,, hooses,,,, k, k,,, and b, b, o solve max log k b s.. k b w ( r ) k ( r ) b,,, 9
( ) ( ), k, k k w k k,,, r ( k k ),,, k b B, b. k k ( )( k k ) ( k k ),,, b b,,, () Proposon : Suppose ha p, p, ; w, w, ; r, ;,, ;,, ; k k b b k, k, ; k, k, s an Arrow-Debreu equlbrum. Then r, ; r, ; w, w, ;,, ;,, ; k, k, ; k, k, ; b, b, ; b, b, s a sequenal markes equlbrum where k r r p b k r r w w p b b k ( ) b ( ) w r k r b k,,,. k k b b Proposon : Suppose ha r, ; r, ; w, w, ;,, ;,, ; k, k, ; k, k, ; b, b, ; b, b, s a sequenal markes equlbrum. Then p, p, ; w, w, ; r, ;,, ;,, ; k, k, ; k, k, s an Arrow-Debreu equlbrum where p r,,, r pr k w pw. p k (Noe ha, aordng o he way n whh we have done hngs, we need o use separae symbols for he Arrow-Debreu wage, he pre of labor serves n perod n erms of perod goods, and he sequenal markes wage, of labor serves n perod n erms of perod goods.)
k k (d) A sequenal markes equlbrum s sequenes of renal raes r,, neres b b raes r,, wages w, w,, onsumpon levels,, ;,,, labor levels,, ;,, apal soks k, k, ; k, k, and bond holdngs b, b, ; b, b, suh ha k k b b Gven r,,, and w, w,, he onsumer hooses,, ;,, ; k, k, ;and b, b, o solve log ( ) log( ) k b s.. k b w ( r ) k ( r ) b,,,, k,, B k k, b. b r ( k k ) ( ),,, k w k k,,, ( ) ( ) ( ) k k ( )( k k ) ( k k ) ( ),,, b b,,,