Problem Set 2 Suggested Solutions

Transcription

1 4.472 Prolem Set 2 Suggested Solutions Reecc Zrutskie Question : First find the chnge in the cpitl stock, k, tht will occur when the OLG economy moves to the new stedy stte fter the government imposes one time lump sum tx on the current young genertion to permnently lower pulic per cpit det, g. The two equtions we need for this exercise re the government uet constrint relting the lump sum tx on the young, T, to the per cpit det level nd the mrket clernce eqution (which sttes tht the cpitl stock in given period equls the svng done y the current old genertion). These equtions re: T (r n)g () ( + n)(k + g) w( t) T c [w( t) T + + r, r] + f (2) To find dk, we cn just implicitly differentite the mrket clernce eqution (2) (we ssume first period mrginl propensity to consume,c Y, is positive nd less thn one). dt ( + n)(dk + ) + c dt Y dk dt ( c + n Y ) dk r n + ( c + n Y ) < This expression sys tht the mount of cpitl per young worker in the new stedy stte increses in two

2 possile wys. First, since the government is offering fewer onds, young workers will put their svings into privte cpitl insted of government onds. This effect lone increses privte cpitl one-for-one with decrese in the pulic det (the thought experiment keeps svings y the young the sme s it ws in the old stedy stte). Second, lowering the lump sum tx gives young workers higher income, of which they sve ( c Y ) (i.e. one minus the mrginl propensity to consume). This second effect cuses the new stedy stte cpitl per young worker to increse y more thn the decrese in the pulic det per young worker, since it increses the level of sving y the young. To find the effects of lowering the pulic det on utility, we cn just differentite the indirect lifetime utility functions of people in our economy. Cll Vu(c, (w( t) T c )( + r) + ) the lifetime indirect utility function of people in our economy. There re two groups of people we need to consider when exmining the effect of the chnge in government policy on utility - () the current young genertion who must finnce the one time decrese in the det nd (2) ll the future young genertions. (The current old genertion is unffected y the policy chnge.) The chnge in lifetime utility of the current young is (for ll derivtives of utility wrt g, I m ctully tking the negtive of these derivtives ecuse we re decresing pulic det): du du (r n)c Y ( + r)(r n)( c Y ) < 0 dc dc 2 The chnge in lifetime utility of the future young is dv fy du du (r n)c Y + ( + r)(r n)( c Y ) > 0 dc dc 2 To nswer the normtive question of should the government implement the policy, we d need to know how the future sum of young genertions utility compred with the loss in utility of the current young genertion in the socil welfre function. SW F δ i V i dsw F dv fy + δ i dv fy dv fy δ dv fy 2δ dv fy + δ δ 2

3 If dsw F > 0 then the government should implement the policy if the government s ojective is to mximize the socil welfre function. Now let s look t wht hppens when the government temporrily increses the lump sum tx on the current young to permnently increse funding of socil security. Agin, to find the chnge in the stedy stte cpitl stock we cn differentite the mrket clernce eqution (2). d ( + n)dk c + r Y + dk d c ( + n) + r Y Given the socil security uet constrint: tw( + n) + (r n)f (3) We cn get dk r n r n + (r n)( c Y ) + n + r c Y + n ( + n)( + r) ( + n)( + r) + (r n)( c Y ) + (r n)( c Y ) + r ( + n)( + r) + r + n dk + r We cn see tht oth policies (decresing the pulic det nd incresing socil security funding) hve similr effects on the new stedy stte cpitl stock. Incresing funding of socil security The +r just picks up timing difference. To fund n increse in the funding of socil security, the lump sum tx must increse y +r, not, since the lump sum tx will e invested in privte cpitl nd ccrue interest for one period, giving n increse in funding of. By incresing the enefits workers receive, future workers now hve more 3

4 lifetime income. This cuses them to consume more in the first period; they will lso consume more in the second period s well. Now consider the chnge in lifetime utility of the current young genertion nd future young genertions. du r n du du du dc + r c Y dc2 (r n)( c Y ) + dc2 (r n) + r + dc2 (r n) < 0 dv fy du r n c Y + du dv fy (r n)( c dc + r dc Y ) 2 + r > 0 Notice tht the utility loss to the current young genertion is less thn when the government decreses the pulic det. This is ecuse the current young genertion gets extr consumption in retirement s result of the higher socil security enefits from the increse in funding. The future young genertions re unequivoclly etter off. Question 2: Now only frction α of ech young genertion sves in lifetime consistent mnner. The rest consume their fter-tx income in the first period nd consume only socil security enefits when old. The mrket clernce eqution is now given y: ( + n)(k + g) α w( t) T c [w( t) T +, r] + f (4) + r The government uet constrints remin the sme s in equtions () nd (3). To find the effect on stedy stte cpitl of decrese in the pulic det we differentite the new mrket clernce constrint: dt + c dt ( + n)(dk + ) α Y dk α(r n)( c Y ) + n 4

5 The cpitl stock increses with decrese in pulic det, ut y less thn in question ecuse there re fewer svers (i.e. the second effect of decrese in pulic det cusing n increse in the level of privte sving is mitigted y fctor of -α). In clculting utility effects we now need to consider four groups of people - () current young svers; (2) current young non-svers; (3) future young svers; (4) future young non-svers. The lifetime indirect utility function of svers is V sve u(c, (w( t) T c )( + r) + ). The lifetime indirect utility function of non-svers is V non sve u(w( t) T, ). The effect on current young svers is: du du sve (r n)c Y ( + r)(r n)( c Y ) < 0 dc dc 2 The effect on current young non-svers is: non sve du (r n) < 0 dc The effect on future young svers is: dv sve fy dv sve cy The effect on future young non-svers is: dv fy non sve non sve Now let s consider permnent increse in socil security funding: 5

6 d ( + n)dk αc Y + ( + n) dk αc r n α(r n) + α(r n)( c Y ) + αn + ( α)r + α(r n)( c Y ) Y + r + r + r + r + r dk dk ( + r)( + n) { + αn + ( α)r + α(r n)( c Y )} > (r > n) + r Now incresing socil security funding increses the cpitl stock more thn decresing the pulic det ecuse incresing funding effectively forces non-svers to sve more. The utility effect on the four groups: sve du r n du du dv sve cy du dc + r c Y (r n)( c Y ) + (r n) + (r n) < 0 dc2 dc2 dc2 non sve du r n du dc + r + (r n) >< 0 dc 2 dv sve fy du r n du dc + r c Y + (r n)( c Y ) > 0 dc2 dv fy non sve dc 2 du (r n) > 0 The welfre effect on current young non-svers is miguous. Forcing them to sve more my ctully mke them etter off; however, it my mking n lredy d sitution worse, it these people y not sving re mximizing lifetime utility (this would require tht non-svers hd different u(, ) thn svers). Svers hve the sme utility chnges s in question. The current svers re worse off, while ll future svers re etter off. Question 3: Now we hve Leontief production function, so the per cpitl cpitl stock is fixed in equilirium t y, y nd output per cpit is q min{y, } y. The equilirium equtions re given y (ssume perfect competition nd price of output is ): 6

7 w q(k) rk y ry (5) T (r n)g (6) t(y ry )( + n) + (r n)f (7) ( + n)(k + g) (y ry )( t) T c [(y ry )( t) T + ] + f (8) + r We wnt to find the effect decrese in the pulic det will hve on the stedy stte interest rte (which is endogenous since cpitl is fixed y the production technology). We cn gin implicitly differentite the mrket clernce eqution (8). ( + n) ( t) y dr gdr (r n) c { ( t) y dr gdr (r n) + d Y + r dr ( + r) 2 dr} c r dr d t( + n) y dr + fdr dr { [ ] [ ] } { + n + ( c Y )(r n)} ( c Y ) ( t) y g + c Y + r (t( + n) y f ) + c ( + r) 2 r dr dr + n + ( c [ ] [ Y )(r n) ] >< 0 (I think) ( c Y ) ( t) y + g + c Y +r (t( + n) y f ) + (+ r) c 2 r The lifetime indirect utility function is V u(c, (y r y )( t) T c + ). Welfre effects re miguous. There re now fewer government onds nd the privte cpitl stock is fixed, so there re fewer plces to put svings. This my cuse the interest rte to increse nd the wge to decrese (wge decrese only for future workers). The current genertion who is txed my enefit in the second period from the higher interest rte. The future genertion will e hppier if the interest rte is higher, ut lower wge will mitigte this hppiness. 7