Economics 6130 Cornell University Fall 2016 Macroeconomics, I - Part 2

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1 Economics 6130 Cornell Universiy Fall 016 Macroeconomics, I - Par Problem Se # Soluions 1 Overlapping Generaions Consider he following OLG economy: -period lives. 1 commodiy per period, l = 1. Saionary endowmens: Saionary preferences: Passive fiscal policy: ω 1 0 = B > 0 for = 0 (ω, ω +1 ) = (A, B) > 0 for = 1,,... u 0 (x 1 0) = D log(x 1 0) for = 0 u (x, x +1 ) = C log(x ) + D log(x +1 ) for = 1,,... m 1 0 = ms = 0 oherwise Goods price of money is p m 0. For each of he following cases: Calculae he offer curve for Mr. 1. Precisely plo (use graph paper if necessary) he offer curve in excess demand space (x ω, x +1 ω +1 ) for Mr. 1. Plo he refleced offer curve, and analyze he global dynamics. (a) A = 10, B = 1, C = 1, D =

2 (b) A = 15, B = 10, C =, D = 3 (c) A = 40, B = 30, C = 0.5, D = 0.5 (d) A = 8 B = 4, C = 1.9, D = 0.95 Is here a paern? Derive he condiions on he MRS for a Samuelson verses a Classical (or Ricardo ) economy and relae hem o he above. Soluion: The problem facing an agen born in dae 1 is (x,x+1 max,x,m,x +1,m ) s.. p x + p +1 x +1 C log x + D log x +1 + p m (x,m + x +1,m ) p A + p +1 B Taking firs order condiions and dividing hrough implies ha p p +1 = Cx+1. Dx We hen divide he budge consrain of a dae household by p +1 and plug his in o ge ha x +1 B = Cx+1 Dx (A x ) For he res of he problem, le z +1 be he excess demand of an old household. Le z be he excess demand of a young household, and s = z be he excess supply of a young household. Plugging his in above yields ha z +1 = C(z+1 + B) D(z + A) z Solving his for z +1 yields he offer curve: z +1 = CBz (D + C)z + DA To solve for he inverse offer curve we use ha s = z, which yields z +1 = CBs DA (D + C)s Below we calculae and plo for each case, he offer curve in excess demand space, he refleced offer curve, and he resource consrain.

3 (a) A = 10, B = 1, C = 1, D = 0.98 The offer curve: z +1 = 1z 1.98z z +1 OC 1 1 z 1 The refleced offer curve: z +1 = 1s s 1 z +1 ROC 1 1 s 1 (b) A = 15, B = 10, C =, D = 3 The offer curve: z +1 = 3 0z 5z + 45

4 5 z +1 OC z 5 The refleced offer curve: z +1 = 0s 45 5s 5 z +1 ROC s 5 (c) A = 40, B = 30, C = 0.5, D = 0.5 The offer curve: z +1 = 15z z + 0 4

5 6 4 z +1 OC The refleced offer curve: z +1 = z 15s 0 s 6 4 z +1 ROC (d) A = 8 B = 4, C = 1.9, D = 0.95 The offer curve: z +1 = s 7.6z.85z

6 1 z +1 OC z The refleced offer curve: z +1 = 7.6s s 1 z +1 ROC s The global dynamics we ge are ha of Ricardo in (a) and (d). We ge he global dynamics of Samuelson in (b) and (c). The equaion we ge from he MRS is ha we have Ricardo whenever CB DA, and we ge Samuelson whenever CB < DA. Overlapping Generaions Consider he following OLG economy: 6

7 Pure exchange, -period lives, one consumer per generaion. u 0 (x 1 0) = x 1 0 ω 1 0 = 1, for = 0, Money ransfers: u (x, x +1 ) = x + x +1 (ω, ω +1 ) = (1, 1) for = 1,,... m 1 0 =, m 1 1 = 1, m 1 = 1, m s = 0 oherwise. (a) Wha is he non-moneary equilibrium allocaion? prices? Wha are he ineres raes? Wha are he (b) Derive he refleced offer curve for consumer = 1,,... (c) Derive he se of equilibrium money prices. (d) Draw he phase diagram and show he full evoluion of his economy (depending on he price of money). (e) Wha is he Pareo opimal allocaion associaed wih he above (money) ax-ransfer policy? (f) Find an alernaive ax-ransfer policy and associaed allocaion which is no Pareo opimal bu in which everyone is sricly beer off han hey would be in auarky. (g) Find an alernaive ax-ransfer policy and associaed allocaion which is Pareo opimal and in which everyone is sricly beer off han hey would be in he non-moneary equilibrium. Soluion: (a) When here is no money, he iniial old problem is max x 1 x s.. p 1 x 1 0 p 1 7

8 And he problem of a person born in dae is max x (x + x +1,x+1 ) s.. p x + p +1 x +1 p + p +1 Normalizing he price of money in dae 1 o be equal o 1, we ge ha he iniial old choose x 1 0 = 1. Now looking a he dae one budge consrain we mus have ha x 1 1 = 1, which from he budge consrain of he person born in dae 1 implies ha x 1 = 1. And we see ha his argumen goes on ad infinium. So our equilibrium allocaion is (c 1 0, {c, c +1 }) = (1, {1, 1}). And from he firs order condiions of he dae generaion (since we are a an inerior soluion) we can ge he prices, p = 1. Which implies he ineres rae is 1 + r = 1. (b) The problem of he iniial old is now max x 1 x s.. x p m x 1,m p m And he problem of generaion is now max x (x + x +1,x+1 ) s.. p x + p +1 x +1 + p m (x,m + x +1,m ) p + p +1 + p m (m + m +1 ) We know ha p m (x,m + x +1,m ) = x 1,m 0 = 0 in equilibrium. And for every dae we noe also ha (m + m +1 ) = 0. Firs order condiions again imply ha p = 1. We noe ha he soluion for he iniial old is now x 1 0 = 1+pm which implies by he resource consrain ha x 1 1 = 1 pm. We coninue his ad infinium as in par (a) o drive a he soluions (x 1, x ) = (1 + p m, 1 p m ). We now ge ha he refleced offer curve from knowing x +1 = x, which implies our refleced offer curve, z +1 = s. Graphed below 8

9 10 5 z +1 OC s 10 (c) The equilibrium se of money prices mus be such ha no individuals consumpion is negaive. This implies we mus look a he consumpion of he young. So we need ha x = 1 p m 0. Which implies our equilibrium se of money prices is p m [0, 1 ]. (d) The phase diagram is as above, noing ha he offer curve in his case is exacly equal o he resource consrain. Thus, for any price of money in he equilibrium se, his picks ou a saionary poin on he phase diagram. (e) The Pareo opimal allocaion associaed wih he above policy is when p m = 1, corresponding o he allocaion x1 0 = and (x, x +1 ) = (0, ). The uiliy of he iniial old generaion is maximized (u 0 =, up from u 0 = 1 in auarky), and every dae- generaion is made no worse off. (f) Consider he ax and ransfer policy such ha he young in dae are axed an amoun of goods τ = 1 (1 ( 1 ) ) and he ransfer is τ o he old. This allocaion will make everyone sricly beer off, bu will no be Pareo Opimal, as we will see in par (g). (g) We re looking for a soluion similar o (f) bu Pareo Opimal his ime, so we re looking for a ax-ransfer sysem ha converges o τ = 1. Such a ax on he young in dae is τ = 1 ( 1 ) and he ransfer o he old of τ. Every generaion is sricly beer off han hey would be in he non-moneary equilibrium (i.e., auarky). The uiliy of he 9

10 dae- generaion is now ( ( 1 ) ( ( 1 +1 ) u = ) ) ( 1 ) ( 1 +1 = + ) which is an improvemen from our soluion o (f), where ( u = ( 1 ) ) = + 1 ( (1 ( 1 ) ( + ) +1 ), ( 1 ) +1 ) and o auarky, where u =. This also underscores ha our soluion o (f) is sricly Pareo improving relaive o auarky. 10