Economic Growh Coninued: From Solow o Ramsey J. Bradford DeLong May 2008 Choosing a Naional Savings Rae Wha can we say abou economic policy and long-run growh? To keep maers simple, le us assume ha he governmen can by proper fiscal and moneary policies se and keep he economy s savings-invesmen rae s a whaever level i wishes. Wha level should he governmen choose for he economy s savings rae? I seems reasonable o assume ha he governmen s objecive is o maximize he wellbeing of he individuals who make up he sociey by maximizing he amoun of goods and services ha hey can consume. Le us, for he momen, simplify hings furher and say ha consumpion C is equal o oal producion Y minus invesmen I: = Y I Where invesmen I is equal o he savings rae s imes oal producion Y: I = s Y So consumpion per worker C/L is equal o: L = ( 1 s)y If we focus our aenion on seady-saes only, seady-sae consumpion per worker on he long-run growh pah is equal o: L s = ( 1 s) n + g + δ 1 E Maximizing Seady-Sae Consumpion per Worker Wha level of he savings rae s should he governmen choose if i wishes he economy o be on he long-run growh pah ha has he highes level of consumpion per worker?
If we look a he rae of change he derivaive of consumpion per worker as a funcion of he savings rae: d ds d ds L L = ( s) 1 + 1 ( 1 s) ( s 2 1 ) 1 E ( n + g + δ) = 1+ 1 s 1 s ( s) E 1 n + g + δ 1 1 The rae of change of consumpion per worker is zero and he level of consumpion per worker is a is highes when: s = This savings rae is called he golden rule savings rae. The associaed seady-sae growh pah is called he golden rule seady-sae growh pah. Anoher way o look a he golden rule seady-sae is o look a he marginal produc of capial he amoun by which an addiional uni of capial booss oupu. The marginal produc of capial is: dy dk = Y K Which is, a he seady-sae growh pah wih K/Y = s/(n+g+δ) = /(n+g+δ), equal o: dy dk = n + g + δ = n + g + δ A he golden rule seady-sae growh pah, he marginal produc of capial is equal o he sum of he populaion growh rae, he efficiency of labor growh rae, and he depreciaion rae. To make he poin anoher way, suppose ha he economy sars wih some seady-sae capial-oupu raio κ and he governmen considers aking seps o increase he savings rae o boos he capial sock by one uni. The amoun by which he change increases producion is he marginal produc of capial n+g+δ. Bu his increase in he capial sock increases he amoun of savings needed o mainain he new, higher capial-oupu raio: n is needed o keep up wih populaion growh, g o mainain pace wih he increased efficiency of labor, and δ o offse depreciaion on he higher capial sock. Thus when he marginal produc of capial is equal o n+g+δ and he savingsinvesmen rae s is equal o hen he exra oupu produced by an increase in he capial-oupu raio is jus equal o he increase in savings and invesmen required o
mainain ha exra increase in he capial-oupu raio. When he capial-oupu raio is less han he golden rule seady-sae, an increase in he capial-oupu raio raises oupu by more han he required increase in savings and invesmen: hus consumpion per worker can increase. When he capial-oupu raio is more han he golden rule seadysae, an increase in he capial-oupu raio does no raise oupu by enough o offse he required increase in savings and invesmen: hus consumpion per worker mus fall. Implicaions for Economic Policy If an economy begins a a seady sae wih a higher capial-oupu raio han he golden rule seady sae, hen consumpion per worker can be increased by reducing he savings rae. A decline in he savings rae will boos he seady-sae level of consumpion per worker, and hus boos consumpion per worker in he long run. Moreover, by cuing back on savings and increasing he funds available for consumpion, consumpion per worker can be increased in he shor run as well. If he economy begins a a seady sae wih a lower capial-oupu raio han in he golden rule, hen he governmen mus ake seps o raise he savings rae in order o reach he golden rule seady sae. In he long run, his increase in he savings rae will boos he seady-sae level of consumpion per worker, and hus boos consumpion per worker in he long run. However, he increase in he savings rae reduces he funds available for consumpion in he shor run. When he economy begins above he golden rule, reaching he golden rule produces higher consumpion a all momens in ime. Bu when he economy begins below he golden rule, reaching he golden rule requires reducing he level of consumpion now and in he near fuure in order o boos consumpion in he long run. A governmen rying o consider wheher o ry o move he economy oward he golden rule seady sae has o consider wheher he long run boos o consumpion ouweighs he shor run cu in consumpion. The governmen mus decide wheher his radeoff beween he near fuure and he disan fuure is worhwhile. How can we figure his ou? We need an explici framework: a uiliarian framework he Ramsey Model. The Ramsey Model Begin wih an objecive funcion social-welfare or represenaive-agen uiliy, as a funcion of per-capia consumpion over ime: max e ρ = 0 c 1 1 1 Two parameers: and ρ.
Consider increasing savings and hen decreasing i in order o pospone a small amoun of consumpion for a small period of ime, and hen reurning o he previously-scheduled pah for consumpion. Then we have, a he opimal plan: ( δc) ( c ) + δc Solve he algebra: And arrive a: ( e ρ(δ ) )( e n(δ ) )( c +δ ) = 0 1+ r( δ) ( 1+ r( δ) )( e ρ(δ ) )( e n(δ ) ) c +δ c =1 ( 1+ r( δ) )( e ρ(δ ) )( e n(δ ) ) c + δ ( 1+ r( δ) ) 1 (ρ + n)(δ) dc c 1+ 1 dc ( δ) c ( 1+ r( δ) ) 1 (ρ + n)(δ) 1 1 dc ( δ) c =1 =1 =1 r (ρ + n) = c dc This ells us ha he governmen s fiscal policy should be o adjus he naional savings rae so ha per-capia consumpion grows according o: dc = ( r (ρ + n) )c Wha is r? r is he social marginal produc of capial: r = Y δ = Y δ K K So he righ hing for he governmen o do is o change spending and axes unil consumpion is growing a a rae (r-(ρ+n))/. Wha is r? 6% + 3% axes + 3% labor rens + 3% exernaliies = 15% Wha is ρ? Impaience? Or should ρ be n? And wha if we recognize ha n is endogenous? There are unsolved problems in he heory of applied uiliarianism
Wha is? is somewhere beween one and 3 Solving he Ramsey Model Wha do hese equaions imply for he ime pahs of he variables in our economic growh model? To solve he model, sar wih he firs-order condiion for uiliy maximizaion as a funcion of per-capia consumpion in he Ramsey model: 1 dc c = r (ρ + n) scale up o aggregae consumpion: 1 d = r (ρ + n) + n Subsiue in for he marginal produc of capial using he Cobb-Douglass producion funcion: 1 d = 1 Y δ (ρ + n) + n K Now consider he evoluion of he capial sock: 1 dk K = Y K K δ And combine i wih he uiliy-maximizaion condiion o ge he evoluion of he consumpion-o-capial raio: 1 d /K /K 1 d /K /K = 1 d 1 K dk = 1 Y + 1 1 δ K = 1 Y δ (ρ + n) + n Y C δ K K K (ρ + n) + n + K A his poin we can eiher bull ahead hrough he model, or we can aemp o solve he model for a convenien-bu ineresing special case by noicing ha if = hen Y/K vanishes from he equaion above. Le s do he special case firs.
Following William Smih (2006) William Smih (2006) is he firs o noe ha he Ramsey model, wih he ransversaliy condiion imposed, collapses ino a one-dimensional linear sysem if we impose = on he parameers. The firs-order condiion for he consumpion-o-capial raio is: 1 d /K /K = 1 (ρ + n) δ + n + K because he Y/K erms cancel. Thus he only possible non-explosive and hence admissible soluion has he consumpion-o-capial raio consan a: = 1 (ρ + n) δ + n K Recall our equaion for he Solow-model evoluion of he capial-oupu raio, which ells us ha he rae of change of he capial-oupu raio is linear in he capial-oupu raio iself and he savings rae: d(k /Y ) = dκ = (1 )(s (n + g + δ)κ ) If he consumpion-o-capial raio is consan, hen he savings rae is linear in he capial-oupu raio: s =1 Y =1 C κ =1 1 (ρ + n) δ + n κ K And so we are down o a one-dimensional linear sysem, which we can solve exacly: d(k /Y ) dκ = dκ = (1 ) 1 1 (ρ + n) δ + n κ (n + g+δ )κ = (1 ) 1 δ + ρ + n +g κ κ = κ 0 + δ + n + ρ +g κ (1 )(δ + n + ρ +g) 0 exp And he complee soluion o he model is:
L = L 0 exp(n) E = E 0 exp(g) Y /L = E ( κ ) 1 K /L = E ( κ ) 1 1 /L = ( /K )(K /L ) = 1 δ + ρ + n n E κ s =1 /Y =1 1 δ + ρ + n n κ wih balanced growh-pah values: 1 κ* = δ + n + ρ + g ( /K ) * = 1 δ + ρ + n n ( s* =1 ( /Y ) * =1 1 )δ + ρ + n n ρ + n + δ + g ( s* =1 1 )δ + n + ρ n (n + g + δ) = ρ + n + δ + g ρ + n + δ + g Compare his o he Golden Rule values: κ gr * = n + g + δ s gr * = n + g + δ Le s consider some sample parameers: = = 2/3, δ = 0.05, n = 0.01, g = 0.03, ρ = 0.03: κ* = δ + n + ρ + g = 6.06 ( /K ) * = 1 δ + ρ + 1 n =.075 ( s* =1 ( /Y ) * =1 1 )δ + n + ρ n ρ + n + δ + g ( s* =1 1 )δ + ρ n (n + g + δ) = ρ + δ + g ρ + n + δ + g =.06.11 =.545 1
Compare his o he Golden Rule values: ω* =.045 κ gr * = n + g + δ = 7.41 s gr * = =.67 Of paricular ineres is he behavior of he savings rae relaive o is Golden Rule value: Savings Rae: Ramsey and Solow-Golden Rule 90% 80% Savings as a Share of Oupu 70% 60% 50% 40% 30% 20% 10% 0% 0 10 20 30 40 50 60 70 80 90 100 Time s s (golden rule) And he behavior of he capial-oupu raio:
Capial-Oupu Raio: Ramsey and Solow-Golden Rule 800% 700% Savings as a Share of Oupu 600% 500% 400% 300% 200% 100% 0% 0 10 20 30 40 50 60 70 80 90 100 Time kappa kappa (golden rule) Bulling Ahead Alernaively, we can bull ahead. Le s analyze his model in erms of he wo sae variable pair made up of he capial-oupu raio κ and he consumpion-capial raio, which we will call ω: The laws of moion of hese wo variables are: 1 dω ω dκ = ω 1 δ + ρ + 1 n + 1 1 κ = ( 1 ) ( 1 (ω + n + g + δ)κ ) wih seady-sae values if we impose he ransversaliy condiion of: ( ω * = 1 )δ + ( 1 )n + ( )g + ρ κ * = δ + g + ρ + n [ ]
The seady-sae growh pah for he model is: L = L 0 exp(n) E = E 0 exp(g) Y * /L = E δ + g + ρ + n K * /L = E δ + g + ρ + n 1 C * /L = ω *(K /L ) = 1 s* = δ + g + n δ + g + ρ + n 1 1 δ + ρ + g + 1 n δ + g + ρ + n 1 1 E Rewriing he algebra around he seady-sae produces he somewha ineresing form: 1 dω ω dκ + 1 = ω ω * = 1 1 1 κ κ * ( δ + g + ρ + n) ( κ κ * ) ( 1 ) ( ω ω * )( κ κ * ) + κ * ( ω ω * ) Compare his o he speed of convergence in he Solow model: dκ ( n + g + δ) ( κ κ * ) = 1 The ne raes-of-reurn in he Solow model along he Golden Rule balanced growh pah and in he Ramsey model are: r gr * = n + g Discussion r R * = n + ρ + g References: William T. Smih (2006), "A Closed Form Soluion o he Ramsey Model," Conribuions o Macroeconomics 6:1 hp://www.bepress.com/bejm/conribuions/vol6/iss1/ar3