Economics 202a, Fall 1998 Lecure 3: Solow Model II Handou Basics: Y = F(K,A ) da d d d dk d = ga = n = sy K The model soluion, for he general producion funcion y =ƒ(k ): dk d = sƒ(k ) (n + g + )k y* = ƒ(k*) Y * y * K * k * K * = z* = k * Y * y * Y * y * K * k * The model soluion, for he specific producion funcion y =k : K = k s = z Y y = n+ g + + K 0 s 1)(n +g + ) e( Y 0 n + g + Y Y ( ) 1 z z ( ) 1 y = ( z ) 1 There are sill oher hings ha we wan o look a: for example, he real wage. If we assume ha facors of producion are paid heir marginal producs, hen he real wage a any ime is:
w = d(f(k, A L )) = A dl ƒ(k ) K ƒ'(k L ) r = d(f(k,al )) = ƒ'(k dk ) w +r K ƒ(k ) = Y As i should from consan reurns o scale. An alernaive way o wrie he real wage is: w [ƒ(k ) k ƒ'(k )] r = ƒ'(k ) which makes i more clear ha he real wage is growing a rae g, and he real reurn o capial is consan in seady-sae growh. If we resric our aenion o Cobb-Douglas producion funcions, hen: w [ƒ(k ) k ƒ'(k )] [k k k 1 ] w = (1 )A k = (1 )A z 1 r = ƒ'(k ) = k 1 = z 1 In oher wor, a share α of oupu is going o capial--and is evenly spli as he reurn o capial. A share 1 α of oupu is going o labor--and is evenly spli as well. And now le s look a wha happens when we change he parameers of he model: I. A change in he savings rae s: Le s ake α =.5; n=g=.01; δ=.03; s sars a.15 and jumps o.20. Sar a ime zero wih A 0 =L 0 =1... Our seady-sae growh pah hen jumps from z* = 3, wih Y growing a.02 per year, o z* = 4, wih Y growing a.02 percen per year--a 33.3% boos o seady-sae capial per effecive worker, and a 33.3% boos o oupu per effecive worker. How fas does his economy converge o is seady sae? (1- α)(n+g+δ) =.025 equals 2.5 percen per year. So he 1/e ime--he ime afer he ime 0 jump in he savings rae for he capial-oupu raio o close he gap o is new seady-sae value o 1/e of is iniial value--is 40 years.
Capial-Oupu Raio: Convergence 5 Capial-Oupu Raio 4 3 2 1 0-20 0 20 40 60 80 100 Time
Convergence of Oupu per Worker o Seady Sae 12 10 8 Oupu 6 4 2 0-20 0 20 40 60 80 100 Time
Consumpion Per Worker 9 8 Consumpion per Worker 7 6 5 4 3 2 1 0-20 0 20 40 60 80 100 Time
Upward Shif in Savings Rae 1 0.9 0.8 0.7 s*ƒ(k), (n+g+d)k 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2.5 5 7.5 10 12.5 15 17.5 20 k Level effec bu no a growh effec... Does consumpion rise when he savings rae rises? Answer: a he beginning, cerainly no. Evenually--i depen. Consumpion per effecive worker along a seady-sae growh pah: c* = y*(1-s) = y* - (n+g+δ)k* Wha happens if we differeniae c*(s, n, g, δ) as a funcion of he model parameers wih respec o he savings rae s? dc * = d ƒ(k*) d (n + g + )k * dc * dc * = ƒ'(k*) dk * (n + g + ) dk * = ƒ'(k*) (n + g + ) [ ] dk *
where, as I said before, he derivaives are aken undersanding he seady-sae values o be funcions of he model parameers. For he sign of he effec of an increase in he savings rae on seady-sae consumpion, we can ignore he las erm. I will be posiive. Wheher he whole righ-hand side is posiive depen on comparing ƒ (k*) o (n+g+δ)--depen on wheher he marginal produc of capial a he seady sae (for ha is wha ƒ (k*) is) is greaer or less han he sum: (n+g+δ). I can go eiher way. If ƒ (k*) < (n+g+δ), hen he economy is called dynamically inefficien--consumpion could be raised a all fuure daes by lowering saving oday. If ƒ (k) = (n+g+δ), he economy is said o be a he Golden Rule,which we will alk abou more laer on. Can we ge any furher in evaluaing he effec of an increase in savings on seady-sae consumpion? We can if we assume ha he producion funcion is Cobb-Douglas. In inensive form, hen: y* = k * = (z * y*) y * y * = z * s y* = (z*) 1 = n + g + 1 And seady-sae consumpion per effecive worker: s c* = (1 s) n + g + 1 So consumpion per worker along he seady-sae growh pah is: C * s = A L (1 s) n + g + 1 And he derivaive of c*, aken o be a funcion of he parameers of he model, wih respec o s is: dc * = s n + g + 1 This will be posiive if and only if: 1 + (1 s) 1 1 S > 0 s < α + 1 s (1 s) n + g + n + g + 1 1 1 This gives you a definie es. And i ells you ha: When s is sufficienly low, increases in s increase seady-sae consumpion.
When s his he magic number of he capial share α, you are a he Golden Rule. When s is higher, increases in s decrease seady-sae consumpion and he economy is dynamically inefficien. Increases in α increase he level of s a which he economy becomes dynamically inefficien. II. A change in he populaion growh rae n: Le s ake α = 1/3; n=g=.01; δ=.03; s =.25.; and suppose suddenly n drops o 0 Sar a ime zero wih A 0 =L 0 =1... Our seady-sae growh pah hen jumps from z* = 5, wih Y growing a.02 per year, o z* = 6.25, wih Y growing a.02 percen per year--a 25% boos o seady-sae capial per effecive worker, and an 11.8% boos o seady-sae oupu per effecive worker. How fas does his economy converge o is seady sae? (1- α)(n+g+δ) =.026667 equals 2.67 percen per year. So he 1/e ime--he ime afer he ime 0 jump in he savings rae for he capial-oupu raio o close he gap o is new seady-sae value o 1/e of is iniial value--is 37.5 years.
Speed of convergence II Le me noe equaion (1.25) on page 22 of Romer s Advanced Macroeconomics: in he viciniy of he balanced growh pah, capial per uni of effecive labor converges oward k* a a speed proporional o is disance from k*... [wih] λ = (1-α k )(n+g+δ). Why develop his when we already have our exac soluion for z? Because our soluion for z is only for he Cobb-Douglas case. Wha Romer produces is much more general: ha if he economy is near is seady-sae growh pah, is convergence behavior is nearly he same as Cobb-Douglas. The Solow Model and he Cenral Quesions of Growh Theory We ahve seen ha only growh in he effeciveness of labor can lead o permanen growh in oupu per worker, and ha for reasonable caes he impac of changes in capial per worker on oupu per worker is modes. Thus only differences in he effeciveness of labor have any reasonable hope--in his framework--of accouning for vas differences in wealh across ime and space. If he reurns ha capial comman in he markes are a rough guide o is conribuions o oupu, hen variaions in he accumulaion of physical capial do no accoun for a significan par of eiher worldwide economic growh or cross-counry income differenials. differences in savings raes oo small o accoun for cross-secion differences in raes of reurn on capial observed in he world are no large enough for capial play a big role: a en-fold difference in oupu per worker arising from differences in capial per worker implies a hundredfold difference in he (gross) marginal produc of capial. he Solow model does no idenify wha he effeciveness of labor is. Growh accouning: d d ln Y = d k () d ln K + R() Convergence: if he A s are he same (or become he same), and if s s are similar, han counries should become more alike. Felein-Horioka: does i sill hold? Mankiw, Romer, and Weil--implied α of 0.60 for heir economies in cross-secion (neglecing reverse causaliy). II. Classical and NeoClassical Growh Models Thursday Jan 22: The Solow Growh Model Lecure 2 handous: Solow model and problem se 1
David Romer, Advanced Macroeconomics, pp. 5-15. Rober Solow (1956), "A Conribuion o he Theory of Economic Growh," Quarerly Journal of Economics 70 (February), pp. 65-94.{Yes} Tuesday Jan 27: The Solow Growh Model II David Romer, Advanced Macroeconomics, pp. 15-33. Rober Solow (1957), "Technical Change and he Aggregae Producion Funcion," Review of Economics and Saisics 39: pp. 312-20. {Yes} Thursday Jan 29: The Solow Growh Model III J. Bradford De Long (1988), "Produciviy Growh, Convergence, and Welfare: Commen" {Yes} Marin Felein and Charles Horioka (1980), "Domesic Saving and Inernaional Capial Flows," Economic Journal 90 (June): pp. 314-329. {Yes} Xavier Sala-i-Marin (1997), "I Jus Ran Four Million Regressions" (NBER working paper 6252). {Yes}