A Three-period Samuelson-Diamond Growth

Blomgren-Hansen, Niels Working Paper A Three-period Samuelson-Diamond Growth Working paper, No. 15-2005 Provided in Cooperation with: Department of Economics, Copenhagen Business School (CBS) Suggested Citation: Blomgren-Hansen, Niels (2005) : A Three-period Samuelson-Diamond Growth, Working paper, No. 15-2005, Copenhagen Business School (CBS), Department of Economics, Frederiksberg, https://hdl.handle.net/10398/7575 This Version is available at: https://hdl.handle.net/10419/208508 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/3.0/ A Three-period Samuelson-Diamond Growth Model Niels Blomgren-Hansen Department of Economics, Copenhagen Business School November 2005 Abstract Samuelson (1958) analyses a three-period model, whereas Diamod (1965) considers a two-period model. This difference poses the question whether the insights derived by analysing the simple two-period model carry over in the more complicated three-period case. They do. The Samuelson model (no productive capital) has only one positive solution (r = n); however, this root is unstable. The Diamond model (no nonproductive abode of purchasing power) has also only one positive solution; the root is stable but inefficient. In a model with both productive capital and a non-productive abode of purchasing power, the inefficient Diamond solution becomes unstable and the socially optimal solution becomes stable. 1Householdandfirm behavior Assumethateachpersonlivesnotjusttwobutthreeperiods.Heworksintwo periods and lives from his saving for yet another period. As in the two-period case (Niels Blomgren-Hansen 2005), a person belonging to generation tmaximizes a simple log-linear utility function U(t)=lnc1t+1 1+ρ·ln c2t+1 (1 + ρ)2·ln c3t subject to his budget constraint, c1t+1 1+rt+1 ·c2t+1 (1 + rt+1)·(1 + rt+2)·ct3≤wt+1 1+rt+1 ·wt+1 1 The f.o.c. is readily derived c1t=µwt+1 1+rt+1 ·wt+1¶·Ã(1 + ρ)2 1+(1+ρ)+(1+ρ)2! c2t=µwt+1 1+rt+1 ·wt+1¶·µ(1 + ρ)·(1 + rt+1) 1+(1+ρ)+(1+ρ)2¶ c3t=µwt+1 1+rt+1 ·wt+1¶·µ(1 + rt+1)·(1 + rt+2) 1+(1+ρ)+(1+ρ)2¶ The firms’ problem is unaffected of the number of periods. They maximize their profit per worker with respect to the capital stock per worker in each period, πt=yt−kt·rt Assuming a C-D production yt=kα t the f.o.c. may be expressed in each of the following forms rt=α·kα−1 t;kt=µα rt¶1 1−α ;kt=α·yt rt wt=(1−α)·kα t=(1−α)·µα rt¶α 1−α 2 Diamond solution The size of the population ’born’ in period tis normalized to 1. The population increases by a constant rate of growth, n. Totalproductioninperiodtis Yt=yt·(1 + 1 1+n)=kα t·(1 + 1 1+n)=µα rt¶α 1−α ·(1 + 1 1+n) The crucial assumption made by Diamond (1965) is that the stock of productive capital in period t+1is equal to the wealth (accumulated purchasing power) at the end of period t Kt+1 =Wt 2 Kt+1 =kt+1 ·(1 + (1 + n)) = µα rt+1 ¶1 1−α ·(1 + (1 + n)) Wt=s1t+s2t−1·1+rt 1+n+s1t−1·(1 + rt)·(1 + rt−1) (1 + n)2 =Ãwt−µwt+1 1+rt+1 ·wt+1¶·Ã(1 + ρ)2 1+(1+ρ)+(1+ρ)2!! +µwt−µwt−1+1 1+rt ·wt¶·µ(1 + ρ)·(1 + rt) 1+(1+ρ)+(1+ρ)2¶¶·1 1+n +Ãwt−1−µwt−1+1 1+rt ·wt¶·Ã(1 + ρ)2 1+(1+ρ)+(1+ρ)2!! ·(1 + rt) (1 + n) or - by collecting terms - as kt+1 ·(1 + (1 + n)) = −wt+1 ·1 1+rt+1 ·Ã(1 + ρ)2 1+(1+ρ)+(1+ρ)2! +wt·⎛ ⎝1−³(1+ρ)2 1+(1+ρ)+(1+ρ)2´+ 1 1+n³1−³(1+ρ)+(1+ρ)2 1+(1+ρ)+(1+ρ)2´´⎞ ⎠ +wt−1·(1 + rt) (1 + n)·Ã1−Ã(1 + ρ)+(1+ρ)2 1+(1+ρ)+(1+ρ)2!! In steady state the stocks of (desired) capital and wealth as fractions of total production and as functions of rreduce to K/Y =α r·(1 + n) W/Y =Ã(1 −α) 1+ 1 1+n!·"3+n+r 1+n−1 1+(1+ρ)+(1+ρ)2· ³(1+ρ)2 1+r+(1+ρ)2+(2+r) 1+n·((1 + ρ)+(1+ρ)2)´# The two functions are sketched in figure A1. 3 1.51.2510.750.50.250 1 0.75 0.5 0.25 0 r W/Y K/Y r W/Y K/Y Figure A1: Wealth and capital as functions of the rate of interest (for α=0.3333; n=0.10; ρ=0.05) The three-period Diamond model has three roots. For α=0.3333; n=0.10; ρ=0.05, the solution is r=−0.843 ,−5.237,0.825.Only the positive root, r=0.825, makes economic sense (as a negative rate of interest implies a negative stock of capital). 3 Ordinary macro-economic equilibrium solution Total consumption is the consumption of the three generations living in period t. Ct=c1t+c2t−1·1 1+n+c3t−2·1 (1 + n)2 =µwt+1 1+rt+1 ·wt+1¶·Ã(1 + ρ)2 1+(1+ρ)+(1+ρ)2! +µwt−1+1 1+rt ·wt¶·µ(1 + ρ)·(1 + rt) 1+(1+ρ)+(1+ρ)2¶·1 1+n +µwt−2+1 1+rt−1 ·wt−1¶·µ(1 + rt−1)·(1 + rt) 1+(1+ρ)+(1+ρ)2¶·1 (1 + n)2 4 which may be written as Ct=wt−2·µ(1 + rt−1)·(1 + rt) (1 + n)2¶·1 1+(1+ρ)+(1+ρ)2 +wt−1·(1 + rt) (1 + n)·⎛ ⎝³1 1+(1+ρ)+(1+ρ)2´·1 1+n +³(1+ρ) 1+(1+ρ)+(1+ρ)2´⎞ ⎠ +wt·⎛ ⎝³(1+ρ) 1+(1+ρ)+(1+ρ)2´·1 1+n+ ³(1+ρ)2 1+(1+ρ)+(1+ρ)2´⎞ ⎠ +wt+1 ·Ã(1 + ρ)2 1+(1+ρ)+(1+ρ)2·1 1+rt+1 ! In steady state this expression reduces to Ct=(1 −α)·y 1+(1+ρ)+(1+ρ)2·⎛ ⎝³(1+r)2 (1+n)2+(1+r) (1+n)·1 1+n´+ ³(1+r) (1+n)+1 1+n´·(1 + ρ)+³1+ 1 1+r´·(1 + ρ)2 ⎞ ⎠ As we assume instant adjustment and perefect foresigfht Investments is the increase in the desired stock of capital from period tto period t+1, It=kt+1 ·((1 + n)+1)−kt·(1 + 1 1+n) whichinsteadystatereducesto It=k·(1 + 1 1+n)·n=α·y r·n·(1 + 1 1+n) From this exercise it becomes obvious that r=nis a solution, as Ct+It=(1−α)·y·(1 + 1 1+n)+α·y·(1 + 1 1+n)=y·(1 + 1 1+n)=Yt However, there may be more solutions, and the question of the stability of the roots remains. In figure A2 S/Y =1−C/Y and I/Y are sketched as functions of r(for technical reasons with rplotted along the abscissa). S/Y = ⎡ ⎢ ⎢ ⎢ ⎣ 1−(1−α) 1+(1+ρ)+(1+ρ)2·1 (1+ 1 1+n)· ⎛ ⎝³(1+r)2 (1+n)2+(1+r) (1+n)·1 1+n´+³(1+r) (1+n)+1 1+n´·(1 + ρ) +³1+ 1 1+r´·(1 + ρ)2 ⎞ ⎠ ⎤ ⎥ ⎥ ⎥ ⎦ I/Y =α·n r 5 Contrary to the two-period model, S/Y is here not a linear function of r. 10.750.50.250 1 0.75 0.5 0.25 0 r S/Y I/Y r S/Y I/Y Figure A2: Saving and investment of functions of the rate of interest (for α=0.3333; n=0.10; ρ=0.05) Expressed in terms of the rate of interest, the model has four roots: −5.234, −0.843 ,0.1,0.825. The three roots are identical with the roots derived above when closing the model as Diamond proposes. The fourth root is the socially optimal root, r=n=0.1. Again, only the two positive roots make economic sense. As the graph illustrates, r=nis a stable root. In case of r>n=0.1,saving (supply of loanable funds) exceeds investment (demand for loanable funds); the rate of interest is competed down. If, on the contrary, the rate of interest exceeds the higher root, r=0.825, the demand for loanable funds exceeds the supply of loanable funds; the rate of interest rises further. 4 The Samuelson case Samuelson (1958) analyses a consumption-loan model with no real capital (and, consequently, α=0). In this setting the rate of interest, r, may be negative. However, the model makes only economic sense if the present value of saving is positive, i.e if .r≥−1.The model has three roots. For n=0.10 and ρ=0.05, 6 the equilibrium condition reduces to S/Y =0.224 96 −0.137 32r2−0.183 19 r+1 −0.570 56r=0 the solution of which is r=−4.593; −0.662; 0.10. The model is sketched in figures A3(i) and A3(ii) below 0.50.250-0.25-0.5-0.75 0.25 0.125 0 -0.125 -0.25 r S/Y r S/Y Figure A3(i): The Samuelson model for n=0.10 and ρ=0.05 in the range r>−1 7