This appendix derives Equations (16) and (17) from Equations (12) and (13).

Transcription

1 Capital growt pat of te neoclaical growt model Online Supporting Information Ti appendix derive Equation (6) and (7) from Equation () and (3). Equation () and (3) owed te evolution of pyical and uman capital by firt-order approximation. Tey were: α β α β { α δ } ( β ) = ( + + ) ( ) + ( ), (S.) n g α β α β ( α ) { β δ } = ( ) + ( + + ) ( ). (S.) n g Bot and are te teady-tate value of te correponding capital, ence tey are contant. Ti mean tat Equation (S.) and (S.) can be implified into a ytem of two differential equation = a + a + b, (S.3) = a + a + b, (S.4) were a, a, a, a, b, b are all contant parameter. In te matrix form, te above ytem can be written a = A + b. (S.5) a a b + a a b Te olution to te ytem of linear, autonomou, firt-order differential equation wit form (S.5) i widely nown. Te ytem a a omogenou olution

2 rt rt = e + e J J, (S.6) were vector J i i a olution to equation [ ] A I J = 0. (S.7) r i i Here, r i (i =,) i an eigenvalue of matrix A and I i te unit matrix. Te a term in Equation (S.3) i te coefficient on in Equation (S.). It follow tat a = n+ g+. (S.8) α β α ( δ) Equation (4) and (5) in te main text gave te teady-tate value of capital. Tey were: β β α β = n+ g+ δ, (S.9) α α α β = n+ g+ δ Subtituting tee into Equation (S.8) give ( ) ( ). (S.0) α β β β α β α α α β a = α ( n+ g+ δ) n+ g+ δ n+ g+ δ α β α β α β = α n+ g+ δ n+ g+ δ ( n+ g+ δ) = N( α) were N (n + g +δ). Similarly, te oter term are derived a, (S.) a = β = β α β N, (S.3) a = α = α α β N, (S.4)

3 a n g α β = β ( + + δ ) = N( β ). (S.5) Becaue 0 < α < and 0 < β <, it follow tat a < 0, (S.6) a > 0, (S.7) a > 0, (S.8) a < 0. (S.9) Te caracteritic equation of matrix A i expreed a r + = tr( A) A 0, (S.0) were tr( A) = a + a = N( α) N( β ), (S.) = N( α β) a a a a = A = N ( α)( β) αβn N ( α β) =. (S.) Ten te caracteritic root, or te eigenvalue, of te matrix i given by N( α β ) ± N ( α β ) 4 N ( α β ) r =. (S.3) Inide te quare root of te rigt and ide of Equation (S.3) can be implified a N ( α β ) 4 N ( α β ) = N ( α + αβ + β ). (S.4) = N ( α + β)

4 Terefore, Equation (S.3) i reduced to N( α β ) ± N( α + β ) r =. (S.5) ( α β) ( α + β) = N It follow tat te ditinct root of te caracteritic equation are r = N( α β ), (S.6) r = N. (S.7) Becaue α β > 0 and N (n + g +δ) > 0, te two root ave te relationip 0 > r > r. (S.8) Extrapolating from te fact tat te two root are bot negative, it i concluded tat te ytem of differential equation (S.5) a te table node. Tat i, te teady-tate olution and are aymptotically table. Te next tep i to olve te ytem of equation (S.7) for r and r. Te ytem i equivalent to a ri a i = a a r J 0. (S.9) i Terefore, for r = N( α β), te eigenvector J a to be a olution for N( α) + N( α β) β N J α N N( β) + N( α β) β β = N J α α = 0 Becaue N > 0, Equation (S.30) can be implified to. (S.30)

5 β β j = 0. (S.3) α α j Ti i te linear relationip between j and j. Solving Equation (S.3) yield j = j. (S.3) Similarly, for r = N, te eigenvector J a to be a olution for N( α) + N β N α N N( β) + N α β = N J α β = 0 Ten, olving Equation (S.33) yield J. (S.33) β j = α j. (S.34) Finally, from Equation (S.3) and (S.34), te omogenou olution for te ytem (S.5) i derived. Te omogenou olution i: t β N( α β) t Nt = C e + C e t α. (S.35) Te general olution for a ytem of differential equation i given by te um of it omogenou olution and it particular olution. Te particular olution for te ytem (S.5) i te value of t and t wen tey are at te teady tate. Tey were already given a and in Equation (S.9) and (S.0): β β α β = n+ g+ δ,

6 α α α β = n+ g+ δ. Hence, te general olution to te ytem (S.5) i t N( ) t Nt C e C β α β e = t + + α. (S.36) Te remaining ta i to impoe te initial condition about and on Equation (S.36) and eliminate te contant term C and C. Suppoe tat it old at time zero tat = 0 and = 0. Ten, te neceary condition for Equation (S.36) to be a correct olution at time zero i β, (S.37) 0 C C = α wic can be rewritten a C β 0 C α 0 + =. (S.38) Now, we define te ditance between te initial value and te teady-tate value of eac capital a K = 0, (S.39) H = 0. (S.40) Ten, Equation (S.38) can be furter implified a β K C C + = α H, (S.4) wic, again, i a ytem of two equation wit two unnown. Note tat K and H are

7 exogenou in ti model becaue bot te initial value and te teady-tate value are exogenou. Solving te ytem of equation (S.4) give te olution α K β H C = + β α α β C = K H α β + β α, (S.4). (S.43) Terefore, te complete olution of te original ytem of differential equation (S.5) i given by t α K β H N( α β) t e = + t β α α β. (S.44) K H β Nt + + e + α β β α α Or eparately, t K H Nt = + β e α β β α, (S.45) α K β H e N( α β) t β α α β t K H Nt = + αe α β β α. (S.46) α K β H e N( α β) t β α α β Tee are identical to Equation (6) and (7) in te main text. From Equation (S.45) and (S.46), te optimal growt pat of te income per effective

8 labour can alo be derived. It i given by y = α β t t t K H Nt α K β H N ( α β) t = + β e + + e + α β β α β α α β K H Nt α K β H N ( α β) t + αe + + e + α β β α β α α β β α (S.47). Taing te logaritm of bot ide of Equation (S.47) yield ln y = α ln + β ln t t t K H Nt α K β H N ( α β) t = αln + βe + + e +. α β β α β α α β K H Nt α K β H N ( α β) t + β ln + αe + + e + α β β α β α α β (S.48) Note tat ti i an empirically tetable equation regarding te growt of national income of te country, provided tat te data on te abolute level of capital, K and H, are available.