New Technology, Human Capital and Growth for Developing Countries.

Transcription

1 New Technology, Human Capital an Growth for Developing Countries Cuong Le Van, Manh-Hung Nguyen an Thai Bao Luong Centre Economie e la Sorbonne, Université Paris-1, Pantheon-Sorbonne B e l Hôpital, Paris Ceex 13, France December 7, 2006 Abstract We consier a eveloping country with three sectors in economy: consumption goos, new technology, an eucation Prouctivity of the consumption goos sector epens on new technology an skille labor use for prouction of the new technology We show that there might be three stages of economic growth In the first stage the country concentrates on prouction of consumption goos; in the secon stage it requires the country to import both physical capital to prouce consumption goos an new technology capital to prouce new technology; an finally the last stage is one where the country nees to import new technology capital an invest in the training an eucation of high skille labor in the same time Keywors: Optimal growth moel, New technology capital, Human Capital, Developing country JEL Classification: D51, E13 The authors thank Tu-Anh Nguyen for thoughtful remarks 1

2 1 Introuction Technology an aoption of technology have been important subjects of research in the literature of economic growth in recent years Sources of technical progress might be omestic or/an international though there always exists believes amongst economic professionals that there is an important ifference between evelope an eveloping countries, ie the first one innovates an exports technology while the secon one imports an copies For eveloping countries, the aoption of technology from international market is vital since it might be the only way for them to improve their prouctivity growth an technical progress (Romer [1997, 1990]) But it is even more important to stress that these countries also nee to care about their human capital (Lucas [1988]) which might be the key factor that etermines whether a country, given their level of evelopment, can take off or might fall into proverty trap This line of argument comes from the fact that the eveloping countries toay are facing a ilemma of whether to invest in physical, technological, an human capital As abunantly showe in literature (eg Barro [1997], Barro & Sala-i-Martin [1995], Eaton & Kortum [2000], Keller [2001], Kumar [2003], Kim & Lau [1994], Lau & Park [2003]) eveloping countries are not convergent in their growth paths an in orer to move closer to the worl income level, a country nees to have a certain level in capital accumulation In their recent work, Bruno et al [2006] point out the conitions uner which a eveloping country can optimally ecie to either concentrate their whole resources on physical capital accumulation or spen a portion of their national wealth to import technological capital These conitions are relate to the nation s stage of evelopment which consists of level of wealth an enowment of human capital an threshols at which the nation migh switch to another stage of evelopment However, in their moel, the role of eucation that contributes to accumulation of human capital an effecient use of technological capital is not fully explore In this paper we exten their moel by introucing an eucational sector with which the eveloping country woul invest in to train more skille labor We show that the country once reaches a critical value of wealth will have to consier the import of new technology At this point, the country can either go on with its existing prouction technology or improve it by purchasing new technology capital in orer to prouce new technology But when the level of wealth passes this value it is always optimal for the country to use new technology which requires high skille workers We show further that with possibility of investment in human capital an given goo conitions on the qualities of the new technology, prouction process, an/or the number of skille workers 2

3 there exists alternatives for the country either to purchase new technology an spen money in training high skille labor or only purchase new technology but not to spen on formation of labor Following this irection, we can etermine the level of wealth at which the ecision to invest in training an eucation has to be mae In this context, we can show that the critical value of wealth is inversely relate to prouctivity of the new technology sector, number of skille workers, an inicator of the impact of the new technology sector on the consumption goos sector but proportionally relate to price of the new technology capital In the whole, the paper allows us to etermine the optimal share of the country s investment in physical capital, new technology capital an human capital formation in the long-run growth path Two main results can be pointe out: (1) the richer a country is, the more money will be investe in new technology an training an eucation, (2) an more interestingly, the share of investment in human capital will increase with the wealth while the one for physical an new technology capitals will ecrease In any case, the economy will grow without boun The paper is organize as follows Section 2 is for the presentation of one perio moel Section 3 eals with ynamic properties with infinitely live representative consumer 2 The one perio moel Consier an economy where exists three sectors: omestic sector which prouces an aggregate goo Y, new technology sector with output Y e an eucation sector characterize by a function h(t ) where T is the expeniture on training an eucation The output Y e is use by omestic sector to increase its total prouctivity The prouction functions of two sectors are Cobb-Douglas, ie, Y = Φ(Y e )K α L1 α an Y e = A e Ke αe L 1 αe e where Φ() is a non ecreasing function which satisfies Φ(0) = x 0 > 0, K, K e, L, L e an A e be the physical capital, the technological capital, the low-skille labor, the high-skille labor an the total prouctivity, respectively, 0 < α < 1, 0 < α e < 1 We assume that this country imports capital goo, the price of which is consiere as numeraire The price of the new technology sector is higher an equal to λ such that λ 1 Assume that labor mobility between sectors is impossible an wages are exogenous Let S be available amount of money enote to the capital goos purchase We have: K + λk e + p T T = S For simplicity, we assume p T = 1, or in other wors T is measure in capital goos 3

4 Thus, the buget constraint of the economy can be written as follows K + λk e + T = S where S be the value of wealth of the country in terms of consumption goos The social planner maximizes the following program subject to max Y = max Φ(Y e )K α L1 α Y e = A e Ke αe L 1 αe e, K + λk e + T = S, 0 L e L eh(t ), 0 L L where h is the eucation technology Assume that h() is an increasing concave function an h(0) = h 0 > 0 Let = {(θ, µ) : θ [0, 1], µ [0, 1], θ + µ 1} From the buget constraint, we can efine (θ, µ) : λk e = θs, K = (1 θ µ)s an T = µs Observe that since the objective function is strictly increasing, at the optimum, the constraints will be bining Let L e = L eh, L = L, then we have the following problem where r e = Ae λ αe L 1 αe e Let max Φ(r eθ αe S αe h(µs) 1 αe )(1 θ µ) α S α L 1 α (θ,µ) ψ(r e, θ, µ, S) = Φ(r e θ αe S αe h(µs) 1 αe )(1 θ µ) α L 1 α The problem now is equivalent to max ψ(r e, θ, µ, S) (θ,µ) (P) Since the function ψ is continuous in θ an µ, there will exist optimal solutions Denote F (r e, S) = max ψ(r e, θ, µ, S) (θ,µ) 4

5 Then by Maximum Theorem, F is continuous an F (r e, S) x 0 L 1 α Suppose that function Φ(x) is a constant in an initial phase an increasing linear afterwars: { x 0 if x X Φ(x) = x 0 + a(x X) if x X, a > 0 Define B = {S 0 : F (r e, S) = x 0 L 1 α }, Lemma 1 B is a nonempty compact set Proof: B is a nonempty since 0 B Of course B is close since the function F is continuous Let us prove that B is boune If not, take a sequence S n in B an converging to + when n + Fix some (θ 0, µ 0 ) Since {S n } B, we have ψ(r e, θ 0, µ 0, S n ) F (r e, S n ) = x 0 L 1 α Let n + then ψ(r e, θ 0, µ 0, S n ) + A contraiction Therefore, B is boune Remark 1 Observe that F (r e, S) x 0 L 1 α If the optimal value for θ equals 0 then the one for µ is also 0 an F (r e, S) = x 0 L 1 α The following proposition shows that if S is small, then the country will not invest in new technology an human capital When S is large, then it will invest in new technology Proposition 1 i) There exists S > 0 such that if S S then θ = 0 an µ = 0 ii) There exists S such that if S > S then θ > 0 Proof: For any S, enote by θ(s), µ(s) the corresponing optimal values for θ an µ (i) Let S satisfies r e S αe h(s) 1 αe = X, Then for any (θ, µ), for any S S, r e θ αe S αe h(µs) 1 αe X an (θ(s), µ(s)) = (0, 0) (ii) Fix µ = 0 an θ (0, 1) Then ψ(r e, θ, 0, S) + when S + Let S satisfy ψ(r e, θ, 0, S) > x 0 L 1 α Obviously, F (r e, S) ψ(r e, θ, 0, S) > 5

6 x 0 L 1 α, an θ(s) > 0 If not, then µ(s) = 0 an F (r e, S) = x 0 L 1 α Remark 1) Now, let us efine S c = max{s 0 : S B} It is obvious that 0 < S c < +, since S c S > 0 an B is compact Proposition 2 If S < S c then θ(s) = 0 an µ(s) = 0, an if S > S c then θ(s) > 0 Proof: Note that for any S 0 we have If S < S c then for any (θ, µ), which implies Thus, F (r e, S) x 0 L 1 α ψ(r e, θ, µ, S) ψ(r e, θ, µ, S c ) F (r e, S) F (r e, S c ) = x 0 L 1 α F (r e, S) = x 0 L 1 α Let S 0 < S c Assume there exists two optimal values for (θ, µ) which are (0, 0) an (θ 0, µ 0 ) with θ 0 > 0 We have F (r e, S 0 ) = x 0 L 1 α = ψ(r e, θ 0, µ 0, S 0 ) We must have r e θ αe 0 Sαe 0 h(µ 0S 0 ) 1 αe > X (if not, Φ(r e, θ 0, µ 0, S 0 ) = x 0 an θ 0 = 0, µ 0 = 0) Since θ 0 > 0, we have r e θ αe 0 (Sc ) αe h(µ 0 S 0 ) 1 αe > r e θ αe 0 Sαe 0 h(µ 0S 0 ) 1 αe > X Hence x 0 L 1 α = F (r e, S c ) ψ(r e, θ 0, µ 0, S c ) which is a contraiction Therefore, if S > S c then which implies θ(s) > 0 > ψ(r e, θ 0, µ 0, S 0 ) = x 0 L 1 α F (r e, S) > x 0 L 1 α The following proposition shows that, when the quality of the training technology (measure by the marginal prouctivity at the origin h (0)) is very high then for S > S c the country will invest both in new technology an in human capital (see 6

7 Proposition 3 If h (0) = +, then for all S > S c, we have θ(s) > 0, µ(s) > 0 Proof: Take S > S c From the previous proposition, θ(s) > 0 Assume µ(s) = 0 For short, enote θ = θ(s) Define F 0 (r e, S, θ, 0) = max ψ(r e, θ, 0, S) = Φ(r e θ αe S αe h(0) 1 αe )(1 θ ) α L 1 α 0 θ 1 an consier a feasible couple (θ, µ) in which satisfies θ = θ + µ Denote We then have F 1 (r e, S, θ, µ) = Φ(r e θ αe S αe h(µs) 1 αe )(1 θ ) α L 1 α F 1 (r e, S, θ, µ) F 0 (r e, S, θ, 0) (1 θ ) α L 1 α = Φ(r e θ αe S αe h(µs) 1 αe ) Φ(r e θ αe S αe h(0) 1 αe ) = r e S αe [θ αe h(µs) 1 αe θ αe h(µs) 1 αe + θ αe h(µs) 1 αe θ αe h(0) 1 αe ] By the concavity of h(x) an f(x) = x αe, we obtain F 1 (r e, S, θ, µ) F 0 (r e, S, θ, 0) r e S αe µh(µs) αe [ α e h(µs)(θ µ) αe 1 + S(1 α e )θ αe h (µs)] Let µ 0 We have h (µs) + The expression in the brackets will converge to +, an we get a contraiction with the optimality of θ When h (0) is finite, we are not ensure that the country will invest in human capital when S > S c But it will o if it is sufficiently rich Proposition 4 Assume h (0) < + Then there exists S M such that µ(s) > 0, θ(s) > 0 for every S > S M Proof: Assume that µ(s) = 0 for any S {S 1, S 2,, S n, } where the infinite sequence {S n } n is increasing, converges to + an satisfies S 1 > S c For short, enote θ = θ(s) Then we have the following FOC: ar e θ αe 1 S αe h(0) 1 αe α e x 0 + a[r e θ αe S αe h(0) 1 αe X] = α 1 θ, (1) an Equation (1) implies ar e θ αe S αe+1 h (0)h(0) αe (1 α e ) x 0 + a[r e θ αe S αe h(0) 1 αe X] α 1 θ (2) ar e θ αe 1 h(0) 1 αe α e x 0 S + a[r αe e θ αe h(0) 1 αe ] α 1 θ (3) 7

8 If θ 0 when S +, then the LHS of inequality (3) converges to infinity while the RHS converges to α : a contraiction Thus θ will be boune away from 0 when S goes to infinity Combining equality (1) an inequality (2) we get h (0)(1 α e )S h 0 α e θ 1 (4) When S +, we have a contraiction since the LHS of (4) will go to infnity while the RHS will be boune from above That means there exists S M such that for any S S M, we have µ(s) > 0 Remark 2 Let S > S c For short, we enote µ an θ instea of µ(s) an θ(s) If µ > 0 then we have the FOC: an ar e θ αe 1 S αe h(µs) 1 αe α e x 0 + a[r e θ αe S αe h(µs) 1 αe X] = ar e θ αe S αe+1 h (µs)h(µs) αe (1 α e ) x 0 + a[r e θ αe S αe h(µs) 1 αe X] α 1 θ µ, (5) = α 1 θ µ (6) The following proposition shows that, if h (0) is low, then the country will not invest in human capital when S belongs to some interval (S c, S m ) Proposition 5 There exists α > 0 such that, if h (0) < α, then there exists S m > S c such that µ(s) = 0, θ(s) > 0 for S [S c, S m ] Proof: Let θ c an S c satisfy the following equations an ar e (θ c ) αe 1 (S c ) αe h(0) 1 αe α e x 0 + a[r e (θ c ) αe (S c ) αe h(0) 1 αe X] = α 1 θ c, (7) (x 0 + a[r e (θ c ) αe (S c ) αe h(0) 1 αe X])(1 θ c ) α = x 0 (8) Equality (7) is the FOC with respect to θ, while equality (8) states that ψ(r e, θ c, 0, S c ) = x 0 L 1 α Teious computations show that equations (7) an (8) are equivalent to an a r e (S c α ) αe h(0) 1 αe x 0 α e = (θ c ) αe 1 (1 θ c ) 1+α x 0 (1 + α α e )θ c = x 0 (x 0 a X)(1 θ c ) 1 α (10) The LHS of equation (10) is linear, increases from 0 when θ c = 0 to x 0 (1 + α when θ c = 1 The RHS of this equation is a non-increasing function, when 8 (9) α e )

9 x 0 ax, from ax when θ c = 0 to x 0 when θ c = 1 In this case, there exists a unique solution for θ c in (0, 1) When x 0 > ax, the RHS is a convex function which increases from ax when θ c = 0 to x 0 when θ c = 1 Again, there exists a unique solution for θ c in (0, 1) Once θ c is etermine, S c is given by equation (9) Now, if h (0) < α = h(0) 1 α e θ c S c 1 α e, then we get ar e (θ c ) αe (S c ) αe+1 h (0)h(0) αe (1 α e ) x 0 + a[r e (θ c ) αe (S c ) αe h(0) 1 αe X] < α 1 θ c (11) Relations (7), (8) an (11) give the the values of S c an θ(s c ) = θ c an µ(s c ) = µ c = 0 When S > S c an close to S c, equality (7) an inequality (11) still hol That means µ(s) = 0 for any S close to S c Remark 3 Let τ be efine by T = τλ, ie τ is the measure of T using the new technology capital as numeraire In this case, conition h (0) < h(0) 1 θ c S c is equivalent to Ye K e > ( Ye L e )( Le τ ) T =0 α e 1 α e Proposition 6 Assume h (0) < + Let S 1 > S c If µ(s 1 ) = 0, then for S 2 < S 1, we also have µ(s 2 ) = 0 Proof: If S 2 S c then µ(s 2 ) = 0 since θ(s 2 ) = 0 (see Proposition 2) For short, we write θ 1 = θ(s 1 ), θ 2 = θ(s 2 ), µ 1 = µ(s 1 ), µ 2 = µ(s 2 ) Observe that (θ 1, S 1 ) satisfy (1) an (2), or equivalently (1) an (4) Equality (1) can be written as h 1 αe 0 ar e [α e θ αe 1 1 (α e + α )θ αe 1 ] = α (x 0 ax) S 1 α e (12) If x 0 ax = 0, then θ 1 = αe α e+α Take θ 2 = θ 1 If S 2 < S 1 then (θ 2, S 2 ) satisfy (1) an (4) That means they satisfy the FOC with µ 2 = 0 Observe that the LHS of equation (12) is a ecreasing function in θ 1 Hence θ 1 is uniquely etermine When x 0 > ax, if (θ 2, S 2 ) satisfy (12), with S 2 < S 1, then θ 2 < θ 1 In this case, (θ 2, S 2 ) also satisfy (4), an we have µ 2 = 0 When x 0 < ax, write equation (12) as: h 1 αe 0 ar e [α e θ 1 1 (α e + α )] = α (x 0 ax) (θ 1 S 1 ) αe (13) If (θ 2, S 2 ) satisfy (12), with S 2 < S 1, then θ 2 > θ 1 Since x 0 < ax, from (13), we have θ 2 S 2 < θ 1 S 1 Again (θ 2, S 2 ) satisfy (12) an (4) That implies µ 2 = 0 From Proposition 6, we have as corollary, the following proposition 9

10 Proposition 7 Assume h (0) < + Then there exists Ŝ Sc such that: (i) S Ŝ µ(s) = 0, (ii) S > Ŝ µ(s) > 0 Proof: Let S = max{s m : S m > S c, an S S m µ(s) = 0}, an S = inf{s M : S M > S c, an S > S M µ(s) > 0} From Propositions 4 an 5, the sets {S m : S m > S c, an S S m µ(s) = 0} an {S M : S M > S c, an S > S M µ(s) > 0} are not empty From Proposition 6, we have S S If S > S, then take S ( S, S) From the efinitions of S an S, there exist S1 < S, S 2 > S such that µ(s 1 ) > 0 an µ(s 2 ) = 0 But that contraicts Proposition 6 Hence S = S Put Ŝ = S = S an conclue Let us recall r e = AeL (1 αe) e λ αe where A e is the prouctivity of the new technology sector, λ is the price of the new technology capital an L e is the number of skille workers Recall also the prouctivity function of the consumption goos sector Φ(x) = x 0 + a(x X) if x X The parameter a > 0 is an inicator of the impact of the new technology prouct x on the this prouctivity We will show in the following proposition that the critical value S c iminishes when r e increases, ie when the prouctivity A e an/or the number of skille workers increase, an /or the price of the new technology capital λ ecreases, an /or the impact inicator a increases Proposition 8 Assume h(z) = h 0 + bz, with b > 0 Let θ c = θ(s c ), µ c = µ(s c ) Then (i) µ c = 0, θ c oes not epen on a an r e (ii) S c ecreases if a or/an r e increases Proof: From Proposition 7, we have µ c = 0 In this case, θ c an S c satisfy equation (5) an, since S c B, we also have F (r e, S c ) = ψ(r e, θ c, 0, S c ) = x 0 L 1 α Explicitly, we have an ar e (θ c ) αe 1 (S c ) αe h 1 αe 0 α e x 0 + a[r e (θ c ) αe (S c ) αe h 1 αe 0 X] = α 1 θ c (x 0 + a[r e (θ c ) αe (S c ) αe h 1 αe 0 X])(1 θ c ) α = x 0 (14) 10

11 Teious computations show that θ c satisfies the equation α e [1 x 0 ax x 0 (1 θ) α +1 ] = θ(α + α e ) If x 0 > ax, then the LHS is a strictly concave function which increases from when θ = 0 to α e when θ = 1 The RHS is linear increasing, equal to 0 at α eax x 0 the origin an to α + α e when θ = 1 Therefore, there exists a unique solution θ c (0, 1) If x 0 < ax, then the LHS is a strictly convex function which ecreases from α eax x 0 when θ = 0 to α e when θ = 1 The RHS is linear increasing, equal to 0 at the origin an to α + α e when θ = 1 Therefore, there exists a unique solution θ c (0, 1) If x 0 = ax, then θ c = αe α e+α In any case, θ c oes not epen on a an r e Equation (14) gives: ar e (S c 1 ) αe = [x 0 ( (1 θ c ) α 1) + ax] 1 (θ c ) αe h 1 αe We see immeiatly that S c is a ecreasing function in a an r e 0 (15) The following proposition shows that the optimal shares θ, µ converge when S goes to infinity an the ratio between physical capital an the total of new technology capital an the amount evote to human capital formation ecreases when S increases Proposition 9 Assume h(z) = h 0 + bz, with b > 0 Then the optimal shares θ(s), µ(s) converge to θ, µ when S converges to + Consier Ŝ in Proposition 7 Then (i) If x 0 < ax, θ(s) ecreases from θ c to θ = θ(ŝ) when S goes from Sc to ŜThe sum θ(s) + µ(s) ecreases when S increases from S c to Ŝ (ii) If x 0 ax, θ(s) increases from θ c to θ = θ(ŝ) when S goes from Sc to ŜThe sum θ(s) + µ(s) increases when S increases from S c to Ŝ (iii)if ar e is large enough, θ(s) ecreases from θ to θ = αe 1+α when S increases from Ŝ to + The sum θ(s) + µ(s) increases with S for S > Ŝ Moreover, µ(s) also increases with S for S > Ŝ Proof: For short, write θ, µ instea of θ(s), µ(s) Consier Ŝ in Proposition 7 When S S c, we have θ = 0, µ = 0 When S c < S Ŝ, then µ = 0 (Proposition 7) But θ satisfies equation (1) which can be written as ar e θ αe 1 h(0) 1 αe α e = α (x 0 ax) S αe + (α + α e )ar e θ αe h(0) 1 αe 11

12 The LHS is a ecreasing function while the RHS is increasing It is easy to check that the unique solution exists an is in (0, 1) Differentiate with respect to S to see that θ increases with S if x 0 > ax an ecreases with S if x 0 < ax When x 0 = ax, we obtain θ = αe α e+α Thus the sum θ + µ increases with S when S (S c, Ŝ) if x 0 > ax an ecreases if x 0 < ax Now consier the case where S > Ŝ Then (θ, µ) satisfy equations (5) an (6) which can be written as follows: θ(α + α e ) = α e µ + [α e α (x 0 ax)α α e e ] (16) ar e S(1 α e ) 1 αe 1 αe b an We obtain θ(1 α e ) = α e µ + α eh 0 bs (17) θ(1 + α ) = [α e α (x 0 ax)α e α e ar e S(1 α e ) 1 αe b 1 αe + h 0α e bs ] (18) an Thus θ + µ = α [1 α α e µ = θ( 1 α e 1) h 0 bs (x 0 ax)α α e e ar e S(1 α e ) 1 αe b ] 1 αe α 1 + α h 0 bs If x 0 ax, then θ + µ increases with S If x 0 < ax, then when ar e is large enough, then θ + µ is an increasing function in S It is obvious to see that θ ecreases with S (equation (18)) if x 0 < ax, or if ar e is large, when x 0 > ax When S converges to +, then θ converges to θ = µ = 1 αe 1+α αe 1+α an µ converges to 3 The Dynamic Moel In this section, we consier an economy with one infinitely live representative consumer who has an intertemporal utility function with iscount factor β < 1 At each perio, her savings will be use to import physical capital or/an new technology capital an/or to invest in human capital We suppose the capital epreciation rate equals 1 12

13 The social planner will solve the following ynamic growth moel max st β t u(c t ) t=0 c t + S t+1 Φ(Y e,t )K α,t L1 α,t Y e,t = A e Ke,t αe L1 αe e,t K,t + λk e,t + T t = S t, 0 L e,t L eh(t t ), 0 L,t L the initial resource S 0 is given The problem is equivalent to max st β t u(c t ) t=0 c t + S t+1 H(r e, S t ), t, with H(r e, S) = F (r e, S)S α where r e = Ae λ αe L 1 αe e,t Obviously, H(r e, ) is continuous, strictly increasing an H(r e, 0) = 0 As in the previous section, we shall use S c efine as follows: where S c = max{s 0 : F (r e, S) = x 0 L 1 α } F (r e, S t ) = max ψ(r e, θ t, µ t, S t ) 0 θ t 1,0 µ t 1 We shall make stanar assumptions on the function u uner consieration H2 The utility function u is strictly concave, strictly increasing an satisfies the Inaa conition: u (0) = +, u(0) = 0 At the optimum, the constraints will be bining, the initial program is equivalent to the following problem st max β t u(h(r e, S t ) S t+1 ) t=0 0 S t+1 H(r e, S t ), t S 0 > 0 given By the same arguments as in Bruno et al [2005], we have the following property Proposition 10 i) Every optimal path is monotonic ii) Every optimal trajectory (S t ) from S 0 can not converge to 0 13

14 Let enote θ t, µ t be the optimal capital shares among technological capital stock an expeniture on training, λk e,t = θ t S t an T t = µ t S t We then obtain the main result of this paper: Proposition 11 Assume h(z) = h 0 + bz, with b > 0 an α e + α 1 If a or/an r e are large enough then the optimal path {S t } t=1,+ converges to + when t goes to infinity Hence: (i) there exists T 1 such that (ii) there exists T 2 T 1 such that θ t > 0 t T 1 θ t > 0, µ t > 0, t T 2 The sum θ t + µ t an the share µ t increase when t goes to infinity an converge to values less than 1 Proof: Let S s be efine by α (S s ) α 1 x 0 L 1 α = 1 β (1) If S 0 > Ŝ (Ŝ is efine in Proposition 7) then θ t > 0, µ t > 0 for every t If S 0 > S c then θ t > 0 for every t If S t converges to infinity, then there exists T 2 where S T 2 > Ŝ an θ t > 0, µ t > 0 for every t T 2 Now consier the case where 0 < S 0 < S c Obviously, θ 0 = 0 It is easy to see that if a or/an r e are large then S c < S s If for any t, we have θ t = 0, we also have K e,t = 0 t, an the optimal path (S t ) will converge to S s (see Le Van an Dana [2003]) But, we have S c < S s Hence the optimal path {S t } will be non ecreasing an will pass over S c after some ate T 1 an hence θ t > 0 when t T 1 If the optimal path {S t } converges to infinity, then after some ate T 2, S t > Ŝ for any t > T 2 an θ t > 0, µ t > 0 It remains to prove that the optimal path converges to infinity if a or/an r e are large enough Since the utility function u satisfies the Inaa conition u (0) = +, we have Euler equation: u (c t ) = βu (c t+1)h s(r e, S t+1) If S t S <, then c t c > 0 From Euler equation, we get H s(r e, S) = 1 β 14

15 We will show that H s(r e, S) > 1 β for nay S > Sc We have H s(r e, S) = F s(r e, S)S α + α F (r e, S)S α 1 From the envelope theorem we get: F s(r e, S)S α = F s(r e, S)S α [ar e θ αe (h (µs)) αe (α e h(µ S) + (1 α e )µ Sh (µ S))S α +α e 1 ] L 1 α (1 θ µ ) α When ar e is large, from Proposition 9, we have θ θ = min{θ c, θ } an θ + µ ζ = max{θ c, θ + µ } We then obtain H s(r e, S) L 1 α (1 θ µ ) α [ar e θ αe (h (µs)) 1 αe α e S α +α e 1 ] L 1 α (1 ζ) α [ar e θ αe (h (0)) 1 αe α e (S c ) α +α e 1 ] since h(x) h(0) an α + α e 1 0 If α + α e = 1, then H s(r e, S) L 1 α (1 ζ) α [ar e θ αe (h (0)) 1 αe α e ], (19) an when ar e becomes very large, the RHS of inequality (19) will be larger than 1 β Now assume α +α e > 1 From equation (15), the quantity ar e (S c ) αe equals 1 γ = [x 0 ( (1 θ c ) α 1) + ax] 1 (θ c ) αe h 1 αe 0 an We now have S c = ( γ ar e ) 1 αe H s(r e, S) L 1 α (1 ζ) α θ αe (h (0)) 1 αe α e γ( γ ) α 1 αe ar e It is obvious that, since α 1 < 0, when ar e is large, we have H s(r e, S) > 1 β Remark 4 To summarize, at low level of economic growth this country woul only invest in physical capital but when the economy grows this country woul nee to invest not only in physical capital but also in first, new technology an then, formation of high skille labor Uner some mil conitions on the quality of the new technology prouction process an on the supply of skille workers, the optimal path (S t ) converges to +, ie the country grows without boun In this case, the share of investment in new technology an human capital (θ t + µ t ) 15

16 will increase while the one in physical capital will ecrease (this is in accorance with the empirical results in Lau an Park (2003)) More interestingly, an in accorance with the results in Barro an Sala-i-Martin (2004), the share µ t will become more important than the one for physical an new technology capitals when t goes to infinity But they will converge to strictly positive values when time goes to infinity References [1] Barro, R, Determinants of Economic Growth: A Cross-countries Empirical Stuy, MIT Press, Cambrige, 1997 [2] Barro, R an Sala-i-Martin, X, Economic Growth, McGraw Hill, New York, secon eition, 2004 [3] Bruno,O, Le Van, C an Masquin, B, When oes a eveloping country use new technologies, Cahiers e la MSE, 2005 [4] Eaton, J an Kortum, S, Trae in Capital Goos, WP, Boston University, 2000 [5] Keller, W, International Technology Diffusion, NBER WP 5873, 2001 [6] Kim, J an Lau, L, The Sources of Economic Growth in the East Asian Newly Inustrial Countries, Journal of Japanese an International Economics, 8, 1994 [7] Kumar, KB, Eucation an Technology Aoption in a Small Open Economy: Theory an Evience, Macroeconomic Dynamics, 7, pp , 2003 [8] Lau, L an Park, J, The Sources of East Asian Economic Growth Revisite, WP, Stanfor University, 2003 [9] Le Van, C an Dana, RA, Dynamic programming in Economics, Kluwer Acaemic Publishers, 2003 [10] Lucas, RE Jr, On the Mechanics of Economic Development, Journal of Monetary Economics, 22, 1 (July), pp 3-42, 1988 [11] Ramsey F, A Mathematical Theory of Saving, Journal of Economic Theory, 38, pp ,

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