The weighted Euler condition: 装-a,E6A医刘 k+1+h CRS=F(k h)k+Fn(k:h)h=F(kh) =A(K+)汇k+1+h+] 6(K)≡ +1A-8- 1+K kt1+h,+1 u(c)=Bl+A(a)-8(K) u(c) 电子科大经管学院马捷 11 元元元兄元
电子科大经管学院 马捷 11 [1 ( ) ( )] ( ) ( ) ( ) 1 1 1 ( ) ( )[ ] ( , ) ( , ) ( , ) ( , ) ( , ) 1 ( ) ( ) The weighted Euler condition : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 t t t t t t t k t h t k h t t t k t t t h t t t t t t t t k t t k t h t t h t t A u c u c k h k h A k h CRS F k h k F k h h F k h k h k F k h h F k h u c u c
The Optimal Ratio x* Euler conditionsF(kh)-=F (kh)-6n CRS->F and F,are both functions of the ratio and F is decreasing in x and F,is increasing in x.>There is a unique optimal ratio x*solves the following problem: maxF(k+l,h,+i)-δk1-δh+ s.t.k+h=constant K=arg max[1+A(K)-8()] 电子科大经管学院马捷 12 元元
The Optimal Ratio κ CRS Fk and Fh are both functions of the ratio κ t+1 and Fk is decreasing in κ and Fh is increasing in κ. There is a unique optimal ratio κ ∗ solves the following problem: 电子科大经管学院 马捷 12 argmax [ 1 ( ) ( )] . . constant max ( , ) * 1 1 1 1 1 1 A s t k h F k h k h t t t t k t h t k t t k h t t h Euler conditions F ( k 1,h 1 ) F ( k 1, h 1 )
Proposition 4.2 Consider the economy with physical and human capital described above,let K=arg max[1+4()-6(K)] and suppose(B,0,F,δ，o)satisfy B[1+A(k*)- k*)]>1>f[1+A(k)-(K*)]e-1.Then,the economy exhibits a balanced growth path. Physical capital,human capital,output,and consumption all grow at a constant rate given by kin=ha=ya =cun =(oll+A(k)-6()>1 h y 电子科大经管学院马捷 13
Proposition 4.2 Consider the economy with physical and human capital described above, let and suppose (β, θ, F, δk,δh) satisfy β [1 + A(κ∗) − δ(κ∗)] > 1 >βθ [1 + A(κ∗) − δ(κ∗)]θ−1 . Then, the economy exhibits a balanced growth path. Physical capital, human capital, output, and consumption all grow at a constant rate given by 电子科大经管学院 马捷 13 argmax[1 ( ) ( )] * A [1 ( ) ( )] 1 1 1 1 1 * * A cc yy hh kk t t t t t t t t
(Cont.)while the investment rate out of total resources is given by s=Bo[1+4(x")-6(K")]0-1 and the optimal ratio of physical to human capital isk=K.The growth rate is increasing in the productivity of either type of capital,increasing in the elasticity of intertemporal substitution,and decreasing in the discount rate. 电子科大经管学院马捷 14
(Cont.)while the investment rate out of total resources is given by s = β θ [1 + A ( κ ∗) − δ ( κ ∗)] θ−1 and the optimal ratio of physical to human capital is kt+1 / h t+1 = κ ∗. The growth rate is increasing in the productivity of either type of capital, increasing in the elasticity of intertemporal substitution, and decreasing in the discount rate. 电子科大经管学院 马捷 14
Market Allocations The household budget: c,+i++(b,+1-b,)≤rk,+Rb,+w,h, k41=(1-6)k,+i h1=(1-66)h,+功 C,+k+1+h,+1+b+1≤(1+r-6)k,+(1+w,-6)h,+(1+R)b, c）=1+R=1+片-d=1+w,-0,→ Bu(c) 7-d=w,-⊙h 电子科大经管学院马捷 15
Market Allocations 电子科大经管学院 马捷 15 t k t h t t k t h t t t t t t t k t t h t t t h t h t t k t k t t t t t t t t t t h t k t t r w R r w u c u c c k h b r k w h R b h h i k k i c i i b b r k R b w h 1 1 1 ( ) ( ) ( 1 ) ( 1 ) ( 1 ) ( 1 ) ( 1 ) ( ) The household budget : 1 1 1 1 1 1 1