电子科技大学：《高级宏观经济学 Advanced Macroeconomics》课程教学资源（课件讲稿）Lecture 03a Ramsey Model

Proposition 3.1 The plan c is a solution to the social planner's problem if and only if U(c,2)1Uc(C,2,)=F(k,) (3.1) U.(c,)=1-6+Fx(k+1,l) (3.2) BUc(C+1,+1) k1=F(k,l,)+(1-δ)k-C, (3.3) ko>0 given,and lim B'U(C,)k=0 (3.4) t>00 电子科大经管学院马捷 11

Proposition 3.1  The plan is a solution to the social planner’s problem if and only if 电子科大经管学院 马捷 11     0 , , t t t t c l k 0 , lim ( , ) 0 ( 3.4 ) ( , ) ( 1 ) ( 3.3 ) 1 ( , ) ( 3.2 ) ( , ) ( , ) ( , ) / ( , ) ( , ) ( 3.1 ) 0 1 1 1 1 1 1                   c t t t t t t t t t t K t t c t t c t t z t t c t t L t t k given and U c z k k F k l k c F k l U c z U c z U c z U c z F k l    

Decentralized Competitive Equilibrium Households Preference:U=>BU(c,=) t=0 UΨ=U(c,z)+UH Time constraint:=1-1/ Budget constraint: c+i+x,≤y=rk+Rb+w,l+Π 电子科大经管学院马捷 12 元元元

Decentralized Competitive Equilibrium  Households 电子科大经管学院 马捷 12 t j j t t j t t j t t j t j t j t j t j t j t j t j t j t j t t j t j t j t c i x y r k R b w l z l U c z U c z                     Budget constraint : Time constraint : 1 ( , ) Preference : ( , ) 1 0 0 U U U

Household j accumulates capital and bonds according to k41=(1-6)k(+4 bi=b+xi Assume CRS technology,so firm profits are Zero:Πo=0 Rewrite the household budget: c+k+b≤(1-6+：)k,+(1+R)b,+w, 电子科大经管学院马捷 13

 Household j accumulates capital and bonds according to  Assume CRS technology, so firm profits are zero:  Rewrite the household budget: 电子科大经管学院 马捷 13 j t j t j t j t j t j t b b x k k i        1 1 (1  ) 0 0  j t t j t t j t t j t j t j t c  k  b    r k   R b  w l   (1 ) (1 ) 1 1 

More Constraint The non-negativity constraint: kH1≥0 -The borrowing constraint: l+Bb≤0-6+i)kk+∑9w, x=t+1 1 9:≡ (1+R)1+R).(1+R) =(1+R)9+l 电子科大经管学院马捷 14 元元

More Constraint  The non-negativity constraint:  The borrowing constraint: 电子科大经管学院 马捷 14  1  0 j t k 1 0 1 1 1 1 1 1 1 ( 1 ) ( 1 )( 1 )...( 1 ) 1 ( 1 ) ( 1 )                       t t t t t t j t t j t t R q R R R q w q q R b r k    

Capital and Bonds Simple arbitrage between bonds and capital implies:R,=y-δ ·Letting a=b,+k,→ Budget:c+a<(1+R,)a+w,l Borrowing constraint 42a4=-12g.w 00 9t t=t+1 电子科大经管学院马捷 15

Capital and Bonds  Simple arbitrage between bonds and capital implies:  Letting 电子科大经管学院 马捷 15 R t  rt   a t j  b t j  kt j                1 1 1 1 1 Borrowing constraint : Budget : ( 1 ) t t t j t j t t j t t j t j t q w q a a c a R a w l   