Kt+1 + Kt Kt+1
(1 + rt+1 )Kt+1 Kt+2 (1 + rt+2 )Kt+2 Kt+3 + + ··· 1 + rt+1 (1 + rt+1 )(1 + rt+2 ) Kt+2 Kt+2 Kt+3 Kt+1 + Kt+1 + ··· 1 + rt+1 1 + rt+1 (1 + rt+1 )(1 + rt+2 ) Kt+1 +
↵ = ↵Kt
1 @ V (Kt+1 ) @ Kt+1 1 + rt+1 @ Kt+1 @ Kt 1 1 ( At L t ) 1 ↵ + V 0 (Kt+1 )(1 ) 1 + rt+1
1
( At L t ) 1
↵
+
(8)
Note that we do not choose controls to make Gs (xt , yt ) = 0, so we need more steps. Rearrange (7), 1 @ V (Kt+1 ) =1 1 + rt+1 @ Kt+1 V 0 (Kt+1 ) = 1 + rt+1 Then substitute into (8),
1
Ts
Marginal cost of labour.
5
which means the issued public debt of year t Bs+1 should cover the di↵erence between outstanding debt Bs+1 (including interest rBs+1 ) and tax. And transversality condition 2 is lim ✓Y s ◆ 1 Bs+1 = 0 1 + rs 0
Since Kt+1 is state variable, we use Benveniste-Scheinkman envelope theorem to find the express for V 0 (Kt+1 ), V 0 (xt ) = Fx (xt , yt ) + V 0 G(xt , yt ) Gx (xt , yt ) @ V ( Kt ) ↵ = ↵Kt @ Kt
Since it is competitive economy, Yt is firm’s real revenue in period t, thus dividends for shareholders in t is, D t = Yt w t Lt It (3)
where wt is real wage. The reason of setting up a dividends equation is to establish the value function of representative firm, ◆ 1 ✓ Y s X 1 Vt = Ds 0 1 + rs 0 s=t
(5)
subject to,
↵ Y t = Kt ( At L t ) 1 ↵
Kt+1 = (1 F.O.C. w.r.t. Lt , (1
↵ 1 ↵ ) Kt At ↵
) Kt + I t
Lt
↵
wt = 0
(6)
F.O.C. w.r.t. It , and due to the second constraint, 1+ 1 @ V (Kt+1 ) =0 1 + rt+1 @ Kt+1 (7)
↵ ) Yt
= K t ( rt + ) = K t rt + K t = K t rt + K t = (1 + rt )Kt Resubstitute back to equation (4), Vt = (1 + rt )Kt = (1 + rt )Kt = K (1 + rt )
Kt + Kt ) Kt
s =t+1
(4)
= Dt + Dt+2 · · · 1 + rt+1 1 + rr+1 1 + rr+2
Set up Bellman equation, with Lt , It being controls, ✓ ◆ 1 V (Kt ) = max Dt + V (Kt+1 ) Lt ,It 1+r ✓ ◆ 1 = max Yt wt Lt It + V (Kt+1 ) Lt ,It 1+r ✓ ◆ 1 ↵ 1 ↵ = max Kt (At Lt ) w t Lt It + V (Kt+1 ) Lt ,It 1+r 2
微观基础
⺫目目标
文文本
1
The Decentralised Economy
What we have seen previously, single individual economy or Robinson Crusoe model is called centralised economy, representative agent make decisions on consumption, saving, leisure and works, which means it can be replicated by a central planner making decision for all individuals. There is no need for market structure, all decisions automatically coordinate. In order to coordinate decision in decentralised economy, we need to introduce markets, such as labour market, capital market, assets market and etc. Neoclassical growth model is a decentralised economy model, it is the baseline model for DSGE modeling. Households and firms will optimise their preferences based on dynamic budget constraints. A general equilibrium will be reached between representative household, representative firm and government.
1
2
Model Setup
We assume a competitive economy with identical
Representative Firm
firms, then we use representatives to build the model. Production function of representative firms is Cobb-Douglas form with Harrod-neutral,
(11)
The modified final F.O.C.s are equations (9) and (11), Yt = rt + Kt Yt ↵) = wt Lt ↵ 4 (5.9) (5.10)
(1
In a competitive economy, M P L = M CL1 and M P K = M CK where M CK equals rent interest r and depreciation . In order to find the equilibrium value of representative firm, substitute F.O.C.s (9), (11) and law of motion of capital (2) into dividend expression (3), D t = Yt = Yt = Yt = ↵ Yt w t Lt (1 (1 It Kt+1 (Kt+1 (1 ) Kt ↵) It Yt Lt Lt It It
This is equilibrium time path of value of firms, which is a function of initial capital stock and real interest rate. Government The dynamic budget constraint is, Bt+1 = Bt (1 + rt ) + Gs
(9) (10)
We have the second F.O.C. to manipulate (6), substitute production function into it, (1
↵ 1 ↵ ) Kt At ↵
Lt
↵
wt = 0
(1
↵)
↵ A1 ↵ L 1 ↵ Kt t t = wt Lt Yt (1 ↵) = wt Lt
↵ V 0 (Kt ) = ↵Kt ↵ = ↵Kt ↵ = ↵Kt
1 V 0 (Kt+1 )(1 1 + rt+1 1 1 ( At L t ) 1 ↵ + 1 + rt+1 (1 1 + rt+1