高级宏观经济学讲义,经济学诺贝尔奖得主萨金特(Sargent)版本,北京大学版lecture5
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1
Overlapping Generation Model with Production
Model Setup Dynamic Ine¢ ciency
I I
Ricardian Equivalence
In…nite Horizon Model Overlapping Generation Model
Overlapping Generation Model with Production
Model Setup Dynamic Ine¢ ciency
Ricardian Equivalence
In…nite Horizon Model Overlapping Generation Model
I
The optimal allocation
1. Introduce production into the overlapping generation model. Study the model by Diamond (1965), in which he …nds that the OLG model can be dynamic ine¢ cient 2. Whether the timing of taxes matters (LS Chapter 10). e.g., If the government cuts the tax rate to stimulate the economy, what is its e¤ect on the aggregate consumption? If the government increases its spending (building railways, infrastructure), will it have a big impact on the aggregate consumption?
Road Map Overlapping Generation Model with Production
Model Setup Dynamic Ine¢ ciency
I I
Ricardian Equivalence
In…nite Horizon Model Overlapping Generation Model
ct t +1
(1 + r t +1 )2 st 1 + r t +1
Decentralized Problem: Firm
Macroeconomic Theory II B. Zhao Road Map
I
The …rm rents capital from households, hence rt+1 = Fk (Kt+1 , Nt+1 ) = f 0 (k t+1 ) wt = Fn (Kt , Nt ) = f (k t ) k t f 0 (k t )
Overlapping Generation Model with Production
Model Setup Dynamic Ine¢ ciency
Ricardian Equivalence
In…nite Horizon Model Overlapping Generation Model
and
t ∂u ct t , c t +1 ∂ r t +1 t = u1 c t t , c t +1 t = u1 c t t , c t +1
Macroeconomic Theory II B. Zhao Road Map
Macroeconomic Theory II
Lecture 5 Zhao Bo1
CCER, PKU
Overlapping Generation Model with Production
Model Setup Dynamic Ine¢ ciency
The idea of Golden-rule is to …nd the optimal allocation that maximizes the welfare of households along the BGP. This criteria ignores the initial conditions and the transition from the initial condition to the BGP. We look at the social planner’ s problem L = u (cy , co ) + µ f (k) co 1+n cy nk
where f (k ) = F (Kt / Nt , 1)
I
Focs f 0 (k) = n u1 = (1 + n ) u2
Social Planner’ s Problem
Macroeconomic Theory II B. Zhao Road Map Overlapping Generation Model with Production
1I
Outline
I
Macroeconomic Theory II B. Zhao
Today we study two things:
Road Map Overlapping Generation Model with Production
Model Setup Dynamic Ine¢ ciency
ds dwt
Decentralized Problem: HH
Macroeconomic Theory II B. Zhao Road Map
I
Using the envelop theorem, it can be shown that
t ∂u ct t , c t +1 t = u1 c t t , c t +1 ∂ wt
Ricardian Equivalence
In…nite Horizon Model Overlapping Generation Model
I
I
Decentralized Problem: HH
I
Macroeconomic Theory II B. Zhao Road Map
The households problem is standard ct t + s t +1 = w t ct t +1 = (1 + r t +1 ) s t +1
t t t u1 c t t , c t +1 = (1 + r t +1 ) u 2 c t , c t +1
with the associated saving function st+1 = s (wt , rt+1 )
I
> 0, drds 7 0. This is because the changes in the t +1 interest rate have substitution e¤ect/income e¤ect on consumption.
Model Setup Dynamic Ine¢ ciency
I
First of all, the social planner will choose MPK = n, which is the standard result on the nature of the Golden rule de…ned by Phelps. (δ = 0, g = 0) If MPK > n, then the economy is in dynamic ine¢ cient. In the in…nite horizon model, it means that if the planner discards capital, it will increase the interest rate and households obtain a higher consumption level for ever. Second, it looks as if households maximize utility according to the interest rate n.
I
Ricardian Equivalence
In…nite Horizon Model Overlapping Generation Model
The Keynesian believe it has big impact on consumption/output, i.e., the so-called tax/government-expenditure multiplier e¤ect. However, it is not obvious...
Road Map Overlapping Generation Model with Production
Model Setup Dynamic Ine¢ ciency
I
Ricardian Equivalence
In…nite Horizon Model Overlapping Generation Model
I
Dividing the feasibility constraint by Nt y co 1+n cy = nk
I
LHS is the in‡ow into the capital stock. RHS is the out‡ow.
Social Planner’ s Problem
I
Macroeconomic Theory II B. Zhao
Ricardian Equivalence
In…nite Horizon Model Overlapping Generation Model
March 24, 2013
should thank Gianluca Violante and Fabrizio Perri for sharing their lecture notes. All the errors are my own. Email: zhaobo@nsd.edu.cn.
t 1 1 ct
+ Nt ct t
Balanced Growth Path
I
Macroeconomic Theory II B. Zhao
Because there is no productivity growth. In the balanced growth path (BGP), the output/labor ratio is constant. (also the consume/labor ratio) Therefore, K t +1 Kt = Kt+1 / Nt+1 Nt+1 = kNt Nt = nKt Nt+1 1 Kt / Nt Nt
Technology
I
Macroeconomic Theory II B. Zhao Road Map
The production function Yt = F (Kt , Nt ) where Kt is determined one-period ahead. The capital depreciation rate δk = 0 Individuals live for two periods, working in the …rst period and retire in the second period. The utility function is given by u ci i
, ci i
and tБайду номын сангаасe
labor force Nt = N0 (1 + n)t where Nt is the total number of young at the beginning of period t.
I
The feasibility constraint now becomes Yt = Kt+1 Kt + Nt
Overlapping Generation Model with Production
Model Setup Dynamic Ine¢ ciency
Ricardian Equivalence
In…nite Horizon Model Overlapping Generation Model
I
We can express wt = φ (rt ) dφ drt d2 φ 2 drt
Overlapping Generation Model with Production
Model Setup Dynamic Ine¢ ciency
I I
Ricardian Equivalence
In…nite Horizon Model Overlapping Generation Model
Overlapping Generation Model with Production
Model Setup Dynamic Ine¢ ciency
Ricardian Equivalence
In…nite Horizon Model Overlapping Generation Model
I
The optimal allocation
1. Introduce production into the overlapping generation model. Study the model by Diamond (1965), in which he …nds that the OLG model can be dynamic ine¢ cient 2. Whether the timing of taxes matters (LS Chapter 10). e.g., If the government cuts the tax rate to stimulate the economy, what is its e¤ect on the aggregate consumption? If the government increases its spending (building railways, infrastructure), will it have a big impact on the aggregate consumption?
Road Map Overlapping Generation Model with Production
Model Setup Dynamic Ine¢ ciency
I I
Ricardian Equivalence
In…nite Horizon Model Overlapping Generation Model
ct t +1
(1 + r t +1 )2 st 1 + r t +1
Decentralized Problem: Firm
Macroeconomic Theory II B. Zhao Road Map
I
The …rm rents capital from households, hence rt+1 = Fk (Kt+1 , Nt+1 ) = f 0 (k t+1 ) wt = Fn (Kt , Nt ) = f (k t ) k t f 0 (k t )
Overlapping Generation Model with Production
Model Setup Dynamic Ine¢ ciency
Ricardian Equivalence
In…nite Horizon Model Overlapping Generation Model
and
t ∂u ct t , c t +1 ∂ r t +1 t = u1 c t t , c t +1 t = u1 c t t , c t +1
Macroeconomic Theory II B. Zhao Road Map
Macroeconomic Theory II
Lecture 5 Zhao Bo1
CCER, PKU
Overlapping Generation Model with Production
Model Setup Dynamic Ine¢ ciency
The idea of Golden-rule is to …nd the optimal allocation that maximizes the welfare of households along the BGP. This criteria ignores the initial conditions and the transition from the initial condition to the BGP. We look at the social planner’ s problem L = u (cy , co ) + µ f (k) co 1+n cy nk
where f (k ) = F (Kt / Nt , 1)
I
Focs f 0 (k) = n u1 = (1 + n ) u2
Social Planner’ s Problem
Macroeconomic Theory II B. Zhao Road Map Overlapping Generation Model with Production
1I
Outline
I
Macroeconomic Theory II B. Zhao
Today we study two things:
Road Map Overlapping Generation Model with Production
Model Setup Dynamic Ine¢ ciency
ds dwt
Decentralized Problem: HH
Macroeconomic Theory II B. Zhao Road Map
I
Using the envelop theorem, it can be shown that
t ∂u ct t , c t +1 t = u1 c t t , c t +1 ∂ wt
Ricardian Equivalence
In…nite Horizon Model Overlapping Generation Model
I
I
Decentralized Problem: HH
I
Macroeconomic Theory II B. Zhao Road Map
The households problem is standard ct t + s t +1 = w t ct t +1 = (1 + r t +1 ) s t +1
t t t u1 c t t , c t +1 = (1 + r t +1 ) u 2 c t , c t +1
with the associated saving function st+1 = s (wt , rt+1 )
I
> 0, drds 7 0. This is because the changes in the t +1 interest rate have substitution e¤ect/income e¤ect on consumption.
Model Setup Dynamic Ine¢ ciency
I
First of all, the social planner will choose MPK = n, which is the standard result on the nature of the Golden rule de…ned by Phelps. (δ = 0, g = 0) If MPK > n, then the economy is in dynamic ine¢ cient. In the in…nite horizon model, it means that if the planner discards capital, it will increase the interest rate and households obtain a higher consumption level for ever. Second, it looks as if households maximize utility according to the interest rate n.
I
Ricardian Equivalence
In…nite Horizon Model Overlapping Generation Model
The Keynesian believe it has big impact on consumption/output, i.e., the so-called tax/government-expenditure multiplier e¤ect. However, it is not obvious...
Road Map Overlapping Generation Model with Production
Model Setup Dynamic Ine¢ ciency
I
Ricardian Equivalence
In…nite Horizon Model Overlapping Generation Model
I
Dividing the feasibility constraint by Nt y co 1+n cy = nk
I
LHS is the in‡ow into the capital stock. RHS is the out‡ow.
Social Planner’ s Problem
I
Macroeconomic Theory II B. Zhao
Ricardian Equivalence
In…nite Horizon Model Overlapping Generation Model
March 24, 2013
should thank Gianluca Violante and Fabrizio Perri for sharing their lecture notes. All the errors are my own. Email: zhaobo@nsd.edu.cn.
t 1 1 ct
+ Nt ct t
Balanced Growth Path
I
Macroeconomic Theory II B. Zhao
Because there is no productivity growth. In the balanced growth path (BGP), the output/labor ratio is constant. (also the consume/labor ratio) Therefore, K t +1 Kt = Kt+1 / Nt+1 Nt+1 = kNt Nt = nKt Nt+1 1 Kt / Nt Nt
Technology
I
Macroeconomic Theory II B. Zhao Road Map
The production function Yt = F (Kt , Nt ) where Kt is determined one-period ahead. The capital depreciation rate δk = 0 Individuals live for two periods, working in the …rst period and retire in the second period. The utility function is given by u ci i
, ci i
and tБайду номын сангаасe
labor force Nt = N0 (1 + n)t where Nt is the total number of young at the beginning of period t.
I
The feasibility constraint now becomes Yt = Kt+1 Kt + Nt
Overlapping Generation Model with Production
Model Setup Dynamic Ine¢ ciency
Ricardian Equivalence
In…nite Horizon Model Overlapping Generation Model
I
We can express wt = φ (rt ) dφ drt d2 φ 2 drt