V.Models with Overlapping Generations(Continued) (II)Government Borrowing
•Government issues one-period riskless bonds.
•B t=number of units of bonds sold in period t.
•p t=price of a bond.
•Government budget constraint in period t is
N t t1t+N t−1t2t+p t B t=B t−1.
•A member of generation t faces the quence of budget constraints
s t=y1t−c1t−t1t−p t b t
c2t+1=y2t+1−t2t+1+(1+r t)s t+b t
where b t=quantity of government bond purchad by a member of generation t.
•The prent-value budget constraint is
c1t+c2t+1
1+r t
=y1t−t1t+
y2t+1−t2t+1
1+r t
−b t
p t−
1
1+r t
.
•Since1-period government bonds have the same risk characteristics as the1-period private borrowing and lending,arbitrage implies1+r t=1/p t.
•Prent-value budget constraint is unaffected by the addition of government bor-rowing in the model⇒consumption and saving functions are also unaffected.
•One conquence of having two identical asts in the economy(government bonds and private borrowing and lending)is that we are unable to identify the particular mix of private borrowing/lending and government bonds that an individual choos (i.e.we cannot write out individual demand functions for particular asts).
•Competitive equilibrium condition is
S t(r t)=p t B t or S t(r t)=
B t
1+r t
.
•Note that p t B t=number of units of the time-t good the government wishes to borrow.
A Numerical Example
•N t=100for all t.
•Utility of an agent born in period t:
u(c1t,c2t+1)=ln c1t+0.95ln c2t+1.
•Endowment stream{y1t,y2t+1}={1,1.25}.
•Government policy:
–In period1,the government borrows5units of the time1good and transfers tho units to the old individuals alive in period1.
–The government will payoffits debt by taxing the members of generation2 in their young age and will issue no debt after that.
–Therefore we have
p1B1=5,t21=−5
100
=−0.05,t12=
B1
100
.
•Period1
–Aggregate savings function is
S1(r1)=48.72−64.10 1+r1
–Equilibrium condition is
48.72−64.10
1+r1
=5⇒r1=0.47.
–Since B1/(1+r1)=5we have B1=5×1.47=7.35.
–Using r1=0.47in the consumption functions we get c11=0.95and c22=
1.33.
–An old agent alive in period1consumes c21=1.25+0.05=1.3.
•Period2
–Taxes impod on period2young are t12=7.35/100=0.0735.
–Aggregate savings function is
S2(r2)=45.14−64.10 1+r2
–No government borrowing in period2implies
45.14−64.10
1+r2
=0⇒r2=0.42
–Using r2=0.42in the consumption functions we get c12=0.93and c23=
1.25.
•Period t≥3
–No government involvement in the economy.
–All cohort members are identical so
c1t=1,c2t+1=1.25,t≥3.
•Competitive equilibrium prices and consumption allocation are
{r t}∞
t=1
={0.47,0.42,0.32,0.32,...}
{c1t}∞
t=1
={0.95,0.93,1,1,...}
{c2t}∞
t=1
={1.30,1.33,1.25,1.25,...}.
Ricardian Equivalence
google英语在线翻译
Given an initial equilibrium under some pattern of lump-sum taxation and government borrowing,alternative(intertemporal)patterns of lump-sum taxation that keep the prent value(at the initial equilibrium’s interest rate)of each individual’s total tax liability equal to that in the initial equilibrium are equivalent in the following n.Corresponding to each alternative taxation pattern is a pattern of gov-ernment borrowing such that the initial equilibrium’s consumption allocation,including consumption of the government,and the initial equilibrium’s gross interest rates are an equilibrium under the alternative taxation pattern.
Diamond’s Model(OLG Model with Production)
•Production economy where aggregate output is given by
Y t=A t Kα
t N1−α
t
,0<α<1
The difference between Endowment and Production economies:
∗Individuals are endowed with time rather than units of goods.
∗There is still only one good but it is storable,so it can be ud for consumption and investment.
∗The model includesfirms who employ workers and capital to produce.
•No capital depreciation.
•Agents are endowed with one unit of labour time when young and0units when old.
•Labour time supplied inelastically.Wage rate is w t.
•Houholds savings are invested in physical capital.Rate of return(real interest rate)is r t.
•Houhold’s quence of budget constraint
c1t=w t−s t
c2t+1=(1+r t)s t •Prent-value budget constraint
c2t+1=(1+r t)(w t−c1t)•Assuming log utility,houhold problem is
max ln c1t+βln[(1+r t)(w t−c1t)]
⇒c1t=
jamario moon1
1+β
w t,s t(w t)=
β
1+β
bjc
w t
•Since production technology has CRS,the number offirms is indeterminate.As-sume there is a singlefirm.
•Firm maximizes(aggregate)profits
Πt=A t Kα
t N1−α
the spice girls
t
−w t N t−r t K t
⇒r t=αA t Kα−1
t N1−α
t
,w t=(1−α)A t Kα
t
N−α初二数学辅导
t
•Population grows at rateη≥0
N t+1=(1+η)N t⇒N t=(1+η)t N0•Technology grows at rateγ≥0
A t+1=(1+γ)A t⇒A t=(1+γ)t A0•In summary we have
S t(w t)=
β
1+β
侯思思
N t w t(1)
r t=αA t Kα−1
t N1−α
t
(2)
w t=(1−α)A t Kα
dulles
t N−α
t
(3)
K t+1=S t(w t)(4)
•Nonlinear difference equation in K –Combine(1)and(4)
K t+1=
β
1+β
N t w t
英汉翻译–Plug in(3)
K t+1=
β
1+β
N t(1−α)A t Kα
t
N−α
t
–Simplifying,we get the transition equation
K t+1=(1−α)β
1+β
A t Kα
普渡大学西拉法叶校区t
N1−α
t
Ca1:No population Growth,No Technological Growth •η=γ=0;N t=N and A t=A for all t.
•Transition equation becomes
K t+1=
(1−α)β
1+β
AN1−α
school gyrlsKα
t
≡zKα
t
.
•Steady-state capital stock K∗is found by tting K t+1=K t in transition equation
⇒K∗=z11−α
•The model without growth in technology and in population eventually reaches a steady-state where all variables are constant over time.
•In the model with logarithmic utility,the saving rate isβ/(1+β).
•To e the effect of a change in the savings rate,suppo agents become less patient (β <β).
∂z ∂β=
(1−α)AN1−α
(1+β)2
≥0
