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Advanced MacroeconomicsThe Ramsey Model
Micha l Brzoza-Brzezina/Marcin Kolasa
Warsaw School of Economics
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 1 / 47
Introduction
Authors: Frank Ramsey (1928), David Cass (1965) and TjallingKoopmans (1965)
Basically the Solow model with endogenous savings - explicitconsumer optimization
Probably the most important model in contemporaneousmacroeconomics, workhorse for many areas, including business cycletheories
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 2 / 47
Basic setup
Closed economy
No government
One homogeneous final good
Price of the final good normalized to 1 in each period (all variablesexpressed in real terms)
Two types of agents in the economy:
FirmsHouseholds
Firms and households identical: one can focus on a representativefirm and a representative household, aggregation straightforward
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 3 / 47
Firms
Final output produced by competitive firmsNeoclassical production function with Harrod neutral technologicalprogress
Yt = F (Kt ,AtLt) (1)
Capital and labour inputs rented from householdsTechnology is available for free and grows at a constant rate g > 0:
At+1 = (1 + g)At
Maximization problem of firms:
maxLt ,Kt
F (Kt ,AtLt)−WtLt − RK ,tKt
Firms maximize their profits, taking factor prices as given(competitive factor markets)First order conditions:
Wt =∂F
∂LtRK ,t =
∂F
∂Kt= f ′(kt) (2)
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 4 / 47
Households I
Own production factors (capital and labour), so earn income onrenting them to firms
Labour supplied inelastically, grows at a constant rate n > 0:
Lt+1 = (1 + n)Lt
Capital is accumulated from investment It and subject to depreciation:
Kt+1 = (1− δ)Kt + It (3)
Total income of households can be split between consumption orsavings (equal to investment):
WtLt + RK ,tKt = Ct + St = Ct + It (4)
Make optimal consumption-savings decisions
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 5 / 47
Households II
Households maximize the lifetime utility of their members (presentand future):
U0 =∞∑t=0
βtu(Ct)Lt (5)
where:
Ct = CtLt
- consumption per capita
β - discount factor (0 < β < 1)u(Ct) - instantaneous utility from consumption:
u(Ct) =Ct
1−θ
1− θ(6)
where:
θ > 0If θ = 1 then u(Ct) = ln Ct
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 6 / 47
Households III
Remarks:
Literally: household members live forever
Justification: intergenerational transfers, people care about utility oftheir offspringDiscounting: households are impatient
Remarks on the utility function:
u(Ct) is a constant relative risk aversion function (CRRA):
− Ctu′′(Ct)
u′(Ct)= θ
u(Ct) is a constant intertemporal elasticity of substitution function:
−∂ ln
(C1
C2
)∂ ln
(u′(C1)
u′(C2)
) =1
θ
CRRA form essential for balanced growth (steady state)
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 7 / 47
Households IV
Households’ optimization problem: maximize (5) subject to themodel’s constraints:Capital law of motion (capital is the only asset held by households),incorporating income definition and savings-investment equality (4)
Transversality condition:
limt→∞
(Kt+1
t∏s=1
1
1 + rs
)= 0 (7)
where: rt = RK ,t − δ = f ′(kt)− δ is the market rate of return oncapital (real interest rate)
Interpretation of the transversality condition:
Analog of a terminal condition in a finite horizonNon-negativity constraint on the terminal (net present) value of assetsheld by households (capital)No-Ponzi game condition: proper lifetime budget constraint onhouseholds
Optimization by households implies (7) holds with equalityMicha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 8 / 47
General equilibrium
Market clearing conditions:
Output produced by firms must be equal to households’ totalspending (on consumption and investment):
Yt = Ct + It (8)
Labour supplied by households must be equal to labour inputdemanded by firmsCapital supplied by households must be equal to capital inputdemanded by firms
Definition of competitive equilibrium
A sequence of Kt ,Yt ,Ct , It ,Wt ,RK ,t∞t=0 for a given sequence ofLt ,At∞t=0 and an initial capital stock K0, such that (i) the representativehousehold maximizes its utility taking the time path of factor pricesWt ,RK ,t∞t=0 as given; (ii) firms maximize profits taking the time path offactor prices as given; (iii) factor prices are such that all markets clear.
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 9 / 47
Households’ optimization problem
Lifetime utility rewritten:
U0 =∞∑t=0
βtCt
1−θ
1− θLt =
= L0
∞∑t=0
βtCt
1−θ
1− θ(9)
where:β = β(1 + n) β(1 + g)1−θ(1 + n) < 1
the last inequality is assumed and ensures that utility is bounded
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 10 / 47
Households’ optimization problem II
Capital accumulation rewritten in per capita terms (using (4)):
Kt+1 =1− δ1 + n
Kt +1
1 + n(Wt + RK ,tKt − Ct) (10)
Capital accumulation rewritten in intensive form (using (4) anddefining wt = Wt
At):
kt+1 =1− δ
(1 + g)(1 + n)kt +
1
(1 + g)(1 + n)(wt + RK ,tkt − ct) (11)
Transversality condition rewritten in intensive form:
limt→∞
(kt+1
t∏s=1
(1 + n)(1 + g)
1 + rs
)= 0 (12)
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 11 / 47
Lagrange function and FOCs
Lagrange function (normalizing L0):
LL =∞∑t=0
βt
(C 1−θt
1− θ+ λt
[(1− δ)Kt + Wt + RK ,tKt − Ct−
−(1 + n)Kt+1
])
First order conditions (FOCs):
∂LL
∂Ct
= 0 =⇒ C−θt = λt (13)
∂LL
∂Kt+1
= 0 =⇒ βλt+1 (1− δ + RK ,t+1)) = (1 + n)λt (14)
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 12 / 47
Euler equation I
Equations (13) and (14) imply:(Ct+1
Ct
)θ= β
RK ,t+1 + 1− δ(1 + n)
(15)
Using the definition of β and rewriting in intensive form:(ct+1
ct
)θ= β
RK ,t+1 + 1− δ(1 + g)θ
(16)
For consumption per capita, using also the definition of the interestrate rt : (
Ct+1
Ct
)θ=
(ct+1At+1
ctAt
)θ= β (1 + rt+1) (17)
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 13 / 47
Euler equation II
Interpretation of the Euler equation (17):
For θ > 0: Ct+1 > Ct ⇐⇒ 1 + rt+1 > β−1
Interpretation: For (per capita) consumption to grow the (market)interest rate must exceed households’ rate of time preferenceInterpretation: It is optimal for households to postpone consumption(i.e. save in the current period and consume more in the next period)iff the related utility loss is more than offset by the rate of return onsavings
Role of θ:
The higher θ the less responsive consumption to changes in the interestrateIn other words: The higher θ the stronger the consumption smoothingmotive (the lower intertemporal substitution)
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 14 / 47
Equilibrium dynamics in intensive form - summary
The equilibrium dynamics of the model at any time t can becharacterized by 3 equations: (16), (11) and (12). They are (aftersome rewriting and using (4) and (8) in intensive form):(
ct+1
ct
)θ= β
f ′(kt+1) + 1− δ(1 + g)θ
(18)
kt+1
kt=
1− δ(1 + g)(1 + n)
+1
(1 + g)(1 + n)
f (kt)− ctkt
(19)
limt→∞
(kt+1
t∏s=1
(1 + n)(1 + g)
f ′(kt+1) + 1− δ
)= 0 (20)
At t = 0 capital is fixed. For given initial k0 and c0, equations (18)and (19) describe the future evolution of these variables: kt and ct .The transversality condition (20) pins down the initial level ofconsumption c0.
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 15 / 47
Steady state equilibrium I
In the steady state equilibrium kt and ct must be constant.
Using (18) and (19), the long-run solution to the Ramsey model:
f ′(k∗) =(1 + g)θ
β− 1 + δ (21)
c∗ = f (k∗)− (n + g + δ + ng)k∗ (22)
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 16 / 47
Steady state equilibrium II
Plotting (21) and (22) in the (k , c) space:
kt
ct
k*
c*
ct+1=ct
kt+1=kt
kG
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 17 / 47
Modified golden rule I
How do we know that k∗ < kG?
From (22):f ′(kG ) = n + g + δ + ng
The transversality condition (20) written in the steady-state:
limt→∞
k∗(
(1 + n)(1 + g)
f ′(k∗) + 1− δ
)t
= 0
This implies:f ′(k∗) > n + g + δ + ng
Since f ′′(k) < 0 for any k > 0
f ′(k∗) > f ′(kG ) =⇒ k∗ < kG (23)
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 18 / 47
Modified golden rule II
Equivalently, we can show that the steady-state savings rate s∗ fallsshort of the savings rate consistent with the golden rule:
From (22), the steady-state savings rate is:
s∗ = 1− c∗
f (k∗)= (n + g + δ + ng)
k∗
f (k∗)
Using (23):
s∗ < f ′(k∗)k∗
f (k∗)= α(k∗)
Intuitive explanation: households are impatient (β < 1) and smoothconsumption (θ > 0, relevant if g > 0).
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 19 / 47
The role of the discount factor
Higher β implies more patient consumers
From (21): if β goes up, f ′(k∗) goes down, which means that k∗
goes up
The ct+1 = ct locus on the (k , c) chart shifts right
Steady-state consumption goes up
Intuition: if households are more patient, they save more, whichbrings them closer to the standard golden rule
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 20 / 47
Phase diagram
From (18): k ≶ k∗ =⇒ ∆c ≷ 0From (19): c ≶ f (k)− (n + g + δ + ng)k =⇒ ∆k ≷ 0
kt
ct
k*
c*
ct+1=ct
kt+1=kt
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 21 / 47
Saddle path (stable arm)
Transversality condition (20) pins down the inital level of c0 for anyinitial k0, so that the system converges to the steady-state:
kt
ct
k*
c*
ct+1=ct
kt+1=kt
k0
c0
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 22 / 47
Uniqueness of equilibrium
How do we know that the saddle path is a unique equilibrium?
If the initial level of consumption were below c0:
capital would eventually reach its maximal level k > kG
this implies:f ′(k) < f ′(kG ) = n + g + δ + ng
which violates the transversality condition (20) since:
limt→∞
k
((1 + n)(1 + g)
f ′(k) + 1− δ
)t
=∞
informally (but more intuitively): at the end of their planning horizon,households would hold very valuable assets, which cannot be optimal
If the initial level of consumption were above c0:
capital would eventually reach 0 but consumption would stay positive,which is clearly not feasible
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 23 / 47
Speed of convergence
Compared to the Solow model, the speed of convergence in theRamsey model depends additionally on the behaviour of the savingsrate along the transition path
For very small time intervals, the following implications hold (seeBarro and Sala-i-Martin, 2004, ch. 2.6.4):
1θ < s∗ =⇒ st − s∗ depends positively on kt − k∗1θ = s∗ =⇒ st = s∗1θ > s∗ =⇒ st − s∗ depends negatively on kt − k∗
Intuition (suppose the economy starts from k0 < k∗, so c0 < c∗):
if households care much about consumption smoothing (θ is high),they wll try to shift consumption from the future to the presentif households care little about consumption smoothing (θ is low), theywill try to postpone consumption to reach steady-state sooner
For standard parameter values 1θ > s∗, so the Ramsey model predicts
relatively fast pace of convergence
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 24 / 47
Wealth and consumption
Consider the lifetime budget constraint of the household
C0 +∞∑t=1
Ct
Rt= (1 + r0)K0 + W0L0 +
∞∑t=1
WtLtRt
≡ Ωt
where Ωt is lifetime wealth. Substitute recursively from Euler
Ct+1
Ct= (β (1 + rt))
1θ (1 + n)
C0+C0(β (1 + r1))
1θ (1 + n)
R1+C0
(β2 (1 + r1) (1 + r2)
) 1θ (1 + n)2
R2+. . . = Ωt
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 25 / 47
Wealth and consumption cont’d
so that
C0 = Ωt
[1 +
∞∑t=1
(βtRt)1θ (1 + n)t
Rt
]−1
For log utility this boils down to
C0 = [1− β (1 + n)] Ωt
So current consumption depends on lifetime wealth. Current incomehas a small impact on current consumption!
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 26 / 47
Efficiency of equilibrium
A very important question in economics: are markets efficient?
What does it mean? If markets are efficient the equilibrium allocationmaximizes households welfare.
Then government intervention is pointless (see Adam Smith’s invisiblehand)
But we know many situations where markest fail. E.g.:
externalitiesimperfect informationprincipal-agent problem
How about the Ramsey model?
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 27 / 47
Social planer’s problem
Imagine a social planer - allocates resources and wishes to maximizehousehold welfare
Constrained by technology and available resources
For simplicity consider a model without labor or technology growth
maxct ,kt+1,yt
U0 =∞∑t=0
βtu(ct) (24)
subject toyt = f (kt)
and
ct + kt+1 = yt + (1− δ)kt
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 28 / 47
Lagrangean and FOC
L =∞∑t=0
βtu(ct)−∞∑t=0
λt [ct + kt+1 − f (kt)− (1− δ)kt ] (25)
FOC:
ct : βtuc,t = λt
kt+1 : −λt + λt+1
[f ′(kt+1) + 1− δ
]= 0
TVC (imposed) : limt→∞
λtkt+1 = 0
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 29 / 47
Optimal allocation
The optimal alocation is given by
uc,tβuc,t+1
= f ′(kt+1) + 1− δ
TVC : limt→∞
βtuc,tkt+1 = 0
yt = f (kt)
ct + kt+1 = yt − (1− δ)kt
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 30 / 47
Comparison to decentralized (competitive) equilibrium
The competitive equilibrium is given by (16), (11) and (12), whichafter ignoring labor and population growth become:
uc,tuc,t+1
= β (RK ,t+1 + 1− δ)
kt+1 = (1− δ) kt + (wt + RK ,tkt − ct)
limt→∞
(kt+1
t∏s=1
1
1 + rs
)= 0
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 31 / 47
Competitive equilibrium cont’d
Use zero profit condition:
kt+1 = (1− δ) kt + yt − ct
Note that
uc,t+1
uc,t= β(1 + rt+1)
so thatt∏
s=1
1
1 + rs=βuc,1uc,0
βuc,2uc,1
· · · βuc,tuc,t−1
=βtuc,tuc,0
Hence TVC condition is equivalent to
limt→∞
(βtuc,tkt+1
)= 0
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 32 / 47
Welfare theorems
Hence, social planner’s and competitive allocations are equal. Ramseyeconomy reflects two fundamental welfare theorems.
1 First welfare theorem: every competitive equilibrium is efficient.
2 Second welfare theorem: any efficient allocation can be supported asa competitive equilibrium (possibly with lump-sum transfers).
First states that resources are not wasted in a competitve economy(Adam Smith would like it).
Second gives the equivalence beween an efficient and competitiveallocation and thus, the possibility to solve the (simpler) plannersproblem.
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 33 / 47
The government
In modern economies governments are big playersGY is 40-50% (once all government expenditure is accounted for)
Some problems worth analysing:
difference/ similarity between tax and debt financing of gov.expenditurereaction of economy to government shocks (spending, taxes)
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 34 / 47
Ricardian equivalence
In the Ramsey model Ricardian equivalence holds:
Government plans sustainable - initial debt plus net present value ofexpenditures equals net present value of revenues
T0 +∞∑t=1
Tt
Rt= (1 + r0)B0 + G0 +
∞∑t=1
Gt
Rt
where
Rt ≡t∏
i=1
(1 + ri )
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 35 / 47
Ricardian equivalence cont’d
Household lifetime budget (with government)
C0 +∞∑t=1
Ct
Rt= (1 + r0)K0 + (1 + r0)B0 + W0L0 +
∞∑t=1
WtLtRt−T0 −
∞∑t=1
Tt
Rt
Substitute from government budget for∑∞
t=1TtRt
C0 +∞∑t=1
Ct
Rt= (1 + r0)K0 + W0L0 +
∞∑t=1
WtLtRt− G0 −
∞∑t=1
Gt
Rt
What matters for the household is the present value of expenditures
Financing expenditures with lump-sum taxes equivalent to financingexpenditures with debt
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 36 / 47
Ricardian equivalence in practice
Empirical research shows that Ricardian equivalence holds partly
Why? Some explanations can be given within our modelingframework.
finite horizon of householdsdifferent interest ratescredit constrained householdsdistortionary taxes
Good to know the theoretical benchmark where Ricardian equivalenceholds
For instance to be able to construct models where gov. deficits havemacro effects
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 37 / 47
Distortionary taxation
The government is assumed to run a balanced budget each period:
Gt = Vt+τwWtLt+τk (RK ,t − δ)Kt+τcCt+τi It+τf (Yt −WtLt − δKt)(26)
where:
Gt - government purchases (exogenous)τw , τk , τc , τi , τf - proportional tax rates on wage income, capitalincome, consumption, investment and firms’ taxable profits,respectively (all exogenous)Vt - lump-sum taxes (net of lump-sum transfers) from households,adjusted so that the balanced budget constraint (26) holds
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 38 / 47
Modified firms’ problem
Maximization problem of firms:
maxLt ,Kt
F (Kt ,AtLt)−WtLt − RK ,tKt − τf (F (Kt ,AtLt)−WtLt − δKt)
First order conditions (using definition rt = RK ,t − δ):
Wt =∂F
∂Lt
rt1− τf
+ δ =∂F
∂Kt= f ′(kt)
Firms are competitive so earn zero profits:
Yt = WtLt + RK ,tKt + τf (Yt −WtLt − δKt) (27)
Market clearing on the product market:
Yt = Ct + It + Gt (28)
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 39 / 47
Modified households’ problem
We assume that government actions do not affect utility directly, sohouseholds’ lifetime utility is still given by (5)
Households’ budget constraint (4) becomes:
WtLt+RK ,tKt−τwWtLt−τk (RK ,t − δ)Kt−Vt = (1+τc)Ct+(1+τi )It(29)
Note that, by (27), households’ factor income is no longer equal tooutput
Modified transversality condition:
limt→∞
(Kt+1
t∏s=1
1
1 + (1− τk)rs
)= 0 (30)
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 40 / 47
Modified households’ optimization problem
Lifetime utility (identical to (9)):
U0 = L0
∞∑t=0
βtC 1−θt
1− θ(31)
Capital accumulation (substituting for investment from (29)):
Kt+1 =1− δ1 + g
Kt +1
(1 + g)(1 + τi )[(1− τw )Wt + (1− τk)RK ,tKt + τkδKt
−(1 + τc)Ct − Vt
](32)
Transversality condition:
limt→∞
(Kt+1
t∏s=1
1 + n
1 + (1− τk)rs
)= 0 (33)
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 41 / 47
Equilibrium dynamics
The equilibrium dynamics of the model at any time t is given by thefollowing equations:
Euler equation (maximizing (31), subject to (32) and using firms’FOC):(
ct+1
ct
)θ= β
1 + τi (1− δ) + (1− τk)(1− τf )(f ′(kt+1)− δ)
(1 + g)θ(1 + τi )(34)
Capital accumulation equation (32) (merged with government budgetconstraint (26) and firms’ zero profit condition (27), with gt = Gt
AtLt):
kt+1
kt=
1− δ(1 + g)(1 + n)
+1
(1 + g)(1 + n)
f (kt)− ct − gtkt
(35)
Transversality condition (33):
limt→∞
(kt+1
t∏s=1
(1 + n)(1 + g)
f ′(kt+1) + 1− δ
)= 0 (36)
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 42 / 47
Simplified variant - only capital tax
Euler equation becomes:(ct+1
ct
)θ= β
1 + (1− τk)(f ′(kt+1)− δ)
(1 + g)θ
Capital accumulation equation:
kt+1
kt=
1− δ(1 + g)(1 + n)
+1
(1 + g)(1 + n)
f (kt)− ct − gtkt
Steady state:
f ′(k∗) =(1 + g)θ − ββ(1− τk)
+ δ
c∗ = f (k∗)− (n + g + δ + ng)k∗ − g∗
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 43 / 47
Simplified variant - simulate shocks
Permanent increase in government expenditure (financed withlump-sum taxes)
Temporary increase in government expenditure (financed withlump-sum taxes)
Permanent increase in capital tax
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 44 / 47
Conclusions from model with government
In the Ramsey model:
Ricardian equivalence holds as long as taxes are nondistortionaryTaxes on capital, investment and firm profits are distortionary (changethe equilibrium allocation)Taxes on consumption and labor income are nondistortionary (but onlyas long as labor supply is exogeneous)Government expenditure crowds out private consumption one-to-onebut does not affect capital or output (if financed by nondistortionarytaxes)
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 45 / 47
Some simulations with the Ramsey model
Several interesting numerical simulations can be done with theRamsey model
find initial consumption for given k0 that fulfils the equilibriumconditionssimulate convergence from c0, k0 to steady statesimulate a change (temporary or permanent) of:
rate of time preferencegovernment expenditurecapital tax
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 46 / 47
Main implications of the Ramsey model
As in the Solow model, long-run growth (of output per capita)possible only with technological progress (exogenous in both models)
We should observe conditional, but not necessarily unconditional,convergence (in line with the data)
Shows several important (though ratehr benchmark) features
Ricardian equivalenceKey role of permanent income (lifetime wealth) in driving consuption
Compared to the Solow model:
Explicit optimality criterion - households’ utilityIf there is no distortionary taxation (i.e. if there is no government or alltaxes are lump-sum), allocations are Pareto optimal: decentralizedequilibrium coincides with allocations dictated by a benevolent socialplaner (markets are competitive and complete, so the first welfaretheorem applies)Savings rate endogenous and in the long-run always lower than impliedby the golden rule
Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 47 / 47