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Advanced MacroeconomicsThe Ramsey Model
Marcin Kolasa
Warsaw School of Economics
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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
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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
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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)
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Households I
Own production factors (capital and labour), so earn income on
renting them to firmsLabour supplied inelastically, grows at a constant rate n>0:
Lt+1= (1 +n)Lt
Capital is accumulated from investment Itand subject to depreciation:
Kt+1 = (1 )Kt+It (3)
Total income of households can be split between consumption or
savings (equal to investment):
WtLt+RK,tKt=Ct+St=Ct+It (4)
Make optimal consumption-savings decisions
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Households II
Households maximize the lifetime utility of their members (present
and future):
U0 =t=0
tu(Ct)Lt (5)
where:
Ct= CtLt - consumption per capita - discount factor (0 < 0If = 1 then u(Ct) = ln Ct
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Households III
Remarks:Literally: household members live foreverJustification: 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)=
u(Ct) is a constant intertemporal elasticity of substitution function:
ln
C1C2
ln
u(C1)
u(C2)
= 1
CRRA form essential for balanced growth (steady state)
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Households IV
Households optimization problem: maximize (5) subject to themodels 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+1t
s=1
1
1 +rs 0 (7)
where: rt=RK,t =f(kt) is the market rate of return on
capital (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) holdswithequality
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General equilibrium
Market clearing conditions:
Output produced by firms must be equal to households total spending(on consumption and investment):
Yt=Ct+It (8)
Labour supplied by households must be equal to labour input
demanded 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 of{Lt,At}
t=0 and an initial capital stock K0, such that (i) the representativehousehold maximizes its utility taking the time path of factor prices{Wt,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.
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Households optimization problem
Lifetime utility rewritten:
U0 =
t=0t
Ct1
1 Lt=
= L0
t=0
tCt
1
1 (9)
where:
=(1 +n) (1 +g)1
(1 +n)
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Households optimization problem II
Capital accumulation rewritten in per capita terms (using (4)):
Kt+1= 1
1 +nKt+
1
1 +n(Wt+RK,tKt Ct) (10)
Capital accumulation rewritten in intensive form (using (4) and
defining wt= Wt
At ):
kt+1 = 1
(1 +g)(1 +n)kt+
1
(1 +g)(1 +n)(wt+RK,tkt ct) (11)
Transversality condition rewritten in intensive form:
limt
kt+1
ts=1
(1 +n)(1 +g)
1 +rs
0 (12)
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Lagrange function and FOCs
Lagrange function (normalizing L0):
LL=t=0
t
C1t1
+t
(1 )Kt+Wt+RK,tKt Ct
(1 +n)Kt+1
First order conditions (FOCs):
LL
Ct= 0 = Ct =t (13)
LL
Kt+1= 0 = t+1(1 +RK,t+1)) = (1 +n)t (14)
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Euler equation I
Equations (13) and (14) imply:
Ct+1
Ct
=
RK,t+1+ 1
(1 +n) (15)
Using the definition of and rewriting in intensive form:ct+1
ct
=
RK,t+1+ 1
(1 +g) (16)
For consumption per capita, using also the definition of the interestrate rt: Ct+1
Ct
=
ct+1At+1
ctAt
=(1 +rt+1) (17)
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Euler equation II
Interpretation of the Euler equation (17):For >0: Ct+1 > Ct 1 +rt+1 >
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:
The higher the less responsive consumption to changes in the interest
rateIn other words: The higher the stronger the consumption smoothingmotive (the lower intertemporal substitution)
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Equilibrium dynamics in intensive form - summary
The equilibrium dynamics of the model at any time tcan becharacterized by 3 equations: (16), (11) and (12). They are (aftersome rewriting and using (4) and (8) in intensive form):
ct+1
ct
=
f(kt+1) + 1
(1 +g) (18)
kt+1
kt=
1
(1 +g)(1 +n)+
1
(1 +g)(1 +n)
f(kt) ctkt
(19)
limtkt+1
t
s=1
(1 +n)(1 +g)
f(kt+1) + 1 = 0 (20)Att= 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.
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Steady state equilibrium I
In the steady state equilibrium kt and ctmust be constant.
Using (18) and (19), the long-run solution to the Ramsey model:
f(k) =(1 +g)
1 + (21)
c =f(k) (n+g++ng)k (22)
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S d ilib i II
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Steady state equilibrium II
Plotting (21) and (22) in the (k, c) space:
kt
ct
k*
c*
ct+1=ct
kt+1=kt
kG
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M difi d ld l I
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Modified golden rule I
How do we know that k
n+g++ng
Since f(k)0
f(k) > f(kG) = k
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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
+
+ng
)
k
f(k)
Using (23):
s 0).
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Th l f th di t f t
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The role of the discount factor
Higher implies more patient consumers
From (21): ifgoes 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
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Ph di
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Phase diagram
From (18): k k = c 0From (19): c f(k) (n+g++ng)k = k 0
kt
ct
k*
c*
ct+1=ct
kt+1=kt
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Saddle path (stable arm)
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Saddle path (stable arm)
Transversality condition (20) pins down the inital level ofc0 for anyinitial k0, so that the system converges to the steady-state:
kt
ct
k*
c*
ct+1=ct
kt+1=kt
k0
c0
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Uniqueness of equilibrium
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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 >kGthis implies:
f(k)
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Speed of convergence
Compared to the Solow model, the speed of convergence in the
Ramsey 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):
1
s = st s depends negatively on kt k
Intuition (suppose the economy starts from k0
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The budget constraint of the government
In the Ramsey model Ricardian equivalence holds:Government plans sustainable - initial debt plus net present value of
expenditures equals net present value of revenuesHouseholds live foreverHence: financing expenditures with lump-sum taxes equivalent tofinancing expenditures with debt
Then, without loss in generality, the government is assumed to run a
balanced budget each period:
Gt=Vt+wWtLt+k(RK,t )Kt+cCt+iIt+f(YtWtLt Kt)(24)
where:
Gt- government purchases (exogenous)w, k, c, i, f - 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(24)holds
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Modified firms problem
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Modified firms problem
Maximization problem of firms:
maxLt,Kt
{F(Kt,AtLt)WtLt RK,tKt f(F(Kt,AtLt)WtLt Kt)}
First order conditions (using definition rt=RK,t ):
Wt= FLt
rt1 f
+= FKt
=f(kt)
Firms are competitive so earn zero profits:
Yt=WtLt+RK,tKt+f(YtWtLt Kt) (25)
Market clearing on the product market:
Yt=Ct+It+Gt (26)
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Modified households problem
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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,tKtwWtLtk(RK,t )KtVt= (1+c)Ct+(1+i)It
(27)Note that, by (25), households factor income is no longer equal tooutput
Modified transversality condition:
limt
Kt+1
ts=1
1
1 + (1 k)rs
0 (28)
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Modified households optimization problem
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Modified households optimization problem
Lifetime utility (identical to (9)):
U0 =L0
t=0
tC1t
1 (29)
Capital accumulation (substituting for investment from (27)):
Kt+1 = 1 1 +g
Kt+ 1(1 +g)(1 +i)
(1 w)Wt+ (1 k)RK,tKt+kKt(1 +c)Ct Vt
(30)
Transversality condition:
limt
Kt+1
ts=1
1 +n
1 + (1 k)rs
0 (31)
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Equilibrium dynamics
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Equilibrium dynamics
The equilibrium dynamics of the model at any time t is given by thefollowing equations:
Euler equation (maximizing(29), subject to (30) and using firmsFOC):
ct+1
ct
=
1 +i(1 ) + (1 k)(1 f)(f(kt+1) )
(1 +g)(1 +i) (32)
Capital accumulation equation (30) (merged with government budgetconstraint (24) and firms zero profit condition (25), with gt=
Gt
AtLt):
kt+1
kt=
1
(1 +g)(1 +n)+
1
(1 +g)(1 +n)
f(kt) ct gtkt
(33)
Transversality condition (31):
limt
kt+1
ts=1
(1 +n)(1 +g)
f(kt+1) + 1
= 0 (34)
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Main implications of the Ramsey model
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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)
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 ruleSpeed of convergence higher than in the Solow model (for standardparameter values)
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