Advanced Macroeconomics - The Ramsey...

Advanced MacroeconomicsThe Ramsey Model

Marcin Kolasa

Warsaw School of Economics

Marcin Kolasa (WSE) Ad. Macro - Ramsey model 1 / 30

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


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


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)


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


Households II

Households maximize the lifetime utility of their members (presentand future):

U0 =∞∑t=0

βtu(Ct)Lt (5)

where:

Ct = Ct

Lt- 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


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)= 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)


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 equality


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 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 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.


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


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)


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)


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)


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)


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.


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)


Steady state equilibrium II

Plotting (21) and (22) in the (k , c) space:

kt

ct

k*

c*

ct+1=ct

kt+1=kt

kG


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)


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).


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


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


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


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) < 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


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


The budget constraint of the government

In Ramsey model with lump sum taxes Ricardian equivalence holds:Government plans sustainable - initial debt plus net present value ofexpenditures equals net present value of revenuesHouseholds live foreverNo financial frictions - one interest rateHence: financing expenditures with lump-sum taxes equivalent tofinancing expenditures with debt

With distortionary taxation Ricardian equivalence breaksStill, for simplicity let us assume that the government runs a balancedbudget each period:

Gt = Vt + τwWtLt + τk (RK ,t − δ)Kt + τcCt + τi It + τf (Yt −WtLt − δKt) (24)

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 (24) holds


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) (25)

Market clearing on the product market:

Yt = Ct + It + Gt (26)


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(27)

Note that, by (25), households’ factor income is no longer equal tooutput

Modified transversality condition:

limt→∞

(Kt+1

t∏s=1

1

1 + (1− τk)rs

)≥ 0 (28)


Modified households’ optimization problem

Lifetime utility (identical to (9)):

U0 = L0

∞∑t=0

βtC 1−θt

1− θ(29)

Capital accumulation (substituting for investment from (27)):

Kt+1 =1− δ1 + g

Kt +1

(1 + g)(1 + τi )[(1− τw )Wt + (1− τk)RK ,tKt + τkδKt

−(1 + τc)Ct − Vt

](30)

Transversality condition:

limt→∞

(Kt+1

t∏s=1

1 + n

1 + (1− τk)rs

)≥ 0 (31)


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 firms’FOC):(

ct+1

ct

)θ= β

1 + τi (1− δ) + (1− τk)(1− τf )(f ′(kt+1)− δ)

(1 + g)θ(1 + τi )(32)

Capital accumulation equation (30) (merged with government budgetconstraint (24) and firms’ zero profit condition (25), with gt = Gt

AtLt):

kt+1

kt=

1− δ(1 + g)(1 + n)

+1

(1 + g)(1 + n)

f (kt)− ct − gtkt

(33)

Transversality condition (31):

limt→∞

(kt+1

t∏s=1

(1 + n)(1 + g)

f ′(kt+1) + 1− δ

)= 0 (34)


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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