The Ramsey Model - Southampton · The Ramsey Model: Introduction 2 Main ... (Topic 3, Certainty) ... Deﬁnition of Equilibrium* 12 A competitive equilibrium is deﬁned by sequences

Model SetupEquilibrium, Planner, Steady State & Dynamics

Government Consumption Spending and DynamicsWays of Financing Government Consumption

The Ramsey ModelLectures 11 to 14

Topics in Macroeconomics

November 10, 11, 24 & 25, 2008

Lecture 11, 12, 13 & 14 1/50 Topics in Macroeconomics



The Ramsey Model: Introduction 2

Main Ingredients

◮ Neoclassical model of the firm

(Topics 1 & 2)

◮ Consumption-savings choice for consumers

(Topic 3, Certainty)

◮ “Solow model + incentives to save”

(recall example with taxes)




Model SetupMarkets and OwnershipRepresentative FirmRepresentative Household

Equilibrium, Planner, Steady State & DynamicsDefinition and Characterization of EquilibriumBenevolent Planner’s ProblemSteady StateOff Steady State Dynamics

Government Consumption Spending and DynamicsGovernment Consumption & Lump-Sum TaxesGvmt. Cons, Lump-Sum Taxes: Permanent IncreaseGvmt. Cons, Lump-Sum Taxes: Temporary Increase

Ways of Financing Government ConsumptionRicardian Equivalence: Lump-Sum Taxes vs. DebtDistortionary Taxes on Capital IncomeDistortionary Taxes on Labour Income




Markets and OwnershipRepresentative FirmRepresentative Household

Model: Markets and ownership 4

Agents◮ Firms produce goods, hire labor and rent capital◮ Households own labor and assets (capital),

receive wages and rental payments, consume and save◮ There are Nt households and many firms

Markets◮ Inputs: competitive wage rates, w , and rental rate, R◮ Assets: free borrowing and lending at interest rate, r◮ Output: competitive market for consumption good





Model: Firms / Representative Firm* 5

(Recall equivalence!)Seeks to maximize profits

Profitt = F (Kt , Lt) − RtKt − wtLt

The FOCs for this problem deliver

∂F (t)∂Kt

= Rt∂F (t)∂Lt

= wt

In per unit of labor terms, let f (kt) ≡ F (kt , 1)

f ′(kt ) = Rt f (kt ) − kt f ′(kt ) = wt

Recall Euler’s Theorem: factor payments exhaust outputLecture 11, 12, 13 & 14 5/50 Topics in Macroeconomics




Model: Households / Representative household 6

Preferences

U0 =

∞∑

t=0

βtu(ct)

Budget constraint

ct + at+1 = wt lt + (1 + rt)at ,

for all t = 0, 1, 2, ...

a0 given

Note: labor supplied inelastically, lt = 1, i.e. Lt = Nt





Model: Households / Representative household 7

Intertemporal version of budget constraint

∞∑

t=0

t∏

s=0

(

11 + rs

)

ct = a0 +∞∑

t=0

t∏

s=0

(

11 + rs

)

wt

We rule out that debt explodes (no Ponzi games)

at+1 ≥ −B for some B big, but finite

More compactly, PDV (c) = a(0) + PDV (w)





Model: Household’s problem 8

max(at+1,ct )

∞

t=0

∞∑

t=0

βtu(ct)

s.t.

ct + at+1 = wt + (1 + rt)at , for all t = 0, 1, 2, ...

at+1 ≥ −B for some B big, but finite

a0 given






Euler equationIn general,

u′(ct) = β(1 + rt+1)u′(ct+1)

From here on, CES utility, u(c) = c1−σ

1−σ , Euler eqn. becomes,

(

ct+1

ct

)σ

= β(1 + rt+1)






Budget constraint

ct + at+1 = wt + (1 + rt)at , for all t = 0, 1, 2, ...






Transversality conditionHH do not want to “end up” with positive values of assets

limt→∞

βtu′(ct)at ≤ 0

HH cannot think they can borrow at the “end of their life”

limt→∞

βtu′(ct)at ≥ 0

Hence,

limt→∞

βtu′(ct)at = 0




Definition and Characterization of EquilibriumBenevolent Planner’s ProblemSteady StateOff Steady State Dynamics

Definition of Equilibrium* 12

A competitive equilibrium is defined by sequences of quantitiesof consumption, {ct}, capital, {kt}, and output, {yt}, andsequences of prices, {wt} and {rt}, such that

◮ Firms maximize profits

◮ Households maximize U0 subject to their constraints

◮ Goods, labour and asset markets clear◮ Choices are consistent with the aggregate law of motion

for capitalKt+1 = (1 − δ)Kt + It





Characterizing Equilibrium Quantities* 13

From the equilibrium conditions derived before, we find:

◮ There cannot be arbitrage opportunities in equilibrium

Rt − δ = rt

In equilibrium it does not pay to invest in capital directly.The riskless asset and capital have the same payoff.





Characterizing Equilibrium Quantities* 14

From the equilibrium conditions derived before, we find:

◮ Substituting out all the prices leads to the following set ofnecessary and sufficient conditions for an equilibrium interms of quantities only.

kt+1 + ct = f (kt) + (1 − δ)kt

ct+1

ct= [β(1 + f ′(kt+1) − δ)]1/σ

limt→∞

βtu′(ct )kt = 0

k0 > 0

Prices can be determined from the firm’s problems FOCs.Lecture 11, 12, 13 & 14 14/50 Topics in Macroeconomics




Benevolent planner’s problem* 15

What is the allocation of resources that an economy shouldfeature in order to attain the highest feasible level of utility?

Central Planner’s optimal choice problem

max(kt+1,ct)

∞

t=0

∞∑

t=0

βtu(ct)

s.t.

ct + kt+1 = f (kt) + (1 − δ)kt , for all t = 0, 1, 2, ...

k0 > 0 given





Benevolent planner’s problem 16

Welfare

Socially optimal allocation coincides with the equilibriumallocation.

The competitive equilibrium leads to the social optimum.

Not surprising: no distortions or externalities→ Welfare Theorems hold





Notes: simplifying features* 17

◮ We are considering an economy without population growth.

◮ There is no exogenous technological change, either.





Steady state* 18

Definition

A balanced growth path (BGP) is a situation in which output,capital and consumption grow at a constant rate.

If this constant rate is zero, it is called a steady state.

We can usually redefine the state variable so that the latter isconstant (i.e. the growth rate is zero)

Recall from the Solow model:

aggregate capital stock for (n = 0, g = 0)capital per unit of labor for (n > 0, g = 0)capital per unit of effective labor for (n > 0, g > 0)





Steady state 19

From the Euler equation,

ct+1

ct= [β(1 + f ′(kt+1) − δ)]1/σ , for all t

If consumption grows at a constant rate (BGP), say γ

1 + γ = [β(1 + f ′(kt+1) − δ)]1/σ , for all t

Thus RHS must be constant→ kt+1 = kt = k∗ must be constant along the BGP





Steady state 20

But then, from the resource constraint with kt = kt+1 = k∗:

ct + kt+1 = f (kt) + (1 − δ)kt , for all t

i.e.,ct = f (k∗) − δk∗

ct+1 = f (k∗) − δk∗

We find that consumption must be constant along the BGP,→ ct+1 = ct = c∗ or γ = 0

Hence we have a steady state in per capita variables.





Steady state* 21

Hence from the Euler equation

1 + γ = 1 = [β(1 + f ′(k∗) − δ)]1/σ

or, simplified

f ′(k∗) =1β− (1 − δ) = ρ + δ

we can solve for k∗and from the (simplified) resource constraint

c∗ = f (k∗) − δk∗

we can solve for c∗





Modified golden rule* 22

The capital stock that maximizes utility in steady state is calledthe modified golden rule level of capital

f ′(k∗) = ρ + δ

Using f (k) = kα, we get

k∗ = kMGR =

[

α

ρ + δ

]1

1−α

Compare to golden rule level of capital(max conso in st. st.)

kGR =[α

δ

]1

1−α

(see Problem set 2, Q 2.2, assume A = 1 and set s = α)Lecture 11, 12, 13 & 14 22/50 Topics in Macroeconomics




Modified golden rule 23

Since ρ > 0 and α ∈ (0, 1),

kMGR =

[

α

ρ + δ

]1

1−α

<[α

δ

]1

1−α

= kGR

This result reflects the impatience of agents.

As long as ρ > 0, they’d always prefer to consume earlier ratherthan later, thereby reducing investments for next period andhence the steady state level of capital (and consumption)!

One of Ramsey’s points was that this is the steady state that weshould aim at because it makes people the happiest - not theone that maximizes consumption per se.





Off steady state dynamics* 24

Off the steady state, consumption and capital adjust to reachthe steady state eventually.

To analyze these dynamics, consider the movements of c and kseparately.

Let ∆c = ct+1 − ct and ∆k = kt+1 − kt . See graphical analysis.






We use 2 equilibrium conditions:

◮ Euler equation (EE)

ct+1

ct= [β(1 + f ′(kt+1) − δ)]1/σ

◮ Resource constraint (RC)

ct + kt+1 = f (kt) + (1 − δ)kt






◮ Let ∆c = ct+1 − ct and ∆k = kt+1 − kt

◮ Use EE to determine points where ∆c = 0

◮ Use RC to determine points where ∆k = 0

◮ Look at dynamics left and right of ∆c = 0

◮ Look at dynamics above and below ∆k = 0

◮ Steady state is where ∆c = 0 and ∆k = 0






◮ Consider the set of points such that ∆c = 0, then from theEuler eqn, the optimal k satisfies f ′(k) = ρ + δ

→ draw vertical line at k∗(< kGR)

To the left: kt < k∗ ⇒ f ′(kt) > f ′(k∗) ⇒ ∆c > 0 ⇒ c ↑To the right: kt > k∗ ⇒ f ′(kt) < f ′(k∗) ⇒ ∆c < 0 ⇒ c ↓

◮ Consider the set of points such that ∆k = 0, then from theResource cstrt, the optimal c satisfies c = f (k) − δk

→ draw hump-shaped line from origin, maximized at kGR

cross 0 again for k such that f (k) = δk

Above: ct > f (kt) − δkt ⇒ ∆k = f (kt ) − δkt − ct < 0 ⇒ k ↓Below: ct < f (kt) − δkt ⇒ ∆k = f (kt) − δkt − ct > 0 ⇒ k ↑





Equilibrium path toward steady state 28

◮ Suppose k0 < k∗

Then, what consumption level should the household pick?

◮ above ∆k = 0 – curve?

This has c rising but would eventually lead to k = 0 andfrom RC jump of c to c = 0 → violates EE

→ cannot be an equilibrium decision◮ below ∆k = 0 – curve?

Yes, for some c0 all equilibrium conditions will be satisfied

Intuition:K-stock too low, marginal product high → invest a lot

◮ if too low

HH oversaving → leads to c = 0 and ∆k = 0 → violates TC





Equilibrium path toward steady state 29

◮ Suppose kt > k∗

Then, what consumption level should the household pick?

◮ below ∆k = 0 – curve?

This would lead to k such that f (k) = δk and c = 0(u′(0) = ∞, >< transversality)

→ cannot be an equilibrium decision◮ above ∆k = 0 – curve?

Yes, for some ct all equilibrium conditions will be satisfied

Intuition:K-stock too high, marginal product low → consume a lot

◮ If too high, get to k = 0 and jump to c = 0 again.




Government Consumption & Lump-Sum TaxesGvmt. Cons, Lump-Sum Taxes: Permanent IncreaseGvmt. Cons, Lump-Sum Taxes: Temporary Increase

Fiscal policy (Romer 1996 p.59–72) 30

→ Government cons. spending (per capita), gt

→ Financed by lump-sum taxes τt borne by households

The main modifications to the model are:

◮ Public sector

τt = gt , for all t = 0, 1, 2, ...

◮ Household pays taxes

ct + at+1 = wt + (1 + rt)at − τt , for all t = 0, 1, 2, ...





Fiscal policy (Romer 1996 p.59–72) 31

In equilibrium,

EE :ct+1

ct= [β(1 + f ′(kt+1) − δ)]1/σ

RC : ct + gt + kt+1 = f (kt) + (1 − δ)kt

At the steady state (assume gt = g, constant),

EE : f ∗(k∗) = ρ + δ

RC : c∗ = f (k∗) − δk∗ − g

→ k∗ and output per capita unaffected by g

→ g reduces c on 1 to 1 basis: full crowding out





A permanent increase in g *graph 32

If there is a permanent increase in g and HH perceive it assuch

◮ Graphically, ∆k = 0 shifts down by the magnitude of ∆g

◮ The economy adjusts instaneously

through a downward jump of c

→ wealth effect

◮ No dynamic effect on capital accumulation

Hence no effect on output





A temporary increase in g *graph 33

If there is a temporary increase in g and HH perceive it as such

◮ Consumption falls but by less than the increase in g:

wealth effect but consumption smoothing

◮ Therefore, capital falls initially

Hence output declines initially

◮ g back → k returns to initial path toward steady state





First conclusions 34

Government spending cannot increase the steady state level ofoutput per capita.

It can even decrease it in the short–run,e.g. when changes are temporary.




Ricardian Equivalence: Lump-Sum Taxes vs. DebtDistortionary Taxes on Capital IncomeDistortionary Taxes on Labour Income

Ricardian equivalence: lump sum taxes vs. debt 35

Does it matter whether the government chooses to finance

expenditures through debt or non distortionary taxes?

Two approaches:





Ricardian equivalence: allowing for govmt. debt 36

(1.) The main modifications to the model are◮ The government may borrow or lend

dt+1 + τt = gt + (1 + rt)dt

◮ The HH’s budget constraint includes public debt

ct + at+1 + dt+1 = wt + (1 + rt )(at + dt ) − τt

In equilibrium,

EE :ct+1

ct= [β(1 + f ′(kt+1) − δ)]1/θ

RC : ct + gt + kt+1 = f (kt) + (1 − δ)kt






In equilibrium,

EE :ct+1

ct= [β(1 + f ′(kt+1) − δ)]1/θ

RC : ct + gt + kt+1 = f (kt) + (1 − δ)kt

Method of financing irrelevant for allocation of resources.

Public debt changes the distribution of taxes over time,but not its total value.






(2.) The intertemporal budget constraint of the household is notaffected by the sequence of public debt and taxes.

◮ The HH’s IBC

PDV (c) = a0 + d0 + PDV (w) − PDV (τ)

= k0 + d0 + PDV (w) − PDV (τ)

◮ The Government’s IBC

d0 + PDV (g) = PDV (τ)

◮ Combining

PDV (c) = k0 + PDV (w) − PDV (g)






PDV (c) = k0 + PDV (w)− PDV (g)

All that matters for household’s behaviour is the present valueof government expenditures, irrespective of how thegovernment decides to pay for it.





Key assumptions for Ricardian equivalence to hold 40

The debate on Ricardian equivalence:

◮ Infinite horizon OR OLG framework

◮ Perfect capital markets → liquidity constraints

◮ Lump-sum taxation → distortionary taxes: next

◮ Full rationality → rule of thumb?

are key assumptions underlying Ricardian





Distortionary taxes on capital income 41

Tax rate on capital income, τk , with receipts being used tofinance government consumption, g.

The main modifications to the basic model are:

◮ Government’s budget constraint

τkt rtat = gt

◮ Household’s budget constraint

ct + at+1 = wt + (1 + (1 − τkt)rt )at

The tax rate influences the after-tax rate of return on savings.





Distortionary taxes on capital income 42

The Euler equation becomes:

EE :ct+1

ct= [β(1 + (1 − τkt+1)rt+1)]

1/θ

In equilibrium,

EE :ct+1

ct= [β(1 + (1 − τkt+1)(f

′(kt+1) − δ)]1/θ

RC : ct + gt + kt+1 = f (kt) + (1 − δ)kt

in addition to k0 > 0 and the transversality condition.

Notice that now the EE changes, hence the intertemporalconsumption-savings decision is distorted.





Switching from LS taxes to DT on K to finance g (*) 43

Recall: Permanent increase in g financed with LS taxesIf there is a permanent increase in g and HH did not expect itbefore but perceive it as permanent now

◮ Graphically, ∆k = 0 shifts down by the magnitude of ∆g

◮ The economy adjusts instaneously

through a downward jump of c

→ wealth effect

◮ No dynamic effect on capital accumulation

Hence no effect on output






Switching to distortionary taxes on K income to finance gA higher tax rate reduces steady-state capital per capita.*

k∗τk =

(

(1 − τk )α

ρ + (1 − τk )δ

)1

1−α

<

(

(1 − τk )α

(1 − τk )ρ + (1 − τk)δ

)1

1−α

=

(

α

ρ + δ

)1

1−α

= k∗τLS = k∗

k∗τk < k∗τLS = k∗






Switching to distortionary taxes on K income to finance g

◮ Graphically, the ∆c = 0 curve shifts to the left.*(k∗τk < k∗)

◮ Adjustment features an immediate upward jump ofconsumption to reach the path to the new steady state.

◮ This is because saving is less profitable.

◮ In the long-run, consumption will also be lower bec. outputfalls.





Taxes on labour income 46

Tax rate on labour income, τn, with receipts being used tofinance government consumption, g.

The main modifications to the basic model are:

◮ Government’s budget constraint

τntwt = gt

◮ Household’s budget constraint

ct + at+1 = (1 − τnt)wt + (1 + rt)at

The tax rate influences the after-tax wage.





Distortionary taxes on labour income 47

The Euler equation becomes:

EE :ct+1

ct= [β(1 + rt+1)]

1/θ

In equilibrium,EE :

ct+1

ct= [β(1 + f ′(kt+1) − δ)]1/θ

RC : ct + gt + kt+1 = f (kt) + (1 − δ)kt

in addition to k0 > 0 and the transversality condition.

Notice that now the EE and RC are the same as for LS taxes.This is because labour is supplied inelastically in this model.That means a change in the after-tax return to work (price)does not affect any decision.

Next: Example with elastic labour supply and labour incometaxes → distortionLecture 11, 12, 13 & 14 47/50 Topics in Macroeconomics





Static (one period) modelHousehold solves

maxc,l

u(c) + v(l)

s.t. c = (1 − τn)(1 − l)w

where c is consumption, l is leisure, w wage,τn is labour income tax, 1 is time endowment,(1 − l) is hours worked






Static (one period) model: rewrittenAfter substituting for c in the utility function, household solves

maxl

u((1 − τn)(1 − l)w) + v(l)

The first order condition is:

u′(c)((1 − τn)w) = v ′(l)






Static (one period) model: solutionIn terms of Marginal rate of substitution:

MRSc,l =v ′(l)u′(c)

= (1 − τn)w

Thus if the labor tax goes up,

Substitution effect: leisure becomes cheaper relative toconsumption → work less

Income effect: less after-tax income even if work the same→ consume less, work more


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The Ramsey Model - Southampton · The Ramsey Model: Introduction 2 Main ... (Topic 3, Certainty) ... Deﬁnition of Equilibrium* 12 A competitive equilibrium is deﬁned by sequences