the ramsey model: equilibrium, dynamics and fiscal … · ramsey model: equilibrium, steady state...

Ramsey Model: Equilibrium, Steady state and DynamicsRamsey Model: Government cons. spending and dynamics

Ramsey Model: Ricardian equivalence

The Ramsey Model:Equilibrium, Dynamics and Fiscal Policy

Lecture 13

Topics in Macroeconomics

November 26, 2007

Lecture 13 1/21 Topics in Macroeconomics



Definition of EquilibriumCharacterizing Equilibrium QuantitiesSteady state and dynamics

Definition of Equilibrium* 2

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* 3

◮ From the equilibrium conditions, substituting out all theprices leads to the following set of necessary and sufficientconditions for an equilibrium in terms 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

◮ These are also the conditions for a social optimum(planner’s problem)





Steady state and dynamics* 4

Last time...

... we showed that per capita consumption and capital must beconstant along the BGP

... we analyzed off steady state dynamics graphically usingequilibrium conditions

Today...

... we will recall the analysis of off steady state dynamicstowards the steady state

... analyze the effects of government spending graphicallyusing equilibrium conditions





Modified golden rule* 5

The (feasible) capital stock that maximizes utility in steady stateis called the modified golden rule level of capital and solves

f ′(k∗) = ρ + δ

Recall thatk∗ = kMGR < kGR





Off steady state dynamics* 6

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 10

◮ 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 11

◮ 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) 12

→ 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) 13

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 14

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 15

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 16

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 17

Does it matter whether the government chooses to finance

expenditures through debt or non distortionary taxes?

Two approaches:




Ricardian equivalence: allowing for govmt. debt 18

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

dt+1 − dt + τt = gt + 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/σ

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





In equilibrium,

EE :ct+1

ct= [β(1 + f ′(kt+1) − δ)]1/σ

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.

Next time: distortionary taxes on capital incomeNext time: discussion of Ricardian equivalence


the ramsey model: equilibrium, dynamics and fiscal … · ramsey model: equilibrium, steady state...

Documents