the ramsey model: financing government consumption ... · review: the ramsey model review:...

Review: the Ramsey ModelReview: Government Consumption in the Ramsey Model

Today: Ways of Financing Government Consumption

The Ramsey Model:Financing Government Consumption

Lecture 14

Topics in Macroeconomics

November 27, 2007

Lecture 14 1/24 Topics in Macroeconomics



Definition of EquilibriumCharacterizing Equilibrium QuantitiesSteady state and dynamics

Outline

Review: the Ramsey ModelDefinition of EquilibriumCharacterizing Equilibrium QuantitiesSteady state and dynamics

Review: Government Consumption in the Ramsey ModelGvmt. cons, lump-sum taxes: Permanent increaseGvmt. cons, lump-sum taxes: Temporary increase

Today: Ways of Financing Government ConsumptionRicardian equivalence: lump-sum taxes vs. debtDistortionary taxes on capital incomeDistortionary taxes on labour income





Definition of Equilibrium* 3

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* 4

◮ 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/θ

limt→∞

βtu′(ct )kt = 0 k0 > 0

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





Steady state and dynamics* 5

Last time...

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

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

... analyzed the effects of government consumption spendingfinanced with lump-sum taxes graphically using equilibriumconditions

Recall:distinction between temporary and permanent changes in g




Gvmt. cons, lump-sum taxes: Permanent increaseGvmt. cons, lump-sum taxes: Temporary increase

A permanent increase in g *graph 6

If there is an unanticipated permanent increase in g and HH didnot expect it before 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





A temporary increase in g *graph 7

If there is an unanticipated temporary increase in g and HHperceive it as such(temporary because people know when g returns to its lowervalue)

◮ 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 8

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 9

Does it matter whether the government chooses to finance

expenditures through debt or non distortionary taxes?

Two approaches:





Ricardian equivalence: allowing for govmt. debt 10

(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 14

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 15

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 16

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 (*) 17

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 20

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 21

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 → distortion






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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