A Two-Period Model: The Government and Ricardian ... Government and Ricardian Equivalence ... March and April 2010 Chapter 6, Part 2 1/27 Topics in ... The Government and Ricardian

A Two-Period Model:The Government and Ricardian Equivalence

Chapter 6, Part 2

Topics in Macroeconomics 2

Economics DivisionUniversity of Southampton

March and April 2010

Chapter 6, Part 2 1/27 Topics in Macroeconomics

Competitive EquilibriumThe Ricardian Equivalence Theorem

Credit Market Imperfections and Consumption

Government Budget ConstraintDefinition

Outline

Competitive EquilibriumGovernment Budget ConstraintDefinition

The Ricardian Equivalence TheoremThe TheoremNumerical and Graphical ExamplesThe Ricardian Equivalence in Practice

Credit Market Imperfections and ConsumptionImpact on Budget ConstraintsOptimization





Budget Constraint in the Current Period

G = T + B

I Government spending G is an exogenous variableI Total taxes collected T = mt , as each of the m consumers

pay lump-sum taxes tI The government can borrow by issuing riskless bonds BI The current-period budget deficit T − G is financed by

issuing bonds





Budget Constraint in the Future Period

G′ + (1 + r)B = T ′

I Government spending G′ is an exogenous variableI Total taxes collected T ′ = mt ′

I The government has to pay interest and principal on debtissued yesterday (1 + r)B

I Note: if B < 0, the government is a net lender in the firstperiod and collects (1 + r)B from private agents in thesecond period





Present Value Budget Constraint

G +G′

1 + r= T +

T ′

1 + rI The budget constraint in the future period implies that

B =T ′

− G′

1 + r

I Replace this expression for B in the current period budgetconstraint to get

G = T +T ′

− G′

1 + r︸︷︷︸

B

I Rearranging gives the government present value budgetconstraint above

I The LHS is the present value of spending, which must beequal to the present value of taxes collected on the RHS





Outline








Definition: Competitive Equilibrium

A competitive equilibrium is a set of endogenous variables forconsumers (c, c′, s), aggregate endogenous variables (C, C′,T , T ′, B) and an endogenous real interest rate (r ) such that,given exogenous variables for consumers (y , y ′) and thegovernment (G, G′), the following conditions are satisfied:

1. For each consumer, given (r , y , y ′, t , t ′), the bundle (c, c′)maximizes the consumer’s utility subject to their presentvalue budget constraint

2. The government present value budget constraint holds3. Markets clear:

Credit market:∑m

i=1 s = Sp = B

Period 1 goods market: C + G = Y =∑m

i=1 y

Period 2 goods market: C′ + G′ = Y ′ =∑m

i=1 y ′





Redundance of goods Market Clearing Condition 1

If the credit market clears, so will the period 1 goods market!

I Aggregate private savings is income not consumed:Sp = Y − T − C

I Since the first period budget constraint of the governmentmust hold:B = G − T

I The credit market clearing condition is:Y − T − C︸︷︷︸

Sp

= G − T︸︷︷︸

B

I Rearranging, we get the first period goods market clearingcondition:C + G = Y





Redundance of goods Market Clearing Condition 2

If the credit market clears, so will the period 2 goods market!

I Aggregate consumption in the second period is:C′ = Y ′

− T ′ + (1 + r)Sp

I Since second period budget constraint of the governmentmust hold:G′ + (1 + r)B = T ′

I replace in the first equation:C′ = Y ′

− (G′ + (1 + r)B)︸︷︷︸

T ′

+(1 + r)Sp, or

C′ = Y ′− G′ + (1 + r)(Sp

− B)

I Since the credit market clears (Sp = B), we get the secondperiod goods market clearing condition




The TheoremNumerical and Graphical ExamplesThe Ricardian Equivalence in Practice

Outline








The Ricardian Equivalence

The Ricardian Equivalence TheoremHolding current and future government spending constant, achange in current taxes with an equal and opposite change inthe present value of future taxes leaves the equilibrium interestrate and the consumptions of individuals unchanged

I This theorem suggests that under certain conditions thetiming of taxes does not matter

I What matters is the present value of tax liabilitiesI Key: consumers realize that a tax break today is not free:

taxes tomorrow will be higher, so they save the tax break





Intuition behind the Ricardian Equivalence Theorem

I Suppose the government lowers t by ∆t and increases t ′

by ∆t ′

I Since the present value of government spending remainsthe same, so does the present value of government taxes:T + T ′

1+r remains the same, so ∆T ′ = −(1 + r)∆TI This implies that the present value budget constraint of

consumers is also the samec + c′

1+r = y + y ′

1+r − t − t ′1+r

I At the same interest rate, individuals will choose the sameconsumption bundle (c, c′) as before

I The only changes are in terms of savings for individualsand the government





Intuition behind the Ricardian Equivalence Theorem

I Since neither consumption nor income change, theperiod 1 budget constraint for consumers implies that∆s = ∆t and ∆Sp = ∆T

I Since government spending does not change, the period 1budget constraint of the government implies that∆B = ∆T

I This means that the credit market still clears:∆Sp = ∆B

I Since the taxes are lower in the first period, consumerssave more today as they know they will have to pay moretaxes tomorrow

I Since the government has lower revenues today, it issuesmore debt, which is bought by consumers





Outline








Data for the Numerical Example

Initial ‘parameters’

n G G′ y y ′ t t ′

500 2000 1475 10 12 3 4

Initial equilibrium

c c′ r B Sp

6.00 9.04 0.05 500 500

I we = 10 − 3 + 12−41.05 = 6 + 9.05

1.05 = 14.61I Sp = 500(10 − 6 − 3) = 2000 − 500 × 3 = 500I Y = C + G and the government’s PVBC holds





Change in Taxes

I Suppose the government cuts taxes from 3 to 2 units in thefirst period

I Then for the government PVBC to hold, t ′ must increase by(1+r), from 4 to 5.05

I If the interest rate remains 5%, then consumers will choosethe same consumption since their PVBC is the same:we = 10 − 2 + 12−5.05

1.05 = 14.61I Sp = 500 ∗ (10 − 2 − 6) = 1000 = 2000 − 500 × 2 = B:

so r = 5% still clears the credit marketI Private savings increased by the same amount as the tax

cut, leaving national savings unchanged (at 0)





The Ricardian Equivalence Graphically

I The original endowment is atpoint E1, and the consumerchooses bundle A

I When current taxes fall andfuture taxes increase, theendowment point moves to E2

I Since the change in taxes doesnot affect wealth, the budgetconstraint does not change andbundle A is still optimal

I The consumer simply saves allthe current tax cut to pay highertaxes tomorrow





Outline








The Ricardian Equivalence May Not Hold in Practice

1. All individuals may not pay the same taxes, changing the taxburden across individuals

I We assumed that ∆t ′ = −(1 + r)∆t for all individualsI If some consumers received higher tax cuts than others, they

would change their consumption and the interest rate wouldchange as well

I In practice, the government can redistribute wealth thoughtax policy

2. Debt may not be paid off during the lifetime of all individualswho were alive when it was issued

I Tax cuts could benefit currently old individuals and highertaxes in the future could be paid by the current young

I This scenario involves an intergenerational redistribution ofwealth





The Ricardian Equivalence May Not Hold in Practice

3. Lump-sum taxes are not used in practiceI As we have seen, proportional wage taxation causes

inefficiencies and changes in behaviourI The same is true if the government taxes the return to

savings

4. Credit markets may not be perfectI If you increase taxes today and consumers cannot borrow,

their consumption will go downI The government can generally borrow at a lower rate than

consumers




Impact on Budget ConstraintsOptimization

Outline








The Second Period Budget Constraint

I Suppose that it is costly to intermediate borrowing andlending

I The lending rate (r1) is smaller than the borrowing rate (r2):r2 > r1

I The first period budget constraint is unaffectedI The second period budget constraint now depends on

whether you are a lender or a borrower in the first period:I For lenders (s > 0):

c′ = y ′− t ′ + (1 + r1)s

I For borrowers (s < 0):c′ = y ′

− t ′ + (1 + r2)s





The Lifetime Budget Constraint

I The lifetime budget constraint of a lender:

c +c′

1 + r1= y − t +

y ′− t ′

1 + r1

I The lifetime budget constraint of a borrower:

c +c′

1 + r2= y − t +

y ′− t ′

1 + r2





The Lifetime Budget Constraint Graphically

I To the left of the endowmentpoint E you are a lender and faceinterest rate r1

I To the right of the endowmentpoint E you are a borrower andface interest rate r2 > r1

I The budget constraint is AEFI The slope of the line AE is

−(1 + r1)

I The slope of the line EF is−(1 + r2)





Outline








Consumer Optimization

I Note that many consumers couldend up consuming theirendowment E1

I For this consumer:I The borrowing rate is too high

to make borrowing worthwhileI The lending rate is too low to

make lending worthwhile

I Note that 1 + r1 < MRS < 1 + r2





Impact of a Change in Taxes

I Suppose the government lowerstaxes in period 1 and increasestaxes in period 2, with∆t ′ = −(1 + r1)∆t

I The endowment point moves to E2

I The budget constraint becomesAE2F

I At interest rate r1, the consumerwould like to consume a lot whenyoung (point G )

I Effectively, the government borrowsfor the consumer at rate r1


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A Two-Period Model: The Government and Ricardian ... Government and Ricardian Equivalence ... March and April 2010 Chapter 6, Part 2 1/27 Topics in ... The Government and Ricardian