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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
Chapter 6, Part 2 2/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
Government Budget ConstraintDefinition
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
Chapter 6, Part 2 3/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
Government Budget ConstraintDefinition
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
Chapter 6, Part 2 4/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
Government Budget ConstraintDefinition
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
Chapter 6, Part 2 5/27 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
Chapter 6, Part 2 6/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
Government Budget ConstraintDefinition
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 ′
Chapter 6, Part 2 7/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
Government Budget ConstraintDefinition
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
Chapter 6, Part 2 8/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
Government Budget ConstraintDefinition
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
Chapter 6, Part 2 9/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
The TheoremNumerical and Graphical ExamplesThe Ricardian Equivalence in Practice
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
Chapter 6, Part 2 10/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
The TheoremNumerical and Graphical ExamplesThe Ricardian Equivalence in Practice
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
Chapter 6, Part 2 11/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
The TheoremNumerical and Graphical ExamplesThe Ricardian Equivalence in Practice
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
Chapter 6, Part 2 12/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
The TheoremNumerical and Graphical ExamplesThe Ricardian Equivalence in Practice
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
Chapter 6, Part 2 13/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
The TheoremNumerical and Graphical ExamplesThe Ricardian Equivalence in Practice
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
Chapter 6, Part 2 14/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
The TheoremNumerical and Graphical ExamplesThe Ricardian Equivalence in Practice
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
Chapter 6, Part 2 15/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
The TheoremNumerical and Graphical ExamplesThe Ricardian Equivalence in Practice
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)
Chapter 6, Part 2 16/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
The TheoremNumerical and Graphical ExamplesThe Ricardian Equivalence in Practice
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
Chapter 6, Part 2 17/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
The TheoremNumerical and Graphical ExamplesThe Ricardian Equivalence in Practice
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
Chapter 6, Part 2 18/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
The TheoremNumerical and Graphical ExamplesThe Ricardian Equivalence in Practice
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
Chapter 6, Part 2 19/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
The TheoremNumerical and Graphical ExamplesThe Ricardian Equivalence in Practice
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
Chapter 6, Part 2 20/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
Impact on Budget ConstraintsOptimization
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
Chapter 6, Part 2 21/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
Impact on Budget ConstraintsOptimization
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
Chapter 6, Part 2 22/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
Impact on Budget ConstraintsOptimization
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
Chapter 6, Part 2 23/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
Impact on Budget ConstraintsOptimization
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)
Chapter 6, Part 2 24/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
Impact on Budget ConstraintsOptimization
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
Chapter 6, Part 2 25/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
Impact on Budget ConstraintsOptimization
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
Chapter 6, Part 2 26/27 Topics in Macroeconomics
Competitive EquilibriumThe Ricardian Equivalence Theorem
Credit Market Imperfections and Consumption
Impact on Budget ConstraintsOptimization
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
Chapter 6, Part 2 27/27 Topics in Macroeconomics