ECO 2021 Intermediate Macroeconomic Theory Professor C. K. Yip

1
A Two Period ModelConsumption-Savings Decision
Ricardian Equivalence

• Chapter 8

2
Two-Period Model of the Economy

• Focus on the consumers and governments
  behavior.
• A consumers consumption-savings decision
  (intertemporal choice) involves a trade-off
  between current and future consumption.
• By saving, a consumer gives up consumption in
  exchange for assets in the present, in order to
  consume more in the future.
• The governments decision concerning the
  financing of government expenditure, involves a
  trade-off between current and future taxes.
• Policy Implication Ricardian equivalence theorem

3
Consumers

• Assumptions
• N consumers, where N is a large number
• Each consumer lives for two periods, current and
  future period.
• Each consumer receives exogenous income in both
  periods.
• Put aside the work-leisure decision.
• Income can differ across consumers.
• Zero wealth endowment (no assets) in the current
  period.
• Each consumer pays lump-sum taxes in both periods.

4
Consumers

• Assumptions
• Only bond is traded in financial market.
• Consumers can lend and borrow at the same real
  interest rate, r (Perfect credit market).
• No risk for holding bonds.
• Define bonds
• A bond issued (by the government) in the current
  period is a promise to pay a certain amount, say
  1 r units, of consumption good in the future
  period.
• ? 1 unit of c can be exchanged for 1 r units
  of c' in the credit market.
• r is the real interest rate on each bond.

5
Consumers

• Notations
• y and t denote real income and tax in first
  period.
• y' and t' denote real income and tax in second
  period.
• Current period budget constraint
• c s y t (1)
• Consumers after-tax income can either be saved
  (s) or consumed (c).
• Saving gt 0 (lt 0)
• ? Buys (sells) a bond with part of his income.
• ? The consumer is a lender (borrower).

6
Consumers

• Future period budget constraint
• c' y' t' (1 r)s (2)
• Apart from the after-tax income, the consumer
  receives the interest and principal on savings.
• For only two periods, no incentive to save in the
  second period.
• ? Consumer will consume everything he has in the
  second period.
• Next, to combine the two constraints into one
  lifetime budget constraint
• From (2), we have

7
Consumers

• Substitute (3) into (1), we have
• RHS The present value of lifetime disposable
  income or lifetime wealth (we).
• LHS The present value of lifetime consumption.
• Note We can think of (1 r )1 as relative
  price of c' in terms of c, while the price of c
  is normalized to one.
• Plot this in a (c , c') graph c' (we c)(1
  r)

8
Consumers Lifetime Budget Constraint

• The slope of the lifetime budget constraint is
  (1 r).
• E is the endowment point (i.e. where s 0).
• Points on BE ? s ? 0 ?consumer is a lender.
• Points on EA ? s ? 0 ?consumer is a borrower.

c' Future Consumption
B
we(1 r)
E
y' t'
A
we
y t
c Current Consumption
9
Consumers Preferences

• Consumers utility function is given by U(c , c')
• Properties of Preferences
• 1) More is preferred to less
• U(. , .) is increasing in both arguments.
• 2) The consumer likes diversity in consumption
  bundle.
• To smooth consumption over time.
• i.e. consumer dislikes consuming a lot in a
  single period but very few in another.

10
Consumers Preferences

• 3) Current and future consumption are normal
  goods.
• Let A be the original optimal choice.
• Suppose income increases, the budget constraint
  shifts upwards in parallel.
• The new optimal choice must lie on BD.
• Also implies consumption smoothing.

c'
B
D
A
c
11
Indifference Curves

• Slope of indifference curve MRSc,c'
• The following are equivalent
• A preference for diversity
• Diminishing MRSc,c'
• Convexity of indifference curve
• Consumption smoothing

c' Future Consumption
Slope MRSc,c'
A
c Current Consumption
12
Consumers Problem

• The consumers optimal consumption bundle is the
  point at which an indifference curve is tangent
  to the budget constraint.
• This implies the following condition
• 1 r MRSc,c'

c' Future Consumption
B
we(1 r)
A
c'
E
y' t'
A
y t
we
c
c Current Consumption
13
Consumers Problem

• The consumer chooses c and c' to maximize U(c ,
  c') subject to the lifetime budget constraint.
• max U(c , c')
• subject to
• Lagrangian

14
Consumers Problem

• First-order conditions
• From the first two conditions, we obtain

15
Comparative Statics

• To determine the effects of changes in y, y' and
  r on c, c' and s.
• Totally differentiate the following system
• Assuming dt dt' 0, then in matrix form, we
  have

16
Comparative Statics

• Consider the bordered Hessian matrix
• The determinant of A is
• ? Ucc 2(1 r)Ucc (1 r)2Ucc
• Given that U(. , .) is strictly quasiconcave, ? gt
  0.

17
Graphical IllustrationsLenders and Borrowers

• Consumers as lenders
• ?Endowment point E1
• ?Consumption bundle A
• ?Consume only c at current period and save the
  amount cF
• Consumers as borrowers
• ?Endowment point E2
• ?Consumption bundle A
• ?Consume at c currently by borrowing the amount
  Dc

c' Future Consumption
B
we(1 r)
E2
H
A
c'
E1
J
we
c
F
D
c Current Consumption
18
1) Increase in Current-Period Income

• By Cramers rule,
• c and c' are normal goods
• ? Ucc (1 r)Ucc gt 0 and Ucc (1 r)Ucc
  gt 0
• ?

19
1) Increase in Current Income for
lendersGraphical Illustration

• Suppose y ?. Then the budget constraint shifts to
  the right.
• The slope of budget constraint remains unchanged,
  as r remains the same.
• New endowment point E2
• New optimal choice B (the consumer is a lender).
• ?y AD gt ?c AF
• ? ?s gt 0.

c' Future Consumption
we2(1 r)
we1(1 r)
B
c2'
A
c1'
D
F
I2
E2
E1
I1
c2
c1
we2
we1
c Current Consumption
20
Consumption smoothing

• When current income ?, consumer will consume more
  during the current period, but will also save
  some of the increase so as to consume more in the
  future as well.
• Prediction Real aggregate consumption should be
  less variable than real GDP.
• This is consistent with the data.

21
2) Increase in Future Income

• By Cramers rule,
• Qualitatively, the effects of a change in y' is
  identical to those of a change in y.
• Since s y c t,

22
2) Increase in Future Income Graphical
Illustration

• Suppose y' ?. Again the budget constraint shifts
  to the right.
• New optimal choice B (the consumer is a
  borrower).
• Rather than spending all the increase in the
  future, the consumer saves less in the current
  period so that c ?.
• ?c FB and ?y 0
• ? ?s lt 0.

c' Future Consumption
we2(1 r)
D
we1(1 r)
B
F
c2'
A
c1'
I2
I1
c1
c2
we2
we1
c Current Consumption
23
Temporary vs. Permanent Changes in Income

• Permanent income hypothesis (Milton Friedman)
• A primary determinant of a consumers current
  consumption is his/her permanent income (lifetime
  wealth).
• Temporary changes in income
• ? Yield small changes in permanent income
• ? Small effects on current consumption
• Permanent changes in income
• ? Large effects on permanent income and current
  consumption
• In our model,
• Temporary changes ? Only y ?
• Permanent changes ? Both y ? and y' ? (to the
  same extent)

24
3) Increase in Real Interest Rate

• By Cramers rule,
• The signs of both derivatives cannot be
  determined due to the presence of opposing income
  and substitution effects.
• Since (1 r)1 is the relative price of c' in
  terms c, a change in r effectively change this
  intertemporal relative price.

25
3) Increase in Real Interest RateGraphical
Illustrations
c' Future Consumption

• Suppose r ?. The budget constraint becomes
  steeper.
• Assume y' t' gt 0, then r ? ? we ? and we(1
  r) ?
• Since it is always possible to have c y
  and c' y', the budget constraint must pivot
  around point E.
• r ? ? Return on saving ?
• ? More c' can be obtained for a given sacrifice
  of c
• ? Intertemporal substitution effect

we2(1 r2)
we1(1 r1)
E
we1
we2
c Current Consumption
26
3) Increase in Real Interest RateGraphical
Illustrations
c' Future Consumption

• For lenders
• Initial optimal choice A.
• Suppose the consumer chooses B after r ?.
• Draw an artificial budget constraint FG.
• Pure substitution effect A ? D (c ? and c' ?).
• Pure income effectD ? B (c ? and c' ?).
• Conclusion c' ? but c (?), s (?).

we2(1 r)
I2
I1
F
B
D
we1(1 r)
A
E
we2
we1
G
c Current Consumption
27
3) Increase in Real Interest RateGraphical
Illustrations
c' Future Consumption

• For borrowers
• Initial optimal choice A.
• Suppose the consumer chooses B after r ?.
• Draw an artificial budget constraint FG.
• Pure substitution effectA ? D (c ? and c' ?).
• Pure income effectD ? B (c ? and c' ?).
• Conclusion c ?, s ? but c' (?).

F
we2(1 r)
I1
I2
we1(1 r)
D
E
B
A
we1
we2
G
c Current Consumption
28
Example Perfect Complements

• Assume that the utility function is of the form
• U(c , c') minac , c'
• where a gt 0 is a constant.
• This is an extreme case of a desire for
  consumption smoothing.
• The consumer will always choose
• c' ac.

c'
I2
I1
A
c' ac
D
B
c
29
Example Perfect Complements

• Using this and the budget constraint, we get
• Since there are no substitution effects in this
  case, the effects of r ? depends only on whether
  the consumer is a lender or a borrower.

c'
I2
I1
A
c' ac
D
B
c
30
Government

• Notations
• G and G' denote current and future government
  purchases of consumption goods.
• T and T' denote current and future aggregate
  quantity of taxes.
• T Nt and T' Nt'.
• B denote the quantity of government bonds issued
  in the current period.
• B lt 0
• ? Government was a lender to the private sector.

31
Government

• Governments budget constraints
• G T B
• G' (1 r)B T'
• Combining the two into a single government
  present-value budget constraint
• which states that the present value of
  government purchases must equal the present value
  of taxes.

32
Competitive Equilibrium

• In a competitive equilibrium, the following must
  hold
• 1) Each consumer chooses c, c' and s optimally
  given r.
• 2) The governments present-value budget
  constraint holds.
• 3) The credit market clears,
• i.e. Sp B where Sp is the aggregate quantity
  of private saving.
• This means the aggregate quantity of private
  savings is equal to the quantity of debt issued
  by the government.
• Since Sp Y T C and B G T, so we have
• Y C T G T or Y C G
• so that the goods market also clears.

33
Ricardian Equivalence Theorem

• Definitions
• The Ricardian Equivalence theorem states that
  changes in current taxes by the government that
  leave the present value of taxes constant have no
  effect on consumers consumption choices or on
  the equilibrium real interest rate.
• Key message A tax cut is not a free lunch
  because the costs of a tax cut (future tax ?)
  exactly offset the benefits.

34
Ricardian Equivalence Theorem

• Suppose the economy is initially in a competitive
  equilibrium with a consumer facing the budget
  constraint
• and the government facing

35
Ricardian Equivalence Theorem

• Since government spending is constant, the full
  amount of the tax cut must be financed by higher
  future taxes.
• With full information, then the consumers will
  save the full amount of the tax cut in order to
  pay for high future taxes.
• Government and private savings have to change by
  equal and opposite amount, so that national
  saving is unaffected.

36
Ricardian Equivalence Theorem
c' Future Consumption

• Suppose t ? (tax cut).
• As we is unaffected, the budget constraint
  remains unchanged.
• The optimal choice A is unaffected.
• The endowment point moves from E1 to E2.
• (i.e. the consumer has more disposable income in
  the current period and less disposable income in
  the future period.)

B
we(1 r)
I
E1
E2
A
we
c Current Consumption
37
Burden of Government Debt

• In the above discussion, we have made the
  following assumptions
• 1) Taxes change by the same amount for all
  consumers.
• 2) Debt issued by the government is paid off
  during the lifetimes of the people alive when the
  debt was issued.
• 3) Taxes are lump sum.
• 4) Credit markets are perfect.
• If any of the above assumptions are violated,
  Ricardian Equivalence may not hold.
• e.g., credit markets are imperfect in the sense
  that
• There are limits on how much a consumer can
  borrow
• Consumers usually borrow at higher interest rates
  than they can lend at

38
Credit Market Imperfections
c' Future Consumption

• Consider a consumer who lends at a rate r1 and
  borrows at r2, where r2 gt r1.
• The budget constraint is AEF, where E is the
  endowment point.
• The slope of AE is (1 r1)
• The slope of EF is (1 r2)

we2(1 r2)
we1(1 r1)
A
E
y' t'
F
we1
we2
y t
c Current Consumption
39
Credit Market Imperfections
c' Future Consumption

• Suppose now the period 1 tax change by ?t lt 0
  with a corresponding change of t(1 r1) in
  future tax.
• Assume that the government pays on its debt at a
  rate r1.
• Effect of tax change
• Shift in endowment point from E1 to E2.

A
I2
we1(1r1)
I1
E1
E2
B
we2
c Current Consumption
40
Credit Market Imperfections
c' Future Consumption

• Optimal choice before the tax cut E1
• Optimal choice after the cut E2
• c increases by ?t
• Reason
• The government is effectively making a low
  interest loan available to the consumer via the
  tax cut scheme.

A
I2
we1(1 r1)
I1
I3
E1
E2
G
B
we2
c Current Consumption