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A Two-Period Model:
The Consumption-Saving Decision
and Ricardian Equivalence
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Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Slide 1
Two-period Model
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Slide 3
Consumers
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Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Slide 2
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(Discounted) Life time total consumption can not
exceed (discounted) life time total income
However in period by period terms, income does not
necessarily has to match consumption
Real interest rate determines the relative price future
C in terms of current C
i.e.
c
y −t
c1 + 2 = y1 − t1 + 2 2
1+ r
1+ r
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Slide 4
Consumers
m consumers (m being large)
Live only two periods (current and future)
no work-leisure decision, but receive
exogenous income (apple trees)
Focus on consumption/saving decision
Consumer budget constraint at period 1
c1 + s1 = y1 - t1
Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
This week: focus on a some dynamics
Intertemporal decisions: economic trade-offs across
time
Dynamic consumption-saving decision
Consumption smoothing
Ricardian equivalence
Two period model
Basic Idea
Trade off between consuming now or future
Saving/borrowing
Simple model, but captures important aspects
of dynamic decision making
Several issues to be addressed among others
consumption smoothing
Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Sofar: one-period (static) model
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If s1>0, consumer is lender at period 1
If s1<0, consumer is borrower
trade one type of financial asset, bond (that can be
issued by government or consumer)
Assumptions regarding bond holding:
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Slide 5
No risk associated with holding bonds, no default risk
Direct trading of bonds in the financial market, no
intermediaries
Bond pays of 1+r units of consumption good next
period
r, real interest rate is the same for lenders and
borrowers
Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Slide 6
1
Consumers
Life-time Budget constraint
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Consumer budget constraint at period 2
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c2 = y2 – t2 + (1+r) s1
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Solve for s1 in period 2 BC
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s1 =
Then substitute s1 into period 1 BC and rearrange to
obtain life-time BC
Present value of lifetime consumption is equal to PV
of lifetime disposable income!!
c
y −t
c1 + 2 = y1 −t1 + 2 2
1+r
1+r
c2 − y2 + t2
1+ r
Slide 7
Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Lifetime wealth
Slide 8
Next step:
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y − t2
W = y 1 − t1 + 2
1+ r
or
W = c1 +
Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
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Bring intertemporal budget constraint together
with the preferences of the consumers!!
The same logic as earlier lectures will apply
c2
1+ r
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Slide 9
Recall from Week 4 assumptions for consumers!
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Slide 10
Figure 6 Consumer’s Lifetime Budget Constraint
A1. More is always preferred than less!
A2. Diversity is important
A3b. Current and future consumption are normal goods!
Write the Wealth equation in slope-intercept form!
c2
1+ r
= = > c 2 = − (1 + r ) c1 + W (1 + r )
W = c1 +
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Slide 11
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Slide 12
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Figure 2 A Consumer’s Indifference Curves
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Table 1
Slide 13
Consumer Optimization
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Optimal consumption bundle is at the tangency of
indifference curve to budget constraint
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Consumer lends distance DB that is s1=y1-t1-c1*>0
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Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Slide 15
A Consumer Who Is a Borrower
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Slide 14
Figure 3 A Consumer Who Is a Lender
At Figure 3, endowment point is at E, but consumer
chooses point A..
At Point A
MRSc1,c2=1+r
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Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Slide 16
Figure 4 A Consumer Who Is a Borrower
s1=y1-t1-c1*<0
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Slide 17
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Slide 18
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The Effects of an Increase in Current Income for a Lender
The Effects of an Increase in Current Income for a Lender
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In a static setting we showed earlier that an
increase in income (dividend income or a
reduction in taxes) will increase consumption
and will reduce labour supply (pure income
effect)
Here, how does an increase in income affect
intertemporal consumption/saving decisions??
W
Slide 19
Figure 5 The Effects of an Increase in Current
Income for a Lender
Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Slide 21
Theory meets Data
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=
y
=
y
−
1
t
y
+
1
−
2
t
+
1
2
r
2
t h e
∆
W
'
1
−
t
c h a n g e
=
W
2
−
1
y
+
i n
W
t h e c h a n g e i n
∆ s =
∆ y 1 − ∆ t
2
1
w
1
=
−
t 2
r
e a l t h
+
y
'
1
−
y
1
s a v i n g s
0
− ∆ c >
i.e. an increase in y1 leads to an increase in
consumption in both periods and an increase in
savings
Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Slide 20
Figure 6 Percentage Deviations from Trend in
GDP and Consumption, 1982-1999
Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Slide 22
Some explanations
Theory predicts consumption smoothing
Aggregate consumption data very strongly
favours smoothing
If you take out durable goods (say cars, fridges
that yield utility for very long periods of time)
results are even stronger
Still data exhibits a bit more consumption
variability than the theory would predict.
Why?
Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
1
t o
W
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Lifetime wealth increases from
Slide 23
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1 Imperfect credit markets
2. Market prices move along the business
cycle
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Slide 24
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Figure 7 An Increase in Future Income
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Figure 8 Temporary Versus Permanent
Increases in Income
Slide 25
Figure 11 An Increase in the Real Interest Rate
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Slide 27
Figure 13 An Increase in the Real Interest Rate
for a Borrower
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Slide 29
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Slide 26
Figure 12 An Increase in the Real Interest Rate
for a Lender
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Slide 28
Table 2
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Slide 30
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Table 3
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Figure 14 Example with Perfect Complements
Preferences
Slide 31
Current consumption is increasing in the
increase in income (current and future)
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Slide 33
Consumer’s Demand for Current Consumption
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Figure 15 A Consumer’s Demand for Current Consumption
Goods, cd, as a Function of Current Income
Consumer’s Demand for Current Consumption
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Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Slide 34
Figure 16 A Shift in a Consumer’s Demand for
Current Consumption
Current consumption demand will shift in
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Real interest rates
Future income
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Slide 35
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Slide 36
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Incorporating Government: Assumptions
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Given current and future G, T, assume we are
initially at the competitive eq’m with r
∆T is put upon initial T
Current Period Government BC
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Government Present Value Budget Constraint
Solve for B1 in G2+(1+r)B1=T2
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G1=T1+B1
B1=(T2-G2)/(1+r)
Substitute into current period Gov. BC to obtain
intertemporal Gov BC
Future Period Government BC
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G1 +
G2+(1+r)B1=T2
Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Slide 37
Competitive Equilibrium
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Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
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RepCon: Optimal consumption/saving decision
given real interest rates
Government BC holds
Credit markets clear! (Sp=B)
Recall from National Accounts
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Sp+Sg=I+CA
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Here I=0, CA=0 Î Sg=-B
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In equilibrium
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Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Slide 39
Government
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Y=C+G
Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Slide 40
Ricardian Equivalence
Remember form one-period model that
government expenditures crowds out private
consumption
Need to make a distinction btw decrease in
taxes and an increase in government spending
Now it is possible as we allow governments to
borrow
Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Slide 38
In Equilibrium
Three conditions
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G2
T
= T1 + 2
1+ r
1+ r
Slide 41
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Def: If current and future government
spending are held constant, then a change in
current taxes with an equal and opposite
change in the present value of future taxes
leaves the equilibrium real interest rate and
the consumptions of individuals unchanged
Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Slide 42
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Step 1
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Step 2
Indicate competitive eq’m with *
Life time BC of the rep. consumer and government
c2
y2 − t 2
= y 1 − t *1 +
1 + r *
1 + r *
G 2
T 2*
*
+
= T 1 +
w h ere
1 + r *
1 + r *
c1 +
G
1
in
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*
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T
*
= m t
A lump sum increase in taxes ∆t so that t1**=t1*+∆t in
aggregate T1**=T1*+m∆t
Then Government intertemporal BC is
*
c o m p e titiv e e q ' m c r e d it m a r k e t c le a r s
G1 +
+ G
Y = C
a g g r e g a te p r iv a te s a v in g
*
G2
T **
= T **1 + 2 *
*
1+ r
1+ r
S p* = Y − C * − T
g o v 't b o r r o w in g is
B * = G − T
*
Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Slide 43
Step 3: showing that r* is still eq’m real interest
rate
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Present value of taxes paid by each consumer is
t1** +
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t 2
G
1
= (G1 + 2 * )
1 + r* m
1+ r
Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Slide 45
c2
y2
G2
1
= y1 +
− (G1 +
)
1 + r*
1 + r* m
1 + r*
Substituting present value of G (and remember
mt*=T* and mt2*=T2*
c2
y
t*
= y1 + 2 * − t *1 + 2 *
*
1+ r
1+ r
1+ r
Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Slide 46
Figure 6-17 Ricardian Equivalence with a Cut in
Current Taxes for a Lender
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Since consumer’s consumption is unchanged in each
period before and after the ∆T, it must be
Y=C*+G
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Î Changes in taxes leave the r* unchanged
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Change in private savings thus
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∆Sp=T*-T**=-m∆t
Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
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c1 +
Step 3 cont’d
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After substituting
c1 +
c2
y − t **
= y1 − t **1 + 2 * 2
*
1+ r
1+ r
Slide 44
Step 3 cont’d
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Present value of taxes for each consumer is the per
capita present value of G.
c1 +
Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Slide 47
Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Slide 48
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Ricardian Equivalence and the Burden of Government
Debt: Critically Recalling Assumptions
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Tax changes are identical across households: (in
reality life is more distortionary)
All government debt is cleared within the lifetime (in
reality Gov’t can reallocate the tax burden across
generations)
Lump-sum taxation affects everybody in an identical
way (reality is more distortionary taxation)
Credit markets are perfect (in reality there are credit
market imperfections, limited participation)
There is no inflation (no money)
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Slide 49
Figure 8.18 Pay-As-You-Go Social Security for
Consumers Who Are Old in Period T
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Social Security Programs
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Two types
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pay as you go
Fully funded
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Slide 50
Figure 8.19 Pay-As-You-Go Social Security for
Consumers Born in Period T and Later
Slide 51
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Slide 52
Fully Funded Social Security
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As long as population growth is larger than the
real interest rate pay as you go will deliver a
better outcome for everybody
The smaller the burden on younger generation
(due to population increase) the higher the rate
of return of the system.
Aging population problem in Europe!
Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Slide 53
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Government invests the proceeds from social
security taxes in the private credit market.
Or people simply save now and invest in the
financial markets to fund their retirement
For simplicity we deal with only one interest
rate r
Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Slide 54
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Figure 8.20 Fully Funded Social Security When
Mandated Retirement Saving Is Binding
Credit Market Imperfections and Consumption
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Assume r2>r1, where r2 is borrowing rate, and r1is lending
rate
Complication in the life time BC
c2 = y2 – t2 + s(1+r1) if s≥0 (consumer is lender)
c2 = y2 – t2 + s(1+r2) if s≤0 (consumer is borrower)
Life time BC then
W 1 = c1 +
W
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Slide 55
Figure 6-18 A Consumer Facing Different
Lending and Borrowing Rates
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Slide 57
if s ≤ 0
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Slide 56
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Slide 58
In Sum
Ricardian equivalence is violated!
Tax cut is effecteively low interest loans from
the government to the consumers
If there are credit market imperfections
government deficits maybe useful to give
‘access’ to credit markets
Monitoring issues
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Two period macroeconomic model
Intertemporal consumption/saving decisions
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Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
c2
y 2 − t2
= y 1 − t1 +
1 + r2
1 + r2
if s ≥ 0
Figure 6-19 Effects of a Tax Cut for a Consumer
with Different Borrowing and Lending Rates
Effects of a Tax Cut
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= c1 +
2
c2
y 2 − t2
= y 1 − t1 +
1 + r1
1 + r1
Slide 59
Lifetime BC
Consumption smoothing
Y1 increasesÎ both C, C increase and S incraeeses
Y2 increases Î both C, C increase and S decreases
Permamnet income increase has larger consumption
implications than temporary income increase
r changes, triggers both substititon and income effect
r changes affect borrowers and lenders differently
Copyright © 2002 Pearson Education, Inc. and Dr Yunus Aksoy
Slide 60
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In sum
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Consumption smoothing
Initial conditions (borrower or lender) matter if
there is a change in the real interest rate
Fiscal policy: Ricardian equivalence
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Slide 61
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