Chapter 8
A Two-Period
Model: The
Consumption–
Savings Decision
and Credit Markets
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Chapter 8 Topics
• Consumer’s consumption/savings decision –
responses of consumer to changes in income and
interest rates.
• Government budget deficits and the Ricardian
Equivalence Theorem.
• Social Security.
• Credit market imperfections.
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Two-Period Model
•
•
•
•
•
N consumers (not representative)
Live for two periods (current, future).
Receive endowment in each period (y, y’).
Pay lump-sum tax in each period (t, t’).
Decision: how much to spend as consumption in
current period, and how much to save (dissave)
to finance future consumption.
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Equation 8.1
The consumer’s current-period budget constraint:
If s > 0, consumers save in current period.
That is, sacrifice c and consume more c’.
If s < 0, consumers dissave in current period.
That is, sacrifice c’ and consume more c, borrow against future income.
If s = 0, consumers consume his endowment net of tax in current period.
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Bonds
• If s > 0, consumers are lenders and buy bonds in the
credit market.
• If s < 0 , consumers are borrowers and sell bonds in the
credit market.
• Abstract from some complications.
– No default risk, direct trade in credit markets, and equal
borrowing and lending rates.
• A bond issued in the current period is a promise to pay
(1+r) unit of future consumption good.
r: real interest rate.
• The relative price of c’ to c is 1 / (1+r).
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Equation 8.2
The consumer’s future-period budget constraint:
(1+r)s : interest payment and principal of saving (dissaving)
carried forward from last period.
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Equation 8.3
Solve the future-period budget constraint for s:
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Equation 8.4
Substitute in the current-period budget constraint
obtaining lifetime budget constraint:
LHS: PV of lifetime consumption.
RHS: PV of lifetime income net of tax.
PV: value in terms of current-period consumption goods.
1 / (1+r) : relative price of future goods to current goods.
Tradeoff between c and c’.
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Equation 8.5
Consumer’s lifetime wealth:
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Equation 8.6
Simplified lifetime budget constraint for the
consumer:
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Equation 8.7
Lifetime budget constraint in slope-intercept form:
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Figure 8.1 Consumer’s Lifetime
Budget Constraint
1.
2.
3.
4.
5.
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Slope is –(1+r).
E: endowment point.
Points on BE are lenders.
Points on AE are borrowers.
Optimal bundle is on BC frontier.
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Figure 8.2 A Consumer’s
Indifference Curves
More is better than less.
Diversity is preferred.
MRS c, c’ = - slope of a tangent to IC.
Diminishing MRS c, c’ (convexity of IC)
implies desire for consumption
smoothing.
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Table 8.1 Sara’s Desire for
Consumption Smoothing
Dislike large difference in consumption between 2 periods.
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Equation 8.8
Marginal condition that holds when the
consumer is optimizing:
1+r : relative price of c to c’.
Recall Ch.4, MRS l , C = w, where w is relative price of l to C.
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Figure 8.3 A Consumer Who Is a
Lender
Optimal bundle:
Tangent point between
BC frontier and IC.
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Figure 8.4 A Consumer Who Is a
Borrower
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An Increase in Current Income
for the Consumer
• Current and future consumption increase.
• Saving increases.
• The consumer acts to smooth consumption over
time.
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Figure 8.5 The Effects of an Increase
in Current Income for a Lender
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Equation 8.9
Decomposing the change in saving for the
consumer:
y  c  s  0
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Observed ConsumptionSmoothing Behavior
• Aggregate consumption of non-durables and
services is smooth relative to aggregate income,
but the consumption of durables is more volatile
than income.
• This is because durables consumption is
economically more like investment than
consumption.
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Figure 8.6 Percentage Deviations
from Trend in GDP and Consumption,
1947–2006
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An Increase in Future Income
for the Consumer
• Current and future consumption increase.
• Saving decreases.
• Again, these results are explained by the
consumer’s motive to smooth consumption over
time.
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Figure 8.7 An Increase in Future
Income
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Temporary and Permanent
Increases in Income
• As a permanent increase in income will have a
larger effect on lifetime wealth than a
temporary increase, there will be a larger effect
on current consumption.
• A consumer will tend to save most of a purely
temporary income increase.
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Figure 8.8 Temporary Versus Permanent
Increases in Income
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Figure 8.11 An Increase in the
Real Interest Rate
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An Increase in the Market Real
Interest Rate
An increase in the market real interest rate
decreases the relative price of future
consumption goods in terms of current
consumption goods – this has income and
substitution effects for the consumer.
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Figure 8.12 An Increase in the
Real Interest Rate for a Lender
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Figure 8.13 An Increase in the
Real Interest Rate for a Borrower
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Table 8.2 Effects of an Increase in
the Real Interest Rate for a Lender
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Table 8.3 Effects of an Increase in
the Real Interest Rate for a Borrower
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Equation 8.17
The government’s current-period budget
constraint:
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Equation 8.18
The government’s future-period budget constraint:
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Equation 8.19
The government’s present-value budget
constraint:
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Equation 8.20: Credit Market
Equilibrium Condition
Total private savings is equal to the quantity of
government bonds issued in the current period.
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Equation 8.21
Credit market equilibrium implies that the incomeexpenditure identity holds.
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Ricardian Equivalence
If current and future government spending are
held constant, a change in current tax with an
equal and opposite change in the present value
of future taxes leave the equilibrium real interest
rate and consumption of individual unchanged.
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Intuition of RE Theorem
• When consumers receive a tax cut in current
period, and this change will be offset by an
equal and opposite tax increase in future period,
the present value of tax burdens are unchanged,
which leaves the consumers’ lifetime wealth
unchanged. So the equilibrium optimal
consumption bundle and real interest rate all
remain the same as before.
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Some Issues about RE
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•
•
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Intergeneration issues
Different tax rates faced by consumers.
Assumption of lump-sum taxes.
Perfect credit market.
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