Consumption
Anthony Murphy
Nuffield College
anthony.murphy@nuffield.ox.ac.uk
Outline
• Consumption – the biggest component of GDP/national
expenditure; a good deal smoother than income.
• The two period model.
• Friedman’s permanent income hypothesis PIH infinitely lived representative agent etc.
• Modligiani’s life cycle hypothesis LCH – finite life,
saving for retirement, population dynamics.
• Hall’s consumption function – uncertainty, rational
expectations and the consumption Euler equation.
• Euler equations versus (approx.) solved out consumption
functions – pros and cons.
• Example of solved out consumption function for US.
Basic Two Period Model (1)
Diagram:
• Axes - c1’y1 on horizontal axis (the present) and c2,y2 on
vertical axis (the future).
• Intertemporal preferences: Regular shaped indifference
curves (as opposed to linear or L shaped ones).
• Less than perfect trade-off between c1 and c2 so want to
smooth consumption over time.
• Intertemporal budget line:
c1+c2/(1+r) = y1 + y2/(1+r)
(You can add an initial endowment a0(1+r) if you want to
the RHS of the budget.)
Fig. 6.02(a)
Consumption tomorrow
Indifference curves: Normal case
0
Consumption today
Fig. 6.02(b)
Consumption tomorrow
Indifference curves: Zero substitution
0
Consumption today
Fig. 6.02(c)
Consumption tomorrow
Indifference curves: Constant substitution
0
Consumption today
Two Period Model (2)
• Budget constraint is a straight line thru’ (y1,y2)
point with slope equal to minus 1/(1+r).
• No borrowing or lending restrictions.
• Borrowing and lending rates are the same.
• Intertemporal budget constraint got by
combining period 1 and period 2 budget
constraints:
c1 + a1 = y 1
c2 = a1(1+r) + y2
Equilibrium in Two Period Model
• Equilibrium where highest attainable indifference
curve is tangential to the budget line.
• You may be a borrower (c1 > y1) or lender (c1 <
y1) in period 1.
• First order condition (FOC):
slope of indifference curve
= slope of budget line
ie. marginal rate of substitution (MRS) between
c1 and c2 = 1/(1 + r).
Fig. 6.03
Consumption tomorrow
Optimal consumption: borrower
D
(i) Consumption today
financed on credit
M
Y2
(ii) Consumption loan
repayment (including
interest)
(ii)
R
C2
(i)
0
Y1
IC1
-(1+r)
C1
B
IC2
IC3
Consumption today
Fig. 6.03
Consumption tomorrow
Optimal consumption: lender
D
(i) Saving from this period’s
income
(ii) Additional consumption
next period
R
C2
(ii)
Y2
A
(i)
-(1+r)
0
C1
Y1 B
IC1
IC2
IC3
Consumption today
FOC and Euler Equation*
• Suppose preferences are additive over
time so U(c1,c2) = u(c1) + u(c2) where 0 <
 < 1 is a discount factor.
• MRS = -dc2/dc1 holding U constant = u'(c1)
/ (u'(c2)), where u'(c1) is the marginal
utility of c1 etc.
• Thus FOC may be re-written as:
u'(c1) = (1+r)u'(c2)
FOC and Euler Equation*
u'(c1) = (1+r)u'(c2)
• This is just a non-stochastic Euler
equation!
• Note intuition – indifferent between shifting
one unit of consumption between the
present and the future.
• Complete smoothing of consumption (c1 =
c2) when  = 1/ (1+r).
CRRA Preferences*
• CRRA preferences appealing – constant savings
rate & fixed allocation of wealth across assets
when interest rates constant.
• u(c) = c1-γ/(1-γ) with γ positive; u'(c) = c-γ so
Euler equation is:
c1-γ = (1+r)c2-γ
• Take natural logs and note that ln(1+r) is approx.
equal to r so:
lnc2 = (ln )/γ + r/γ
CRRA Preferences (2)*
• The elasticity of intertemporal substitution EIS is
the coeff. on r in the Euler equation.
• The EIS is 1/γ, the inverse of the constant coeff.
of relative risk aversion.
• The Euler equation implies that a higher interest
rate increases savings (c1 falls and c2 rises).
• However, need to examine this effect in more
detail. (Why? Only looking at slope of budget
line not position of line).
Playing Around with the
Basic Two Period Model
• Rise in permanent income (both y1 and y2 rise) –
outward parallel shift in budget line. c1 and c2
both rise.
• Rise in current or future income – budget line
shifts out parallel but not by as much as above.
Ditto for c1 and c2.
• Current consumption is higher if future income
rises even if current income is unchanged!
• A transitory rise in income may be represented
by a small rise in y1 (and possibly a offsetting
small fall in y2?). c1 and c2 only rise by a small
amount.
Real Interest Rate Effects (1)
• Suppose r rises.
• Budget line swivels around (y1,y2) and is
steeper.
• Need to look at substitution and wealth
effects.
• Substitution effect given by Euler equation.
• Substitution effect on c1 is negative.
Real Interest Rate Effects (2)
• For borrower, wealth effect on c1 is also
negative.
• For lender, wealth effect on c1 is positive.
• Overall, the effect of a rise in real interest rate on
current consumption is not clear cut.
• Empirical consensus is that interest rate effect is
small and negative.
• Size of effect depends on incidence of credit
constraints and initial wealth, inter alia.
Fig. 6.09
D
A
R
R´
B´
B
Consumption today
(a) Student Crusoe
(borrower)
Consumption tomorrow
Consumption tomorrow
Effect of an increase in the interest rate: negative
income effect for borrowers, positive for lenders
D
R´
R
A
B´ B
Consumption today
(b) Professional athlete
(lender)
Credit Constraints (1)
• Assume that representative agent cannot
borrow in period 1.
• Budget line is now discontinuous at (y1,y2).
• Budget line same as before in lending
region i.e. to left of (y1,y2).
• Budget line drops down to horizontal axis
in borrowing region i.e. to right of (y1,y2).
Credit Constraints (2)
• Now a corner solution at (y1,y2) is a distinct
possibility.
• A rise in future income y2 has no effect on
current consumption if credit constrained.
• A permanent or transitory rise in current income
has a large effect if credit constrained (marginal
propensity to consume is one).
• Interest rate effects smaller or zero if credit
constrained.
Fig. 6.11
Consumption tomorrow
With a credit constraint, the choice set is
reduced.
C
A
R
0
B
D
Consumption today
Permanent Income & Life Cycle
Hypotheses (1)
• Can generalize analysis from two periods to
many or an infinite number of periods.
• Standard model often called PIH–LCH model.
• Original permanent income model of
consumption uses a rational, infinitely lived,
representative consumer.
• Emphasis on different response of consumption
to permanent and transitory changes in income
etc.
Fig. 6.05
Consumption tomorrow
Temporary vs. permanent income change
D´
Temporary: R to R´
D
Permanent: R to R´´
A´´R´´
Y2´
Y2
0
R´
A´
A=R
Y1
Y1´ B
B´ B´´
Consumption today
PIH and LCH (2)
• In the life cycle model, aggregate consumption
derived from behaviour of individual consumers
(of different ages) with finite lifespans.
• Consumption smoothing and the life cycle
pattern of income mean that the young borrow,
the middle aged save and the retired dis-save.
• Obviously, aggregate consumption depends
positively on population and income growth.
• The level of savings also depends on length of
retirement relative to length of working life.
Stochastic Income & Interest Rates
• Solved-out consumption functions useful e.g.
c1 = k(r)W
where wealth W = a0(1+r) + y1 + y2/(1 +r) +….
and k(.) is a known function of the real interest
rate r.
• Difficult to derive exact results in PIH-LCH model
when income and interest rates are random.
• Interest rates often assumed constant and point
expectations of future income used.
• Hall’s (1978) insight – look at Euler equation.
Hall’s Consumption Equation
• The stochastic Euler equation for the
infinitely lived representative consumer is:
u'(c1) = E1(1+r1)u'(c2)
where Et is the conditional expectation at
time t given the information set It.
• Aside: Can rearrange Euler equation to
get pricing kernel / stochastic discount
factor.
• Rational expectations assumed.
When Does Consumption Follow
A Random Walk?
• Under very special and unrealistic
assumptions, Euler equation implies that
consumption is a random walk.
• When u(c) is quadratic and  = 1/(1+r),
then ct = ut with Et(ut|It) = 0 so both ct
and ut are innovations (unpredictable).
• Since Et(ct|It) = 0, Et(ct|It) = ct-1.
Stochastic Euler Equations (1)
• Hall’s Euler equation is only a FOC, as
noted already.
• It does not tell you anything about the
effects of income shocks, uncertainty etc.
• To examine these sorts of issues, you
need to embed it in a bigger model!
• This is one reason why some argue that
approximate solved out consumption
functions are more useful.
Stochastic Euler Equations (2)
• Assuming CRRA preferences, joint normality of
rt and ct etc., the best we can do is:
Etlnct = (ln )/γ + rt/γ+½(σct)2/γ
which shows that uncertainty increases savings
(since ct rises and ct-1 falls as the variance of ct
rises).
• Long list of assumptions – rational expectations,
representative agent, no credit constraints etc.
Testing Consumption Euler
Equations
• Consumption Euler equations do not fare very
well empirically.
• For example, if the basic model is correct, then
variables in the information set at time t-1 should
not help in predicting lnct.
• A natural test of this hypothesis is to include the
prediction of lnyt, using variables dated t-1. or
even t-2, in the regression of lnct on a constant,
rt and a proxy for (σct)2.
• Predicted income growth is always highly
significant.
Solved Out Consumption
Function for US
• See separate note for example of solved
out consumption function for US.
• Source: Muellbauer (1994), Consumers
Expenditure, Oxford Review of Economic
Policy.
Summary (1)
• Rational consumers attempt to smooth
consumption over time, by borrowing in bad
times (or when young) and saving in good times
(or in middle age) .
• Consumption is primarily driven by the present
discounted value of current and future nonlabour income and initial assets.
• Financial market imperfections generate credit
constraints. Current income matters more for
credit constrained consumers.
Summary (2)
• The effect of a change in the real interest rate is
ambiguous since wealth effects differ for lenders
& borrowers. Overall the effect is probably small
and negative.
• Over the life cycle, consumption is smoothed by
borrowing when young, saving in middle age
and dis-saving when retired.
• Temporary changes in income (or other
disturbances) have small effects. Permanent
changes or shocks have large effects.