Consumption

1
Consumption

• Anthony Murphy
• Nuffield College
• anthony.murphy_at_nuffield.ox.ac.uk

2
Outline

• Consumption the biggest component of
  GDP/national expenditure a good deal smoother
  than income.
• The two period model.
• Friedmans permanent income hypothesis PIH -
  infinitely lived representative agent etc.
• Modligianis life cycle hypothesis LCH finite
  life, saving for retirement, population dynamics.
  
• Halls 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.

3
Basic Two Period Model (1)

• Diagram
• Axes - c1y1 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
• c1c2/(1r) y1 y2/(1r)
• (You can add an initial endowment a0(1r) if you
  want to the RHS of the budget.)

4
Figure 6.2(a)
Fig. 6.02(a)
Indifference curves Normal case
Consumption tomorrow
0
Consumption today
5
Figure 6.2(b)
Fig. 6.02(b)
Indifference curves Zero substitution
Consumption tomorrow
0
Consumption today
6
Figure 6.2(c)
Fig. 6.02(c)
Indifference curves Constant substitution
Consumption tomorrow
0
Consumption today
7
Two Period Model (2)

• Budget constraint is a straight line thru
  (y1,y2) point with slope equal to minus 1/(1r).
• 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 y1
• c2 a1(1r) y2

8
Equilibrium in Two Period Model

• Equilibrium where highest attainable indifference
  curve is tangential to the budget line.
• You may be a borrower (c1 gt y1) or lender (c1 lt
  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).

9
Figure 6.3(a)
Fig. 6.03
Optimal consumption borrower
Consumption tomorrow
0
Consumption today
10
Figure 6.3(b)
Fig. 6.03
Optimal consumption lender
D
Consumption tomorrow
Y2
A
IC3
IC2
IC1
Y1
B
0
Consumption today
11
FOC and Euler Equation

• Suppose preferences are additive over time so
  U(c1,c2) u(c1) ?u(c2) where 0 lt ? lt 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) (1r)?u'(c2)

12
FOC and Euler Equation

• u'(c1) (1r)?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/ (1r).

13
CRRA Preferences

• CRRA preferences appealing constant savings
  rate fixed allocation of wealth across assets
  when interest rates constant.
• u(c) c1-?/(1-?) with ? positive u'(c) c-? so
  Euler equation is
• c1-? (1r)?c2-?
• Take natural logs and note that ln(1r) is
  approx. equal to r so
• ?lnc2 (ln ?)/? r/?

14
CRRA Preferences (2)

• The elasticity of intertemporal substitution EIS
  is the coeff. on r in the Euler equation.
• The EIS is 1/?, 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).

15
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.

16
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.

17
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.

18
Figure 6.9
Fig. 6.09
Effect of an increase in the interest rate
negative income effect for borrowers, positive
for lenders
D
D
A
Consumption tomorrow
Consumption tomorrow
R
R
A
B
B
Consumption today
Consumption today
(a) Student Crusoe(borrower)
(b) Professional athlete(lender)
19
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).

20
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.

21
Figure 6.11
Fig. 6.11
With a credit constraint, the choice set is
reduced.
C
Consumption tomorrow
A
R
D
0
Consumption today
22
Permanent Income Life Cycle Hypotheses (1)

• Can generalize analysis from two periods to many
  or an infinite number of periods.
• Standard model often called PIHLCH 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.

23
Figure 6.5
Fig. 6.05
Temporary vs. permanent income change
D
Temporary R to R Permanent R to R
D
Consumption tomorrow
Y2
R
Y2
A
AR
Y1
Y1
B
B
B
0
Consumption today
24
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.

25
Stochastic Income Interest Rates

• Solved-out consumption functions useful e.g.
• c1 k(r)W
• where wealth W a0(1r) 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.
• Halls (1978) insight look at Euler equation.

26
Halls Consumption Equation

• The stochastic Euler equation for the infinitely
  lived representative consumer is
• u'(c1) E1(1r1)?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.

27
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/(1r), then ?ct
  ut with Et(utIt) 0 so both ?ct and ut are
  innovations (unpredictable).
• Since Et(?ctIt) 0, Et(ctIt) ct-1.

28
Stochastic Euler Equations (1)

• Halls 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.

29
Stochastic Euler Equations (2)

• Assuming CRRA preferences, joint normality of rt
  and ct etc., the best we can do is
• Et?lnct (ln ?)/? rt/?½(sct)2/?
• 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.

30
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 (sct)2.
• Predicted income growth is always highly
  significant.

31
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.

32
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 non-labour
  income and initial assets.
• Financial market imperfections generate credit
  constraints. Current income matters more for
  credit constrained consumers.

33
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.