Consumption
Why studying consumption?
• Important for the study of both growth and cycles
• Has been at the center of many important empirical analysis
• Plan: Romer (2012) Chapter 8
Consumption under Certainty: Life-Cycle/Permanent-Income Hypothesis
Modigliani-Brumberg (1954) and Friedman (1957)
Developed in reaction to the empirical problems coming out of
Keynes (1936) consumption function
Presented in Romer with many simplifying assumptions
• Individual living for T period
• Initial wealth of A0
• Interest rate and discount rate set to zero (see fn 1, page 366)
• Labour income of Y1 , Y2 ,..., YT
Lifetime utility function:
T
U   u (Ct ),
u '()  0, u ''()  0
t 1
Individual can save and borrow
8.1
Individual budget constraint:
T
C
t
t 1
T
 A0   Yt ,
8.2
t 1
The Lagrangian is:
T
T


  u(Ct )    A0   Yt   Ct .
t 1
t 1
t 1


T
8.3
First-order condition for Ct is:
u '(Ct )  
This hold for any period (r-ρ=0) so Ct is constant and from 8.2:
T
1

Ct   A0   Yt 
T
t 0

for all t ,
8.5
Consumption is a function of permanent income YP
the right-hand side of 7.5. Transitory income is the difference
between current income Yt and permanent income.
One time income of Z raises consumption by Z/T
Temporary income tax cut has a small effect on consumption.
The time pattern of income is not important for consumption
But it is fundamental for Saving S:
St  Yt  Ct
C YP
St  Yt  YP
8.6'
Empirical applications: Understanding Estimated Consumption Function
The traditional Keynesian consumption function is:
C  a  bY , with 0<b  1
Where Y is current income. The stable and consistent relationship
was rejected in some important empirical studies. Three cases to consider
Ci
1) The Keynesian C function
was not rejected in pure
cross-sectional studies
(across household, at one
point in time)
45o
Yi
Ct
But rejected in time-series
of one country over a long
period of time
45o
Ci
Yt
Whites
Blacks
45o
Yi
And the cross-section
consumption function
differs across groups
In a celebrated paper, Friedman (1957) demonstrate that the
permanent income hypothesis provides and explanation for the
three sets of facts.
Suppose that the true model is C = YP and current income Y=YP + YT
Suppose that YT has a mean zero and is uncorrelated with permanent
income
Consider a regression based on the wrong model:
Where, in OLS:
C  a  bY  e .
8.7
i
i
i
P
T
P
Cov
(
Y
,
C
)
Cov
(
Y

Y
,
Y
)
ˆ
b

Var (Y )
Var (Y P  Y T )
Given that: Cov( X  Y , X )  Var ( X )  Cov( X , Y )
here Cov(Y P , Y T )  0
P
Var
(
Y
)
ˆ
b
Var (Y P )  Var (Y T )
8.8
An increase in current income is associated with an increase in cons
if only it reflects an increase in permanent income
Furthermore, the estimated constant equals:
ˆ where the X stands for the mean of X
ˆ  C  bY
a
P
T
ˆ
aˆ  Y  b(Y  Y )
 (1  bˆ)Y P ,
8.9
P
1) For cross-section studies, Var(YT) (coming from unemployment, H,
and differences in their life-cycle) is comparable with Var(YP),
consequently, b is smaller than 1 and a is positive.
2) In time series, most of the Var(Y) is associated with long-run
growth and b is close to one and a close to zero.
3) As for the differences between blacks and whites, their relative
Var are the same but: Y P whites  Y P blacks
Read carefully the discussion on page 371.
Consumption under Uncertainty: Hall’s Random-Walk Hypothesis
Introducing uncertainty and suppose the u function is quadratic. Then,
the individual maximizes:
T 
a 2 
E (U )  E   Ct  Ct   ,
2 
 t 1 
a0
Under budget constraint 8.2
Here Romer use the same approach than he used for the Euler equation.
A consumer on its optimal path:
(1  aCt 1 )dC  Et 1 1  aCt  dC
8.11'
Where E1 stands for the expectation based on information available
at time 1.
Then, we get:
Ct 1  Et 1 Ct  ,
for t  2,3,..., T .
8.12
And taking the expectation on both sides of the budget constraint
and using 8.12, we get:
T
1

C1   A0   E1 Y t 
T
t 0

8.14
Since by definition of expectations:
Ct  Et 1 Ct   et ,
8.15
Where the expectation of et is zero, 8.15 and 8.12 imply:
Ct  Ct 1  et ,
8.16
Ct  Ct 1  et ,
8.16
Hall’s (in)famous result (see fn 6 page 375)
Permanent income hypothesis imply that only unexpected event
can affect consumption. Expected event are smoothed by for-sighted
agents.
Empirical applications: testing the random-walk hypothesis
1) Hall’s approach: regress the changes in C (t-1,t) on variables
that are known at time t-1. For example lag changes in income.
This could be easily done in EViews with Granger causality test
Pairwise Granger Causality Tests
Date: 11/06/03 Time: 16:25
Sample: 1953:1 1984:4
Lags: 4
Null Hypothesis:
DLY does not Granger Cause DLCS
DLCS does not Granger Cause DLY
Obs
F-Statistic
Probability
123
1.04969
6.77287
0.38489
6.3E-05
Simple test using consumption and income data in Green, 4th edition
databank (CD) from US data. The null is not rejected in the first row.
Consequently, the null of Random Walk is not rejevted.
Similarly, with 5 lags:
Pairwise Granger Causality Tests
Date: 11/06/03 Time: 16:28
Sample: 1953:1 1984:4
Lags: 5
Null Hypothesis:
DLY does not Granger Cause DLCS
DLCS does not Granger Cause DLY
Obs
122
F-Statistic
1.18578
5.57381
Probability
0.32076
0.00013
But this approach was criticized by Campbell and Mankiw (1989b).
AS Romer (2001) puts it: “Hall’s result that lagged income does not
have strong predictive power for consumption could arise not because
predictable changes in income do not produce predictable changes
in consumption but because lagged values of income are of little use
in predicting income movements.”
Campbell and Mankiw suppose that the fraction λ of households
consume their current income and the other follow Hall’s consumption.
Consequently, they want to test:
Ct  Ct 1   (Yt  Yt 1 )  (1   )et
  Zt  t
8.22
Where et is the change in households’ estimate of their permanent
income between t-1 and t. But Zt and et are correlated!! (explanation, when
current income increase, permanent income also usually increase)
Then the right-hand side independent variable is correlated with the error
term. In this case, OLS estimates might be biased (and the variance not
accurately estimates). In such a case, the approach is to use IV.
Instruments are variablea correlated with the right hand side variable but
not correlated with the residual.
This is simply done with two-stage least squares (as done in the book).
The first regression is a regression of Zt on the instruments. The
second regression is a regression of the left-hand side variable on
the fitted values of Z:
Ct  Ct 1   Zˆt   ( Z t  Zˆ )   t
  Zˆ 
t
8.23
t
Often, as in this case, the instruments are lagged variables.
Campbell and Mankiw find that lagged changes in income has no
predictive power for future changes (putting doubts on Hall’s
empirical analysis). As a base case, they use lagged changes in
consumption as instruments. Romer reports estimates of λ of
0.42 (0.16) and 0.52 (0.13) for three and five lags respectively.
Dependent Variable: DLCS
Method: Two-Stage Least Squares
Date: 11/04/03 Time: 17:02
Sample(adjusted): 1954:1 1984:4
Included observations: 124 after adjusting endpoints
Instrument list: C DLCS(-1) DLCS(-2) DLCS(-3)
Variable Coefficient
C
0.006187
DLY
0.294314
Std. Error
0.001710
0.178857
t-Statistic
3.618526
1.645522
Prob.
0.0004
0.1024
Dependent Variable: DLCS
Method: Two-Stage Least Squares
Date: 11/04/03 Time: 17:02
Sample(adjusted): 1954:3 1984:4
Included observations: 122 after adjusting endpoints
Instrument list: C DLCS(-1) DLCS(-2) DLCS(-3) DLCS(-4) DLCS(-5)
Variable Coefficient
Std. Error
t-Statistic
Prob.
C
DLY
0.001671
0.166591
2.668920
2.924062
0.0087
0.0041
0.004460
0.487123
Read carefully the remaining of 8.3
The interest rate and consumption.
What happen if interest rate and discount rate non null We already know,
follow Romer analysis in 8.4. (redo it yourself) when
he get the now well known Euler equation (in discreet time):
Ct 1  1  r 


Ct  1   
1

8.27
They could be a trend in the consumption depending on r - ρ
The interest rate and saving
• Although an increase in the interest rate at period t reduce the
ratio of consumption at period t + 1 over consumption at
period t , it does not mean that saving is increased at period 1.
• Change in the interest rate have a substitution and an income
effect.
• Best illustrated graphically with a 2 period model.
• Assume the individual has no initial wealth.
• The individual budget constraint go through the point Y1 –
Y2.
• The slope of the budget constraint is –(1+r).
1- No initial saving: pure substitution effect, C1 rises
and S increases
2 – Saving is initially positive: both a substitution and
an income effect – effect on saving ambiguous (effect
on C1 ambiguous)
3 – Borrowing initially: both substitution and income
effect work in the same direction – C1 decreases and
saving rises
Consumption and risky assets (Romer 4th section 8.5)
Consider an individual living for two periods who invest in a
risky asset witch bear an uncertain return 1 + 𝑟𝑡+1 .
Suppose he reduces consumption in period t and purchase the
asset. The Euler equation for this problem is again:
𝑢′ 𝐶𝑡 =
1
𝐸
1+𝜌 𝑡
1 + 𝑟𝑡+1 𝑢′(𝐶𝑡+1 )
8.28
Using the covariance rule, 8.28 becomes
𝑢′ 𝐶𝑡 =
1
1+𝜌
𝐸𝑡 1 + 𝑟𝑡+1 𝐸𝑡 𝑢′(𝐶𝑡+1 ) + 𝐶𝑜𝑣𝑡 (1 +
• If we assume a quadratic utility function
• 𝑢 𝑐 =𝐶
𝑎𝐶 2
−
2
and marginal utility is 1 − 𝑎𝐶 .
• Bring this result in 8.29 simplifies the analysis of the
covariance term. Using 2 simple covariance rules:
• (8.30)𝑢′ 𝐶𝑡 =
1
1+𝜌
𝐸𝑡 1 + 𝑟𝑡+1 𝐸𝑡 𝑢′(𝐶𝑡+1 ) −
• Suppose the agent has the choice between holding the risky
asset or a risk free asset with equal expected returns r.
• The agent will be better off to buy the risky asset if its
expected return is high when the marginal utility of C is high,
or when consumption is low (when the cov in 8.30 is negative)
• But as the agent invest more in the risky asset the covariance
in 8.30 becomes less negative since the agent portfolio (and
consumption) is more dependent on the return of the risky
asset.
• In the case we consider (expected risky return is the risk-free
rate), the agent invest in the risky asset up to the point that the
covariance become 0.
• Consequently, the returns of the assets of individual should not
be positively correlated with the sources of risk of
consumption (Example in the book a US steelworker should
not hold assets of US steel industries)
The Consumption Capital-asset pricing model
• In practice asset returns are determined by markets. If an asset
payoff is positively correlated with consumption, its market
returns will be driven down. We could isolate the expected
return in 8.30:
• 𝐸𝑡 1 + 𝑟𝑡+1 =
1
𝐸𝑡
𝑢′(𝐶𝑡+1 )
(1 + 𝜌)𝑢′ 𝐶𝑡 +
• In the case of a risk-free asset, 8.32 simplifies to
If we subtract 8.33 from 8.32, we get
The expected risk premium is proportional to the covariance
Between the return and consumption. This is the Consumption
asset pricing model CAPM. The coefficient from a regression
of an asset returns time series on consumption growth is a
consumption beta.
Empirical Application: The Equity-Premium Puzzle
• We could use the analysis to explain the difference between
the returns of stocks i and risk-free bonds j which is the equity
premium. We start from 8.28 in the case of a CRRA function
where 𝜃 is the coefficient of risk aversion:
After some manipulations (see Romer equations 8.36 to 8.41)
we get the equation for the equity premium for our two assets:
𝐸 𝑟 𝑖 − 𝐸 𝑟 𝑗 = 𝜃𝐶𝑜𝑣(𝑟 𝑖 − 𝑟 𝑗 , 𝑔𝑐 ) 8.41
The facts
• Mehra and Prescott (1985) test this prediction based on US
data for the average return of stocks versus bonds. The equity
premium averages 6 % (.06). The standard deviation of the
growth of consumption is 0.036, the standard deviation of the
excess return on the market is 0.167, and the correlation
between the two is 0.40.
• Consequently, the Cov between the excess return and
consumption growth is 0.40(0.036)(0.167) or 0.0024
• With this number, 𝜃 in 8.41 should be around 25. This
number is far too big, benchmark number in calibration around
2.