Lecture 2: Consumption
Professor Eric Sims
University of Notre Dame
Fall 2009
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Consumption
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Environment
Assume there exists a representative household who lives for two
periods – the present and the future
This is an endowment economy – we abstract from production for
now
This means that household income is exogenous
Helpful to think about income and consumption as being in the same
units
Call these units fruit
Households derive utility from their consumption of fruit
For now, partial equilibrium
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Utility
Utility is how much “happiness” the household derives from its
consumption
The utility function is a function which maps the amount of
consumption into the amount of utility (“happiness”). Denote this
function u (c ).
c denotes consumption in the present, and c 0 denotes future
consumption (similar notation for income)
Utility function is the same in either period
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Properties of Utility
Increasing: marginal utility (…rst derivative) is positive
More is better
Concave: second deriviative is negative
Utility is increasing at a decreasing rate
Diminishing returns: …rst unit of fruit increase utility by more than the
second unit of fruit
Draw example utility function and marginal utility schedule
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Example Utility Functions
Log:
u (c ) = ln(c )
Linear:
u (c ) = c
General iso-elastic form:
u (c ) =
c1 σ
1 σ
Log and linear special cases of general form: σ = 0 is linear, σ = 1 is
log (use L’Hopital’s rule to show that)
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Marginal Utility
First derivative of utility function
Basic rules for derivatives
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Lifetime Utility
Lifetime utility is equal to a weighted sum of utilities in both periods
of life
U = u (c ) + βu (c 0 )
β
1 and is called the discount factor
People put less weight on future utility. Why?
The objective of the household is to maximize its lifetime utility – to
make itself as happy as possible
Can’t just consume as much as it wants – consumption decisions are
constrained by income (how much fruit falls from the tree)
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Endowments and Prices
Let y and y 0 denote how much fruit falls from the household’s tree in
the present and future, respectively
Households can buy other households’fruit or sell their own fruit to
other households at prices, p and p 0 . Households are price-takers
Households also have a savings vehicle – risk-free nominal bonds,
which we denote by b. Bonds are IOUs which pay an interest rate, i
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Within Period Budget Constraints
In the …rst period, dollar value of consumption plus saving must equal
dollar value of endowment (i.e. income):
In period 1, dollar value of consumption must equal dollar value of
endowment plus accrued interest:
Why is there no bond-holding in the future?
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Intertemporal Budget Constraint
What links the two within period budget constraints?
Nominal saving
Solve for b from either within period constraint and plug into the
other. Show derivation.
Simpli…cation yields:
c + c0
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1 p0
1+i p
= y + y0
Consumption
1 p0
1+i p
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Real Rate of Return
i is the nominal rate of return: it tells you how many dollars you get
next period if you save $1 today
r is the real rate of return: it tells you how many fruits you get
tomorrow if you save one fruit today
What is the real rate of return?
1+r
r
= (1 + i )
i
p
p0
π
r is the intertemporal price of consumption: summarizes the tradeo¤
between present and future
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Real Intertemporal Budget Constraint
Plug the Fisher equation into the intertemporal budget constraint:
y0
c0
=y+
1+r
1+r
What does this say in words?
c+
The present discounted stream of consumption is equal to the present
discount stream of income (endowment)
Present discounted value: the amount today that makes you
indi¤erent between that amount and some future amount
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Household Problem
The household’s objective is to maximize its lifetime utility subject to
the real intertemporal budget constraint:
max
c+
U
c0
1+r
= u (c ) + βu (c 0 )
s.t.
y0
= y+
1+r
We can characterize the solution to the household’s problem
graphically using an indi¤erence curve/budget line diagram
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Indi¤erence Curve
What is an indi¤erence curve?
It’s a plot showing di¤erent consumption bundles which hold total
utility constant
In intertemporal setting, just think of consumption in each period as
di¤erent goods
What is the slope of an indi¤erence curve?
This has a special name – the marginal rate of substitution (MRS)
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Graphical Representation
Graph sample indi¤erence curves
Why is the curve steep near the origin and ‡atter as we move further
to the right?
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Budget Line
Budget line: a line showing the feasible set of consumption bundles,
taking the endowment and interest rate as given
Derivation
Connect these with a straight line and that’s the budget line. What
is the slope?
(1 + r )
Why?
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Budget Line Graphically
Graph budget line
What does budget line show us?
It shows us the attainable set of consumption bundles. All points on or
inside the line are feasible. All points beyond the line are not
Budget line must pass through the endowment point – it is always
feasible to simply consume the endowment and save or borrow at all
It is helpful to show this when graphing budget lines
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Optimal Consumption Bundle
The household optimizes where it achieves the highest utility level
subject to the budget constraint
This occurs where the indi¤erence curve just touches the budget line
– i.e. where the two curves are tangent (slopes are equal)
Show graphically
Mathematically, we know the two slopes. Optimality condition (or
“tangency condition” or “Euler equation”) is:
MRS =
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u 0 (c )
= 1+r
βu 0 (c 0 )
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Comparative Statics
Exogenous variables: y , y 0 , and r (for now). β is a parameter
Endogenous variables: c and c 0
Change the exogenous variables, and see how the values of the
endogenous variables change
Show graphically
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Discussion
Increasing either current or future income leads to higher
consumption both today and in the future
This is important – current consumption depends not only on current
income, but on expectations about future income
An increase in the interest rate increases future consumption, has an
ambiguous e¤ect on current consumption
Substitution e¤ect
Income e¤ect
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Algebraic Discussion
Indi¤erence curve/budget line diagram is useful, but it is di¢ cult to
say much speci…c
We can algebraically solve the household problem using a little bit of
calculus. Assume the within period utility is log, and that the
discount factor is 1. The household problem is then:
max
U
c+
c0
1+r
= ln(c ) + ln(c 0 )
s.t.
y0
= y+
1+r
How do we solve this problem?
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The Consumption Function
The Euler equation/tangency condition is a condition characterizing
the optimal consumption pro…le
But we would like to derive a “consumption function” – optimal
consumption in each period as a function of things the household
takes as given
How do we do this?
Take the tangency condition, solve for either c or c 0 , plug back into the
budget constraint, and simplify
Show derivation
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Savings
Sometimes it is useful to think about a savings function
(Real) Savings is just de…ned as current income minus current
consumption: s = y c
Savings is increasing in current income, decreasing in future income,
and ambiguously related to the real interest rate, but probably
increasing in r
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Marginal Propensity to Consume
The marginal propensity to consume (MPC) is the …rst derivative of
the consumption function with respect to current income – it tells us
how much consumption today responds to an increase in income
today, holding everything else …xed
For the example utility function, we see that the MPC is 12 . The
MPC is always going to be bound between 0 and 1
This means that people like to “smooth” consumption out over time –
an increase in current income causes people to increase both current
consumption and future consumption (savings)
The MPC is not a “deep” parameter – it doesn’t show up in the utility
maximization problem. Rather, it depends on the form of the utility
function, how long households live, how households discount the
future, etc.
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Application: Permanent and Transitory Tax Cuts
Let’s introduce a government to this framework
Each period, the government con…scates some of the household’s
fruit. Denote these taxes by T and T 0
For now, assume that the government is completely wasteful – it does
nothing with the fruit it con…scates, and simply allows it to spoil
The only change to the household’s problem is the intertemporal
budget constraint, which now re‡ects the net endowment
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Application: Permanent and Transitory Tax Cuts
Consider two “stimulus” plans: one cuts taxes today, with no
expected cut in taxes tomorrow. The other cuts taxes in both
periods. Which has the bigger e¤ect on desired household
consumption?
Why?
Is a policy of “targeted, timely, and temporary” tax cuts a good one to
stimulate?
No, but there are caveats
We’ll come back to the “targeted” part
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Application: Ricardian Equivalence
Instead of assuming that the government does nothing with the fruit it
con…scates, now assume that the government consumes it (G and G 0 )
Household IBC is the same
Now the government has an intertemporal budget constraint of the
same form:
G0
T0
G+
=T+
1+r
1+r
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Application: Ricardian Equivalence
Household IBC can be simpli…ed using the government budget
constraint (show):
What is “missing”?
Taxes. Taxes dissappear from the household budget constraint
Ricardian equivalence: the timing of taxes does not matter for
household decisions. Only the time path of government purchases does
What happens if government temporarily cuts current taxes through
“de…cit …nance”?
Consumers don’t adjust their consumption at all. Since disposable
income is higher, they just save the whole tax cut
They do this because they know they’ll face higher taxes in the future
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Permanent Income/Life Cycle Hypothesis
Assume now that households live for a number of periods, not just two
Household dies at some known date, works for a known number of
periods with a known income stream until retirement
Denote time periods by t, t + 1, t + 2, . . . etc.
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The Generalized Intertemporal Budget Constraint
IBC: PDV of consumption equals PDV of income. This can be
expanded to any number of periods:
Summation notation (explain and show):
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The Permanent Income Hypothesis
There will be an Euler equation / MRS = price ratio / tangency
condition for each two adjacent periods of time
Simplifying assumptions: β(1 + r ) = 1 (“subjective rate of time
preference = objective intertemporal price”). This implies perfect
consumption smoothing
Log utility tangency condition:
ct + 1
ct
ct + 1
ct
=
β (1 + r )
= 1 ) ct + 1 = ct = c
This means that consumption is constant and the same in each period
of life
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Solving for Consumption
Make these substitutions into the IBC to solve for the amount of
consumption in each period of life:
Loosely speaking, consumption in each period is equal to the average
income over life
Basic result:
Households save during years of work, draw down savings during
retirement, fully depleting savings at death
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Picture: Life-Cycle Model
Draw in consumption/income pro…le
Also do it for more realistic scenario where income pro…le is growing
over time
How does saving change throughout the life cycle?
If income pro…le is su¢ ciently steep, you borrow during early years of
work, save during late years of work
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Permanent vs. Transitory Income
Milton Friedman (1957): current income comprised of permanent and
transitory components:
Y = YP +YT
Permanent income: the “annuity value of wealth”
What does that mean?
Loosely, average income. With non-zero interest rate, take a stream of
income and …gure out the constant payout during each period that
depletes the stream by the end of life
Friedman: c = Y P . Consumption equal to permanent income.
Transitory ‡uctuations in income are saved
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Permanent vs. Transitory Income
Simple example: go back to life-cycle setup
Suppose income goes up in just one period of life. What e¤ect does
this have on average income?
Now suppose income goes up by the same amount both in the current
period and in every future period. What happens to consumption?
MPC out of permanent income should be close to 1, MPC out of
transitory income should be close to zero
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Precautionary Saving
What is the impact of uncertainty on optimal household behavior?
MRS: equate ratio of expected marginal utilities of consumption to
one plus interest rate
Discussion of expected value
In general, expected marginal utility is greater than marginal utility of
expected future consumption
“Bad” state of the world hurts consumers more than the “good”
state helps them, so they save for the bad state.
More uncertainty – more savings, less consumption
Is this empirically relevant?
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Random Walk Hypothesis (Hall (1978))
Assumes quadratic utility, so uncertainty has no e¤ect on consumption
Basic idea: changes in consumption should be unpredictable
Why?
Consumption is forward-looking, and is a “su¢ cient statistic” for all
information about future wealth
Example: you know that your salary is going to go up a year from
now. Should your consumption in a year move when the salary goes
up?
No, consumption should go up today when you get the news
Best predictor of future consumption is current consumption:
c 0 = c + ε0
ε0 represents news about lifetime wealth revealed in the future and
unknown in the present. Its mean is zero.
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Summary of Predictions of Model of Consumption
Consumption depends on current and future income
E¤ect of changes in interest rate on consumption is technically
ambiguous, probably negative
Consumption should respond little to temporary changes in income,
and roughly one for one with permanent changes in income
People should save during their working years
Consumption should not fall at retirement
Changes in consumption should be unpredictable
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What Does the Evidence Say?
A theory is only good if it can help explain the data
How well does our theory of consumption explain the data?
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Evidence: Tax Cuts
Shapiro and Slemrod (2003):
2001 tax rebates of $300 or $600. First installment of a 10 year
reduction, so more or less permanent
Few people planned to spend it. MPC should be close to 1 if it’s
permanent
Ricardian motivations? Maybe they didn’t believe it to be permanent?
Shapiro and Slemrod (2008):
2008 tax rebates as part of stimulus plan. Most people planned to
save the rebates or pay down debt (e¤ectively saving)
Tax rebates were clearly transitory, so this is in accord with our theory
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Evidence: Retirement
Fact: consumption of food drops at retirement (also during
unemployment spells)
Shouldn’t happen in our model unless retirment is unexpected
How do we measure consumption? By expenditure
But what happens when people retire (or become unemployed)? They
have more time – more to shop for bargains, more time to prepare food
at home
Hurst and Aguilar (2005): consumption of food measured by calories
does not change at retirement or drop by much during unemployment
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Evidence: Anticipated Tax Changes
Parker (1999): looks at consumption responses to predictable changes
in take home pay from Social Security taxes
Only pay social security taxes on …rst $50,000 or so of income at the
time, now $108,600
Once yearly income has hit threshold, take home pay goes up by about
7 percent (now 7.65%)
Consumption should not change when take home pay does go up
But it does
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Predictable Changes in Income
Shea (1995): unionized workers know something about the future
time path of their earnings
Pre-speci…ed wage increases, etc.
Consumption should not respond to this – it should be set at the
moment workers get knowledge of future time path of earnings
But consumption does respond
Consumption is excessively sensitive to predictable changes in income
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Reconciling Theory and Evidence
Some instances where PIH seems to …t the data; others where it fails
How can we reconcile theory and evidence?
Possibility 1: liquidity constraints
e.g. consumers are restricted from borrowing, so c y (or more
generally there is a limit on how much they can borrow)
Draw in indi¤erence curve/budget line
With liquidity constraints, consumption will be more tied to current
income
Application: transitory and targeted tax cuts
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Rule of Thumb Consumers
Campbell and Mankiw (1989): propose idea of “rule of thumb”
consumers
These consumers just set consumption equal to current income (or
equal to some fraction of income)
Why? Liquidity constraints. Costs of optimization
Utility cost of failing to optimize is pretty small; so this makes sense
Evidence suggests that there is little or no excess sensitive to large
predictable changes in income; gives some credence to this story
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