Consumption - Savings Problems
Spring 2013
• Solow: exogenous savings rate
• In general, what determines the consumption level? (for an individual household / in the
aggregate)
– Level of income
– Accumulated wealth
– Expectations about the future
– Age (for an individual)
– Family structure
– Risk aversion
– Goverment policies (e.g. Social Security, insurance)
– Demographics (in the aggregate)
– Access to credit
– Interest rates
– Self-control
– Advertising for stuff
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The Keynesian Consumption Function
• Keynes (1936):
“The fundamental psychological law, upon which we are entitled to depend with great
confidence both a priori from our knowledge of human nature and from the detailed
facts of experience, is that men are disposed, as a rule and on the average, to increase
their consumption as their income increases but not by as much as the increase in the
income.”
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ECON 52, Spring 2013
2 A TWO-PERIOD WORLD
Ct = a + bYt
• b is the “marginal propensity to consume”
• Keynes: b < 1
• What are the long run implications (if Yt is growing)?
– Note on early Keynesians worrying about lack of demand in the long term
• Evidence on savings rates in the long run rejects this
2
A two-period world
• Assume
– Perfectly rational forward-looking behaviour
– Two-period life
– No uncertainty
– Additively separable utility. Discount rate β
– Unconstrained lending and borrowing at interest rate r
• Consumer problem:
max u (c1 ) + βu (c2 )
c1, a,c2
s.t.
a = y 1 − c1
c2 = y2 + (1 + r) a
• Rearrange budget constraint:
c2 = y2 + (1 + r) (y1 − c1 )
1
1
c1 +
c2 = y 1 +
y2
1+r
1+r
• Only Present Value of income matters, not timing
W = y1 +
is lifetime wealth.
2
1
y2
1+r
ECON 52, Spring 2013
2 A TWO-PERIOD WORLD
• What is a Present Value? (or “Net Present Value”)
– Imagine you are selling your future income in the market
– “Next year, I will give you y2 . How much are you willing to give me today in exchange?”
Call this amount x
– For somebody giving you x, the alternative is to put it in the bank, and therefore obtain
(1 + r) x next year
– They will only be willing to give you x today if what you promise them tomorrow is at
least (1 + r) x
– Hence
y2 = (1 + r) x
y2
x=
1+r
–
y2
1+r
is the present value of y2 goods delivered next year
–
1
1+r
is the relative price of t = 2 goods
• Graph
– Saving-for-retirement example
– Optimism example
– Effect of interest rates
– Borrowing constraints
• Lagrangian:
1
1
c2 − y 1 −
y2
L (c1 , c2 , λ) = u (c1 ) + βu (c2 ) − λ c1 +
1+r
1+r
• FOC:
u0 (c1 ) − λ = 0
1
βu0 (c2 ) − λ
=0
1+r
⇒ u0 (c1 ) = β (1 + r) u0 (c2 )
• “Euler equation” . Interpretation
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ECON 52, Spring 2013
2 A TWO-PERIOD WORLD
• Example with CRRA utility:
c1−σ
1−σ
0
u (c) = c−σ
u (c) =
so Euler equation becomes
−σ
c−σ
1 = β (1 + r) c2
1
c2
= [β (1 + r)] σ
c1
• Role of β, r and σ in determining the time path of consumption
• Explicit solution: use budget constraint
1
1
c2 = y 1 +
y2
1+r
1+r
1
1
1
c1 [β (1 + r)] σ = y1 +
y2
c1 +
1+r
1+r
h
i
1
1
1
y2
c1 1 + β σ (1 + r) σ −1 = y1 +
1+r
1
y1 + 1+r
y2
c1 =
1
1
1 + β σ (1 + r) σ −1
c1 +
• Exercises:
– increase in y1 holding y2 fixed. Relate to Keynesian consumption function
– Increase in both y1 and y2 .
– Increase in y2 holding y1 constant
– change in interest rates. Relate equation to graphs.
Adding taxes: Ricardian equivalence and its limitations
• A government wants to make purchases of goods and services G1 and G2 and needs to collect
taxes to pay for them
– For now, we don’t ask why they choose G1 and G2 : we take them as given
– Assume taxes are lump-sum
• The government can also borrow and save at interest rate r
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ECON 52, Spring 2013
2 A TWO-PERIOD WORLD
• Government budget constraint
B = G1 − T1
T2 = G2 + (1 + r) B
⇒ T2 = G2 + (1 + r) (G1 − T1 )
1
1
G2 = T1 +
T2
G1 +
1+r
1+r
• NPV of taxes must equal NPV of spending
• Consumer now has to pay taxes, so budget constraint
c1 +
1
1
c2 = (y1 − T1 ) +
(y2 − T2 )
1+r
1 +r
1
1
y2 − T1 +
T2
= y1 +
1+r
1+r
1
1
= y1 +
y2 − G1 +
G2
1+r
1+r
• Budget constraint of consumer depends on NPV of taxes
• NPV of taxes has to equal NPV of government spending (due to government budget)
• Therefore consumer budget depends on NPV of government spending
• Consumer budget does NOT depend on the timing of taxes: “Ricardian equivalence”
• Examples:
– Effect of a temporary tax rebate
– Effect of a temporary tax rebate plus the announcement of lower government spending
in the future
• Sources of non-equivalence in practice
– Borrowing constraints
– Lack of foresight
– Expectations of future G changing
– Distortionary taxation
• What Ricardian equivalence does NOT say:
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ECON 52, Spring 2013
3 MANY OR INFINITE PERIODS
– An increase in G implies a simultaneous decrease in C
– G has no effect on the economy
– Anything about G
3
Many or infinite periods
• Household solves
T
X
max
ct ,at+1
β t u (ct )
t=0
s.t.
at+1 = yt − ct + (1 + r) at
some limitation on aT +1
• Here I have a constant interest rate, you could also have interest rates changing over time
• “Life-cycle” model:
– T finite, e.g. T = 85
– Some pattern of yt , e.g. lower yt after t = 65
– aT +1 ≥ 0
• Infinite-horizon model
– T =∞
– Interpretation as an infinitely-lived family
– limt→∞
at+1
(1+r)t
≥ 0 (no Ponzi games condition)
• Uses of each type of model
• Lagrangian:
L (c0 , c1 , ..., a1 , a2 , ..., λ0 , λ1 , ...) =
T
X
β t u (ct ) −
t=0
T
X
t=0
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λt [at+1 − yt + ct − (1 + r) at ]
ECON 52, Spring 2013
3 MANY OR INFINITE PERIODS
• FOC:
β t u0 (ct ) − λt = 0
β t+1 u0 (ct+1 ) − λt+1 = 0
−λt + (1 + r) λt+1 = 0
⇒ β t u0 (ct ) = (1 + r) β t+1 u0 (ct+1 )
u0 (ct ) = β (1 + r) u0 (ct+1 )
• Same Euler equation
• Sequence of budget constraints is equivalent to single PV budget constraint:
a1 = y0 − c0 + (1 + r) a0
a2 = y1 − c1 + (1 + r) a1
= y1 − c1 + (1 + r) (y0 − c0 ) + (1 + r)2 a0
a3 = y2 − c2 + (1 + r) a2
= y2 − c2 + (1 + r) (y1 − c1 ) + (1 + r)2 (y0 − c0 ) + (1 + r)3 a0
...
aT +1 =
aT +1
T
(1 + r)
=
T
X
t=0
T
X
t=0
(yt − ct ) (1 + r)T −t + (1 + r)T +1 a0
yt − ct
+ (1 + r) a0
(1 + r)t
• Imposing no-debt-after-death condition or no-Ponzi condition:
T
X
t=0
T
X yt
ct
≤
+ (1 + r) a0 ≡ W
(1 + r)t
(1 + r)t
t=0
• Budget depends on total wealth: PV of lifetime income plus initial savings
• Consumption function
c (W )
instead of
c (yt )
• Exact consumption function depends on β, r and σ.
– This was also true in the two-period model
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ECON 52, Spring 2013
• For β =
1
,
1+r
3 MANY OR INFINITE PERIODS
exact solution is easy:
– Euler equation implies
u0 (ct ) = u0 (ct+1 )
ct = ct+1
constant consumption
– Budget constraint:
T
X
t=0
c
≤W
(1 + r)t
– T = ∞:
c
1+r
≤W
r
r
c≤
W
1+r
• What is the marginal propensity to consume out of wealth?
• What is the marginal propensity to consume out of income?
– If income is temporary
– If income is permanent
• The “Permanent Income Hypothesis”
– What happens if you win the lottery?
– What happens if you get an unexpected promotion?
– What happens if you get an expected promotion?
• How does this explain the facts?
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