MACROECONOMICS
LECTURE 7
Alexander Rubin
HSE University in Saint-Petersburg
September 30, 2022
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Consumer Choice
Consumers do not supply only labour. They also decide how much to
consume and how much to save.
Suppose that income of consumer is given {Yt }∞
t=0 . A part of this
income comes from supply of labour, but it also can include other
categories including financial wealth.
Then we want to know how this income is divided on consumption and
savings. Savings is a supply of loanable funds in the economy. It’s a
source of investments which forms capital of the firms.
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Keynesian Consumption Function
The most simple way to model consumer choice is to use Keynesian
consumption function.
C = Ca + M P C(Y − T ),
where C – consumption, Ca – autonomous consumption,
M P C ∈ (0, 1) – marginal propensity of consume, Y – income, T –
lump-sum tax.
Here consumption (and savings) depends only on current incomes.
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Keynesian Consumption Function
The idea had been expressed by John Meynard Keynes in General
Theory of Employment, Interest and Money (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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Keynesian Function: Graph
C
C = Ca + M P C(Y − T )
1
The graph for Keynesian
function starts above zero
because of the autonomous
consumption.
2
The slope is less than one
because a part of consumer’s
income is saved.
3
But what the data says?
45o
Y
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Keynesian Function: Over Time
C
1
If use use data within a
country over time, then
aggregate consumption is
proportional to aggregate
income.
2
It differs from the estimation
with cross-sectional data.
Long Run
45o
Y
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Keynesian Function: Across Groups
C
Whites
1
Estimations of consumption
function give different results
for different groups: e.g.
blacks and whites.
2
It depends on the average
income which is more or less
constant within a group.
3
Permanent income varies
substantially across groups.
Blacks
45o
Y
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Permanent Income Hypothesis
Friedman, M. (1957). The permanent income hypothesis. In A theory
of the consumption function (pp. 20-37). Princeton University Press.
See also Romer’s textbook, Chapter 8 “Consumption”:
Romer, D. (2012). Advanced Macroeconomics, 4e. New York:
McGraw-Hill.
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Permanent Income Hypothesis: Model
Total consumer’s income is divided into permanent Y P and transitory
Y T part. Transitory income is zero-mean and uncorrelated with
permanent part:
Yit = YiP + YitT
Consumption depends mostly on the permanent part:
Cit = YiP + eit
Transitory consumption eit is also zero-mean and uncorrelated with
permanent income.
Both transitory income and consumption are unpredictable.
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Permanent Income Hypothesis: OLS
Now let’s estimate linear regression model using OLS:
Ci = a + bYi + ui
After minimizing the sum of squared errors we get:
b̂ =
cov(C, Y )
cov(Y P + e, Y P + Y T )
var(Y P )
=
=
var(Y )
var(Y P + Y T )
var(Y P ) + var(Y T )
â = C̄ − b̂Ȳ = (1 − b̂)Y P
We see that b̂, aka M P C, depends on the proportion of variance
permanent income in total income variance.
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Permanent Income Hypothesis: OLS
If variation in transitory income YitT is small enough, then:
b̂ → 1
â → 0
Over long period of time variation in permanent income plays the major
role. Therefore, change in come trnslates into change in consumption
almost one-to-one.
The relative variances of permanent and transitory income are similar
for the groups but average incomes are different. Therefore,
consumption also different.
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Intertemporal Choice
Marginal propensity of consume is not a structural parameter. It
depends on interest rates and other macreconomic conditions
including future anticipated income. We need to endogenize M P C.
Economic agents live in time and they have to decide how much to
consume, save, invest and work taking the long-run perspective into
the account, i.e. make intertemporal choice.
Therefore, they decide how much to consume today and tomorrow. As
in case of labour market consumer sovles optimization problem. This
problem is now dynamic.
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Discounting
To study intertemporal choice we need to use a concept of
discounting.
We need to compare two types of values:
monetary values, e.g. income and consumption
utilities from consumption.
Therefore, we need two distinct ways to compare them.
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Discounting of Monetary Flows: One Period
Assume that agent has a right for some monetary value CF which will
arrive at the end of period. The alternative is to get it right now as S
and invest at some interest rate r.
It means that opportunity costs of waiting are rS and consumer will be
indifferent between these two options if:
S(1 + r) = CF
This is no-arbitrage condition again. Then discounted value, or
presetn value of CF is equal to:
S = DV =
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CF
1+r
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Discounting of Monetary Flows: Many Periods
Now assume that agent waits for a flow of payments
CFt , t = 0, 1, 2, ...T during time T . By the same argument we can write
for some t:
S(1 + r)t = CFt , t = 0, 1, 2, ...T
Then we discount the future value of CFt in the following way:
S = DV =
CFt
, t = 0, 1, 2, ...T
(1 + r)t
Again, consumer is indifferent between these two options if the
equality above holds.
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Discounting of Monetary Flows: Cash Flow
The present, or discounted, value of the whole flow CFt , t = 0, 1, 2, ...T
is equal to:
T
X
CFt
DV =
(1 + r)t+1
t=0
If payments arrive at the beginning of the period, then the formula is
slightly different:
T
X
CFt
DV =
(1 + r)t
t=0
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Compounding Monetary Values
The inverse operation to discounting is called compounding.
If helps to compute future value of present monetary holdings at the
end of the period t:
CF = F V = S(1 + r)t
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Comparing Monetary Flows
Now we can make comparisons of monetary flow over time.
The rule is that one should discount all the payments taking their
values and timing into the account and them sum them up and
compare.
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Console Bond
Consider an infinite flow of equal payments
CF = CF0 = CF1 = CF2 = ... Then we can obtain simple formula for
present value.
Use the fact the we have geometric series with factor q ∈ (0, 1):
!
∞
X
CF
1
1
1
= CF
=
DV = CF
1
t+1
(1 + r)
1 + r 1 − 1+r
r
t=0
If payments arrive at the beginning of every period, then:
!
∞
X
1
1
1+r
DV = CF
= CF
= CF
1
t
(1 + r)
r
1 − 1+r
t=0
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Subjective Discount Factor
Subjective discount factor is a number between zero and one:
β ∈ (0, 1).
It characterizes relative utility of tomorrow and today consumption:
β=
u(Ct+1 )
,
u(Ct )
where u(·) – utility function at time t and Ct – aggregate consumption
at time t.
Discount factor is a deep, or structural, parameter that characterizes
consumer’s behavior. It doesn’t depend on government policy
interventions while M P C does.
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Subjective Discount Rate
We also can imagine (just imagine!) that discount factor can be
computed using some discount rate ρ:
β=
1
1+ρ
It follows that “discount rate” ρ can be computed as follows:
ρ=
1
−1
β
It’s easy to see that ρ > 0.
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Budget Constraint: First Period
Now assume that agent lives for two periods: t = 0, 1 and receives
incomes Y0 and Y1 .
Then budget constraint in the first period is:
C0 + B1 ≤ Y0 + B0
where B0 – initial financial wealth, C0 is consumption in the first period;
B0 are net assets in the first period.
B0 < 0 corresponds to net borrowing and B0 > 0 net lending (saving).
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Budget Constraint: Second Period
Assume that agents borrow and save at the same interest rate r. Then
budget constraint in the second period is:
C1 + B2 ≤ Y1 + (1 + r)B1
Assuming monotonicity of preferences we conclude that both budget
constraints hold as equalities.
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Transversality Condition
B2 ≤ 0 because it’s optimal to consume everything at the end (agent
doesn’t derive any utility from savings or leaving bequest). This is
transversality condition.
B2 ≥ 0 because we prohibit Ponzi scheme, when consumer borrows to
consume more and then repay principal and interest with new
borrowings. This is No-Ponzi-game condition.
Transvesality condition prevents overaccumulation of wealth, while
No-Ponzi-game condition prevents overaccumulation of wealth. Two
ineqaulities together give B2 = 0.
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Intertemporal Budget Constraint
Express B1 from the first budget constraint:
B 1 = B 0 + Y0 − C 0
Now substitute it into the second budget constraint:
C1 = Y1 + (1 + r)(B0 + Y0 − C0 )
Divide by (1 + r) and rearrange terms:
C0 +
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Y1
C1
= B0 + Y0 +
1+r
1+r
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Intertemporal Budget Constraint: General Case
Here we see that discounted money value of consumption is no more
than discounted value of income.
C0 +
C1
Y1
= B0 + Y0 +
1+r
1+r
This budget constraint can be generalized for any number of periods:
T
X
t=0
T
X Yt
Ct
=
B
+
0
(1 + r)t
(1 + r)t
t=0
We can also consider the infinite horizon (T → ∞) if series on the leftand right-hand side converge. In other words, consumption and
income do not grow fast.
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Intertemporal Budget Constraint: Intuition
Here we see that discounted money value of consumption is no more
than discounted value of income.
C0 +
Y1
C1
= B0 + Y0 +
1+r
1+r
It means that current consumption is not necessary equal to current
income.
Rather consumer compares present values and may allocate his
lifetime income over time.
Gross interest rate 1 + r is a “price” of current consumption.
Consumption of one unit today costs 1 + r units of tomorrow
consumption.
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Intertemporal Budget Constraint: Graph
C1
1
Let B0 = 0 for simplicity.
2
The ratio of “prices” is (1 + r).
3
The point (Y0 , Y1 ) is always
on the budget line.
4
Budget line rotates around
(Y0 , Y1 ) when r changes.
C0 +
Y1
Y0
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C1
Y1
= Y0 +
1+r
1+r
C0
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Intertemporal Preferences
Agent has to choose between today and tomorrow consumption given
the subjective discount factor.
Preferences are represented by utility function:
U (C0 , C1 ) = u(C0 ) + βu(C1 )
Here u(·) is an instantaneous utility function, i.e. utility function for
only one period of time.
We assume that preferences are stable over time and only β makes
difference.
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Intertemporal Preferences: General Case
Consumer’s preferences can be generalized for any periods of time:
U (C0 , C1 , ...CT ) =
T
X
β t u(Ct )
t=0
We can also consider the infinite horizon (T → ∞) if series
β t u(Ct ), t = 0, 1, 2, .. converge.
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Consumer’s Problem: Two Periods
Consider two-period model of consumption:
max U (C0 , C1 ) = U (C0 ) + βU (C1 ) s.t.
C0 ,C1
C0 +
Y1
C1
= B 0 + Y0 +
1+r
1+r
Lagrange function for this problem:
C1
Y1
max L = U (C0 ) + βU (C1 ) + λ C0 +
− B0 − Y0 −
C0 ,C1 ,λ
1+r
1+r
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Consumer’s Problem: Solution
First-order conditions for the problem are:
U ′ (C0 ) + λ = 0
λ
=0
βU ′ (C1 ) +
1+r
Exclude λ and write:
U ′ (C0 )
= β(1 + r)
U ′ (C1 )
This is known as Keynes-Ramsey rule, or Euler equation.
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Consumer’s Problem: Logarithmic Utility
Consider two-period model of consumption:
max U (C0 , C1 ) = log C0 + β log C1
C0 ,C1
C0 +
s.t.
Y1
C1
= B 0 + Y0 +
1+r
1+r
Lagrange function for this problem:
C1
Y1
max L = log C0 + β log C1 + λ C0 +
− B 0 − Y0 −
C0 ,C1 ,λ
1+r
1+r
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Consumer’s Problem: Lagrangian
Fist-order conditions are:
1
+λ=0
C0
β
λ
+
=0
C1 1 + r
Exclude λ and write
C1
1+r
= β(1 + r) =
C0
1+ρ
Again, this is Keynes-Ramsey rule for the logarithmic utility function.
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Keynes-Ramsey Rule
Keynes-Ramsey Rule relates consumption values in two adjacent
periods of time. Therefore, it brings dynamics into models.
We can generalize it to the situation when consumer lives more than
two periods t = 1, 2, ...T . Even infinite number of them t = 1, 2, ...
1+r
Ct+1
= β(1 + r) =
Ct
1+ρ
Intuitively, this means that consumption/savings choice depends on the
preferences, interest rates and this choice is made over the whole
horizon, i.e. agents are forward-looking.
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Consumer’s Problem: Graphical Solution
‘
Net borrower (C0∗ ≥ Y0 )
Net lender (C0∗ ≤ Y0 )
C1
C1
Y1
C1∗
C1∗
Y1
C0∗
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Y0
C0
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Y0
C0∗
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Increase in Interest Rate
C1
1
Rational consumer obeys
WARP.
2
If agent is willing to save
under lower interest rate he
will save under higher one.
3
Net lender remains net lender
and may lend more.
4
Net borrower may become
net lender or borrow less.
Y1
Y0
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C0
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Decrease in Interest Rate
C1
Y1
Y0
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1
Rational consumer obeys
WARP.
2
If agent is willing to borrow
under higher interest rate he
will borrow under lower.
3
Net borrower remains net
borrower and may borrow
more.
4
Net lender may become net
borrower or lend less.
C0
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Current Consumption
Consider Keynes-Ramsey Rule for logarithmic utility and combine it
with budget constraint.
C1
= β(1 + r)
C0
C1
Y1
C0 +
= B0 + Y0 +
1+r
1+r
Substitute C2 from the rule into budget constraint:
C0 +
C0 β(1 + r)
Y1
= C0 (1 + β) = B0 + Y0 +
1+r
1+r
Or finally we have:
1
Y1
C0 =
B0 + Y0 +
1+β
1+r
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Marginal Propensity to Consume
It’s easy to see that 0 < 1/(1 + β) < 1. Therefore, only a part of future
income is consumed today.
!
Y1
1
B0 + Y0 +
C0 =
1+β
1+r
Now assume that income Y1 increases by some ∆Y1 . Then only a part
of this increase will be consumed:
∆C0 =
1
∆Y0
1+β
Factor 1/(1 + β) is marginal propensity to consume. It’s a share of
current income spent to consumption.
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Solow Revisited
Compare intertemporal budget constraint with the one from the Solow
model:
Ct + Bt+1 = Bt (1 + rt ) + wt Lt
| {z }
Yt
In case of Solow model income is not purely exogenous. Agent takes
wage rate wt as given and supplies labour inelastically, but at the end
wage rate is determined at the equilibrium inside the model.
The only difference is that we allowed variable interest rate.
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Intertemporal Budget Constraint: First Step
Take first two constraints:
C0 + B1 ≤ B0 + Y1
C1 + B2 ≤ B1 (1 + r) + Y2
Divide the second constraint by 1 + r and sum them up. B1 cancels out
and we get:
B2
Y1
C1
+
≤ B 0 + Y0 +
C0 +
1+r 1+r
1+r
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Intertemporal Budget Constraint: Second Step
Now add the third constraint:
C0 +
B2
Y1
C1
+
≤ B 0 + Y0 +
1+r 1+r
1+r
C2 + B3 ≤ B2 (1 + r) + Y2
Divide the third constraint by (1 + r)2 and sum them up. B2 /(1 + r)
cancels out and we get:
C0 +
C1
C1
B2
Y1
Y2
+
+
≤ B0 + Y0 +
+
2
2
1 + r (1 + r)
(1 + r)
1 + r (1 + r)2
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Intertemporal Budget Constraint: Other Steps
We can continue in the same way and get:
∞
X
t=0
∞
X Yt
Bt+1
Ct
+ lim
≤ B0 +
t
t
t→∞ (1 + r)
(1 + r)
(1 + r)t
t=0
Again we have to restrict Bt+1 to prevent overaccumulation of debt and
call it No-Ponzi-game condition:
lim
t→∞
Bt+1
≥0
(1 + r)t
Intuitively, it means that debt should not grow too fast (exponentially).
What it it does?
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Infinite Horizon Model: Maximization Problem
Consumer maximizes:
max
∞
{Ct }t=0
U (C) =
∞
X
β t u(ct ) s.t.
t=0
Ct + Bt+1 ≤ Bt (1 + r) + Yt ,
t = 0, 1, 2, ...
Lagrange function for this problem:
max
Ct ,Bt ,λ
L=
Alexander Rubin (HSE)
∞
X
t=0
β t u(ct ) +
∞
X
λt (Bt (1 + r) + Yt − Ct − Bt+1 )
t=0
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Infinite Horizon Model: First-Order Condition
First-order condition for consumption:
β t u′ (Ct )λt = 0
β t+1 u′ (Ct+1 )λt+1 = 0
First-order condition for savings:
λt (1 + r) − λt+1 = 0
If follows that:
u′ (Ct )
λt
β t u′ (Ct )
=
=
=1+r
t+1
′
′
β u (Ct+1 )
βu (Ct+1 )
λt+1
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Infinite Horizon Model: Solution
Again we have Euler equation:
u′ (Ct )
= β(1 + r)
u′ (Ct+1 )
For example, for linear-quadratic utility u(Ct ) = Ct − 2b Ct2 :
1 − bCt
= β(1 + r)
1 − bCt+1
We use this result later to discuss Hall hypothesis.
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Consumption Smoothing
Let’s assume for a moment that β(1 + r) = 1. Then
C0 = C1
Or in general case:
C0 = C1 = ... = Ct = ...
Consumption smoothing happens even if β(1 + r) ̸= 1. Consumers
doesn’t like sharp fluctuations in consumption.
Consumption smoothing is positive if we expect such behavior from
rational consumer.
Consumption smoothing is normative because it’s rational to do so.
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Consumption Smoothing: Graph
u(C)
1
Special case β(1 + r) = 1.
2
Consumption of average C̄
brings more utility than
average of utilities.
3
The result follows from the
concavity of instantaneous
utility function u(·).
4
See also Jensen inequality.
u(C2 )
u(C̄)
ū(C)
u(C1 )
C1
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C̄
C2
C
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Linear-Quadratic Preferences
Assume linear-quadratic preferences:
b
U (Ct ) = Ct − Ct2
2
Then utility maximization gives us:
1 − bC0
= β(1 + r)
1 − bC1
Assuming that β(1 + r) = 1 we get:
C0 = C1
This is a version of Friedman (1957) and Hall (1978) Permanent
Income Hypothesis.
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Linear-Quadratic Preferences: Uncertainty
Now assume that future income is uncertain and consumer maximizes
expected utility:
"
!#
b 2
b 2
max U = E0 C0 − C0 + β C1 − C1
s.t.
C0 ,C1
2
2
C0 +
C1
Y1
= Y0 +
1+r
1+r
Income is arbitrary stationary process. On can show that in this case:
C0 = E0 [C1 ]
Or generally speaking:
Ct = Et [Ct+1 ]
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Linear-Quadratic Preferences: Consumption
Now take budget constraint and substitute Keynes-Ramsey rule:
"
#
"
#
C0
Y1
E0 C0 +
= E0 Y0 +
1+r
1+r
Use properties of mathematical expectation:
1+r
E[Y1 ]
C=
Y0 +
2+r
1+r
!
In case of uncertainty consumption depends on expected income.
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PIH: Intertemporal Choice Model
Assume that consumer lives infinitely long t = 1, 2, ... then:
C
∞
X
t=0
∞
X Yt
1
=
(1 + r)t
(1 + r)t
t=0
The sum of geometric series on the left-hand side is (1 + r)/r.
C=
∞
r X Yt
1+r
(1 + r)t
t=0
This is permanent income and share of this income spent on
consumption is equal to r/(1 + r) ∈ (0, 1).
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Permanent and Average Lifetime Income
It’s easy to see that permanent income is a weighted average of the
whole life-time income with weights 1/(1 + r)t , t = 0, 1, 2....
∞
r X Yt
C=
1+r
(1 + r)t
t=0
If case of uncertainty, permanent income is a weighted average of the
expected income:
#
" ∞
X Yt
r
E0
C=
1+r
(1 + r)t
t=0
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Permanent Shock
Assume that shock affects permanent income changing it once and
forever. This is permanent shock. Then new incomes are Yt + ∆Y :
∞
r 1+r
r X ∆Y
∆Y = ∆Y
=
∆C =
1+r
(1 + r)t
1+r r
t=0
Change in permanent income translates into change in income
one-to-one.
Alexander Rubin (HSE)
Macroeconomics
September 30, 2022
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Transitory Shock
Assume that shock affects transitory income in the first period, i.e. this
is transitory shock. Then new income is Y1 + ∆Y :
∆C =
r
∆Y
1+r
Therefore, only a part of increase in income will be spent on
consumption.
Assume that r = 0.01, then (r/(1 + r) = 0.01 and only 1% of transitory
income will be spent on consumption. The other part is saved.
Alexander Rubin (HSE)
Macroeconomics
September 30, 2022
56 / 64
Anticipated Future Shock
Assume that a consumer anticipates a permanent shock in the next
period t = 1. For example, an income tax cut which increases
permanent income.
What will happen to consumption right now?
∆C =
∞
r 1
∆Y
r X ∆Y
∆Y =
=
t
1+r
(1 + r)
1+rr
1+r
t=1
Consumption is affected immediately before actual increase in income,
but this increase is slightly smaller than before 1/(1 + r) ≈ 0.99.
In case of unanticipated shock consumption will be affected next
period only.
Alexander Rubin (HSE)
Macroeconomics
September 30, 2022
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Expected and Unexpected Shocks
It follows that consumers react differently on different types of shocks.
Anticipated shocks affect consumption and savings before the shock
happens while unanticipated shocks affect only when it occurs.
Permanent shocks mostly affect consumption while transitory shocks
mostly affect savings.
Alexander Rubin (HSE)
Macroeconomics
September 30, 2022
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The Role of Expectations
We saw that reaction of an agent depends on his expectations, i.e. the
process of thinking about the future and ability to do it correctly.
Different types of expectations may be embedded into a model:
constant expectations, i.e. agent thinks something about future
and these thoughts may be correct but he doesn’t change his
mind with time in response to any shocks
adaptive expectations, i.e. agent corrects his perception of future
when new information arrives
rational expectations, i.e. agent uses all the available information
and builds correct prediction.
In case of rational expectations an agent is forward-looking. In case
of adaptive – backward-looking.
Alexander Rubin (HSE)
Macroeconomics
September 30, 2022
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Why PIH May not Work?
In reality PIH may be violated due to different reasons.
1
Bounded rationality, i.e. departure from rational model and/or
from rational expectations.
2
Precautionary savings, i.e. additional savings due to
risk-aversion.
3
Liquidity constraints, i.e. inability to lend or borrow.
4
Presence of durable goods in consumption bundle, i.e. we need
to take into the account depreciation of durable goods instead on
expenditures on them.
5
Habit formation, i.e. proclivity to consume the same amount.
Alexander Rubin (HSE)
Macroeconomics
September 30, 2022
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Precautionary Savings
Take Euler equation for general (non-quadratic) utility fiunction:
u′ (C0 )
= β(1 + r)
E0 [u′ (C1 )]
Assume again that β(1 + r) = 1. Then:
u′ (C0 ) = E0 [u′ (C1 )]
It’s easy to see that if expectation increases, then consumption
decreases, because marginal utility is a decreasing function:
↑ E0 [u′ (C1 )] =⇒ ↓ C0
Let u′′′ > 0 and variance in consumption increases while expected
value remains the same.
Alexander Rubin (HSE)
Macroeconomics
September 30, 2022
61 / 64
Precautionary Savings: Graph
u′ (C1 )
1
Mean-preserving spread, i.e.
expectation is the same, but
variance is larger.
2
Marginal utility is concave, i.e.
u′′′ > 0.
3
E0 [u′ (C1 )|σ1′ ]
4
E0 [u′ (C1 )|σ1 ]
Today consumption
decreases, i.e. agent makes
precautionary savings.
C1
E0 [C1 ]
Alexander Rubin (HSE)
E0 [u′ (C1′ )|σ1′ ] > E0 [u′ (C1 )|σ1 ].
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September 30, 2022
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Liquidity Constraints: Not Bounding
C1
1
Due to liquidity constraint
consumer cannot borrow in
the first period.
2
Red part of budget set is now
unavailable for him.
3
Due to parameters and
preferences liquidity
constraint is not active and
bounding.
C1∗
Y1
C0∗
Alexander Rubin (HSE)
Y0
C0
Macroeconomics
September 30, 2022
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Liquidity Constraints: Bounding
C1
C1∗ = Y1
C0∗ = Y0
Alexander Rubin (HSE)
1
Due to liquidity constraint
consumer cannot borrow in
the first period.
2
Red part of budget set is now
unavailable for him.
3
Due to parameters and
preferences liquidity
constraint is active and
bounding consumer choice.
4
In the absence of liquidity
constraint optimal
consumption may be larger
C0∗ > Y0 .
C0
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September 30, 2022
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