INFINITE-PERIOD FRAMEWORK
CHAPTER 8
2BASICS
Introduction
 Modern macroeconomic frameworks feature an infinite number of
periods
 Especially useful for thinking about asset accumulation and asset
pricing
 The intersection of modern macro theory and modern finance theory
 Here, suppose just one real asset
 Call it a “stock” – i.e., a share in the S&P 500
 (In monetary analysis, two nominal assets: bonds and money)
 Index time periods by t, t+1, t+2, etc.
 Important: all analysis conducted from the perspective of the very
beginning of period t…
 …so an “infinite future” (period t+1, period, t+2, period t+3, …) for
which to save
3BASICS
Introduction
 Timeline of events
 Notation
 ct: consumption in period t
 Pt: nominal price of consumption in period t
 Yt: nominal income in period t (“falls from the sky”)
 at-1: real wealth (stock) holdings at beginning of period t/end of period t-1
 St: nominal price of a unit of stock in period t
 Dt: nominal dividend paid in period t by each unit of stock held at the start of t
 πt+1: net inflation rate between period t and period t+1
 yt: real income in period t ( = Yt/Pt)
Period t Period t+1
at-1 Economic events during
period t: income,
consumption, savings
Economic events during
period t+1: income,
consumption, savings
at at+1
Period t+2
Economic events during
period t+2: income,
consumption, savings
at+2
…
1 1
1 1
t t t
t
t t
P P P
P P
  
 
   
 
The “definining
features” of
stock
4BASICS
Introduction
 Timeline of events
 Notation
 ct+1: consumption in period t+1
 Pt+1: nominal price of consumption in period t+1
 Yt+1: nominal income in period t+1 (“falls from the sky”)
 at: real wealth (stock) holdings at beginning of period t+1/end of period t
 St+1: nominal price of a unit of stock in period t+1
 Dt+1: nominal dividend paid in period t by each unit of stock held at the start of
t+1
 πt+2: net inflation rate between period t+1 and period t+2
 yt+1: real income in period t+1 ( = Yt+1/Pt+1)
Period t Period t+1
at-1 Economic events during
period t: income,
consumption, savings
Economic events during
period t+1: income,
consumption, savings
at at+1
Period t+2
Economic events during
period t+2: income,
consumption, savings
at+2
…
2 1 2
2
1 1
1t t tt
t t
P P P
P P
   
 
 
   
 
The “definining
features” of
stock
5SUBJECTIVE DISCOUNT FACTOR
Macro Fundamentals
 Subjective discount factor
 β (a number between zero and one) measures impatience
 The lower is β, the less does individual value future utility
 Simple assumption about how “impatience” builds up over time
 Multiplicatively: i.e., discount one period ahead by β, discount two periods
ahead by β2, discount three periods ahead by β3, etc.
6UTILITY
Model Structure
 Preferences v(ct, ct+1 , ct+2 , …) with all the “usual properties”
 Lifetime utility function
 Strictly increasing in ct, ct+1, ct+2, ct+3 , …
 Diminishing marginal utility in ct, ct+1, ct+2, ct+3 , …
 Lifetime utility function additively-separable across time (a
simplifying assumption), starting at time t
ct ct+1
v(ct,ct+1,ct+2,ct+3, …)v(ct,ct+1,ct+2,ct+3, …)
etc.
1 2 3 1 2 3
2 3( , , , ,...) ( ) ( ) ( ) ( ) ...t t t t t t t tv c c c c u c u c u c u c           
7BUDGET CONSTRAINT(S)
Model Structure
 Suppose again Y “falls from the sky”
 Yt in period t, Yt+1 in period t+1, Yt+2 in period t+2, etc.
 Need infinite budget constraints to describe economic
opportunities and possibilities
 One for each period
 Period-t budget constraint
 Period t+1 budget constraint
1 1t t t t t t t t tPc S a Y S a Da    
Total income in period t: period-t Y
+ income from stock-holdings
carried into period t (has value St
and pays dividend Dt)
Total expenditure in period t:
period-t consumption + wealth
to carry into period t+1
1 1 1 1 1 1 1t t t t t t t t tP c S a Y S a D a         
Total income in period t+1: period-
t+1 Y + income from stock-
holdings carried into period t+1
(has value St+1 and pays dividend
Dt+1)
Total expenditure in period t+1:
period-t+1 consumption +
wealth to carry into period t+2
can rewrite as
can rewrite as
1 1( )t t t t t t t tPc S a a Y Da    
1 1 1 1 1 1( )t t t t t t t tP c S a a Y D a        
Savings during
period t (a flow)
Savings during
period t+1 (a
flow)
Dividend income
during period t (a
flow)
Dividend income
during period t+1
(a flow)
And identical-looking budget
constraints for t+2, t+3, t+4, etc…
8LAGRANGE ANALYSIS: SEQUENTIAL APPROACH
Infinite-Period Model: Sequential Formulation
 Sequential formulation highlights the role of stock holdings (at)
between period t and period t+1
 Accords better with the explicit timing of economic events than the
lifetime approach…
 …but yields the same result
 Advantage: allows us to think about interaction between asset prices
and macroeconomic events (intersection of finance theory and macro
theory)
 Apply Lagrange tools to consumption-savings optimization
 Objective function: v(ct, ct+1 , ct+2 , …)
 Constraints:
 Period-t budget constraint:
 Period-t+1 budget constraint:
 Period-t+2 budget constraint:
 etc…
 Sequential Lagrange formulation requires infinite multipliers
1 1 0t t t t t t t t tY S a Da Pc S a     
1 1 1 1 1 1 1 0t t t t t t t t tY S a D a P c S a          
2 2 1 2 1 2 2 2 2 0t t t t t t t t tY S a D a P c S a            
INFINITE
constraints
9LAGRANGE ANALYSIS: SEQUENTIAL APPROACH
Infinite-Period Model: Sequential Formulation
 Step 1: Construct Lagrange function (starting from t)
 Step 2: Compute FOCs with respect to ct, at, ct+1, at+1, ct+2, …
with respect to ct:
with respect to at:
with respect to ct+1:
 
 
 
 
2 3
2
1 2 3
1
1 1 1 1 1 1 1 1
2 2 2 2 1 2 2 2 2
3 3 3 3 2 3 3
3
3 3
...( ) ( ) ( ) ( )
( )
( )
(
(
..
)
)
t t t t
t t t t t t t t t
t t t t t t t t t
t t t t t t t t t
t t t t t t t t t
u c u c u c u c
Y S D a Pc S a
Y S D a P c S a
Y S D a P c S a
Y S D a P c S a
 








  

       
        
        
   
    
    
    
    
 .
First the lifetime utility function….
…then the period t constraint…
…then the period t+1 constraint…
…then the period t+2 constraint…
…then the period t+3 constraint…
Infinite number of terms
IMPORTANT:
Discount factor β
multiplies both
future utility and
future budget
constraints
Everything (utility
and income) about
the future is
discounted
'( ) 0t t tu c P 
1 1 1'( ) 0t t tu c P    
1 1 1( ) 0t t t t tS S D       
Identical
except for
time
subscripts
Equation 1
Equation 2
Equation 3
10
THE BASICS OF ASSET PRICING
Finance Fundamentals
 Equation 2 
 Much of finance theory concerned with pricing kernel
 Pricing kernel where macro theory and finance theory intersect
 To take more macro-centric view
 Solve equations 1 and 3 for λt and λt+1
 Insert in asset-pricing equation
'( ) 0t t tu c P 
1 1 1'( ) 0t t tu c P   
1 1 1( ) 0t t t t tS S D       
Equation 1
Equation 2
Equation 3
1
1 1( )
t
t
t ttS S D





 
 
 
 BASIC ASSET-PRICING EQUATION
Future
return
Pricing
kernel
xPeriod-t stock
price
=
Two components:
1. Future price of stock
2. Future dividend payment
11
MACROECONOMIC EVENTS AFFECT ASSET PRICES
Macro-Finance Connections
 Consumption across time (ct and ct+1) affects stock prices
 Fluctuations over time in aggregate consumption impact St
 Inflation affects stock prices
 Fluctuations over time in inflation impact St
 ANY factor (monetary policy, fiscal policy, globalization, etc.) that
affects inflation and GDP in principle impacts stock/asset markets
1 1
1
1'(
'( )
( )
)t
t
t
t
t
t
tS
u c
Pu
P
S D
c





 
 
 
  


 
1
1 1
1
1
(
'( )
'( )
)
1
t
t
t
t t
t
u c
u c
S S D






 
 
 
  



Using definition of inflation: 1+πt+1 = Pt+1 / Pt
VIEW AS A CONSUMPTION-SAVINGS
OPTIMALITY CONDITION
12
CONSUMER OPTIMIZATION
Consumption-Savings View
1 1
1
1'(
'( )
( )
)t
t
t
t
t
t
tS
u c
Pu
P
S D
c





 
 
 
  


 
1 1
1 1
'( ) 1
'( ) 1
t t t
t t t
u c S D
u c S 
 
 
  
   
  
Move u’(ct) and βu’(ct+1) terms to left-hand-side,
and St to right-hand-side
Some sort of price ratio…MRS between period t
consumption and
period t+1
consumption
i.e., ratio of
marginal
utilities
13
CONSUMER OPTIMIZATION
Consumption-Savings View
 Recover Chapter 3 & 4 framework by setting t =1 and β = 1
1 1
1
1'(
'( )
( )
)t
t
t
t
t
t
tS
u c
Pu
P
S D
c





 
 
 
  


 
1 1
1 1
'( ) 1
'( ) 1
t t t
t t t
u c S D
u c S 
 
 
  
   
  
Move u’(ct) and βu’(ct+1) terms to left-hand-side,
and St to right-hand-side
Analogy with Chapters 3 & 4:
must be (1+rt)
MRS between period t
consumption and
period t+1
consumption
i.e., ratio of
marginal
utilities
Recall real interest
rate is a price
CONSUMPTION-SAVINGS
OPTIMALITY CONDITION
14
CONSUMER OPTIMIZATION
Consumption-Savings View
 Recover Chapter 3 & 4 framework by setting t =1 and β = 1
 Infinite-period framework is sequence of overlapping two-period
frameworks
1 1
1
1'(
'( )
( )
)t
t
t
t
t
t
tS
u c
Pu
P
S D
c





 
 
 
  


 
1 1
1 1
'( ) 1
'( ) 1
t t t
t t t
u c S D
u c S 
 
 
  
   
  
Move u’(ct) and βu’(ct+1) terms to left-hand-side,
and St to right-hand-side
Analogy with Chapters 3 & 4:
must be (1+rt)
MRS between period t
consumption and
period t+1
consumption
i.e., ratio of
marginal
utilities
Recall real interest
rate is a price
CONSUMPTION-SAVINGS
OPTIMALITY CONDITION
c1
c2
slope = -
(1+r1)
optimal choice between
p1 and p2
c2
c3 optimal choice between
p2 and p3
c3
c4 optimal choice between
p3 and p4
etc.
slope = -
(1+r2)
slope = -
(1+r3)
15
A LONG-RUN THEORY OF MACRO
Modern Macro
 Consumption-savings optimality condition at the heart of modern
macro theories
 Emphasize the dynamic nature of aggregate economic events
 Foundation for understanding the periodic ups and downs (“business
cycles”) of the economy
 (Chapter 14: business cycle theories)
1
'( )
1
'( )
t
t
t
u c
r
u c 
 
NEXT: Impose “steady state”
and examine long-run
relationship between interest
rates and consumer impatience
1
1 r

 
STEADY-STATE OF INFINITE-PERIOD
ECONOMY: WHY ARE LONG-RUN REAL
INTEREST RATES POSITIVE?
17
STEADY STATE
Macro Fundamentals
 Steady state
 A concept from differential equations
 (Optimality conditions of economic models are differential equations…)
 Heuristic definition: in a dynamic (mathematical) system, a steady-
state is a condition in which the variables that are moving over time
settle down to constant values
 In dynamic macro models, a steady state is a condition in which all
real variables are constant.
 But nominal variables (i.e., price level) may still be moving over time
(will be important in monetary models – Chapter 15)
 Simple example
 Suppose Mt/Pt = ct is an optimality condition of an economic model (ct is
consumption, Pt is nominal price level, Mt is nominal money stock of
economy)
 Even if ct eventually becomes constant over time (i.e., reaches a steady-
state), it is possible for both Mt and Pt to continue growing over time (at the
same rate of course…)
 Bottom line: in ss, real variables do not change over time, nominal
variables may change over time
18
REAL INTEREST RATE
Macro Fundamentals
 Recall earlier interpretation of r
 Price of consumption in a given period in terms of consumption in the
next period
 (Chapter 3 & 4: r measures the price of period-1 consumption in terms
of period-2 consumption)
 Now a second interpretation of r: long-run (i.e., steady state) real
interest rate simply a reflection of degree of impatience of
individuals in an economy
 The lower is β, the higher is r
 The more impatient a populace is, the higher are interest rates
 Which came first, β or r?
 Modern macro view: β < 1 causes r > 0, not the other way around
 A deep view of why positive real interest rates exist in the world
1
1 r

 
Inverse of subjective
discount factor
(one plus) real
interest rate

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