CONSUMPTION-SAVINGS
FRAMEWORK
CHAPTER 3
2BASICS
Introduction
 Consumption-Savings Framework – provides foundation for
 Goods-market demand function (again…but w/different interpretation)
 Financial-market supply function
 Two time periods
 Important: all analysis will be conducted from the perspective of the very
beginning of period 1…
 …so a “future” (period 2) for which to save
 Dynamic models are the staple of modern macroeconomic analysis
 An explicit accounting of time
 Two periods are sufficient to illustrate the basic principles
 Soon will extend beyond two periods (Chapter 8)
3BASICS
Introduction
 Timeline of events
 Notation
 c1: consumption in period 1
 c2: consumption in period 2
 P1: nominal price of consumption in period 1
 P2: nominal price of consumption in period 2
 Y1: nominal income in period 1 (“falls from the sky”)
 Y2: nominal income in period 2 (“falls from the sky”)
 A0: nominal wealth at the beginning of period 1/end of period 0
 A1: nominal wealth at the beginning of period 2/end of period 1
 A2: nominal wealth at the beginning of period 3/end of period 2
 i: nominal interest rate between periods
 r: real interest rate between periods
 π2: net inflation rate between period 1 and period 2
 y1: real income in period 1 ( = Y1/P1)
 y2: real income in period 2 ( = Y2/P2)
2 1 2
2
1 1
1
P P P
P P

 
   
 
Period 1 Period 2
A0 A2Economic events during
period 1: income,
consumption, savings
A1 Economic events during
period 2: income,
consumption, savings
Start of the
world
End of the
worldStart of economic
planning horizon
End of economic
planning horizon
4STOCKS VS. FLOWS
Macro Fundamentals
 Stock variables, aka accumulation variables
 Quantity variables whose natural measurement occurs at a particular
moment in time
 Checking account balance
 Credit card indebtedness
 Mortgage loan payoff
 Flow variables
 Quantity variables whose natural measurement occurs over the course
of a given interval of time
 Income
 Consumption
 Savings
 The two broad categories of income
 Labor income
 Asset income (generated by interest rate(s) on (components of)
wealth)
Economic
examples
Economic
examples
Interpret A in our framework as net
wealth ( = total assets – total debts)
All income is a
FLOW
regardless of
source
5BASICS
Macro Fundamentals
 Building blocks of consumption-savings framework
 Utility
 Describes the benefits of engaging in financial market (and other)
activities
 Budget constraint
 Describes the costs of engaging in financial market (and other)
activities
 Utility and budgets two DISTINCT concepts
 As in basic consumer analysis (Chapter 1)
 Only after describing utility and budgets separately do we bring
the two together to obtain predictions from the framework
6UTILITY
Model Structure
 Preferences u(c1, c2) with all the “usual properties”
 Lifetime utility function
 Strictly increasing in c1
 Strictly increasing in c2
 Diminishing marginal utility in c1
 Diminishing marginal utility in c2
 Plotted as indifference curves
 Utility side of consumption-savings
framework identical to Chapter 1
framework
c1
u(c1,c2)
c2
u(c1,c2)
c2
c1
7BUDGET CONSTRAINT(S)
Model Structure
 Suppose again Y “falls from the sky”
 Y1 in period 1, Y2 in period 2
 Need two budget constraints to describe economic opportunities
and possibilities
 One for each period
 Period-1 budget constraint
 Period-2 budget constraint
1 1 1 1 0(1 )Pc A Y i A   
Total expenditure in period 1:
period-1 consumption +
wealth to carry into period 2
Total income in period 1:
period-1 Y + income from
wealth carried into period 1
(inclusive of interest)
2 2 2 2 1(1 )Pc A Y i A   
Total expenditure in period 2:
period-2 consumption +
wealth to carry into period 3
Total income in period 2:
period-2 Y + income from
wealth carried into period 2
(inclusive of interest)
1 1 1 0 1 0Pc A A Y iA   
2 2 2 1 2 1Pc A A Y iA   
Savings during
period 1 (a flow)
Savings during
period 2 (a flow)
Asset income
during period 1 (a
flow)
Asset income
during period 2 (a
flow)
DEFINITION: A consumer’s savings
during a given period is the change in
his wealth during that period
can rewrite as
can rewrite as
8BUDGET CONSTRAINT(S)
Model Structure
 Adopt a lifetime view of the budget constraint(s)
 All analysis conducted from perspective of beginning of period 1
 Period-1 budget constraint
 Period-2 budget constraint
1 1 1 1 0(1 )Pc A Y i A   
2 2 2 2 1(1 )Pc A Y i A   
9BUDGET CONSTRAINT(S)
Model Structure
 Adopt a lifetime view of the budget constraint(s)
 All analysis conducted from perspective of beginning of period 1
 Period-1 budget constraint
 Period-2 budget constraint
1 1 1 1 0(1 )Pc A Y i A   
Asset position at end of
period 1/beginning of
period 2 the key link
- will think further about
this soon…2 2 2 2 1(1 )Pc A Y i A   
Assume = 0 (no bankruptcies + strictly increasing utility)
10
BUDGET CONSTRAINT(S)
Model Structure
 Adopt a lifetime view of the budget constraint(s)
 All analysis conducted from perspective of beginning of period 1
 Period-1 budget constraint
 Period-2 budget constraint
 Combine into lifetime budget constraint (LBC)
 Solve period-2 budget constraint for A1…
 …and substitute into period-1 budget constraint
1 1 1 1 0(1 )Pc A Y i A   
Asset position at end of
period 1/beginning of
period 2 the key link
- will think further about
this soon…2 2 2 2 1(1 )Pc A Y i A   
2 2 2
1 1 1 0(1 )
1 1
Pc Y
Pc Y i A
i i
    
 
Present discounted
value (PDV) of all
lifetime expenditure
Present discounted value (PDV)
of all lifetime income
Assume = 0 (no bankruptcies + strictly increasing utility)
11
BUDGET CONSTRAINT(S)
Model Structure
 Adopt a lifetime view of the budget constraint(s)
 All analysis conducted from perspective of beginning of period 1
 Period-1 budget constraint
 Period-2 budget constraint
 Combine into lifetime budget constraint (LBC)
 Solve period-2 budget constraint for A1…
 …and substitute into period-1 budget constraint
1 1 1 1 0(1 )Pc A Y i A   
Asset position at end of
period 1/beginning of
period 2 the key link
- will think further about
this soon…2 2 2 2 1(1 )Pc A Y i A   
2 2 2
1 1 1 0(1 )
1 1
Pc Y
Pc Y i A
i i
    
 
Present discounted
value (PDV) of all
lifetime expenditure
Present discounted value (PDV)
of all lifetime income
For graphical simplicity, will often assume A0 = 0 (i.e., consumer begins planning horizon with zero net wealth).
Note this is a different assumption than A2 = 0.
Assume = 0 (no bankruptcies + strictly increasing utility)
12
LIFETIME BUDGET CONSTRAINT
Model Structure
Graphically
2 2 2
1 1 1
1 1
Pc Y
Pc Y
i i
  
 
Solve for c2
c1
c2
13
LIFETIME BUDGET CONSTRAINT
Model Structure
2 2 2
1 1 1
1 1
Pc Y
Pc Y
i i
  
 
2 2 2
1 1 1
1 1
Pc Y
Pc Y
i i
   
 
subtract P1c1
2 1 1 2
1
2 2 2
1
1 1
c P Y Y
c
i P P i P
 
    
  
divide by P2
1 1 2
2 1
2 2 2
(1 ) (1 )P i i Y Y
c c
P P P
  
    
 
multiply by (1+i)
14
LIFETIME BUDGET CONSTRAINT
Model Structure
Graphically
2 2 2
1 1 1
1 1
Pc Y
Pc Y
i i
  
 
Solve for c2
c1
c2
2
2 1 1
22 2
1(1 1) Yic c
P i
P
Y
P P
 
 
 
 
 
 
 
15
LIFETIME BUDGET CONSTRAINT
Model Structure
Graphically
2 2 2
1 1 1
1 1
Pc Y
Pc Y
i i
  
 
Solve for c2
c1
c2
2
2 1 1
22 2
1(1 1) Yic c
P i
P
Y
P P
 
 
 
 
 
 
 
Rearrange further using
definition of inflation:
2 1
2
1 2 2
1
1
1
P P
P P


   

2
2 1 1
22 2
1
1
1 Yi
c c Y
P P
i

 
 
 
 


 
slope = -(1+i)/(1+π2)
The larger is (1+i)/(1+π2), the
steeper is the budget line
IMPORTANT: Changes in nominal
interest rates (Fed) and/or
inflation affect the slope of the LBC
16
CONSUMER OPTIMIZATION
Model Structure
 Consumer’s decision problem: maximize lifetime utility subject to
lifetime budget constraint – bring together both cost side and
benefit side
 Choose c1 and c2 subject to
 Plot budget line
 Superimpose indifference map
 At the optimal choice
2 2 2
1 1 1
1 1
Pc Y
Pc Y
i i
  
 
c1
c2
slope = -(1+i)/(1+π2)
optimal choice
* *
1 1 2
* *
2 1 2 2
( , ) 1
( , ) 1
u c c i
u c c 



CONSUMPTION-SAVINGS
OPTIMALITY CONDITION
- key result in modern macro
analysis
MRS (between
consumption in
consecutive time periods)
price ratio (across
consecutive time
periods)
17
LAGRANGE ANALYSIS
The Mathematics of the Consumption-Savings Model
 Apply Lagrange tools to consumption-savings optimization
 Objective function: u(c1,c2)
 Constraint:
 Step 1: Construct Lagrange function
 Step 2: Compute first-order conditions with respect to c1, c2, λ
 Step 3: Combine (1) and (2) (with focus on eliminating multiplier)
2 2 2
1 2 1 1 1( , ) 0
1 1
Y Pc
g c c Y Pc
i i
    
 
2 2 2
1 2 1 2 1 1 1( , , ) ( , )
1 1
Y Pc
L c c u c c Y Pc
i i
 
 
       
* *
1 1 2
* *
2 1 2 2
( , ) 1
( , ) 1
u c c i
u c c 



CONSUMPTION-SAVINGS
OPTIMALITY CONDITION
MRS (between
consumption in
consecutive time periods)
price ratio (across
consecutive time
periods)
1 1 2 1( , ) 0u c c P 
2
2 1 2( , ) 0
1
P
u c c
i

 

LBC(1) (2) (3)
18
SAVINGS AND ASSET POSITIONS
Macro Fundamentals
 Definition: A consumer’s savings during a given time period is the
change in his wealth during that time period
 Assets/wealth (whether positive or negative) are a means for
“transferring income over time”
19
SAVINGS AND ASSET POSITIONS
Macro Fundamentals
 Definition: A consumer’s savings during a given time period is the
change in his wealth during that time period
 Assets/wealth (whether positive or negative) are a means for
“transferring income over time”
c1
c2
slope = -(1+i)/(1+π2)
Y1/P1
Y2/P2
(continuing to assume A0 = 0)
optimal choice
Optimal c1 > Y1/P1  consumer
has negative wealth at end of
period 1  period-1 savings is
negative (due to A0 = 0)
20
SAVINGS AND ASSET POSITIONS
Macro Fundamentals
 Definition: A consumer’s savings during a given time period is the
change in his wealth during that time period
 Assets/wealth (whether positive or negative) are a means for
“transferring income over time”
c1
c2
slope = -(1+i)/(1+π2)
Y1/P1
Y2/P2
optimal choice
c1
c2
slope = -(1+i)/(1+π2)
Y1/P1
Y2/P2
optimal choice
OR
Consumer
dissaved
during period 1
Consumer
saved during
period 1
(continuing to assume A0 = 0)
Optimal c1 < Y1/P1  consumer has
positive wealth at end of period 1 
period-1 savings is positive (due to
A0 = 0)
Optimal c1 > Y1/P1  consumer
has negative wealth at end of
period 1  period-1 savings is
negative (due to A0 = 0)