INFLATION AND INTEREST RATES
IN THE CONSUMPTION-SAVINGS
FRAMEWORK
CHAPTER 4
2FISHER EQUATION
Macro Fundamentals
 Nominal interest rate – measured in dollars
 Real interest rate – measured in goods
 Fisher equation: a link between the nominal interest rate, inflation
rate, and real interest rate
 Exact Fisher equation
 Approximate Fisher equation (intro macro)
1
1
1
i
r


 

(1 )(1 ) 1
1 1
r i
r r i

 
   
    
In advanced economies, r and π are both generally small 
rπ ≈ 0
≈ 0
r i   A useful rule of thumb
More useful for our
analytical framework
3REAL INTEREST RATE
Macro Fundamentals
 r a key variable for macroeconomic analysis
 Unit Analysis (i.e., analyze algebraic units of variables)
 Economic decisions depend on real interest rates (r), not nominal
interest rates (i)
 Measures the cost of borrowing/lending in terms of goods…
 …which is presumably what people most care about
c2
slope = - (1+i)/(1+π2)
= - (1+r) BY FISHER EQUATION
Slope measures how much c2 must be given
up in order to obtain one more unit of c1 (“rise
over run”) when saving or dissaving at
market interest rates
1+r is the price of period-1 consumption in
terms of period-2 consumption
More generally: r measures the price of
current goods in terms of future goods
c1
4TWO-PERIOD FRAMEWORK IN REAL TERMS
From Nominal to Real
 Depending on application, may be useful to work with framework
in nominal terms or in real terms
2 2 2
1 1 1
1 1
Pc Y
Pc Y
i i
  
 
LBC in nominal terms (assuming A0 = 0)
2 1 2
1 2
1 1 1(1 ) (1 )
P Y Y
c c
P i P P i
 
   
  
Divide by P1
Multiply and divide last term on right-hand-side by P2
2 1 2 2
1 2
1 1 1 2(1 ) (1 )
P Y P Y
c c
P i P P i P
   
     
    
Use definitions: y1 = Y1/P1, y2 = Y2/P2, and 1+π2 = P2/P1
2 2
1 2 1 2
1 1
1 1
c c y y
i i
     
     
    
Use Fisher equation: (1+π2)/(1+i) = 1/(1+r)
2 2
1 1
1 1
c y
c y
r r
  
 
LBC in real terms (assuming A0 = 0)
5CONSUMPTION-SAVINGS OPTIMALITY CONDITION
The Mathematics of the Consumption-Savings Model
 Emphasizing i and π
 Emphasizing r
* *
1 1 2
* *
2 1 2
( , ) 1
( , ) 1
u c c i
u c c 



* *
1 1 2
* *
2 1 2
( , )
1
( , )
u c c
r
u c c
 
Fisher equation
6MICRO-LEVEL SAVINGS
Savings Behavior in the Micro
How do micro-level consumption/savings choices change
as the real interest rate changes (continue assuming A0 = 0 for simplicity)?
c1
c2
slope = -(1+r1)
y2
y1
original optimal choice
7MICRO-LEVEL SAVINGS
Savings Behavior in the Micro
How do micro-level consumption/savings choices change
as the real interest rate changes (continue assuming A0 = 0 for simplicity)?
REAL INTEREST RATE: r1 < r2
c1
c2
y2
y1
original optimal choice
new optimal choice
IMPORTANT: LBC pivots around
the point (y1, y2) because (y1, y2) is
always a possible choice of
consumption
RESULT: optimal choice of c1
falls as r rises 
slope = -(1+r1)
slope = -(1+r2)
8MICRO-LEVEL SAVINGS
Savings Behavior in the Micro
How do micro-level consumption/savings choices change
as the real interest rate changes (continue assuming A0 = 0 for simplicity)?
REAL INTEREST RATE: r1 < r2
c1
c2
y2
y1
original optimal choice
new optimal choice
IMPORTANT: LBC pivots around
the point (y1, y2) because (y1, y2) is
always a possible choice of
consumption
RESULT: optimal choice of c1
falls as r rises  optimal
choice of savings (= y1 – c1)
rises as r rises
c1
c2
slope = -(1+r1)
y2
y1
original
optimal choice
slope = -(1+r2)
slope = -(1+r1)
slope = -(1+r2)
new optimal
choice
OR
Empirical evidence shows that when r rises, period-1 (i.e.,
“short-run”) consumption of all types of consumers falls
implying that when r rises, period-1 (i.e., “short-run”)
savings of all types of consumers rises…
RESULT: optimal choice of c1
falls as r rises  optimal
choice of savings ( = y1 – c1)
rises as r rises
9SAVINGS
Savings Behavior in the Micro and the Macro
 Define private savings function (in period 1) for an individual
s1
r
1 1 2 1 1 1 2( , , ) ( , , )
privs r y y y c r y y  Emphasizing functional relationships
private savings function
-+
Sum over all individuals
Individual-level savings function Aggregate-level savings function
s1
r
private savings function
(No tension between the
micro and macro as there
is for labor supply)
(Recall alternative (equivalent)
definition: savings is the change in
wealth during a period)
10
THE THREE MACRO (AGGREGATE) MARKETS
The Three Macro Markets
 Goods Markets
 Demand derived from C-L framework
 Labor Markets
 Supply derived from C-L framework
 Capital/Savings/Funds/Asset Markets
(aka Financial Markets)
 Supply derived from C-S framework
c
P
labor
wage
capital/
savings
real
interest
rate
S
D
S
EFFECTS OF A “CREDIT CRUNCH”
12
ASSESSING THE EFFECTS OF THE CREDIT CRUNCH
Effects of a Credit Crunch
c1
c2
y1
y2
(continuing to assume A0 = 0)
Optimal choice BEFORE credit crunch:
Consumers BORROWING during period 1
“Credit crunch” in period 1 limits the ability
of consumers to borrow during period 1.
Most extreme case: consumers cannot
borrow AT ALL during period 1.
Consumers’ goal (presumably…) is STILL to
maximize lifetime utility…but now they must
do so WITHOUT going into debt at the end of
period 1.
Period-1 consumption must thus satisfy
c1 = y1
From a lifetime perspective,
consumers are on a lower
indifference curve…
…even though c2 actually
RISES due to the restriction in
the availability of loans in
period 1.
13
ASSESSING THE EFFECTS OF THE CREDIT CRUNCH
Effects of a Credit Crunch
 “Credit crunch” – financial sector has restricted the quantity of loans it is
willing to extend to consumers in the “short run”
 Can analyze macroeconomic consequences of shrinkage of credit
availability using two-period model
 Interpret “short run” to be period 1 (i.e., 2008-2010)
 Consequences
 A large fraction of consumers (though not all) unable to borrow to
pay for their desired period-1 consumption  their period-1 (i.e.,
“short run”) consumption falls
 Consumption ≈ 2/3 of GDP  period-1 (i.e., “short run”) GDP falls
 Consumption in period 2 (i.e., “the long run”) actually rises

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