LECTURE 6
Aggregate demand and its components
Øystein Børsum
21rst February 2006
Overview of forthcoming lectures

Lecture 6: Aggregate demand and its components



Lecture 7: Aggregate demand and aggregate supply


Macroeconomic dynamics in the AS-AD model
Lecture 8: Stabilization policies



Determinants of aggregate investments and consumption,
important and volatile components of aggregate demand
Aggregate demand put together: The AD curve
Goals for stabilization policies: Stable output and inflation
Optimal policy rule: Demand and supply shocks
Lecture 9: Limits to stabilization policies


Rational expectations and the Policy Ineffectiveness Proposition,
the Ricardian Equivalence Theorem and the Lucas Critique
Policy rules versus discretion: Credibility of economic policy
PART 1
Private investment
Overview of Q-theory of investment

The market value of a firm is determined by discounting future
dividends to the owners

By investing in capital, the firm grows and hence its capacity
to generate dividends increases

The cost of investing one unit of capital is exogenous

This provides an incentive for firms with a high market value
per unit of capital to invest

Definition: q = the ratio between the market value of the firm
(V) and the replacement value of its capital stock (K)

Note: Q-theory applied to housing investment (section 15.4) is
self-study
Pricing by arbitrage condition

Arbitrage condition: In every period, stocks and bonds must
yield the same risk-adjusted rate of return
( r   )Vt  Dte  Vt e1  Vt

Vt = real stock market value of the firm at the start of period t
Vet+1 = expected real stock market value of the firm at the start
of period t+1
De = real expected dividend at the end of the period t
r = real interest rate on bonds
 = risk premium on shares
Dte  Vt e1
Vt 
1 r  
The fundamental value of a firm

Successive substitution gives:
Dte
Dte1
Vt e 2
Vt 


2
1 r  
(1  r   )
(1  r   ) 2
Dte
Dte1
Dte 2
Vt e3




2
3
1 r  
(1  r   )
(1  r   )
(1  r   )3
Dte
Dte1
Dte 2
Vt e n



 .... 
1 r  
(1  r   ) 2
(1  r   )3
(1  r   ) n

Assume that the future value of the firm Vet+1 cannot rise faster
than r +  (else it would be of infinite value), i.e.:
Vt e n
lim
0
n  (1  r   ) n
The fundamental value of a firm

Dte n
Vt  
n 1
(1

r


)
n 0

Then the infinite sum can be written as:

Interpretation: The fundamental value of the firm equals the
present value of expected future dividends

Implications: Stock prices may fluctuate because of changes in:
o
o
o
expected future dividends
the real interest rate
the risk premium between stocks and bonds

The role of the interest rate: We only assume that the expected
return on shares is systematically related to the return on bonds

What about investments? The firm must decide whether to pay
out its profits now (as dividends) or invest it in order to increase
profits (dividends) later: Maximize Vt with respect to It
The decision to invest

Definition: qt = Vt / Kt = the ratio between the market value of
the firm and the replacement value of its capital stock

Expected value of the firm tomorrow:
Vt e1  qte1Kt 1  qt  Kt  It 
 where we have used:

Kt 1  Kt  I t and
qte1  qt
Cash flow constraint: Dte   te  I t  c( I t )
e = expected profit
c = installation costs

Assume the following installation cost function: c ( I )  a I 2
t
t
2
Optimal level of investment depends on q
Dte
Vte1
a 2
  I t  I t  qt  K t  I t 
2
Vt 
1 r  
e
t

V
t
Maximization of Vt taking qt as given gives
 0the following

qt
I t
first-order conditions:
Vt
Vt  0
I t  0
I t




qt  1  aI t
qt  1  aI t
expected
expected
capital gain
capital gain
qt
qt


foregone dividend
foregone dividend
dc
 1  dc
 1  dIqt  1  aI t
dI tt
qt  1
It 
a
foregone d
expected
capital gain


1
d
An example of the investment function

Assume that
Dte i  Dte in order to simplify the value of the firm to
e


D
1
1
1
t
Vt  Dte 



...

 r 
2
3
1

r


(1

r


)
(1

r


)



Assume furthermore that Dte   t and
 t  Yt
 = expected dividend pay-out ratio
 = constant profit share

Using the definition of q this gives the investment function

 1    Yt / K t
It    
 1
 a  r  

The general investment function

Abstracting from the functional form the general investment
function is: I  f ( Y , K , r , E )
(  ) ( ) ( ) ( )
E = index of business confidence

Note that the risk premium is omitted

Note that in chpt. 17 the level of capital K is assumed constant
and the notation changes slightly ( is the index of business
confidence)
I  I (Y , r ,  )
I
IY 
 0,
Y
I
Ir 
 0,
r
I
I 
0

PART 2
Private consumption
Overview of intertemporal consumption theory

Diminishing marginal utility of consumption provides an incentive for
consumption smoothing over time.

Through the capital market, consumers can save or borrow and thus
separate consumption from current income.

The discounted value of disposable lifetime income (human wealth) plus
the initial stock of financial wealth represents the consumer’s lifetime
budget constraint.

In optimum the consumer is indifferent between consuming an extra unit
today and saving that extra unit in order to consume it tomorrow.

Current consumption will be proportional with wealth – not income.

Note: Issues on debt-financed tax cuts and ricardian equivalence will be
treated later on in the course.
Intertemporal consumer preferences

Representative consumer with a two-period utility function
u (C2 )
U  u (C1 ) 
,
1 

u '  0, u ''  0,   0
Properties of the utility function:


the marginal utility of consumption in each period is positive, but
diminishing (provides an incentive for consumption smoothing)
the consumer is impatient: the rate of time preference  is positive
Intertemporal budget constraint

Period 1 budget constraint
V2  1  r  V1  Y1L  T1  C1 

Period 2 budget constraint
C2  V2  Y2L  T2

The consumer’s intertemporal budget constraint
L
C2
Y
L
2  T2
C1 
 V1  Y1  T1 
1 r
1 r
V = financial wealth
r = real rate of interest
YL = labour income
T = net tax payment (taxes minus transfers)
C = consumption
Human wealth and financial wealth

V1 represents the consumer’s initial financial wealth

The present value of disposable lifetime income can be thought of
as human wealth (or human capital) H
Y2L  T2
H1  Y  T1 
1 r
L
1

This simplifies the notation of the intertemporal budget constraint
C2
C1 
 V1  H1
1 r
Optimal intertemporal consumption

Utility over the consumer’s life-time becomes (as a function of C1)
u  (1  r )(V1  H1  C1 ) 
U  u (C1 ) 
1 

Maximization of U with respect to C1 gives the following first-order
conditions:
C2


 1 r  
dU
 0  u '(C1 )  
 u ' (1  r )(V1  H1  C1 ) 

dC1
 1    


The Keynes-Ramsey rule:  1  r u '(C1 )
u '(C1 )   u '(C u) /(1
'(C2)  )  1  r
 1   2
Optimal intertemporal consumption

In optimum, the marginal rate of substitution between present and
future consumption (MRS) must equal the relative price of present
consumption (1+r)
Example of the consumption function with CES utility

The constant (intertemporal) elasticity of substitution utility
function
1
u (Ct ) 
Ct11/ 
1  1/ 
for   0,   1

u’(Ct) = Ct-1/ u(Ct )  ln Ct for  =1

Insert this into the Keynes-Ramsey rule
1/ 
(1   )(C2 / C1 )
(1   )(C2 / C1 )1/   1  r
 1  r  1/ 
C1/2   
 C1
 1  
 C1/2 

 1 r
 1  r  1/ 

 C1
 1  

 1 r 
C2  
 C1
1 
(1   )(C2 / C1 )1/   1  r
 1  r  1/ 
C 
 C1
  1  
1/
2

 1 r 
 C2  
 C1
 1  

Example of the consumption function with CES utility

Insert the expression for the optimal C2 in terms of C1 into the
intertemporal budget constraint.
C1  (1  r ) 1 (1   )  C1  V1  H1
C1    (V1  H1 ),
0  
1
 1
1  (1  r )
(1   )

1

Current consumption C1 is proportional to total current wealth
(not current income).

The propensity to consume wealth is positive, but less than one.
The general consumption function


C1  C  Y1d , g , r , V1 
 (  ) (  ) (?) (  ) 
g = growth rate of income (increases human wealth)

Some consumers may be credit constrained, hence Y1d

In chpt. 17 notation is slightly changed:


The value of financial wealth is treated implicitly in r
 is an index of consumer confidence (proxy for expected income
growth)
C  C (Y  T , r ,  )
0  CY T 
C
C 
C
 1, Cr 
0,
C

0


 (Y  T )
r

PART 3
Aggregate demand
Overview over aggregate demand theory with
endogenous monetary policy

Private investments and consumption are sensitive to changes in the
real interest rate, hence there is a potential for stabilization policy

The government cares about stabilizing both output and inflation

In order to achieve the government’s objectives, the central bank sets
the nominal short-term interest rate according to a Taylor rule

The resulting aggregate demand curve will be downwards-sloping in
(y;) space

Important properties of the aggregate demand curve (the exact slope
as well as the shift properties) will depend on the policy priorities
(implied by the choice of coefficients in the Taylor rule)

Note: We will return to questions about fiscal policy (public
consumption and taxes) later in the course
Equilibrium condition in the goods market gives
the aggregate demand function Y

Investments plus consumption = aggregate private demand D
I  I (Y , r ,  )
C  C (Y  T , r ,  )
I
I
I
IY 
0, Equlibrium
Ir 
condition
0,
I  for
 the goods
0
market (closed economy)
Y

r


C
C 
C
0  CY T 

1,
C
0,
C

0
r 


Y

D
(
Y
,
G
,
r
,

)

G
 (Y  T )
r


Properties of the aggregate private demand function
D
0  DY 
 CY  IY  1,
Y
Dr 
D
 Cr  I r  0,
r
D 
D
C
DG 

 CY  0,
G
(Y  T )
D
 C  I   0

Evidence from Denmark seem to confirm a close
relationship between private sector demand and
the real interest rate over time
The real interest rate and the private sector savings surplus in Denmark, 1971-2000. Per cent
Source: Erik Haller Pedersen, ‘Udvikling i og måling af realrenten’, Danmarks Nationalbank, Kvartalsoversigt,
3. kvartal, 2001, Figure 6
The aggregate demand function on log-linearized form

The long run equilibrium values of aggregate demand
Y  D(Y , G, r ,  )  G

The textbook shows how the aggregate demand function Y
can be log-linearized around its long-run equilibrium values to
give this very convenient form
y  y  1 ( g  g )   2 (r  r )  v,
y  ln Y ,
y  ln Y ,
g  ln G,
G
,
Y
 
 2  m 
1  m(1  CY ) 
1  0,
g  ln G,
 Dr 
,
Y


m
2  0
1
1  DY
  D 
v  m
  ln   ln 
Y



The real and the nominal interest rate


1 r
The definition of the expected real interest rate
1 i
1 r 

e
1   1
As long as i and  are close to zero, we can approximate the
1 i
real interest rate
 r  i   e 1
1   e1

Expectations play a central role in macroeconomics

As a first approach we will assume static expectations
 e1  

r  i 
The Taylor rule as a proxy for monetary policy

History shows that governments care about stabilizing both
output and inflation.

As a proxy for these policy motives, we can use the following
interest rate rule proposed by John Taylor
i  r    h  (   *)  b  ( y  y ),



h  0,
b0
With this rule, y,  and r will be on their long-run equilibrium
values on average.
* is interpreted as the inflation target (can be implicit or explicit).
For the stability of this economy, the parameter must be h
positive so that an increase in inflation triggers an increase in the
real interest rate (the Taylor principle).
Evidence from the euro area seems to confirm the
Taylor rule as a proxy for monetary policy
The 3 month nominal interest rate and an estimated Taylor rate for the euro area, 1999-2003. Per cent
Source: Centre for European Policy Studies, Adjusting to Leaner Times, 5th Annual Report of the CEPS
Macroeconomic Policy Group, Brussels, July 2003
Policy priorities implied by the Taylor rule coefficients
seem to vary across countries
Estimated interest rate reaction functions of four central banks
1. Source: Richard Clarida, Jordi Gali and Mark Gertler, ‘Monetary Policy Rules in Practice – Some
International Evidence’, European Economic Review, 42, 1998, pp. 1033–1067.
2. Source: Centre for European Policy Studies, Adjusting to Leaner Times, 5th Annual Report of the CEPS
Macroeconomic Policy Group, Brussels, July 2003.
i  r    h  (   *)  b  ( y  y ),
h  0,
Aggregate demand curve with endogenous
monetary policy

The log-linearized version of the aggregate demand function Y
and the Taylor rule can be combined to an aggregate demand
curve in (y;) space
i  r    h(   *)  b( y  y ),
h  0,
y  y  1 ( g  g )   2 (r  r )  v,
1  0,
r r
r r
b0
2  0
y  y   ( g  g )   [h(   *)  b( y  y )]  v 
y  y  1 (1g  g )   2 [2h(   *)  b( y  y )]  v 
y  y   ( *  )  z
y  y   ( *  )  z
where

2h
0
1   2b
and
z
v  1 ( g  g )
1   2b
The shape of the aggregate demand curve will
depend on the priorities of monetary policy
Illustration of the aggregate demand curve under alternative monetary policy regimes
(indicated by the choice of coefficients h and b in the Taylor rule)
ADDITIONAL MATERIAL
Term structure of interest rates
The expectations theory of the term structure of
interest rates

Investment decisions depend on the expected cost of capital
over the entire life of the asset (easily +10 years)

To what extend does the short-term policy rate influence longterm interest rates?
(1  itl ) n  (1  it )  (1  ite1 )  (1  ite 2 )  ........  (1  ite n 1 )


If short-term and long-term bonds are perfect substitutes (risk
neutral investors) then the following arbitrage condition will hold
1
itl  (it  ite1  ite 2  ......  ite n 1 )
n
Taking logs and using the approximation ln(1+i)  I
itl  it
iff
ite j  it
for all j  1, 2,..., n  1
In 2001, U.S. long-term interest rates kept a steady
level as the short-term policy rate fell
The Federal funds target rate (U.S. policy rate) and the yield on 10 year U.S. government
bonds, 2001-2002. Per cent
Source: Danmarks Nationalbank