LECTURE 6 Aggregate demand and its components

Øystein Børsum

21rst February 2006

Overview of forthcoming lectures

Lecture 6: Aggregate demand and its components Determinants of aggregate investments and consumption,

important and volatile components of aggregate demand Aggregate demand put together: The AD curve

Lecture 7: Aggregate demand and aggregate supply Macroeconomic dynamics in the AS-AD model

Lecture 8: Stabilization policies 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

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

1( ) e et t t tr V D V V 1

1

e et t

t

D VV

r

The fundamental value of a firm

Successive substitution gives:

Assume that the future value of the firm Vet+1 cannot rise faster

than r + (else it would be of infinite value), i.e.:

1 22 2

1 2 32 3 3

1 22 3

1 (1 ) (1 )

1 (1 ) (1 ) (1 )

....1 (1 ) (1 ) (1 )

e e et t t

t

e e e et t t t

e e e et t t t n

n

D D VV

r r r

D D D V

r r r r

D D D V

r r r r

lim 0(1 )

et n

nn

V

r

The fundamental value of a firm

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 expected future dividends o the real interest rate o 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

10 (1 )

et n

t nn

DV

r

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:

where we have used: and

Cash flow constraint: e = expected profitc = installation costs

Assume the following installation cost function:

1et tq q 1t t tK K I

1 1 1e e

t t t t t tV q K q K I

( )e et t t tD I c I

2( )2t t

ac I I

Optimal level of investment depends on q

Maximization of Vt taking qt as given gives the following first-order conditions:

1

2

21

et e

t

DV

et t t t t t

t

aI I q K I

Vr

foregone dividendexpected

capital gain

0 1

1

1

tt

t t

t t

tt

V dcq

I dI

q aI

qI

a

foregone dividendexpected

capital gain

0 1

1

1

tt

t t

t t

tt

V dcq

I dI

q aI

qI

a

foregone dividendexpectedcapital gain

0 1

1

1

tt

t t

t t

tt

V dcq

I dI

q aI

qI

a

An example of the investment function

Assume that in order to simplify the value of the firm to

Assume furthermore that and

= expected dividend pay-out ratio = constant profit share

Using the definition of q this gives the investment function

e et i tD D

2 3

1 1 1...

1 (1 ) (1 )

ee t

t t

DV D

r r r r

et tD t tY

/11t t

t

Y KI

a r

The general investment function

Abstracting from the functional form the general investment function is:

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)

( , , )

0, 0, 0Y r

I I Y r

I I II I I

Y r

( ) ( ) ( ) ( )( , , , )I f Y K r E

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

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

21

( )( ) , ' 0, '' 0, 0

1

u CU u C u u

Intertemporal budget constraint

Period 1 budget constraint

Period 2 budget constraint

The consumer’s intertemporal budget constraint

V = financial wealthr = real rate of interestYL = labour incomeT = net tax payment (taxes minus transfers)C = consumption

2 1 1 1 11 LV r V Y T C

2 2 2 2LC V Y T

2 2 21 1 1 11 1

LLC Y T

C V Y Tr r

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

This simplifies the notation of the intertemporal budget constraint

2 21 1 1 1

LL Y T

H Y Tr

21 1 11

CC V H

r

Optimal intertemporal consumption

Utility over the consumer’s life-time becomes (as a function of C1)

Maximization of U with respect to C1 gives the following first-order conditions:

The Keynes-Ramsey rule:

1 1 11

(1 )( )( )

1

u r V H CU u C

2

1 1 1 11

1 2

10 '( ) ' (1 )( )

1

1 '( ) '( )

1

C

dU ru C u r V H C

dC

ru C u C

1

2

'( )1

'( ) /(1 )

u Cr

u C

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

u’(Ct) = Ct-1/

Insert this into the Keynes-Ramsey rule

1 1/1( ) for 0, 1

1 1/

( ) ln for =1

t t

t t

u C C

u C C

1/2 1

1/ 1/2 1

2 1

(1 )( / ) 1

1 C

1

1 C

1

C C r

rC

rC

1/2 1

1/ 1/2 1

2 1

(1 )( / ) 1

1 C

1

1 C

1

C C r

rC

rC

1/2 1

1/ 1/2 1

2 1

(1 )( / ) 1

1 C

1

1 C

1

C C r

rC

rC

Example of the consumption function with CES utility

Insert the expression for the optimal C2 in terms of C1 into the intertemporal budget constraint.

Current consumption C1 is proportional to total current wealth (not current income).

The propensity to consume wealth is positive, but less than one.

11 1 1 1

1 1 1

1

(1 ) (1 )

( ),

10 1

1 (1 ) (1 )

C r C V H

C V H

r

The general consumption function

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)

1 1 1(?)( )( ) ( )

, , ,dC C Y g r V

( , , )

0 1, 0, 0( )

Y T r

C C Y T r

C C CC C C

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

Equlibrium condition for the goods market (closed economy)

Properties of the aggregate private demand function

( , , )

0, 0, 0Y r

I I Y r

I I II I I

Y r

( , , )

0 1, 0, 0( )

Y T r

C C Y T r

C C CC C C

Y T r

( , , , )Y D Y G r G

0 1, 0,( )Y Y Y G Y

D D CD C I D C

Y G Y T

0, 0r r r

D DD C I D C I

r

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

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

1 2 1 2( ) ( ) , 0, 0y y g g r r v

1 2

1ln , ln , ln , ln ,

1

(1 ) , , ln ln

Y

rY

y Y y Y g G g G mD

DDGm C m v m

Y Y Y

( , , , )Y D Y G r G

The real and the nominal interest rate

The definition of the expected real interest rate

As long as i and are close to zero, we can approximate the

real interest rate

Expectations play a central role in macroeconomics

As a first approach we will assume static expectations

11

11

1e

e

ir r i

1e r i

11

11

1e

e

ir 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

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).

( *) ( ), 0, 0i r h b y y h b

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

( *) ( ), 0, 0i r h b y y h b

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.

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

1 2 1 2( ) ( ) , 0, 0y y g g r r v

( *) ( ), 0, 0i r h b y y h b

1 2( ) [ ( *) ( )]

( * )

r r

y y g g h b y y v

y y z

2 1

2 2

( )0

1 1

h v g gz

b b

1 2( ) [ ( *) ( )]

( * )

r r

y y g g h b y y v

y y z

where and

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 long-term interest rates?

If short-term and long-term bonds are perfect substitutes (risk neutral investors) then the following arbitrage condition will hold

Taking logs and using the approximation ln(1+i) I

Alternative: Static interest rate expectations

1 2 1(1 ) (1 ) (1 ) (1 ) ........ (1 )l n e e et t t t t ni i i i i

1 2 1

1( ...... )l e e e

t t t t t ni i i i in

iff for all 1, 2,..., 1l et t t j ti i i i j n

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