CHAPTER 18INFLATION, UNEMPLOYMENT AND AGGREGATE SUPPLY

Themes of the chapter

• Nominal rigidities, expectational errors and employment fluctuations

• The short-run trade-off between inflation and unemployment

• The relationship between the theory of inflation and the theory of structuralunemployment

• Supply shocks and the ’New Economy’

• The aggregate supply curve

WHERE WE ARE HEADING

The theory of inflation and unemployment developed in this chapter may besummarized in

The expectations-augmented Phillips curve

π = actual rate of inflation

πe = expected rate of inflation

u = actual rate of unemployment

( ) 0e u u ,π = π +α − α >

natural rate of unemploymentu =

(1)

In the following we will derive the expectations-augmented Phillips curve from atheory of wage and price setting

PRICE SETTING

Production function in sector i

1 0 1i iY BL ,−α= < α < (3)

The marginal product of labour

( )1i i i iMPL dY / dL BL−α≡ = −α (4)

The demand curve for the product of sector i

1ii

P YY ,P n

− σ⎛ ⎞= σ >⎜ ⎟⎝ ⎠

(5)

Total revenue is , so according to (5) marginal revenue isi i iTR PY≡

PRICE SETTING

Marginal revenue in sector i

1 i i i ii i i i

i i i i

dTR dP dP YMR P Y PdY dY dY P

⎛ ⎞ ⎛ ⎞≡ = + = + ⇒⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

11i iM R P ⎛ ⎞= −⎜ ⎟σ⎝ ⎠(6)

Marginal cost in sector i

( )1i i

ii i

W WMCMPL BL−α= =

− α

Maximization of profits requires MRi = MCi , implying

PRICE SETTING

Mark-up pricing

From (7) we obtain an expression for Pi /P. Insert this along with (3) into (5) to get

( ), 1

1 1

iMC

p pii

i

WP m mBL−

⎛ ⎞= ≡ >⎜ ⎟⎜ ⎟− −⎝ ⎠

i α

σα σ

(7)

( )1

1

i

i

P / PY

pi

ii

m W / P YB LB L n

− σ

− α− α

⎡ ⎤⎢ ⎥⎛ ⎞⎢ ⎥= ⇔⎜ ⎟− α⎢ ⎥⎝ ⎠⎢ ⎥⎣ ⎦

LABOUR DEMAND IN SECTOR i

Thus labour demand is a declining function of the real wage wi , and thenumerical wage elasticity of labour demand at the sectoral level is givenby ε

( )

( )

1

1 1

/

i ip

ii

BYL w ,n B m

Ww ,P

εε σ− ε⎛ ⎞− α⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠

σ≡ ε ≡

+ α σ −

(8)

WAGE SETTING UNDER PERFECT INFORMATION

Workers in sector i are educated and trained to work in that particularsector, so they cannot move to another sector to look for a job. Hence theiroutside option is simply equal to the real rate of unemployment benefit, b.All workers in sector i are organized in a monopoly trade union whichseeks to maximize

The total rent accruing to workers in sector i (trade union objective)

( ) ( ) ( )i i i iw w b L wη

⎡ ⎤Ω = − ⎣ ⎦ (9)

where the labour demand function Li(wi) is given by (8), and where the parameterη measures the union’s preference for high employment relative to the goal of ahigh real wage for employed members. If the union has perfect information aboutthe current price level P, it will choose the nominal wage rate Wi so as to maximizeΩ(wi) with respect to wi , implying the first-order condition:

WAGE SETTING UNDER PERFECT INFORMATION

Thus the union sets the real wage as a mark-up over the real rate of unemploymentbenefit. The mark-up is lower the higher the values of η and ε

( ) ( )

( )

1

0

1 0

1

i ii i i

i i

i i i

i i i

w wi

d w dLL w b Ldw dw

w b dL ww dw L

w m b, m

η η−

= −ε

Ω ⎛ ⎞= + − η = ⇔⎜ ⎟

⎝ ⎠

η − ⎛ ⎞+ = ⇔⎜ ⎟

⎝ ⎠

ηε= ≡

ηε −(10)

WAGE SETTING UNDER IMPERFECT INFORMATION

Equation (10) assumes that the union has perfect information on the currentprice level. In practice, the union must set the nominal wage rate at the startof the current period, based on the price level expected to prevail over thatperiod (Pe ), so as to achieve an expected real wage equal to the target level mwb.Hence we get

The optimal nominal wage rate under imperfect information

e wiW P m b= ⋅ (11)

Note that

the nominal wage rate is pre-set for one period at a time, so in the short runwe have nominal rigidity

Pe may deviate from P, so there may be expectational errors

THE EXPECTATIONS-AUGMENTED PHILLIPS CURVE

From (11) we get

The actual real wage

ewiW P m b

P P⎛ ⎞

= ⎜ ⎟⎝ ⎠

which may be inserted into (8) to give

Labour demand in sector i

( )1/

i p w e

BY PLnB m m b P

εε σ ⎛ ⎞− α⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

(12)

In a symmetric equilibrium aggregate employment is L=nLi and total output is

THE EXPECTATIONS-AUGMENTED PHILLIPS CURVE

Aggregate output in symmetric equilibrium

1i iY nY nBL −α= =

Substituting this into (12) and using the definition of ε, we get

Aggregate employment in symmetric equilibrium

( ) 11

/

i p w e

B PL nL nm m b P

α⎛ ⎞− α

= = ⋅ ⋅⎜ ⎟⎝ ⎠

(13)

Inserting the long run equilibrium condition Pe=P into (13), we find

The natural level of employment

( ) 11

/

p w

BL n

m m b

α⎛ ⎞− α

= ⎜ ⎟⎝ ⎠

(15)

THE EXPECTATIONS-AUGMENTED PHILLIPS CURVE

Dividing (13) by (15) and denoting the labour force by N, we get

( )( )

111

/

e

u NL PL u N P

α− ⎛ ⎞= = ⎜ ⎟− ⎝ ⎠

Taking logs on both sides of (16), and using the approximation ln(1+x) ≈ x,we obtain

( ) ln lne e ep p u u , p P, p P= +α − ≡ ≡

from which we derive

The expectations-augmented Phillips curve

( ) 1 1 e eu u , p p , p p− −π=π +α − π≡ − π≡ − (17)

(16)

EXPLAINING THE BREAKDOWN OF THE SIMPLEPHILLIPS CURVE IN THE LATE 1960s

Up until the 1960s the price level was reasonably stable in peacetime. In such asituation it is reasonable to assume that πe = 0. From (17) we then get

The simple Phillips curve

( )u uπ α= − (18)

However, in the late 1960s the inflation rate had been positive and rising forseveral years, so people started to expect a positive inflation rate, πe > 0.In accordance with (17), this led to a gradual rise in the rate of inflationassociated with any given rate of unemployment. Thus the apparent trade-offbetween unemployment and inflation discovered by Phillips is only ashort-run trade-off which will hold only as long as the expected rate ofinflation stays constant. When the expected inflation rate increases, theshort-run Phillips curve will shift upwards, and vice versa.

u

π

The expectations-augmented Phillips curve

THE LINK BETWEEN UNEMPLOYMENTAND THE CHANGE IN INFLATION

The natural rate of unemployment u is the rate of unemploymentprevailing in a long-run equilibrium where expectations are fulfilled, πe = π.Suppose we have

Static expectations

1eπ π −= (19)

From (17) we then get

( )1 u uπ π π α−∆ ≡ − = − (20)

which shows that inflation will accelerate when unemployment is below thenatural rate and decelerate when unemployment is above its natural level. Forthis reason the natural rate is sometimes called the Non-Accelerating-Inflation-Rate-of-Unemployment (NAIRU).

WHAT DETERMINES THENATURAL RATE OF UNEMPLOYMENT?

Recall that aggregate employment is L = nLi . We may choose units such thatthe labour force in each sector is 1 so that the total labour force (N) is equal to n.Thus we have

The rate of employment

iLe LN

≡ =

Aggregate output

1 1i iY nY nBL nBeα α− −= = =

Inserting these relationships along with the symmetry condition Wi=W into thelabour demand curve (8) and solving for e (using the definition of ε), we get

Aggregate labour demand

( ) 1/ 1/1p

B Wem P

α αα −−⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

(14)

WHAT DETERMINES THENATURAL RATE OF UNEMPLOYMENT?

By rearranging (14), we obtain the real wage implicitly offered by firms,also termed

The price setting curve

( )1 1M PL

p

W B eP m

αα −⎛ ⎞= −⎜ ⎟⎝ ⎠

In a symmetric equilibrium (Wi=W) where expectations are correct (Pe=P),equation (11) gives the real wage claimed by workers, also termed

The wage setting curve

wW m bP

= ⋅

(PS)

(WS)

WHAT DETERMINES THENATURAL RATE OF UNEMPLOYMENT?

The natural rate of employment is the value of e which makes the real wageclaimed by workers consistent with the real wage implicitly offered by firms.Equating the right-hand sides of (PS) and (WS) and solving for e, we thus get

The natural rate of employment

( ) 1/1p w

Be

m m b

αα−⎛ ⎞

= ⎜ ⎟⎝ ⎠

It is reasonable to assume that unemployment benefits are linked to real incomeper capita which is proportional to total factor productivity in the long run. Hencewe assume that b=cB. Inserting in the expression above, we then obtain

The natural rate of unemployment

1/11 1 p wu em m c

αα−⎛ ⎞≡ − = − ⎜ ⎟⎝ ⎠

(22)

WHAT DETERMINES THENATURAL RATE OF UNEMPLOYMENT?

Implications of (22):

The natural rate of unemployment is higher

the lower the degree of competition in product markets (a lower value ofσ increases mp and mw )

the weaker the union preference for high employment relative to a high realwage (a lower value of η increases mw )

the more generous the level of unemployment benefits (the higher the valueof c )

Note that the natural unemployment rate is independent of the level of productivity(B). This is consistent with empirical evidence.

ALTERNATIVE MICRO FOUNDATIONS FORTHE EXPECTATIONS-AUGMENTED PHILLIPS CURVE

the trade union model without intersectoral labour mobility (see above) orwith labour mobility (see exercise 18.1)

the efficiency wage model (see Chapter 23, section 4)

the worker-misperception model of a competitive labour market (see Chapter18, section 3)

The worker-misperception model does not include nominal rigidities, so toobtain an expectations-augmented Phillips curve it is only necessary to assumeexpectational errors. However, as shown in section 18.3, the existence ofnominal rigidities will amplify the employment fluctuations generated bynominal rigidities.

Our model abstracts from nominal price rigidities, but in practice such rigiditiesmay help to explain the sluggish adjustment of inflation and unemployment totheir long run equilibrium levels.

SUPPLY SHOCKS

In practice, the level of productivity and the wage and price mark-ups willfluctuate around their long-run trend levels (which we denote by bar super-scripts). It is plausible to assume that the rate of unemployment benefit islinked to the trend level of productivity. In that case we may rewrite (13) as

Actual employment

( ) ( ) 11

1/e

p w

B PL u N nm m cB P

α⎛ ⎞−α

≡ − = ⋅ ⋅⎜ ⎟⎝ ⎠

(31)

The long-run equilibrium level of employment is the employment levelprevailing when expectations are fulfilled and when productivity as well asthe mark-ups are at their trend levels. Hence we have

Natural employment

( )111

/

p wL u N nm m c

α− α⎛ ⎞≡ − = ⋅ ⎜ ⎟⎝ ⎠

(32)

THE EXPECTATIONS-AUGMENTED PHILLIPS CURVEWITH SUPPLY SHOCKS

Dividing (31) by (32), we get1

11

/p w

p w e

u Bm m Pu Bm m P

α⎛ ⎞−

= ⋅⎜ ⎟− ⎝ ⎠(33)

Taking logs in (33) and using the approximation ln(1+x) ≈ x plus the definitionsof πe and π, we end up with

( )

ln ln ln

e

p w

p w

u u s ,

m m Bsm m B

π = π + α − +

⎛ ⎞ ⎛ ⎞ ⎛ ⎞≡ + − ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠

(34)

TESTING THE PHILLIPS CURVE THEORY

With static expectations, πe = π-1 , we may write (34) as

[ ], 0u s E s∆ = − + =π α β (35)

A regression analysis based on U.S. data for 1962-1995 yields

The expectations-augmented Phillips curve in the USA

2

s.e.=1.081 s.e.=0.1724.467 0.723 , 0.355u R∆ = − ⋅ =π

(26)

Estimate of the natural unemployment rate in the USA

u = α /β = 4.467/0.723 = 6.2%

PRODUCTIVITY GROWTH, THE PHILLIPS CURVEAND THE ’NEW ECONOMY’

The combination of low unemployment and inflation in the U.S. since 1995may be explained by a positive productivity shock, reflected in an acceleratingrate of productivity growth. When productivity growth accelerates, we have

ln ln 0 0B B s− > ⇒ <Recall that the target real wage of workers is

* ww m cB=so in a period of accelerating productivity growth, the target real wage lagsbehind the growth in actual productivity growth, thereby reducing inflationarypressure.

THE AGGREGATE SUPPLY CURVE

Since

( ) , 1i iY nY L u N nL= ≡ − =

it follows from the production function (3) that

( )1

11LY nB n B u N

n

−α−αα⎛ ⎞ ⎡ ⎤= = −⎜ ⎟ ⎣ ⎦⎝ ⎠

(38)

Taking logs on both sides of (38) and using ln(1-u) ≈ -u, we get

( ) ( )( ) ( )

ln ln ln 1 ln 1

ln ln 1 ln 1

y Y n B u N

n B N u

α

α

⎡ ⎤≡ = + + −α −⎣ ⎦≈ + + −α − −α ⇔

ln lnln1

n B yu Nα + −

= +− α

(39)

THE AGGREGATE SUPPLY CURVE

In parallel to (38), we may specify natural output as

( ) 11Y n B u N

−αα ⎡ ⎤= −⎣ ⎦and take logs to get

ln lnln1

n B yu Nα + −

= +− α

(41)

Substituting (39) and (41) into the expectations-augmented Phillips curve (34),we get

The Short-Run Aggregate Supply (SRAS) curve

( )( )

ln ln ln

1 1

e

p w

p w

y y s,

B/ Bm m, sm m

π=π +γ − +

⎛ ⎞ ⎛ ⎞αγ≡ ≡ + −⎜ ⎟ ⎜ ⎟−α −α⎝ ⎠ ⎝ ⎠

(42)

PROPERTIES OF THE AGGREGATE SUPPLY CURVE

the SRAS curve slopes upwards, because higher output → higheremployment → lower MPL → higher MC → higher prices via themark-up pricing behaviour of firms

the SRAS curve shifts upwards in case of a rise in the expected inflationrate or in case of an unfavourable supply shock (higher mark-ups or a negative productivity shock)

the Long-Run Aggregate Supply (LRAS) curve is obtained when expectationsare fulfilled (πe=π) and mark-ups and productivity are at their trend levels. TheLRAS curve is vertical in (y,π)-space, that is, in the long run there is no trade-offbetween inflation and output (employment)

Note that in a model with intersectoral labour mobility a rise in activity leadsto increased wage pressure which contributes to the positive slope of theSRAS curve.

y

π

Aggregate supply in the short run (SRAS) and in the long run (LRAS)