In⁄ation, Unemployment and Aggregate Supply · dL i = (1 α)BL α i (2) The demand curve for the...

Ination, Unemployment and Aggregate Supply

Prof. Ester Faia, Ph.D.

Goethe University Frankfurt

June 2010

Prof. Ester Faia (Goethe University Frankfurt)Ination, Unemployment and Aggregate Supply 06/10 1 / 33

Phillips curve

The Phillips curve is a statistical relation between ination andunemployment

π = πe + α (u u) , α > 0

π = actual rate of inationπe = expected rate of inationu = actual rate of unemploymentu = natural rate of unemployment

Phillips curve in the UK, 1861 - 1913

Phillips curve in the UK, 1948 - 1957

Phillips curve in the U.S. in the 1960s

Breakdown of the simple Phillips curve in the U.S.

Price setting

Production function in sector i

Yi = BL1αi , 0 < α < 1 (1)

The marginal product of labor

MPLi =dYidLi

= (1 α)BLαi (2)

The demand curve for the product of sector i

Yi =PiP

σ Yn, σ < 1 (3)

Total revenue is TRi = PiYi , so according to (3) marginal revenue is

Price setting

Marginal revenue in sector i

MRi dTRidYi

= Pi + Yi

dPidYi

MRi = Pi

Marginal cost in sector i

MCi =Wi

(1 α)BLαi

Maximization of prots requires MRi = MCi , implying

Price setting

Mark-up pricing

Pi = mp

MCiz | Wi

(1 α)BLαi

, mp σ

σ 1 > 1 (6)

From (6) we obtain an expression for PiP . Insert this along with (1)into (3) to get

Yiz | BL1α

26664Pi/Pz |

mpWi/P(1 α)BLα

37775σ

Labor demand in sector i

Li =YnB

ε/σ B (1 α)

wεi (7)

P, ε σ

1+ α (α 1)Thus labor demand is a declining function of the real wage wi , and thenumerical wage elasticity of labor demand at sectoral level is given by ε.

Wage setting under perfect information

Workers outside option is simply equal to the real rate ofunemployment benet, 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 (tradeunion objective)

Ω (wi ) = (wi b) [Li (wi )]η (8)

where the labor demand function Li (wi ) is given by (7), and wherethe parameter η measures the unions preference for high employmentrelative to the goal of a high real wage for employed members. If theunion has perfect information about the current price level P, it willchoose the nominal wage rate Wi so as to maximize Ω(wi ) withrespect to wi , implying the rst-order condition:

Wage setting under perfect information

dΩ (wi )dwi

= Lηi + (wi b) ηLη1

DLidwi

1+η (wi b)

=εz | dLidwi

wi = mw b, mw ηε

ηε 1 (9)

Thus the union sets the real wage as a mark-up over the real rate ofunemployment benet. The mark-up is lower the higher the values of ηand ε.

Wage setting under imperfect information

Equation (9) assumes that the union has perfect information on thecurrent price level. In practice, the union must set the nominal wagerate at the start of the current period, based on the price levelexpected to prevail over that period (Pe ), so as to achieve anexpected real wage equal to the target level mw b. Hence we get theoptimal nominal wage rate under imperfect information

Wi = Pemw b (10)

Note that:

the nominal wage rate is pre-set for one period at a time, so in theshort run we have nominal rigidityPe may deviate from P, so there may be expectational errors

The expectations-augmented Phillips curve

From (10) we get the actual real wage

which may be inserted into (7) to give the labor demand in sector i

ε/σ B (1 α)

mpmw bPPe

In a symmetric equilibrium aggregate employment is L = nLi andtotal output is

....Aggregate output in symmetric equilibrium

Y = nYi = nBL1αi

Subtituting this into (11) and using the denition of ε, we get theaggregate employment in symmetric equilibrium

L = nLi = nB (1 α)

mpmw bPPe

Inserting the long run equilibrium condition Pe = P into (12), we ndthe natural level of employment

L = nB (1 α)

mpmw b

Dividing (12) by (13) and denoting the labor force by N, we get

LL=(1 u)N(1 u)N =

Taking logs on both sides of (14), and using approximationln (1+ x) x , we obtain

p = pe + α (u u) , p = lnP, pe = lnPe

from which we derive the expectations-augmented Phillips curve

π = πe + α (u u) , π p p1, π pe p1 (15)

The breakdown of the simple PC in the late 1960s

Up until the 1960s the price level was reasonably stable. In such asituation it is reasonable to assume that Πe = 0.

The simple Phillips curve

π = α (u u) (16)

However, in the late 1960s the ination rate had been positive andrising for several years, so people started to expect a positive inationrate, Πe > 0. The trade-o¤ between unemployment and ination isonly a short-run trade-o¤ which will hold only as long as the expectedrate of ination stays constant.

The link between unemployment and the change ination

The natural rate of unemployment u is the rate of unemploymentprevailing in the long-run equilibrium where expectations are fullled,Πe = Π.Suppose we have static expectations

πe = π1 (17)

From (15) we then get

∆π π π1 = α (u u) (18)

[18] shows that ination will accelerate when unemployment is bellowthe natural rate and decelerate when unemployment is above itsnatural level. The natural rate is sometimes called theNon-Accelerating-Ination-Rate-of-Unemployment (NAIRU).

Recall that aggregate employment is L = nLi . Labour force isnormalized to 1 in each sector, so that the total labour force (N) isequal to n. Thus we have the rate of employment

e LN= Li

Aggregate output is given by

Y = nYi = nBL1αi = nBe1α

Inserting these relationships along with the symmetry conditionWi = W into the labour demand curve (7) and solving for e (usingthe denition of ε), we get the aggregate labor demand:

e =B (1 α)

1/α WP

By rearranging (19), we obtain the real wage implicitly o¤ered byrms, also termed as the price setting curve

MPLz | B (1 α) ea (PS)

In a symmetric equilibrium (Wi = W ) where expectations are correct(Pe = P), equation (10) gives the real wage claimed by workers, alsotermed as the wage setting curve

WP= mw b (WS)

What determines the natural rate of unemployment?

The natural rate of employment is the value of e which makes thereal wage claimed by workers consistent with the real wage implicitlyo¤ered by rms. Equating the right-hand sides of (PS) and (WS) andsolving for e, we thus get the natural rate of employment

e =B (1 α)

mpmw b

It is reasonable to assume that unemployment benets are linked toreal income per capita which is proportional to total factorproductivity in the long run, b = cB. The natural rate ofunemployment is given by

u 1 e = 11 α

mpmw b

The natural rate of unemployment

The natural rate of unemployment is higher

the lower the degree of competition in product markets (a lower valueof σ increases mp and mw )the weaker the union preference for high employment relative to a highreal wage (a lower value of η increases mw )the more generous the level of unemployment benets (the higher thevalue of c)

Supply shocks

In practice, the level of productivity and the wage and price mark-upswill uctuate around their long-run trend levels (which we denote bybar super-scripts). It is plausible to assume that the rate ofunemployment benet is linked to the trend level of productivity. Inthat case we may rewrite (12) as actual employment:

L (1 u)N = nB (1 α)

mpmw cBPe

The long-run equilibrium level of employment is the employment levelprevailing when expectations are fullled and when productivity aswell as the mark-ups are at their trend levels. Hence we have thenatural employment:

L (1 u)N = n1 α

mpmw c

The expectations-augmented Phillips curve with supplyshocks

Dividing (21) and (22) we get:

1 u1 u =

BmpmwPPe

Taking logs in (23) and using the approximation ln (1+ x) x plusthe denitions of Πe and Π, we end up with:

π = πe + α (u u) + s,

s lnmp

Testing the Phillips curve theory

With static expectations, Πe = Π1, we may write (24) as:

∆π = α βu + s, E [s ] = 0 (25)

A regression analysis based on the U.S. data for 1962 - 1995 yieldsthe expectations-augmented Phillips curve in the USA:

∆π = 4.467s .e .=1.081

0.723s .e .=0.172

u, R2 = 0.355 (26)

Estimate of the natural rate of unemployment in the USA

u = α/β = 4.467/0.723 = 6.2%

Relationship between unemployment and in ination in theU.S. 1962 - 1995

The shifting short-run Phillips curve in the U.S.

Actual and predicted ination in the U.S.

The Aggregate Supply Curve

SinceY = nYi ,

L (1 u)N = nLiit follows from the production function (1) that:

Y = nBLn

= nαB [(1 u)N ]1α (27)

Taking logs on both sides of (23) and using ln (1 u) u, we get:

y lnY = ln nα + lnB + (1 α) ln [(1 u)N ]

ln nα + lnB + (1 α) lnN (1 α) u ,

u = lnN +ln nα + lnB y

1 α(28)

The aggregate supply curve

In parallel to (27), we may specify natural output as:

Y = nαB [(1 u)N ]1α

and take logs to get:

u = lnN +ln nα + ln B y

1 α(29)

Substituting (28) and (29) into the expectations-augmented Phillipscurve (15), we get the short-run aggregate supply (SRAS) curve

π = πe + γ (y y) + s (30)

1 α, s ln

ln (B/B)

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 behavior of rms

The SRAS curve shifts upwards in case of a rise in the expectedination rate or in case of an unfavorable supply shock (highermark-ups or a negative productivity shock)

The Long-Run Aggregate Supply (LRAS) curve is obtained whenexpectations are fullled (Πe = Π) and mark-ups and productivityare at their trend levels. The LRAS curve is vertical in (y ,π)-space,that is, in the long run there is no trade-o¤ between ination andoutput (employment)

Note that in a model with intersectoral labour mobility a rise in activityleads to increased wage pressure which contributes to the positive slope ofthe SRAS curve.

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

In⁄ation, Unemployment and Aggregate Supply · dL i = (1 α)BL α i (2) The demand curve for the...

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