In ation, Unemployment and Aggregate Supply

In ation, Unemployment and Aggregate Supply Prof. Ester Faia, Ph.D. Goethe University Frankfurt June 2010 rof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 1 / 33

Phillips curve The Phillips curve is a statistical relation between in ation and unemployment π = π e + α (ū u), α > 0 π = actual rate of in ation π e = expected rate of in ation u = actual rate of unemployment ū = natural rate of unemployment Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 2 / 33

Phillips curve in the UK, 1861-1913 Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 3 / 33

Phillips curve in the UK, 1948-1957 Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 4 / 33

Phillips curve in the U.S. in the 1960s Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 5 / 33

Breakdown of the simple Phillips curve in the U.S. Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 6 / 33

Price setting Production function in sector i The marginal product of labor Y i = BL 1 i α, 0 < α < 1 (1) MPL i = dy i dl i = (1 α) BL α i (2) The demand curve for the product of sector i Y i = σ Pi Y P n, σ < 1 (3) Total revenue is TR i = P i Y i, so according to (3) marginal revenue is Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 7 / 33

Price setting Marginal revenue in sector i MR i dtr i dy i Marginal cost in sector i = P i + Y i MC i = dpi dy i MR i = P i 1 W i MPL i = = P i 1 + dp i dy i Y i 1 σ W i (1 α) BL α i Maximization of pro ts requires MR i = MC i, implying P i =) (4) (5) Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 8 / 33

Price setting Mark-up pricing MC i z } { P i = m p W i (1 α) BLi α, m p σ σ 1 > 1 (6) From (6) we obtain an expression for P i P. Insert this along with (1) into (3) to get Y i z } { BL 1 α i = 2 3 P i /P z } { m p W i /P 6 4 (1 α) BLi α 7 5 σ Y n =) Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 9 / 33

Labor demand in sector i L i = Y nb ε/σ B (1 α) m p w i W i P, ε σ 1 + α (α 1) ε w ε i (7) Thus labor demand is a declining function of the real wage w i, and the numerical wage elasticity of labor demand at sectoral level is given by ε. Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 10 / 33

Wage setting under perfect information Workers outside option is simply equal to the real rate of unemployment bene t, b. All workers in sector i are organized in a monopoly trade union which seeks to maximize the total rent accruing to workers in sector i (trade union objective) Ω (w i ) = (w i b) [L i (w i )] η (8) where the labor demand function L i (w i ) is given by (7), and where the parameter η measures the union s preference for high employment relative to the goal of a high real wage for employed members. If the union has perfect information about the current price level P, it will choose the nominal wage rate W i so as to maximize Ω(w i ) with respect to w i, implying the rst-order condition: Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 11 / 33

Wage setting under perfect information dω (w i ) = L η i + (w i b) ηl η 1 DLi i = 0, dw i dw i = ε z 1 + η (w } { i b) dli w i = 0, w i dw i L i w i = m w b, m w ηε ηε 1 Thus the union sets the real wage as a mark-up over the real rate of unemployment bene t. The mark-up is lower the higher the values of η and ε. (9) Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 12 / 33

Wage setting under imperfect information Equation (9) assumes that the union has perfect information on the current price level. In practice, the union must set the nominal wage rate at the start of the current period, based on the price level expected to prevail over that period (P e ), so as to achieve an expected real wage equal to the target level m w b. Hence we get the optimal nominal wage rate under imperfect information Note that: W i = P e m w b (10) the nominal wage rate is pre-set for one period at a time, so in the short run we have nominal rigidity P e may deviate from P, so there may be expectational errors Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 13 / 33

The expectations-augmented Phillips curve From (10) we get the actual real wage W i P P = e m w b P which may be inserted into (7) to give the labor demand in sector i L i Y nb ε/σ B (1 α) m p m w b P ε P e (11) In a symmetric equilibrium aggregate employment is L = nl i and total output is Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 14 / 33

The expectations-augmented Phillips curve...aggregate output in symmetric equilibrium Y = ny i = nbl 1 i α Subtituting this into (11) and using the de nition of ε, we get the aggregate employment in symmetric equilibrium B (1 α) P 1/α L = nl i = n m p m w b P e (12) Inserting the long run equilibrium condition P e = P into (12), we nd the natural level of employment L = n B (1 α) 1/α m p m w (13) b Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 15 / 33

The expectations-augmented Phillips curve Dividing (12) by (13) and denoting the labor force by N, we get L L = (1 u) N (1 ū) N = P 1/α P e (14) Taking logs on both sides of (14), and using approximation ln (1 + x) x, we obtain p = p e + α (ū u), p = ln P, p e = ln P e from which we derive the expectations-augmented Phillips curve π = π e + α (ū u), π p p 1, π p e p 1 (15) Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 16 / 33

The breakdown of the simple PC in the late 1960s Up until the 1960s the price level was reasonably stable. In such a situation it is reasonable to assume that Π e = 0. The simple Phillips curve π = α (ū u) (16) However, in the late 1960s the in ation rate had been positive and rising for several years, so people started to expect a positive in ation rate, Π e > 0. The trade-o between unemployment and in ation is only a short-run trade-o which will hold only as long as the expected rate of in ation stays constant. Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 17 / 33

The expectations-augmented Phillips curve Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 18 / 33

The link between unemployment and the change in ation The natural rate of unemployment ū is the rate of unemployment prevailing in the long-run equilibrium where expectations are ful lled, Π e = Π. Suppose we have static expectations π e = π 1 (17) From (15) we then get π π π 1 = α (ū u) (18) [18] shows that in ation will accelerate when unemployment is bellow the natural rate and decelerate when unemployment is above its natural level. The natural rate is sometimes called the Non-Accelerating-In ation-rate-of-unemployment (NAIRU). Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 19 / 33

NAIRU Recall that aggregate employment is L = nl i. Labour force is normalized to 1 in each sector, so that the total labour force (N) is equal to n. Thus we have the rate of employment e L N = L i Aggregate output is given by Y = ny i = nbl 1 i α = nbe 1 α Inserting these relationships along with the symmetry condition W i = W into the labour demand curve (7) and solving for e (using the de nition of ε), we get the aggregate labor demand: B (1 α) 1/α W 1/α e = m p (19) P Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 20 / 33

NAIRU By rearranging (19), we obtain the real wage implicitly o ered by rms, also termed as the price setting curve W 1 P = m p MPL z } { B (1 α) e a (PS) In a symmetric equilibrium (W i = W ) where expectations are correct (P e = P), equation (10) gives the real wage claimed by workers, also termed as the wage setting curve W P = mw b (WS) Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 21 / 33

What determines the natural rate of unemployment? The natural rate of employment is the value of e which makes the real wage claimed by workers consistent with the real wage implicitly o ered by rms. Equating the right-hand sides of (PS) and (WS) and solving for e, we thus get the natural rate of employment ē = B (1 α) 1/α m p m w b It is reasonable to assume that unemployment bene ts are linked to real income per capita which is proportional to total factor productivity in the long run, b = cb. The natural rate of unemployment is given by ū 1 ē = 1 1 α 1/α m p m w (20) b Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 22 / 33

The natural rate of unemployment The natural rate of unemployment is higher the lower the degree of competition in product markets (a lower value of σ increases m p and m w ) the weaker the union preference for high employment relative to a high real wage (a lower value of η increases m w ) the more generous the level of unemployment bene ts (the higher the value of c) Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 23 / 33

Supply shocks In practice, the level of productivity and the wage and price mark-ups will uctuate around their long-run trend levels (which we denote by bar super-scripts). It is plausible to assume that the rate of unemployment bene t is linked to the trend level of productivity. In that case we may rewrite (12) as actual employment: L (1 B (1 α) P u) N = e 1/α n m p m w (21) c B P The long-run equilibrium level of employment is the employment level prevailing when expectations are ful lled and when productivity as well as the mark-ups are at their trend levels. Hence we have the natural employment: L (1 1 α 1/α ū) N = n m p m w (22) c Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 24 / 33

The expectations-augmented Phillips curve with supply shocks Dividing (21) and (22) we get: 1 u B 1 ū = mp m w B m p m w P 1/α P e (23) Taking logs in (23) and using the approximation ln (1 + x) x plus the de nitions of Π e and Π, we end up with: π = π e + α (ū u) + s, m p m w s ln m p + ln m w ln B B (24) Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 25 / 33

Testing the Phillips curve theory With static expectations, Π e = Π 1, 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 yields the expectations-augmented Phillips curve in the USA: π = 4.467 s.e.=1.081 0.723 u, s.e.=0.172 R2 = 0.355 (26) Estimate of the natural rate of unemployment in the USA u = α/β = 4.467/0.723 = 6.2% Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 26 / 33

Relationship between unemployment and in in ation in the U.S. 1962-1995 Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 27 / 33

The shifting short-run Phillips curve in the U.S. Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 28 / 33

Actual and predicted in ation in the U.S. Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 29 / 33

The Aggregate Supply Curve Since L (1 Y = ny i, u) N = nl i it follows from the production function (1) that: Y = nb L 1 α = n α B [(1 u) N] 1 α (27) n Taking logs on both sides of (23) and using ln (1 u) u, we get: y ln Y = ln n α + ln B + (1 α) ln [(1 u) N] ln n α + ln B + (1 α) ln N (1 α) u, u = ln N + ln nα + ln B y 1 α (28) Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 30 / 33

The aggregate supply curve In parallel to (27), we may specify natural output as: and take logs to get: Ȳ = n α B [(1 ū) N] 1 α ū = ln N + ln nα + ln B 1 α ȳ (29) Substituting (28) and (29) into the expectations-augmented Phillips curve (15), we get the short-run aggregate supply (SRAS) curve γ α 1 α, s ln π = π e + γ (y ȳ) + s (30) m p m p m w + ln m w ln (B/ B) 1 α Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 31 / 33

Properties of the aggregate supply curve The SRAS curve slopes upwards, because higher output! higher employment! lower MPL! higher MC! higher prices via the mark-up pricing behavior of rms The SRAS curve shifts upwards in case of a rise in the expected in ation rate or in case of an unfavorable supply shock (higher mark-ups or a negative productivity shock) The Long-Run Aggregate Supply (LRAS) curve is obtained when expectations are ful lled (Π e = Π) and mark-ups and productivity are 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 in ation and output (employment) Note that in a model with intersectoral labour mobility a rise in activity leads to increased wage pressure which contributes to the positive slope of the SRAS curve. Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 32 / 33

Aggregate supply in the short run (SRAS) and in the long run (LRAS) Prof. Ester Faia (Goethe University Frankfurt)In ation, Unemployment and Aggregate Supply 06/10 33 / 33