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B&W’s derivation of the Phillips curve
Ch 12.3: The Battle of the mark-ups as a framework for understanding inflation.
Wage and price setting. Slides for 26.
August 2003 lecture
Ragnar Nymoen
University of Oslo, Department of Economics
Using the notation in B&W we have
P = (1 + θ)
W
(Y /L)
price setting
(12.4’)
Y
W/P e = (1 + γ) , wage setting
(12.5’)
L
θ represents firms setting prices as a mark-up on unit labour costs, and γ
represents workers and union strive to set the “expected real wage” as a markup on average productivity.
If we substitute W in (12.4’) from (12.5), we obtain
August 26, 2003
P = (1 + θ)(1 + γ)P e,
(12.6)
the price level depends only on the price expected by wage negotiators–which
hence becomes a nominal anchor in this model.
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From level to dynamics:
P
Pe
− 1 = (1 + θ)(1 + γ)(
− 1) + (1 + θ)(1 + γ) − 1
P−1
P−1
Pe
P
− 1 = (1 + θ)(1 + γ)(
− 1) + θ + γ + θγ
P−1
P−1
Introducing:
π=
for inflation, and
π̄ =
for core inflation we obtain:
P
−1
P−1
Pe
−1
P−1
π = (1 + θ)(1 + γ)π̄ + θ + γ + θγ
= π̄ + θ + γ + θπ̄ + γ π̄ + θγ
Next B&W hypothesize that the mark-up moves pro-cyclically:
θ + γ = a(Y − Ȳ ) = −b(U − Ū )
where Ȳ is the trend component of output (or GDP), whereas Ū is the equilibrium level of the unemployment rate. The right hand side equality is referred
to as Okun’s Law (cf Ch 11).
Thus we can draw this Phillips curve either in a π, Y diagram (AS schedule)
or on a π, U (Phillips curve) diagram, see Fig 12.5.
To either version of the story, B&W add (additive supply) shocks, denoted s,
hence we have
π = π̄ + a(Y − Ȳ ) + s
= π̄ − b(U − Ū) + s
≈ π̄ + θ + γ
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(12.7)
4
(12.11)
What is on B&W’s mind? Are the static equations (12.4’) and (12.5) meant
to be interpreted as behavioural relationships or as long-run steady state relationships? Either interpretation has its drawbacks/inconsistencies.
1. Behavioural equations: But then, for given expectations, prices and wages
are reacting without lags to changes in productivity (and changes in the
mark-up). We know that this is not even approximately true: In the real
world there are substantive lags.
2. Long-run steady state relationships: Inconsistent then to include price expectations. In a steady-state, there is no room for expectation errors.
Before we return to B&W we will take a detour into a more coherent approach
to wage-price dynamics, where we build on the insight that relationships between wage and price levels are best interpreted as hypothetical long-run
relationships. At the same time we will concentrate on wage and price setting
of small open economies.
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2.1
2
The Norwegian main-course model
The Scandinavian model of inflation was formulated in the 1960s, by the Norwegian economist Odd Aukrust. It became (an still is!) the framework for
both medium term forecasting and normative judgements about “sustainable”
centrally negotiated wage growth in Norway.
In its day the Scandinavian model and the Phillips curve were views as alternative models. No doubt that the Phillips curve “won”.
Pity, since Odd Aukrust’s (1977) model can be reconstructed as a consistent
set of propositions about long-run relationships and causal mechanisms. The
reconstructed Norwegian model of inflation serves as a reference point for, and
in many respects also as a corrective to, the modern models of wage formation
and inflation in open economies. The Phillips curve is a special case!
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A model of long-run wage and price setting
Central to the model is the distinction between a tradables sector where frims
are price takers, and a non-tradables sector where firms set prices as mark-ups
on wage costs.
Notation: we,t denotes the nominal wage in the tradeable or exposed (e)
industries. qe and ae are the product price and average labour productivity of
the exposed sector.
ws, qs and as are the corresponding variables for the sheltered (s) sector. p is
the consumer price.
mi(i = e, s) are means of the wage shares in the two industries and mes
denotes the mean of the relative wage. φ is a coefficient that reflects the
weight of non-traded goods in private consumption.
Equation (1) has two implications: First, it defines the exposed sector wage
share we,t − qe,t − ae,t as a constant. Second, since both qe and ae show
trendlike growth: the nominal wage we is also trending (upwards).
Thus, we define the main-course variable:
mc = ae + qe
(5)
All variables are in logs, so e.g., we = log(We).
we − qe − ae = me
(1)
ws − qs − as = ms
(3)
we = mes + ws
p = φqs + (1 − φ)qt,
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(2)
0 < φ < 1.
Aukrust clearly meant equation (1) as a long-run relationship between the
e-sector wage level and the main-course made up of product prices and productivity.
(4)
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The relationship between the “profitability of E industries” and the
“wage level of E industries” that the model postulates, therefore, is a
certainly not a relation that holds on a year-to-year basis”. At best it is
valid as a long-term tendency and even so only with considerable slack.
It is equally obvious, however, that the wage level in the E industries is
not completely free to assume any value irrespective of what happens
to profits in these industries. Indeed, if the actual profits in the E
industries deviate much from normal profits, it must be expected that
sooner or later forces will be set in motion that will close the gap.
(Aukrust, 1977, p 114-115).
For example
The profitability of the E industries is a key factor in determining the
wage level of the E industries: mechanism are assumed to exist which
ensure that the higher the profitability of the E industries, the higher
their wage level; there will be a tendency of wages in the E industries
to adjust so as to leave actual profits within the E industries close to
a “normal” level (for which however, there is no formal definition).
(Aukrust, 1977, p 113).
Aukrust goes on to specify “three corrective mechanisms”, namely wage negotiations, market forces (wage drift, demand pressure) and economic policy.
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There are two other main hypothesis contained in (2) and (3): w-sector wage
leadership, and normal cost pricing in the s-sector.
log wage level
"Upper boundary"
Main course
where the ∗ indicates the three endogenous long-run level variables. ae and as
are exogenous productivity trends. qe is also an exogenous variable, reflecting
that e-sector firms are price takers (due to fierce competition from foreign
firms) and that the exchange rate is independent of domestic prices and wages.
"Lower boundary"
0
To summarize, the three basic long-run propositions of the reconstructed maincourse model are:
H1mc we∗ − qe − ae = me,
H2mc we∗ − ws∗ = mes,
H3mc ws∗ − qs∗ − as = ms
time
Figure 1: The ‘Wage Corridor’ in the Norwegian model of inflation.
A plausible generalization of H1mc is represented by
H1gmc
we∗ = me,0 + mc + γe,1u + γe,2D,
where ut is the log of the rate of unemployment and D is a catch-all for other
factors than ut which can lead to shifts in the mean of the wage share, thus in
H1gmc.
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2.2
Causality
The main-course model specifies the following three hypotheses about causation:
H4mc mc → we∗,
H5mc we∗ → ws∗,
H6mc ws∗ → qs∗ → p∗,
where → denotes one-way causation. In his 1977 paper, Aukrust sees the
causation part of the theory (H4mc-H6mc) as just as important as the long
term relationships (H1mc-H3mc).
From a modern viewpoint this seems to be something of a strait-jacket, in
that the steady state part of the model can be valid even if one-way causality
is untenable. For example H1mc, the main course proposition for the exposed
sector, makes perfect sense also when the nominal exchange rate, together with
wage adjustments are stabilizing the wage share around a long-run mean.
2.3
The Norwegian model and the Battle of the Mark-ups
In a way,...,the basic idea of the Norwegian model is the “purchasing power doctrine” in reverse: whereas the purchasing power doctrine
assumes floating exchange rates and explains exchange rates in terms
of relative price trends at home and abroad, this model assumes controlled exchange rates and international prices to explain trends in
the national price level. If exchange rates are floating, the Norwegian
model does not apply (Aukrust 1977, p. 114).
Chapter 12.3 in the book by B&W contains a general framework for thinking
about inflation.
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Hence there is a conflict between workers and firms: both care about the real
wage, but they have imperfect control: Workers influence the nominal wage,
while the nominal price is determined by firms.
The Norwegian model of inflation fits right into this (modern) framework.
Hence, using H1mc−H3mc above we have that
1. firms typically attempt to mark-up up their prices on unit labour costs,
2. workers and unions on formulate real wage claims which are a mark-up on
productivity.
There are however two differences:
First: don’t have the full “circular process” emphasized by B&W (see p 287),
but if the Norwegian model is extended to incorporate also effects of consumer
prices (which would be an average of qe and qs), in e-sector wage setting, full
circularity would result.
w∗ = me + qe + aq ,
qs∗ = ms + w∗ − as.
The desired wage level is a mark up on prices and productivity, exactly as in
equation (12.5’) in B&W. The s-sector price level is a mark-up on unit-labour
costs (as in B&W’s equation (12.4’).
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Second, B&W present the battle of mark-ups model in a static setting.This of
course runs against our main message, namely that actual wage and prices are
better described by a dynamic system.
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2.4
Dynamic adjustment
Using the same 1-1 transformation as explained before:
If e-sector wages deviate too much from the main-corse, forces will begin to
act on wage setting so that adjustments are made in the direction of the maincourse.
We can use the autoregressive distributed lag model, ADL, to represent this:
we,t = β0 + β11mct + β12mct−1 + β21ut + β22ut−1 + αwe,t−1 + εt.
(6)
There are now two explanatory variables (“x-es”): the main-course variable
mct and ut.
Assume (first) that both mct and ut are exogenous variables. We will then see
that corrective forces are at work even at any constant rate of unemployment.
This is a thought provoking contrast to “natural rate models” which dominates
modern macroeconomic policy debate, and which takes it as a given thing
that unemployment has to adjust in order to bring about wage and inflation
stabilization.
(7)
∆we,t = β0 + β11∆mct + β21∆ut
+ (β11 + β12)mct−1 + (β21 + β22)ut−1 + (α − 1)wet−1 + εt
and
(8)
∆we,t = β0 + β11∆mct + β21∆ut
½
¾
β11 + β12
β21 + β22
− (1 − α) we,t−1 −
mct−1 −
ut−1 + εt
1−α
1−α
Using H1gmc, with γe,2 = 0 since equation (6) omits other shift variables than
unemployment, this can be expressed as
∆we,t = β0 + β11∆mct + β21∆ut
n
o
− (1 − α) we.t−1 − mct−1 − γe,1ut−1 + εt
(9)
β11 + β12 = (1 − α)
(10)
as long as also the following restriction is imposed in (6)
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rate of
unemployment
(10) embodies that the long-run multiplier implied by (6) is unity. The shortrun multiplier is β11, which can be considerably smaller than unity without
violating the main-course hypothesis H1gmc.
The formulation in (9) is often called an equilibrium correction model, ECM
for short. In fact, we can write
t
time
0
wage level
0
∆we,t = β0 + β11∆mct + β21∆ut
a
− (1 − α) {we. − w∗}t−1 + εt
b
where we∗ is given by the left hand side of the extended main course relationship
H1gmc (simplified by setting γe,2 = 0).
t0
time
Figure 2: The main course model: A permanent increase in the rate of unemployment, and possible wage responses.
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Two final remarks
Exercise 2.1 Is β22 > 0 a necessary and/or sufficient condition for path b to
occur?
Exercise 2.2 What might be the economic interpretation of having β21 < 0 ,
but β22 > 0?
1. First, looking back at the consumption function example, it is clear that
the dynamics there also has an equilibrium correction interpretation.
Exercise 2.3 Assume that β21 + β22 = 0. Try to sketch the wage dynamics (in
other words the dynamic multipliers) following a rise in unemployment in this
case!
2. Below we will show that also the Phillips curve has an ECM interpretation.
The main difference is the nature of the corrective mechanism: In Aukrust’s
model there is enough collective rationality in the system to secure dynamic
stability of wages setting at any rate of unemployment (also very low rates).
Wage growth and inflation never gets out of hand or out of control. In
the Phillips curve model on the other hand, unemployment has to adjust
to a special level called the “natural rate” and/or NAIRU for the rate of
inflation to stabilize.
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The Phillips curve
3.1
In the 1970s, the Phillips curve and Aukrust’s model were seen as alternative,
representing “demand” and “supply” model of inflation respectively. However,
as pointed at by Aukrust, the difference between viewing the labour market
as the important source of inflation, and the Phillips curve’s focus on product
market, is more a matter of emphasis than of principle, since both mechanism
may be operating together.
Moreover, modern derivations give the Phillips curve a supply side interpetation!. We now show formally how the two approaches can be combined .
The long-run, causality and the Phillips curve natural rate
Without loss of generality we concentrate on the wage Phillips curve, and recall
that according to Aukrust’s theory it is assumed that
1. we∗ = me + mc, i.e., H1mc above.
2. u∗t = mu, i.e., unemployment has a stable long-run mean mu .
3. the causal structure is “one way” as represented by H4mc and H5mc above.
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To establish the natural rate of unemploym,ent in this model (which we can
call the main-course rate of unemployment), rewrite first (11) as
A Phillips curve ECM system is defined by the following two equations
∆wt = βw0 + βw1∆mct + βw2ut + εw,t,
βw2 ≤ 0,
∆ut = βu0 + αuut−1 + βu1(w − mc)t−1 + εu,t
(11)
(12)
0 < αu < 1,
where we have simplified the notation somewhat by dropping the “e” sector
subscript. On the other hand, since we are considering a dynamic system, we
have added a w in the subscript of the coefficients.
Note that compared to (7) the autoregressive coefficient αw is set to unity in
(11)–this is of course not a simplification but a defining characteristic of the
Phillips curve.
∆wt = βw1∆mct + βw2(ut − ŭ) + εw,t,
(13)
where
βw0
(14)
−βw1
is the rate of unemployment which does not put upward or downward pressure
on wage growth.
ŭ =
Next, consider a steady state situation where ∆mct = gmc. From assumption
1, we that ∆w∗ − gmc = 0 where ∆w∗. i Then (13) defines the defines the
main-course equilibrium rate of unemployment which we denote uphil :
0 = βw2[uphil − ŭ] + (βw1 − 1)gmc.
or
Equation (12) represents the basic idea that low profitability causes unemployment. Hence if the wage share is too high relative to the main-course
unemployment will increase in most situations, i.e., βu1 ≥ 0.
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β −1
gmc),
uphil = (ŭ + w1
−βw2
(15)
uphil represents the unique and stable steady state of the dynamic system.
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The slope of the long-run Phillips curve represented one of the most debated
issues in macroeconomics in the 1970 and 1980s (this is not reflected in B&W
Ch 12, though).
wage growth
Long run Phillips curve
∆ w0
g
mc
u
0
u phil
log rate of
unemployment
In the context of an open economy this discussion appears as somewhat exaggerated, since a long-run trade-off between inflation and unemployment in
any case does not follow from the premise of a downward-sloping long-run
curve. Instead, as shown in figure 3, the steady state level of unemployment is
determined by the rate of imported inflation mmc and exogenous productivity
growth. Neither of these are normally considered as instruments (or intermediate targets) of economic policy.
Figure 3: Open economy Phillips curve dynamics and equilibrium.
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It is important to note that the Phillips curve needs to be supplemented by an
equilibrating mechanism in the form of an equation for ut. Without such an
equation in place, the system is incomplete and we have a missing equation.
The question about the dynamic stability of the natural rate (or NAIRU) cannot
be addressed in the incomplete Phillips curve system.
Conversely: The main-course model can be extended with an equation for ut
without inconsistencies! If ut depends on wages as in (12), we are essentially
adding one more of Aukrusts “corrective mechanisms”, meaning that adjustment to the main-course will be faster than in the case with exogenous ut!
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