Mr. Phillips and His Curve

Economics 813a 12/11/2018

 R. H. Rasche 1997

Mr. Phillips and his Curve

The 1958 study of the relationship between the rate of change of the nominal wage rate and the unemployment rate in the UK by Phillips established the framework for discussions of inflation for almost a decade. The initial study is fundamentally an exercise in curve fitting without any economic theory.

Phillips took annual data for the UK for the period 1861 - 1957, but it is important to note that he did not perform his empirical analysis on these data points. Rather he grouped the data into six classes depending upon the level of the unemployment rate and constructed average rates of change of the nominal wage rate and the unemployment rate for each of the six groups. The groups were defined by the following ranges of the unemployment rate: 0-2, 2-3, 3-4, 4-5, 5-7, and 7-11 percent. A curve of the following nonlinear form was fitted to these six grouped observations:

ln(a+y) = ln(b) + cln(x)

where a, b, and c are the parameters to be estimated and y is the average rate of change of the nominal wage rate and x is the average unemployment rate. The parameter a is required because the observed average rate of change of nominal wages at high average rates of unemployment was negative, and log transformations of negative numbers do not exist. In retrospect it is hard to see how a nonlinear regression on six data points attracted as much attention as the Phillips curve achieved.

The remainder of Phillips article is devoted to telling stories about the year-by-year deviations (yt - y*t) where yt is the observed annual rate of change of nominal wages and y*t is the predicted annual rate of change of nominal wages based upon the estimated parameters a, b, and c and the observed annual value of the unemployment rate. There are three themes to these stories:

1) that these deviations are positively affected by import prices, presumably working through something like implicit cost of living adjustments.

2) that these deviations are affected by the rate of change in the unemployment rate. For a given level of unemployment, Phillips suggests that the deviation (yt - y*t) tends to be positive when the unemployment rate is falling (the expansion phase of a business cycle) and negative when the unemployment rate is rising (the contraction phase of a business cycle). Therefore within a business cycle, the actual year-to-year behavior of the rate of change of nominal wages tends to circle around the Phillips curve in a counter clockwise fashion.

3) that these deviations are affected by food price changes through cost-of-living adjustments. This is implicit in Phillips Table I, where

yt = y*t + f(pt)

and

y*t is referred to as "demand-pull" wage inflation

pt is the rate of change of food prices

f(pt) is referred to as "cost-push" inflation.

Lipsey [1960] sought to provide some foundation Phillips' observations in terms of a simple dynamic economic model. He argued that the unemployment rate could be interpreted as a measure of excess demand (low unemployment) or excess supply (high unemployment) in the labor market. He proposed that Mr. Phillips' curve should be interpreted as a simple dynamic model that says the rate of change of a price (the nominal wage rate) is proportional to the excess demand (or excess supply) in a particular market.

When the Phillips curve idea was exported to the U.S. it frequently was expressed in different terms: a relationship between price inflation and the unemployment rate (see Samuelson and Solow [1960]). As such it was viewed as an opportunities locus at which expressed the trade-off between inflation and unemployment that was faced by economic

Source: P.A. Samulson and R.M. Solow, “Analytical Aspects of Anti-Inflation Policy”, American Economic Review, 1960, 50:177-194

policymakers. The consensus view was that there was a cost to maintaining any level of unemployment in the economy in terms of the level of inflation, and that a lower rate of unemployment could only be maintained at a cost of a permanently higher rate of inflation. For example the 1963 Economic Report of the President states: “While the record of the postwar years indicates that wages tend to rise more rapidly in years when unemployment is low, given the present high unemployment rate demand for labor can expand substantially without resulting in much additional pressure on labor markets” (p. 84). This view generated considerable discussion of economic policies (particularly microeconomic policies) that might “improve” the trade-off between inflation and unemployment by permanently shifting the “Phillips curve” down.

Many subsequent empirical studies found that the Phillips curve at least in the U.S. (measured either as a relationship between the rate of price inflation and unemployment or the rate of wage inflation and unemployment) was not particularly stable. This led to a search for the factors that caused the Phillips curve to shift. Eventually a large number of explanations and or shift variables were proposed (see for example Perry [1966]).

Friedman/Phelps and the “Expectations Augmented Phillips Curve” or the “Accelerationist Hypothesis”

In his Presidential Address to the American Economic Association in 1967, Milton Friedman [1968] suggested that the whole conception of a permanent tradeoff between inflation and unemployment was fundamentally wrong. Essentially the same line of inquiry was being followed simultaneously and independently by Phelps [1970]. The part of Friedman's presidential address that deals with the Phillips curve can be viewed as four pages that redirected macroeconomics.

Friedman argued that the alleged theoretical foundations of the Phillips curve were all wrong. He noted that a simple dynamic model of the rate of change of a market price in response to excess demand in that market is fine, but the price that is appropriate in that model is the relative price (real price) of the good or service that is being traded in the market. Therefore, the proposed theoretical foundation for the Phillips curve is a theory not of the behavior of the nominal wage rate, but of the behavior of the real wage rate.

Friedman suggested four propositions:

1) there exists “some level of [un]employment which has the property that it is consistent with equilibrium in the structure of real wages. At that level of [un]employment, real wages are tending on the average to rise ... at a rate that can be maintained indefinitely so long as capital formation, technological improvements, etc. remain on their long-run trends.”

2) Assume that spending [aggregate demand] starts to rise. At this time assume that “people have been expecting prices to be stable, and prices and wages have been set for some time in the future on this basis. It takes time for people to adjust to a new state of demand.”

3) After a increase in spending [aggregate demand] “because selling prices of products typically respond to an unanticipated rise in nominal demand faster that prices of factors of production, real wages received have gone down, though real wages anticipated by employees went up, since employees implicitly evaluated the wages offered at the earlier price level.”

4) “the decline in ex-post real wages will soon come to affect anticipations. Employees will start to reckon on rising prices of the things that they buy and demand higher nominal wages for the future.”

From these propositions he reached the following conclusions:

1) “even though the higher rate of monetary growth [aggregate demand] continues, the rise in real wages will reverse the decline in unemployment, and then lead to a rise, which will then to return unemployment to ifs former level. In order to keep unemployment at its target level ... the monetary authority would have to raise monetary growth still more. Chose a target rate of unemployment that is above the natural rate, and they will be led to produce deflation and an acceleration of deflation at that.”

2) “there is always a temporary trade-off between inflation and unemployment; there is no permanent trade-off. The temporary trade-off comes not from inflation, but from unanticipated inflation.

The operational question that arises from these conclusions is how long the temporary trade-off lasts? This question is still debated today. Friedman's guess was that the initial effects of a permanent increase in the money growth rate would last 2-3 years, but that full adjustment might take as long as a couple of decades.

Testing the Accelerationist Hypothesis

The Lucas and Rapping [1969] article represents one of the first attempts to test the Friedman/Phelps accelerationist hypothesis under the adaptive expectations hypothesis. The basic model is a labor supply curve of the form:

(1) ln(Nt/Mt) = 0 + 1ln(wt/w*t) + 2lnw*t - 3ln(P*t/Pt)

where

Nt = employment (total hours worked)

Mt = population

wt = the real wage rate. Note the change in notation from previous notes

Pt = the price level

P*t = the anticipated (expected) price level for period t+1 based on information available at t. Therefore P*t = tPt+1 in our previous notation.

w*t = the anticipated (expected) real wage rate for period t+1 based on information available at t.

They assume that 1 > 0, 3 > 0 and that the sign of 2 is uncertain. Under these definitions ln(P*t/Pt) = [lntPt+1 - lnPt] = the rate of inflation that is anticipated at time t (see page 343)

Lucas and Rapping introduce a subsidiary hypothesis that defines the “normal labor supply” which represents the amount of labor services that would be supplied as a function of the real wage rate when there is no expectional error (w*t-1 = wt and P*t-1 = Pt). With these assumptions and equation (1) the normal labor supply is defined as:

(2) ln(N*t/Mt) = 0 + 1lnw*t-1 + (2 - 1)lnw*t - 3lnP*t + 3lnP*t-1.

Subtracting equation (1) from (2) gives:

(3) ln(N*t/Nt) = 1ln(w*t-1/wt) + 3ln(P*t-1/Pt).

Hence N*t/Nt depends only on expectational errors. Note that [(N*t - Nt)/Nt]  ln(N*t/Nt), so equation (3) can be thought of as an equation for the unemployment rate, but the variable on the left hand side of that equation wouldn't cover unreported unemployment or the frictional components of measured unemployment. Therefore Lucas and Rapping introduce a second subsidiary hypothesis:

(4) Ut = g0 + g1ln(N*t/Nt) where g0, g1 > 0

and Ut is measured unemployment. Finally they introduce a third subsidiary hypothesis, namely adaptive expectations, to operationalize their expectations variables for real wages and the price level:

(5) lnw*t = lnwt + (1-)lnw*t-1 and

(6) lnP*t = lnPt + (1-)lnP*t-1 where 0 <  < 1.

Note that the adaptive expectations coefficient is the same in both equations (5) and (6). This implies that the expected nominal wage rate is generated by an adaptive expectations mechanism with the same adaptive expectations coefficient, , that generates the expected price level.

If equations (5) and (6) are rewritten in lag operator notation we get:

(7) lnw*t-1 = [1 - (1-)L]-1lnwt-1 and

(8) lnP*t-1 =  [1 - (1-)L]-1lnPt-1.

Equations (4), (7) and (8) can be substituted into equation (3) to get an equation for the measured unemployment rate implied by the primary and subsidiary hypotheses:

(9) Ut = g0 + g11{[1 - (1-)L]-1lnwt-1 - lnwt}

+ g13{[1 -(1- )L]-1lnPt-1 - lnPt}

Factor [1 - (1-)L]-1 from the coefficient on lnPt: [1 -(1- )L]-1lnPt-1 - lnPt =

[1 - (1-)L]-1[LlnPt -{1-(1-)L}lnPt] = [1 - (1-)L]-1[lnPt-1-lnPt+lnPt-1-lnPt-1] =

-[1 - (1-)L]-1[lnPt-lnPt-1] and multiply both sides of (9) by [1 - (1-)L]:

(10) Ut = g0 - g11[1-L]lnwt - g13 [1-L]lnPt + [1-]Ut-1.

Now consider how unemployment responds to the inflation rate, [1-L]lnPt, both in the short run (Ut-1 fixed) and in long-run equilibrium (Ut = Ut-1). In the former case the response is measured by the coefficient of [1-L]lnPt = - g13 < 0. Thus there is a short-run tradeoff between inflation and unemployment such that unemployment rises (employment falls) when the inflation rate falls. In the latter case the effect is measured by setting Ut = Ut-1 = U* and computing the effect of the inflation rate on U*. This coefficient is -g13/ < 0. Therefore their model implies a long-tradeoff (though the implied long-run Phillips curve is steeper than the short-run Phillips curve) between inflation and unemployment, contrary the the Friedman/Phelps accelerationist hypothesis. Note that the parameters 1 and 2 are not relevant to the issue of the slope of the Phillips curve, but only affect shifts in the Phillips curve.

They then consider alternative subsidiary hypothesis for price expectations in the class that they define as “rational distributed lag functions” of the form:

(11) a*(L)lnP*t = b(L)lnPt

where a*(L) = 1 - a1L - ... - asLs = 1 - a(L) with roots outside the unit circle, and b(L) = b0 + b1L + ... + brLr.

lnP*t = b(L)[1-a(L)]-1lnPt is a rational polynomial distributed lag function in lnPt, since the polynomial in the lag operator L which multiplies lnPt can be written as the ratio of two lag polynomials. The numerator polynomial and the denominator polynomial are subject to one restriction a(1) + b(1) - 1 = 0.

There follows (p. 345) a very confusing discussion about conditions that generate a vertical long-run Phillips curve. Consider some very general rational distributed lag expectations mechanisms:

(12) lnw*t = b1(L)/[1-a1(L)]-1lnwt and

(13) lnP*t = b2(L)/[1-a2(L)]-1lnPt such that a2(1) + b2(1) - 1 = 0.

Equations (12), (13) and (4) can be substituted into equation (3) to get:

(14) Ut = g0 + g11{b1(L)[1-a1(L)]-1Llnwt - lnwt}

+ g13{b2(L)[1-a2(L)]-1LlnPt - lnPt}

Factor [1-a2(L)]-1 from the coefficient on lnPt and [1-a1(L)]-1from the coefficient on lnwt to get:

(15) Ut = g0 + g11[1-a1(L)]-1[b1(L)L - 1 + a1(L)]lnwt

+ g13[1-a2(L)]-1[b2(L)L - 1 + a2(L)]lnPt.

Since we have assumed that a2(1) + b2(1) - 1 = 0 we can write [b2(L)L - 1 +a2(L)] =

-[1-L]c(L), i.e. the restriction that Lucas and Rapping place on their rational polynomial distributed lag function is that the rational polynomial has a unit root. Then equation (15) becomes:

(16) Ut = g0 + g11[1-a1(L)]-1[b1(L)L - 1 + a1(L)]lnwt

- g13[1-a2(L)]-1c(L)[1-L]lnPt.

The question of when the long-run Phillips curve is vertical then is the question of the conditions under which c(1) = 0, since 1 - a2(1)- b2(1) = 0. But c(1) = 0 when c(L) has a unit root, i.e. when the polynomial a2(L) + b2(L) -1 has two or more unit roots.

Consider the two following special cases:

a) Adaptive Expectations on the price level lnPt (the initial Lucas and Rapping case) Under this assumption b2(L) =  and a2(L) = (1-) so a2(1) + b2(1) = 1. But b2(L)L - 1 + a2(L) = L -1 + (1-)L = -(1-L) so c(L) = -1. Hence, as shown by Lucas and Rapping, the long-run Phillips curve is not vertical.

b) Adaptive Expectations on the Inflation Rate

Adaptive expectations in the inflation rate are:

tpt+1 - t-1pt = [pt - t-1pt]

where pt is the inflation rate. This can be rewritten in lag operator notation as:

[1 - (1-)L]tpt+1 = pt.

Now substitute for tpt+1 and pt using Lucas and Rapping notation:

and collect all the terms in lnPt on the right hand side:

[1 - (1-)L]ln = [1 - (1-)L + (1-L)]lnPt = [1 +  - L]lnPt

Under this assumption b2(L) = (1+) - L and a2(L) = (1-)L. Again b2(1) + a2(1) = 1 +  -1 + (1-) = 1, so this model falls into the general class of rational polynomial models considered by Lucas and Rapping. But now

b2(L)L - 1 + a2(L) = (1-)L - L2- 1 + (1+)L =

2L - 1 - L2 = -(1-L)2 = (1-L)C(L)

so in this case c(L) = -(1-L) and the long-run Phillips curve is vertical. Also note that c(L) = c0 + c1L where c0 = -1, so the short-run Phillips curve is negatively sloped.

The conclusion from these two examples is that the presence or absence of a vertical long-run Phillips curve under the natural rate hypothesis with adaptive expectations depends upon the subsidiary hypothesis about the expectations mechanism. Adaptive expectations in the inflation rate (as assumed by Friedman) will produce the vertical long-run Phillips curve; adaptive expectations in the price level (as initially assumed by Lucas and Rapping) will not. Both mechanisms fall into the class of restricted polynomial distributed lag hypotheses that are considered by Lucas and Rapping, so the restriction to this class of expectations models is not sufficient to produce the vertical long-run Phillips curve.

Lucas and Rapping estimated empirical “expectations augmented” Phillips curves using annual data for the U.S. over the period 1904-65 and for three separate subperiods: 04-29; 30-45; and 46-65. They concluded that there was no stability of their estimated equation across the subsamples:

Period / Short-Run Phillips Curve / Long-Run Phillips Curve
04-65 / Yes / Yes
04-29 / Yes / No
30-45 / Yes / Yes
46-65` / No / No

where "Yes" in the Short-run Phillips Curve column of the above table indicates a significant negative relationship between the unemployment rate and the current inflation rate and "Yes" in the Long-run Phillips Curve column indicates that the sum of the distributed lag coefficients on the inflation rate is significantly different from zero.

References

Friedman, M. (1968), “The Role of Monetary Policy”, American Economic Review, May, pp. 1-17.

Lipsey, R.G. (1960), “The Relation between Unemployment and the Rate of Change of Money Wage Rates in the United Kingdom, 1862-1957: A Further Analysis”, Economica, February.

Lucas, R.E. and Rapping, L.A. (1969), “Price Expectations and the Phillips Curve”, American Economic Review, June, pp. 342-350.

Perry, G.L. (1969) Unemployment, Money Wage Rates, and Inflation, Cambridge: M.I.T. Press.

Phelps, E.S. (1970), “Money Wage Dynamics and Labor Market Equilibrium”, in E.S. Phelps (ed.), Micoreconomic Foundations of Employment and InflationTheory, NewYork: W.W. Norton and Company.

Phillips, A.W. (1958), “The Relation between Unemployment and the Rate of Change in Money Wages in the U.K. 1892-1957”, Economica, November, pp. 283-99

Samuelson, P.A. and Solow, R. (1960), “Analytical Aspects of Anti-Inflation Policy”, American Economic Review, May

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