I revised "Expectations and the neutrality of interest rates" and presented at the Hoover Economic Policy workshop. Thanks to the great Hoover team, here it is by video. If the embed doesn't work, here's the Hoover webpage with the video. The updated paper and slides are here.

## Tuesday, December 20, 2022

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A. W. Phillips, "The Relation Between Unemployment and the Rate of Change of Money Wage Rates in the United Kingdom, 1861 - 1957", Economica, Vol. 25, No. 100, Nov. 1958.

ReplyDeleteI. Hypothesis: "When the demand for a commodity or service is high relatively to the supply of it we expect the price to rise, the rate of rise being greater the greater the excess demand. Conversely when the demand is low relatively to the supply we expect the price to fall, the rate of fall being greater the greater the deficiency of demand."

II. 1861 - 1913: Phillips examined the relationship between unemployment and the rate of change in money wage rates in the period 1861-1913. He determined that a smooth curve faired through the data points yielded a reproducible first approximation of this relationship in equation form. The equation he derived for the approximate relationship between the rate of change in money wage rates and the rate of unemployment is given by,

y + a = b.x^c

or in logarithms,

log(y + a) = log(b) + c.log(x)

where,

y = the rate of change of wage rates

x = percentage unemployment

a = 0.900

b = 9.638

c = - 1.394

The fit of Phillips's equation is only fair, as the charts in his paper demonstrate. The parameter values are chosen to fit his estimate (or average) values rather than the data per se. The curve can be then said to "faired" or fitted by eye, rather than by regression, to the data.

Given that A. W. Phillips set out to prove the hypothesis described in Section I of his paper by testing earlier historical periods, and he never undertook subsequently further effort to extend the hypothesis into a theory, it is not at all surprising that the equation he developed to fit the data lacks expectations of future period changes in the money wage rate and unemployment. Nevertheless, Phillips provides within the body of his paper the insight relating the model to expectations of wage rate changes relative to expectations of industrial activity (i.e., unemployment) that subsequent theory by other academic economists, chiefly American, developed.

Equation (6) of Phillips.pdf, 2022-12-17 version, reads as

ReplyDeleteE{pi(t+1)|Q(t)} = [pi(t)+sigma.kappa.i(t)]/(1+sigma.kappa).

Eqn (6) places one input on the LHS of the equality sign, and one input and one output on the RHS of the equality sign. It mixes up the timing of E{pi(t+1)|Q(t)} and pi(t) by putting the cart, pi(t), before the horse E{pi(t+1)|Q(t)}.

Consider the original reduced form equation, eqn.(3), i.e.,

pi(t) = (1+sigma.kappa).E{pi(t+1)|Q(t)} - sigma.kappa.i(t).

The conditional expectation is an input, as is the rate of interest, i(t), whereas the term on the LHS of (3) is the output, i.e., the measured variable at the end of period t. The inputs are entered at the start of period t.

At period t+1, eqn.(3) is written,

pi(t+1) = (1+sigma.kappa).E{pi(t+2)|Q(t+1)} - sigma.kappa.i(t+1).

Now, subtract E{pi(t+1)|Q(t)} from pi(t+1):

pi(t+1) - E{pi(t+1)|Q(t)} =

E{pi(t+2)|Q(t+1)} - E{pi(t+1)|Q(t)} - sigma.kappa.[i(t+1) - E{pi(t+2)|Q(t+1)}].

The first term on the RHS of the equals sign is the change in the conditional expectation of inflation in period t+2, while the second term on the RHS represents the conditional expectation of the real rate of interest (approx.) in period t+2. The term on the LHS of the equals sign is the "shock", or difference between the measured rate of inflation in period t+1 and the period t conditional expectation of the rate of inflation in period t+1. Note the sign of the second term on the RHS of the equation. If i(t+1) exceeds E{pi(t+2)|Q(t+1)}, then the "shock" is decreased, ceteris paribus; if i(t+1) is less than E{pi(t+2)|Q(t+1)}, the shock is increased, ceteris paribus. In any case, expectations drive the period to period evolution of the rate of inflation. And, if J. Powell, Chmn. of the Board of the Federal Reserve System, is to be believed, then the time sequence of the rate of interest, {i(t), i(t+1),...} is also driven by expectations of the evolution of the rate of inflation.

The factor that is seldom examined is the process by which the conditional expectations, E{pi(t+1)|Q(t)}, E{pi(t+2)|Q(t+1)}, ..., etc., are arrived at. Absence knowledge of this process, the model is an academic theory that will continue to frustrate every effort to operationalize it.

“The Federal Reserve does not even pretend to control money supply, especially inside money”.

ReplyDeleteIf you can’t define the variables, you can’t apply the math. If you can't apply the math, you know nothing about the variables.

Nice Post.

ReplyDelete