The Complete Hall and Taylor Model

**PLEASE NOTE:** These basic lecture notes on HT Chapter 8 are required
for Econ 302. If possible, students should read these notes prior to
attending class lectures on HT chapter 8. A more advanced detailed version
of these notes is also available on this web site; the advanced notes are
recommended but not required.

- Basic References:
- HT Chapter 8;
- Study Guide, Chapter 8.

In previous chapters, HT develop two of the components of their macro model: labor market equations for the determination of potential GDP Y*(T) in each period T; and the IS-LM model for the determination of actual GDP Y(T) in each period T. In this chapter HT develop the third and last component of their macro model: the "Phillips Curve" describing the way in which the general price level P(T) in period T adjusts over time in response to discrepancies between potential and actual GDP.

Before detailing the nature of this price adjustment process, we need to introduce a more comprehensive definition of equilibrium than the notion of IS-LM equilibrium introduced in HT7.

Recall from HT7 that the IS Curve for an economy is a schedule showing
all combinations of the real GDP level Y and the real interest rate R for
which the product market is in equilibrium, and the LM Curve for an economy
is a schedule showing all combinations of Y and R for which the money market
is in equilibrium. An economy is then said to be in an *IS-LM
equilibrium* if both its product market and its money market are in
equilibrium. Graphically speaking, then, an economy is in a IS-LM
equilibrium if and only if its current Y and R levels constitute an
intersection point of its IS and LM curves.

By construction, the aggregate demand (AD) Curve for an economy is derived from the successive intersection points of the economy's IS and LM curves as the general price level P is varied. Consequently, yet another way to characterize an IS-LM equilibrium for an economy is to say that the economy is on its AD curve.

Note, however, that the fact an economy is in an IS-LM equilibrium says nothing about whether its labor market is in equilibrium. That is, being in an IS-LM equilibrium gives no guarantee that actual employment N is equal to full employment N*, or equivalently (in the Hall and Taylor framework), that the actual GDP level Y equals the potential GDP level Y*.

For the complete HT model, we need a more comprehensive definition of equilibrium that takes the labor market into account.

**DEFINITION**: An economy is said to be in *internal balance* if all
of its domestic markets (product, money, and labor) are in equilibrium.
In particular, then, an economy is in internal balance if two conditions
hold: (i) it is in an IS-LM equilibrium; and (ii) actual GDP Y is equal to
potential GDP Y*.

In economic models in use through the early nineteen seventies, a common assumption was that the price level P(T) for period T would not change immediately if firms found themselves producing above or below the potential GDP level Y*(T). Rather, it was assumed that the existence of a gap in period T between potential and actual GDP only resulted in a change in the price level in the following period T+1 at the earliest.

In particular, it was assumed that the inflation rate from
period T to T+1 was positively correlated with (moved up or down
together with) the GDP gap in period T for each period T. Such a
relation between the inflation rate and the GDP gap is referred to
as a *Phillips Curve* in honor of A. W. Phillips, who first
popularized the relation in a 1957 *Economica* article.

#Example: A Commonly Used Form for the Traditional Phillips Curve#: Y(T)- Y*(T) (8.1) INF(T,T+1) = f [ ------------- ] , Y*(T) where P(T+1) - P(T) INF(T,T+1) = --------------- = inflation rate P(T) from T to T+1 ; Y(T)- Y*(T) -------------- = GDP gap in period T ; Y*(T) f = positive exogenously given coefficient multiplying the GDP gap which reflects the responsiveness of the inflation rate to changes in the GDP gap.

Note that, by manipulating terms, relation (8.1) determines the price level P(T+1) for period T+1 as a function of the period T price level P(T), the period T GDP level Y(T), and the period T potential GDP level Y*(T):

Y(T)- Y*(T) (8.2) P(T+1) = P(T) + P(T) f [-------------] . Y*(T)Consequently, once a relation such as (8.1) is appended to the IS-LM model developed in HT7, augmented by labor market relations for the determination of Y*(T), one has a way of determining the movement over time of P(T) as well as the movement over time of Y(T), R(T), and other endogenous variables such as aggregate household consumption C(T).

The basic underlying motivation for the Phillips Curve (8.1) is a key empirical regularity, known as "Okun's Law."

It is a well-established empirical fact that GDP and unemployment tend to be negatively correlated over time, in the sense that periods of high GDP correspond to periods of low unemployment and conversely. See, for example, HT Chapter 1, Figures (1-3) and (1-5).

As briefly discussed by HT in Chapter 3, however, it is also an
empirical fact that the magnitude of this negative correlation tends to be
fairly stable over time. In particular, for each percentage point that the
unemployment rate U(T) is *above* the natural unemployment rate U*(T),
real GDP Y(T) tends to be about 3 percent *below* potential real GDP
Y*(T). [See HT Chapter 3, Figure (3-5).] This empirical regularity is known
as *Okun's Law* in honor of Arthur Okun, the economist who first
discovered it.

In algebraic terms, Okun's Law takes the following form:

#Algebraic Form of Okun's Law#: Y(T) - Y*(T) (8.3) ---------------- = -3.0 [U(T) - U*(T)] , Y*(T) discrepancy between the actual and the natural GDP gap unemployment rates in in period T period TRelation (8.3) predicts that a

Many economists, including HT, answer that it reflects the tendency of firms to engage in hedging behavior in response to demand shocks. For example, in response to an unfavorable demand side shock (a decrease in Y^D), firms tend to hedge against the possibility that the negative shock is transient by keeping on workers they do not actually need and demanding less work effort from their workers per hour. Thus, measured productivity (average output per paid work hour) declines along with output Y as firms try to bring Y down in line with the lower Y^D, but the percentage increase in unemployment (laid off or fired workers) is less in magnitude than the percentage decline in Y -- in particular, unemployment increases by only about 1 percent in response to a 3 percent decline in Y. A similar story can be told in reverse for favorable demand side shocks.

In this way, firms keep up worker morale by providing increased job security, and they also avoid some of the transactions costs associated with the hiring and firing of workers. Of course, if demand shocks persist, most firms ultimately will resort to changes in their employment levels (firing and hiring) to bring these levels in line with output levels.

THE TRADITIONAL PHILLIPS CURVE

[See HT figure (8-5).]

Consider an economy in internal balance in some period T with an aggregate consumption function given by C = a + b[1-t]Y. Suppose consumer tastes suddenly change -- in particular, suppose the coefficient "a" in the consumption function suddenly decreases, implying that households demand fewer goods and services at each given level of Y.

In terms of the IS-LM model developed in HT7, the economy's IS curve will shift down in period T in response to this change in the consumption function. Assuming that IS-LM equilibrium is quickly restored -- in particular, that firms act to keep their short-run production levels in line with their demands so that product market equilibrium is retained -- actual GDP Y(T) will then decrease to some lower level Y(T)'; for the intersection of the IS and LM curves will now occur at a point where Y and R are both lower.

Note that this decrease in Y(T) in response to the sudden period
T change in the consumption function coefficient "a" occurs for any
value of the current price level P(T), implying that the AD Curve for
period T is *shifted to the left* (less Y for each P). On the
other hand, period T potential GDP is unaffected by the change in the
consumption function coefficient. Consequently, a negative GDP gap
opens up at the current price level P(T), i.e., Y(T) falls below
Y*(T).

By Okun's Law, the opening up of this negative GDP gap
results in an unemployment rate U(T) that is *higher* than the
natural unemployment rate U*(T) as firms start to decrease their
employment levels in response to decreased demand for their goods
and services. Assuming the unfavorable demand shock persists (so
that the AD curve is permanently shifted to the left), these
conditions ultimately lead to falling wage rates and also to a
fall in the prices charged to household for final goods and
services as consumption demand for goods and services weakens in
response to the fall in Y.

This commonly observed sequence of events, from the opening up of a negative (or positive) GDP gap to a fall (or rise) in the prices for newly produced final goods and services, is precisely what is captured by the traditional Phillips Curve (8.1).

[See HT figure (8-5).]

Okun's Law and the traditional Phillips Curve (8.1) together predict a tight negative correlation between the inflation rate INF(T,T+1) and the unemployment rate U(T), in the sense that one goes up if and only if the other goes down. This prediction seemed to fit U.S. data and policy experience fairly well during the nineteen fifties and sixties. In the mid-nineteen seventies, however, things started to go haywire.

Specifically, as can be seen from Figures (3-1) and (3-4) in HT Chapter 3, the inflation rate and the unemployment rate actually exhibited a positive correlation in the nineteen seventies, in the sense that they tended to increase together -- a phenomenon labelled "stagflation." By the end of the nineteen seventies (1979=the second oil price shock), the inflation rate had soared to historically unprecedented heights (over 12 percent) while at the same time the unemployment rate appeared to be trending upwards. This was followed, in 1982, by a sudden drop in the inflation rate and a huge increase in unemployment -- the traditional negative correlation had reappeared.

Numerous researchers subsequently attempted various fix-ups of the basic Phillips curve relation (8.1) that could explain all of these empirical observations.

One economically reasonable modification of the relation (8.1)
that provides a systematic explanation for these events is the
*expectations-augmented Phillips Curve*. The expectations-augmented
Phillips curve was actually developed by Edmund Phelps and Milton Friedman in
the late 1960s in response to perceived theoretical deficiencies of (8.1).
Nevertheless, the importance of their work was not accepted until the
nineteen seventies, when the empirical deficiencies of (8.1) became obvious.

HT postulate and use a simple form of the expectations-augmented Phillips curve, as follows:

Y(T) - Y*(T) (8.4) INF(T,T+1) = INF^e(T,T+1) + f [----------------] Y*(T) where INF^e(T,T+1) = Expected inflation rate from T to T+1 as perceived by firms at the beginning of period T; f = positive exogenously given coefficient multiplying the GDP gap

Under relation (8.4), in contrast to (8.1), firms are assumed to have a
more sophisticated understanding of inflation. If firms expect the inflation
rate to be INF^e(T,T+1) from period T to period T+1, then firms will adjust
their prices to keep them in line with this expected change in the general
price level *even if the GDP gap is zero*; and a *positive* GDP
gap will induce firms to increase their prices at a *faster* pace than
the expected rate of inflation -- that is, the actual inflation rate will
exceed the expected inflation rate.

Recall that internal balance requires the GDP gap to be zero. However,
according to (8.4), having a zero GDP gap does *not* guarantee that the
inflation rate INF(T,T+1) is zero. Rather, a zero GDP gap is consistent with
any positive or negative level for the inflation rate as long as this level
is correctly anticipated.

More generally, as will be clarified in the next section and in later experiments with the complete HT model, (8.4) is consistent with the stagflation experience of the 1970s. That is, under (8.4), it is possible to have a situation in which the inflation rate is increasing at the same time that the GDP gap is becoming more negative -- or in other words, a situation of stagflation in which inflation and unemployment are increasing together.

The simplest postulate concerning how firms form inflation expectations is that they expect the current inflation rate to be the same as it was in the previous period. More generally, they could form some kind of weighted average over the inflation rates they have observed over a number of previous periods. Expectations formed in this way -- as weighted averages over past observations -- are said to be "adaptive."

*Example 1: Adaptive Expectations for the Inflation rate*

Firms determine their expected inflation rate INF^e(T,T+1) for period T to T+1 by taking a simple weighted average over past observed inflation rates:

(8.5) INF^e(T,T+1) = a_1INF(T-1,T) + a_2INF(T-2,T-1) + ...

*Example 2: Rational Expectations for the Inflation Rate*

An alternative hypothesis concerning expectation formation that has become very fashionable in recent years is to assume that agents understand the actual mechanisms generating key economic variables and use this information to form their expectations. In particular, agents understand the structural relations that determine the inflation rate, and they use this understanding in formulating their inflation rate expectations. They do not simply extrapolate over past observations.

Suppose, for example, that the *actual* inflation rate from
T to T+1 is determined by the growth of the money supply from T
to T+1; i.e.,

M^S(T+1) - M^S(T) (8.6) INF(T,T+1) = ------------------- M^S(T) = actual percentage change in the money supply M from T to T+1.

Then the agents in the economy are said to have *rational expectations*
for the inflation rate INF(T,T+1) if these expectations are given by

(8.7) INF^e(T,T+1) = the expected percentage change in M^S from T to T+1.

Various factors might affect in some degree the way in which firms formulate an expected inflation rate: for example, past rates of inflation; monetary policy; and negotiated wage contracts, especially if these wage contracts include cost of living adjustments (COLAs).

Economists making use of the expectations-augmented Phillips curve (8.4) generally assume that, regardless of how firms form their expectations, they tend to adjust their expectations to conform to their observations. In particular, it is assumed that firms continually adjust INF^e to conform more closely to observed values of INF.

Given this assumption, relation (8.4) implies the following "accelerationist property," sometimes also referred to as the "natural rate property":

**NATURAL RATE PROPERTY**: As long as actual GDP Y is maintained above
(below) potential GDP Y*, the inflation rate INF will *perpetually increase
(decrease)* as people keep attempting to adjust their inflation rate
expectations to the actual inflation rate. Conversely, a *zero* GDP gap
is consistent with any inflation rate INF *as long as this inflation rate
is correctly anticipated*.

Economists have estimated the relation (8.4) over various
different time intervals, often relying on the following simple
adaptive assumption for the expected inflation rate: namely, the
expected inflation rate from T to T+1 is proportional to the
actual inflation rate from T-1 to T. The following data is taken
from S. Sheffrin, *The Making of Economic Policy*, Blackwell,
1991, p. 1989. The numbers in parentheses below the estimated
coefficient values denote measured standard deviations.

#Pre-WWII study:# 1873-1914: Y(T) - Y*(T) INF(T,T+1) = 0.22INF(T-1,T) + 0.34 [-------------] Y*(T) (0.14) (0.19) R^2 = 0.14 (goodness of fit measure, 1 = best possible) #Post-WWII study:# 1953-1982: Y(T) - Y*(T) INF(T,T+1) = 0.005 + 0.92INF(T-1,T) + 0.37 [------------] Y*(T) (0.07) (0.003) R^2 = 0.83 (goodness of fit measure, 1 = best possible)

Comparing the two regressions, the coefficient f on output deviations is somewhat higher in the post-war era, indicating a somewhat higher price responsiveness to the GDP gap. The regression for the earlier period does not explain nearly as much variance in inflation as the postwar regression, however, as evidenced by the small goodness of fit measure. Inflation is much more persistent in the postwar error, in the sense that the coefficient on the lagged inflation term is much closer to 1.0 (implying that inflation in one period tends to spill over into the next period).

Combining the labor market equations for the determination of potential GDP with the HT7 IS-LM model and the expectations-augmented Phillips Curve developed above, one obtains the complete HT model. This model will permit us to derive the movement of the economy from one IS-LM equilibrium to the next as the price level adjusts in response to either a nonzero GDP gap or a change in the expected inflation rate.

#PERIOD T MODEL EQUATIONS FOR THE COMPLETE HT MODEL# #Product and Money Markets#: (1) [Adaptive Inflation Expectations] INF^e(T,T+1) = zINF(T-1,T) (2) [IS] R(T) = [a + e + G + g]/[d + n] - (1-b[1-t] + m)/[d + n])Y(T) (3) [LM] R(T) = - ( M/hP(T) + INF^e(T,T+1) ) + [k/h]Y(T) #Labor Market#: (Determination of Full Employment levels for labor employment N and the real wage w) (4) [Potential Labor Supply] N*(T) = h([1-t]w*(T)) (5) [Potential Labor Demand] w*(T) = AF_N(N*(T),K(T)) #Potential GDP and Price Adjustment#: (6) [Potential GDP] Y*(T) = AF(N*(T),K(T)) (7) [Def of Inflation Rate] INF(T,T+1) = [P(T+1) - P(T)]/P(T) (8) [Phillips Curve] INF(T,T+1) = INF^e(T,T+1) + f[Y(T) - Y*(T)]/Y*(T) #Movement in Capital Stock over Time#: (9) [Gross Investment] I(T) = e - dR(T) (10) [Capital Stock Adjustment] K(T+1) = I(T) + K(T) - xK(T) Note: xK(T) denotes the amount of capital depreciation in period T. Capital K(T+1) in period T+1 equals period T capital K(T) plus gross (total) period T investment I(T) minus xK(T). #PERIOD T CLASSIFICATION OF VARIABLES# Ten Period T Endogenous Variables: R(T), Y(T), INF^e(T,T+1), INF(T,T+1), P(T+1), Y*(T), N*(T), w*(T), I(T), K(T+1) Period T Predetermined Variables: P(T), INF(T-1,T), K(T) Exogenous Variables: Positive Policy Variables G, M , t with 0 less than t less than 1 Positive Coefficients a, e, g, d, n, b, m, k, h, z, A, u, f, x #BOX DIAGRAM OF THE MOVEMENT OF THE ECONOMY OVER TIME#: ..................... . . . . . Period T Model . P(T),K(T),INF(T-1,T) --> . Equations (1)-(10).--> P(T+1),K(T+1),INF(T,T+1) . . State of the Economy . . State of the Economy at the Beginning of . . at the Beginning of Period T ..................... Period T+1