Cobweb versus rational expectations models

Cobweb versus rational expectations models

Economics Letters North-Holland COBWEB Lessons VERSUS Umersrty, Received RATIONAL from the Market Gary A. ZARKIN Duke 87 13 (1983) 87-95 MOD...

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Economics Letters North-Holland

COBWEB Lessons

VERSUS

Umersrty,

Received

RATIONAL

from the Market

Gary A. ZARKIN Duke

87

13 (1983) 87-95

MODELS

*

Durham,

11 February

EXPECTATIONS

for Public School Teachers

NC 27706,

USA

1983

A cobweb model of occupational choice is found to fit the data almost as well as a rational expectations model, even though there are large forecastable changes in demand. Striking differences. however, are revealed when we simulate the two models.

1. Introduction

Most previous work on occupational choice makes use of cobweb models for the determination of the number of new entrants. ’ In those models, agents usually make their occupational decision based solely on the expected starting salary, as opposed to the entire sequence of expected future wages. Agents form these expectations using static or adaptive expectations, and are therefore myopic with respect to future demand conditions. However, it is quite possible that even though expected starting salaries are competitive with other jobs, the future may be either very bright or very bleak. This is particularly important in the public education market where expected starting salaries may be competitive but future enrollments may decrease substantially. Myopic models, as usually constructed in the literature, do not incorporate the potentially important effects of future enrollments on the decision to acquire public school certification. In a recent paper, Zarkin (1982) I developed a rational expectations model of the decision to acquire certification as a public school teacher. * I would like to thank George Tauchen for helpful discussions. ’ particular, Richard Freeman (1971. 1975a, b, 1976a, b, 1977), who has done the most extensive

work on occupational

0165-1765/83/$3.00

choice,

frequently

makes use of cobweb

0 1983. Elsevier Science Publishers

models.

B.V. (North-Holland)

88

G.A. Zarkrn / Cobweb us. rational expectations

models

In that model, future forecastable demand conditions, in particular children enrolled in public schools in the future, are explicitly incorporated into the reduced form equation for new entrants. Future forecastable demand conditions are found to be important determinants of the decision to receive secondary school certification for both men and women, whereas only lagged demand conditions are important determinants for women receiving elementary school certification. * In contrast, a cobweb specification of occupational choice is myopic with regard to future forecastable demand conditions as demonstrated in the cobweb specification taken from Freeman and Leonard (1977): GR, = a, + a,GR,_,

+ aZGR,_Z + u~IC_~ + a,OPF-,

+ 7,

(1)

where GR, is the number of degrees conferred in secondary education in period t, K,_, is the number of children enrolled in t - 2, the time of occupational decision, and OPw_, is the opportunity wage in t - 2. This formulation is derived from either a partial adjustment model with static expectations or from a model with adaptive expectations. In either case, there is no provision for incorporating future forecastable demand conditions into eq. (I). Since (1) is a reduced form that reflects the path of secondary school certifications then, clearly, a forecastable shock to future children would not affect new graduates until two periods after the shock begins. A forward-looking model, however, developed from a rational expectations, stock-flow model [see Zarkin (1982)] implies that people react prior to the onset of the shock as demonstrated in the following specification:

GR,=b,+b,7;_,+b,GRAD,+;

b3+,K,+, i=o

+b,OPPy

t b,,DUM+

cr,

where 7;_, is the lagged stock of teachers,

(2) GRAD, is the total number

of

* Neither lagged nor future demand conditions are important determinants for men receiving elementary school certification. Men comprise approximately 15% of elementary school teachers compared to 50% of secondary school teachers. Apparently whatever motivates this small proportion of men to acquire certification is not well captured in the model.

G.A. Zarkrn / Cobweb OS. rational expectatmns

models

89

college graduates in year t, DUM is a dummy variable equal to one from 1967 to 1972 to capture the subsidy to teaching in effect during that time, as well as the effect of the Vietnam War on the decision to receive teaching certification. OPPV is the expected average opportunity present value foregone from choosing teaching. In eq. (2) future children directly enter the reduced form for secondary school certification. We use the actual number of children enrolled, as opposed to the expected number based on information as of time t - 2, because it is assumed that there is perfect foresight in forecasting children up to five years into the future. From an aggregate viewpoint, the number of children in school is essentially deterministic, given that we focus on only those children who are already born. Forecasts of future children enrolled in school can be made by using the feature that as children age they move sequentially through the school system. 3

2. Comparison of the cobweb and rational expectations models Table 1 presents the results of estimating eqs. (1) and (2). In order to investigate the possibility that lagged children (K,_, and K,_,) rather than future children (K, - K,+5) are important determinants of the number of newly-certified teachers, the former variables are included in the estimation of eq. (2) as well. The F6,,4 statistic in column (2) tests the hypothesis that all six future values are jointly equal to zero. We reject the hypothesis at approximately the 1% level. Notice also that lagged children are statistically significant at approximately the 6% level. 4 Using the coefficient estimates from table 1, the predicted values of each of the models were derived based on information known at the time of the occupational decision, t - 2. ’ These two series are compared in 3 For further details on the estimation of this equation, see Zarkin (1982). 4 In Zarkin (1982), future children were significant but lagged children were not in the secondary school regression. The difference from the results here arises from the inclusion of an additional demand variable (the weighted sum of the future path of the expected teacher-student ratio) in the previous paper, which was dated t -2 and which partially captured the effect of lagged children. This demand variable was dropped for the purpose of this paper because it was negative (contrary ta the model) and insignificant. s For both models, we assume that people making occupational decisions in period t -2 know the number of people who received secondary school certification in the previous period.

90

Table I Secondary

G.A. Zarkin

school certification

LA GGR LAG2GR

/ Cobweb us. rational expectumns

regressions.

d

Cobweb

Rational

13090.2 (13300) 1.819 (1.81) (0.090) - 0.902 ( - 0.90) (0.105)

29030 (55599)

LAGT GRAD OPP W/OPP

K-2

v

- 3.02 (- 0.20) (4.98) 0.0011 (0.15) (0.0012)

K -I

KO K, K2 K3

F6.14 Pr z F F 2.14 Pr z F RHO SSE DFE

- 0.44 (0.20) 9.2x lo* 22

models

expectations

-0.806 (- 5.68) (0.169) 0.393 (2.28) (0.071) - 2.74 (2.26) 0.019 (2.69) (0.0077) 0.0002 (0.03) (0.0076) - 0.0030 ( ~ 0.45) (0.0075) 0.0055 (0.84) (0.008) 0.010 (1.56) (0.0079) -0.0017 (-0.27) (0.0079) - 0.0029 ( - 0.47) (0.0078) 0.0071 (1.15) (0.006) 0.0195 (2.71) (0.009) 0.015 (2.36) (0.005) 36874 (7256) 4.19 0.013 3.59 0.055 0.43 (0.28) 6.6 x 10” 14

Footnote

a on next page

G.A. Zarkin

/ Cobweb OS. rational expectations

91

models

fig. 1 to the actual number of people certified. Perhaps the most striking feature is that overall the cobweb model fits the data about as well as the forward-looking, rational expectations model, with the former explaining 0.9736 percent of the total variation and the latter explaining 0.9817 percent. At first blush, the close fit of both of the models is rather surprising in light of the fact that we have demonstrated empirically that future children are important determinants of the decision to acquire secondary school certification. At the same time, there have been large, presumably forecastable movements in secondary school enrollment. Finally, as we have already noted, the cobweb model is myopic and one might think it would perform much worse in this type of environment. On second thought, however, this is not so surprising. In forming the one-step-ahead forecasts using the cobweb model we condition on the actual values of once and twice lagged certifications. Thus, each year we update our forecast using the previous number of people receiving certification. In this way, our fitted model should never miss the actual turning points of the data by more than one period and, in turn, the cobweb model may look fairly reasonable even though it may be a totally inaccurate description of the market. ’ In order to drive this point home, suppose that people receiving secondary school certification have perfect foresight and respond to

6 On the other hand, if we were to condition on past certification data, the fit would deteriorate substantially.

forecasts

rather

than

the actual

____ Table I (continued) d The numbers in parenthesis below the estimates are the standard errors; the numbers to the right are the elasticities evaluated at the means of the relevant variables. The regressions are in levels. The dependent variable for both columns is the number of graduates completing certification requirements for the first time for secondary school. The variables names are: LAGGR LAGZGR LACT GRAD OPP W/OPPV DUM

= = = =

dependent variable, lagged once, dependent variable, lagged twice, lagged stock of teachers, number of college graduates, = the opportunity wage for column one; the opportunity column two, = dummy variable equal to one from 1967-1972.

present

value for

92

G.A. Zarkin / Cobweb OS. rational expectations

models

1’ 1955

1975

190000 1 180000

170000

160000

150000

i’fOOO0

130000

120000

110000

100000

9000-O

80000

70000

60000

50000

‘too00

30000

20000

1950

“““‘,““““‘1”“““‘/““““‘/““““‘/ 1960 1965

1970

1980

DATE LEGEND

:

LG2HS GHSZAR GHSFRE Fig. 1. Actual

GHSFRE

= Number = Estimate = Estimate

------

GHSZAR

Prepored to Teach in Secondary from Forward-Looking Model from Cobweb Model

---

LG2HS

School

data and estimates,

future forecastable demand conditions. If the number of children enrolled in school decreases at some point in the future, then the number of people receiving certification

is predicted

to peak before

the downturn

in

G.A. Zarkin

/ Cobweb OS. rational expectations

93

models

9-

8-

l-

6-

5-

q-

3-

2-

l-

O-

-l-

-2 -

-3 -

-4-

-5 -10

-5

0

5

10

15

20

TIME LEGEND

COB FW

= Cobweb = Forward-

Fig. 2. Simulated

response

:

-

COB

---

FW

Model Looking

Model

to a forecastable

pulse: cobweb

versus forward-looking

model.

enrollment. 7 If we were to form the one-step-ahead forecasts from the estimated cobweb regression coefficients these forecasts should be fairly close to the actual data since we continually update our forecasts based ’ This result is derived from the model in Zarkin (1982). It also is what we actually observe in the raw data. The number of people receiving secondary school certification peaks in 1971, four years before the downturn in secondary school enrollment.

94

GA.

Zm-kin / Cobweb us. ratronal expectatrons models

on the actual lagged values of certifications, which by assumption are responding to future demand conditions. But people are not myopic in this world. On the basis of the fit alone, however, we may conclude that a myopic model accurately describes the data, when in fact people take explicit account of the future. As noted

in Zarkin

(1982)

the cobweb

and

rational

expectations

models imply strikingly different dynamics. In fig. 2 we compare the responses of the two models to a forecastable three-year increase in the number

of children

enrolled

in school,

beginning

This figure records the change in the certification previous steady-state paths and can be interpreted an anticipated

baby

school enrollment anticipated.

boom.

Again,

(conditioned

five years from today. paths relative to their as the implications of

due to the deterministic

on birth),

this boom

should

nature

of

be largely

Consistent with our earlier discussion, the rational expectations, forward-looking model responds before the increase in enrollments. In contrast, the cobweb model responds two years after the initial increase in enrollment and continues to increase beyond the end of the shock. After period twenty-five, the cobweb model continues the oscillatory pattern pictured here, with the amplitude slowly approaching zero. In terms of the scale in the figure, the cobweb model greater than 0.5, fifty-one periods after the transitory

forecasts a change shock ends!

Based on these simulation results, it is difficult to take the cobweb model seriously as an economic model of the decision to acquire secondary

school

sample period.

certification,

even though

it fits the data well over the

The model is based on a myopic

expectation

mechanism

which leads to persistent forecast errors over a long period of time. It is difficult to imagine that people making certification decisions would actually

behave

in such an irrational

manner

for such a long period

of

time.

3. Summary

and conclusions

In Zarkin (1982) we find, using a forward-looking rational expectations model, that future forecastable demand conditions are important determinants of the decision to acquire secondary school certification. At the same time, a myopic cobweb model fits the data only marginally worse. This paper demonstrates that we may be led to incorrectly accept a cobweb model of occupational choice if we look solely at the fit of the

G.A. Zarkin

/ Cobweb L;S.rational expectations

models

95

model. Because of its autoregressive specification, a cobweb model should fit most data well, regardless of the true underlying structure. This same feature makes it difficult in other contexts to distinguish a world in which agents are myopic from one in which they pursue a foward-looking, rational expectations mechanism. In investigating the dynamic properties of the two models, we find that a cobweb model implies a long, oscillatory adjustment path in response to a future anticipated baby boom. If we re-estimate the cobweb regression after the boom occurs, however, the good fit we would (presumably) obtain might lead the unwary into believing it was an accurate description of the secondary school certification decision.

References Freeman, R.B., 1971, The market for college-trained manpower (Harvard University Press, Cambridge, MA). Freeman, R.B., 1975a. Legal cobwebs: A recursive model of the market for new lawyers, Review of Economics and Statistics 57. May, 171-179. Freeman, R.B., 1975b, Supply and salary adjustments to the changing science manpower market: Physics. 1948-1973, American Economic Review 65, March, 27-39. Freeman, R.B., 1976a. A cobweb model of the supply and starting salary of new engineers. Industrial and Labor Relations Review 29, Jan., 236-248. Freeman, R.B., 1976b. The overeducated American (Academic Press, New York). Freeman, R.B. and J. Leonard, 1977, Autoregressive degree patterns: Evidence of endogenous cycles and the market? Industrial Relations Research Assocation, Proceedings of the Thirtieth Annual Winter Meeting, Dec., IO- 19. Zarkin. G., 1982, Occupational choice: An applicat,ion to the market for public school teachers. Duke University working paper series no. 82- 19, Nov.