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863 . .
3 1293 00099 1 Michigan State
University .

 

This is to certify that the

thesis entitled

The Demand For "Free" Goods

presented by

John William Reifel

has been accepted towards fulfillment
of the requirements for

Ph . D. degree in Economics

 

 

214M 8. A4104? '

Major professor
./

Date 4167/ lg] M72, ‘/

 

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ABSTRACT

THE DEMAND FOR "FREE" GOODS

BY

John William Reifel

A significant amount of government goods and
services are provided "free" to their recipients. However,
the demand in many cases has not equaled the expectation of
agency forecasts. This thesis examines the determinants
of the demand for such "free" goods offered to clients or
recipients at an explicit cost of zero.

The model considers the consumption of goods by an
individual to be an activity produced by combining physical
and time inputs. Therefore, all "free" goods have at
least an implicit cost in the form of consumption time
which must be allocated from alternative uses. In addition,
all forms of forfeited income, such as welfare or unemploy—
ment compensation, are included as costs to the individual.
Thus, in the specification of the model, all parameters
influencing the choice of the individual recipient are

included.

John William Reifel

The model was then used to establish explicitly
the individual's demand relationship for "free" vocational
rehabilitation services and his supply relationship of
time. These relationships were then estimated using cross
sectional data from the Division of Vocational Rehabilita-
tion of the Michigan Department of Education. The
statistical results generated by the application suggest

that the variables included in the model are important.

THE DEMAND FOR "FREE" GOODS

BY

John William Reifel

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Economics

1972

This thesis is dedicated
to my parents William and Bertha Reifel
for the opportunities they have given me.

ii

ACKNOWLEDGMENTS

I wish to express my gratitude to Professor Jesse S.
Hixson who guided my work from the proposal stage to the
final draft. He gave freely of his time and ideas and
promptly read the numerous drafts. I also gratefully
acknowledge the other members of my dissertation committee,
Professor Anthony Y. C. Koo and Professor Subbiah Kannappan,
for their helpful comments and suggestions.

I am indebted to the Division of Vocational Rehabili-
tation of the Michigan Department of Education for allowing
me to use data on a portion of their closed vocational
rehabilitation cases. Estimation of the demand model would
have been impossible without such data. I would especially
like to thank Mr. Gabriel Cifor and Dr. Robert D. Struthers
of the Division of Vocational Rehabilitation for their
assistance.

. Special thanks are also due to Mr. Ronald Tracy, my
computer programmer, for his technical assistance and to
Dr. David W. Dunlop for his knowledgeable insights into the
workings of vocational rehabilitation in Michigan.

Also, I express my thanks to the Department of
Economics at Michigan State University for financial support

during my years of graduate study.

iii

I assume full responsibility for any errors or

omissions.

iv

DEDICAT

TABLE OF CONTENTS

ION 0 O O O O O O O O O O O 0

ACKNOWLEDGMENTS . . . . . . . . . . .

LIST OF

Chapter
I.

II.

III.

IV.

BIBLIOG

APPENDI

TABLES O O O O O O O O O O O 0

INTRODUCTION . . . . . . . . . .
THE ANALYTICAL FRAMEWORK . . . . . .
Preliminary Formal Considerations . . .
Static Choice . . . . . . . . . .
Investment Activities . . . . . . .
"Free" Goods . . . . . . . . . .

THE EMPIRICAL APPLICATION . . . . . .

The Supply of Vocational Rehabilitation in

Michigan . . . . . . . .
The Individual' 3 Demand for Vocational
Rehabilitation Services . . . . . .
Formulation of the Hypotheses . . . .
Choice of Sample and Variables . . . .
Empirical Results . . . . . . . .
The Wage Prediction Equations . .
The Demand for Vocational Rehabilitation
Services . . . . . . . .

The Supply of Time . . . . . . .

SUMMARY AND CONCLUSIONS . . . . . .

RAP HY I O O O O O O O O O O O

X 0 O O O O O O O O O O O C O

Page
ii
iii

vi

l4
17

20

21
24
28
29
44
44

47
49

52

58

63

LIST OF TABLES

Table Page

1. Demand for vocational rehabilitation
regression results . . . . . . . . . 48

2. Supply of time to vocational rehabilitation
regression results . . . . . . . . . 50

3. Statistics on raw and transformed vocational
rehabilitation data . . . . . . . . 65

vi

CHAPTER I

INTRODUCTION

The decade of the 1960's witnessed a significant
growth in the amount of government goods and services
provided "free" to their recipients. Rapid funding
increases for established social welfare programs such
as public assistance, education, food stamps, and voca-
tional rehabilitation, combined with the inception of new
antipoverty programs such as job training, community
action, and VISTA, propelled the percentage of all
government spending allocated for social welfare programs
to 47% by 1970.1

However, demand for "free" goods in many cases has

not equalled the expectation of agency forecasts.

Typically, the cause of these discrepancies is the manner

 

1Alfred M. Skolnik and SOphie R. Dales, "Social
Welfare Expenditures, 1969-70," Social SecurityiBulletin,
XXXIII (December, 1970), 8-11.

2See, for example, Marian Gornick, "Use of Medical
Services as Demanded by the Urban Poor," American Journal
of Public Health, LIX (August, 1969), 1304; Mark Lepper,
"Approaches to Meeting Health Needs of Large Poverty
Populations," American Journal of Public Health, LVII
(July, 1967), 1154-55; and U.S., Department of Labor,
Manpower Administration, Manpower Report of the President-—
January 1969 (Washington: Government Printing Office,
1969), p. 201.

 

 

 

 

 

in which agency forecasts are made, usually being set at
the agency's estimation of the population's "need" for such
goods and services. "Need" is defined as the difference
between the population's present consumption and some
arbitrary standard deemed optimal by the agency but having
no economic basis.3

The consequence of this procedure for forecasting
"demand" is that forecasts rarely coincide with effective
demand. The "need" for the goods and services supplied by
a government agency is economically relevant only if
expressed by effective demand. In the case of government
programs where effective demand is less than agency fore-
casts, the implication for resource allocation is that,
if agencies are funded to the level of expected demand and
such demand does not materialize, resources will be wasted.

To overcome this difficulty, it is necessary to
introduce economic considerations into the procedure for
forecasting demand. Consumption of "free" goods and
services, though not requiring explicit money expenditures

by the recipient, will require use of his scarce time that

 

3Bibliographies of journal articles estimating the
"need" for health service programs are found in Abstracts
of Hospital Management Studies (Ann Arbor: The Cooperative
Information Center for Hospital Management Studies)
indexed under the heading "Health Services: Use, Need,
and Demand" and in Hospital Literature Index (Chicago:
American Hospital Association) under "Hospital and Health
Facilities: Need." Estimates of "needs" for manpower
training programs can be found in Manpower Report of the
President (Washington: Government Printing Office).

 

 

 

 

could be allocated to other activities. This generates an
impliCit opportunity cost to the recipient and demand
considerations immediately arise. Any forecast procedure
that ignores these economic considerations will be handi-
capped in achieving results that approximate effective
demand. Therefore, construction of a rational basis for
forecasting demand, consistent with economic theory, is
required.

The purpose of this dissertation is to facilitate
the inclusion of economic considerations in demand fore-
casts by developing a comprehensive model of consumer
behavior incorporating "free" goods and services. The
specification of the model will include all parameters
influencing the individual's choices. The model will
then be employed to estimate the demand for the "free"
good vocational rehabilitation services supplied by the
Department of Education in Michigan.

Several benefit-cost analyses of the "free" good
vocational rehabilitation services have already been made,
but the optimal social allocation of resources approach

has always been taken.4 This dissertation will view the

 

4See, for example, Ronald W. Conley, "A Benefit-
Cost Analysis of the Vocational Rehabilitation Program,"
The Journal of Human Resources, IV (Spring, 1969), 226—52;
Ronald W. Conley, The Economics of Vocational Rehabilita-
tion (Baltimore: Johns Hopkins Press, 1965); U.S.
Department of Health, Education, and Welfare, An Explora-
tory Cost-Benefits Analysis of Vocational Rehabilitation

 

rehabilitation process from both the client's demand for
services vieWpoint and from the client's supply of time
viewpoint.

Knowledge of the magnitude and significance of
the parameters in the model, combined with a knowledge of
the characteristics of the population to be served, will
allow develOpment of "free" good demand forecasts having
a rational economic basis. Also, knowledge about the
model parameters will suggest ways by which agency
officials can choose magnitudes of discretionary variables
so as to either increase or decrease effective demand.

Preliminary considerations and the development of
the comprehensive model of consumer equilibrium will be
handled in Chapter II. In Chapter III, the model will
be applied to the "free" good vocational rehabilitation
services and estimated using data from the Department of
Education of the State of Michigan. Conclusions and

implications of the analysis will be made in Chapter IV.

 

(Washington: Government Printing Office, 1967); and U.S.
Department of Health, Education, and Welfare, Human
Investment Programs: Vocational Rehabilitation
(Washington: Government Printing Office, 1967).

CHAPTER II

THE ANALYTICAL FRAMEWORK

PreliminaryAFormal Considerations

"Free" goods are goods or services supplied by a
dispensing agency to the consumer at a zero money price.
The spectrum of such "free" goods being considered ranges
from "free" money (welfare payments) to "free" services
(inoculations, dental care, birth control clinics, voca-
tional rehabilitation). Some "free" goods, such as food
and welfare payments, are used up in the immediate market
period and thus generate benefits only in that period.
Other "free" goods, such as inoculations and vocational
rehabilitation services, are human capital—enhancing and
hence generate benefits for longer periods of time.

Because of their nature, analysis of the demand
for "free" goods necessitates the use of a more comprehen-
sive model than the traditional static consumer equilibrium
model which envisions the individual allocating a fixed
income to consume that combination of market goods and
services that maximizes his utility. In this simple
model, the individual is constrained in his acquisition

of goods solely by his budget (his total money income),

which is assumed constant. This is reflected in the first-
order equilibrium conditions for optimal consumption which
state that the individual's equilibrium consumption is such
that the marginal utility of each good consumed is equal
to the product of its marginal cost and the marginal
utility of money.1 The application of this analysis to
"free" goods is inappropriate because their explicit money
price is zero and the income constraint is irrelevant.
The equilibrium consumption of a zero priced good is
independent of the income constraint.

While the individual's time constraint is not con-
sidered in the traditional static model, consumption of a
"free" good, or any good, requires the use of a portion of
the individual's limited time. In consuming a "free"
good, the individual pays an opportunity cost in terms of
the time required which must be allocated from alternative
uses. Therefore, considerations of "free" good consumption
require the development of a fully comprehensive model of
consumer choice whiCh includes the crucial implicit cost
of time and the total time constraint.

Furthermore, some "free" goods are human capital-
enhancing commodities, such as scholarships or vocational

rehabilitation. That is, their consumption is an

 

1Paul A. Samuelson, Foundations of Economic
Analysis, Harvard Economic Series, Vol. LXXX (Cambridge:
Harvard University Press, 1947), pp. 90-100.

 

investment which generates a stream of benefits that
extends into future periods. In addition, their con-
sumption may require more than one period. Since such
human capital-enhancing commodities introduce inter-
temporal considerations, the consumer framework must be
general enough to encompass the investment aspects of
consumer behavior.

Such a comprehensive framework for consumer demand
was developed by Becker, who views consumption as a pro-
duction activity and specifically incorporates time as
one of the inputs of the production process.2 In this
model, the individual is envisioned to produce consumed
goods by combining market goods and time subject to his
production functions. The individual produces that combina-
tion of commodities, subject to income and time constraints,
that maximizes his utility.

In viewing consumption as a production activity,
all of the inputs--both physical and time-~required for
the production of a commodity are explicitly included.
Thus, the opportunity cost of producing a given bundle of
commodities is made explicit. This exposes the crucial
nature of the time constraint. Time is needed both to
produce and consume commodities and to earn income in the

market with which to purchase the market good inputs.

 

2Gary S. Becker, "A Theory of the Allocation of
Time," Economic Journal, LXXV (September, 1965), 493-517.

 

Thus, Becker's model is essentially a model of the alloca-
tion of time. £While Becker makes no attempt to include
either commodities with an intertemporal nature or "free".
goods, his model does expose the crucial nature of the
time constraint, as well as the underlying input-output
relations.

Successful incorporation of "free" goods into a
consumer equilibrium model necessitates the development
of a comprehensive theory of consumer choice which
incorporates the possibility of zero priced inputs, con—
sumption of commodities with intertemporal consequences,
and all of the relevant constraints (production possi-
bilities, time, and budget) the individual faces. Since
the role of producer is being attributed to the individual,
it seems plausible that his actions should be describable
within a production theory framework. In the world of
the consumer, the production theory framework can be used
to describe how the individual combines physical inputs
and time, subject to his production functions and to
his time and budget constraints, to produce vectors of
present and future commodities that maximize his utility.

In the remainder of this chapter a model of
static utility maximization under certainty will be
develOped. This model will specify all of the important

decision variables relevant to an individual's utility

maximizing consumption decisions. From this model a

demand function for "free" goods will be derived.

Static Choice

 

In any market period, an individual is assumed to
engage in a set of activities, combining market purchased
goods and time to produce a set of commodities Zi’ i =
l,..., m.3 For example, if the commodity being consumed
is a meal (Ze), its production requires inputs of the dif-
ferent food items eaten as well as the time required to
purchase, prepare, and eat the food.

The production function for a commodity Zi may be
written, in general, as
i

i i i
(x1, x2,..., Xn’ T ) (l)

z.=fl
l

‘where Zi is the amount of commodity i produced by the

individual and x3, j = l,..., n, are the explicit

quantities used of the n physical inputs available for
use by the individual in producing commodity i. Ti is
the individual's time input in producing commodity i.
fi is this individual's production function for combining
the physical and time inputs (xg's and Ti) into the

consumed commodity Zi' The individual's production

technology for commodity Zi is embodied in fl. That is,

 

3Ibid., p. 495.

10

fi determines which market goods (xj's) may be used in
the production activity and how they may be combined with
time to produce Zi' fi is unique for this individual and
depends on the circumstances or characteristics of the
individual at a particular point in time.

Generally, the individual's production possi-
bilities are specified by the set of m economic production
functions

Z. = f1(xl

1 ll X:I°'°l X1: T1), 1 = l,..., m, (2)

n

one for each of the m commodities the individual may
produce. For a given rate of final output Zi = fi, fi,

i = l,..., m, defines an isoquant in the input space
giving the technical range of substitution of the xj and
T in production of 21' However, Zi represents only the
quantitative dimension of commodity i. If factor pro-
portions are continuously variable along the isoquant,
the qualitative dimension of the commodity may be assumed
to be determined by the factor proportions selected. To
the extent that the production function allows substitu—
tion between factors, therefore, the individual can alter

the qualitative nature of the commodity by changing the

relative proportions of the factors used.4 For example,

 

4This method of characterizing the commodity is
analogous to that suggested by Sven Dano, Industrial
Production Models (New York: Springer Verlag New York,
1966), PP. 132-47.

 

 

11

if a housewife is preparing an evening meal (Ze), she
could choose between many different combinations of food
items and her time. With the use of alternative food item
combinations, the meal will vary qualitatively on the
basis of nutritional value, taste, aroma, etc. Also,
since many food items are available in semiprepared
states, the housewife can use more or less time in pre-
paring the meal. In general, the quantitative and quali—
tative properties of a commodity are completely determined
by its input vector (xi, x:,..., xi, Ti) and its produc—
tion function.

The individual's production set G, the activities
(consumed commodities) technically possible for the
individual, is defined by combining the time constraint
(Iii = T) with his production possibilities (2). That is,
since the individual's total time T available in the
market period is fixed, this time constraint must be
incorporated with his production functions (each of which

requires a time input T1) to generate his production set

(G). This defines

m
G = {glg = (fl, f2,..., fm, x1, xl,...,Xn
rl,..., Tm); XI, T1 > 0,
J —
m1 = T} (3)

12

Since the individual is considered to be a pro-
ducer of the commodities he consumes, our theory must
provide solutions for the quantitative final commodity
vector Z = [Zl,..., Zm] produced and for the qualitative
choice of production methods (input proportions) used.
This is facilitated by assuming that the individual's
preferences are defined on the entire production set G,
i.e., that the individual has ordered preferences for
both the quantities of commodities produced (consumed)
and for the production techniques (factor proportions)
used to produce them. Thus, the individual's utility

function is of the form

1 l l m

Xn' T ;...; xl,..., Xm, Tm). (4)

n

The individual's choice of preferred gsG is
restricted by a budget constraint, i.e., expenditures per
period must not exceed receipts. One of the activities
the individual may engage in is work for an employer.

The rate of remuneration W received may be assumed to
depend on the individual's labor skills. Total work
remuneration is therefore WTw where Tw is the amount of
time spent on the job. Also, the individual may be the
recipient of non-wage income V such as interest, dividends,
social security, pensions, disability income, and unemploy-

ment compensation. His total income Y is then defined as

l3

WIW + V, wage income plus non-wage income. To acquire the
market goods (xj's) used as inputs in the production of
commodities, the individual must pay the market prices of
the goods used. Therefore, given the vector of market
good prices P, his obtainable wage rate W, and his non-
wage income V, the individual's choice is constrained to
those productions for which Px i WTw + V. The set of
attainable productions K for a given vector [P,W,V] can

now be defined as
K(p,w,V) = {geGl PX : WTW + V} (5)

For a given [P,W,V], the individual chooses that
production gEK which maximizes his utility as determined
by his preference function. This is his equilibrium con-
sumption relative to [P,W,V]. The individual's demand
correspondence associating equilibrium consumptions with
given parameter values in [P,W,V] is therefore obtained
by varying K over G to obtain the individual's choice
response to changes in P, W, and V. In this way one

obtains the individual's demand correspondence

1 l l l m m

m
Z = [Xll X21000! an T 7-0-1 X1! X21000! an

Tm; 1‘”) = ¢(P,W,Vl{f}). (6)

14

Investment Activities

 

Investment activities are consumptions resulting
in an alteration of one or more of the individual's already
available production functions fi and/or in an expansion of
the set of attainable productions available to the indi-
vidual in present and future periods. These alterations
of the production functions generate an expansion of the
individual's production possibilities set G. Usually,
investment activities require more than one period.
Examples of such investment activities that are human
capital—enhancing include educational training, exercise
programs, and vocational rehabilitation.

Investment activities can be incorporated into the
previous analysis by recognizing that each geG defines a
specific set of present consumptions and therefore a
specific set of attainable productions in future periods,
depending on which, if any, investment activities are
included in 9. That is, each g has an intertemporal
significance in that it defines a future set of G's which

can be expressed as

rt=llooorNI (7)

where G; is the individual's production possibility set
in period t if he selects g in the present period, and
where N is the last period in the individual's time

horizon. If g includes investment activities then the

15

G , t = l,..., N, will be different, expressing the

altered set of production possibilities that become

available through time. If g contains no investment
activities then all GS, t = l,..., N can be assumed
identical.

The motivation for an individual to select a
gEG containing one or more investment activities is that
he prefers some commodity set which will become avail—
able in the expanded production possibility sets resulting
in future periods. Consumption of multiperiod investment
activities that expand an individual's production possi-
bility sets in future periods requires the individual
to allocate some resources from the production of utility
yielding commodities during the investment period. This
implies that the individual's time stream of consumption
20,..., ZH--where Zt = (Z§,..., Z3) and H is the number
of periods required for consumption of the investment
activity—-will be such that if he chooses a multiperiod
investment activity, then the quantity and/or quality of
commodities consumed must be reduced for the duration of
the investment period so that resources can be allocated
to the investment activity.

It is assumed that the individual's preference

function is defined over all alternative time streams of

consumptions that are possible as a result of possible

l6

investment consumptions. To arrive at a price in the
present period for a multiperiod investment, it is
necessary to discount the multiperiod costs of the
investment to the present. In other words, it is
assumed that the individual must, in the present period,
set aside enough money to pay for both the inputs in the
present period and, with interest collected on the
remainder that is put aside, the costs for the entire
investment period when they come due. Since the
individual must pay for his multiperiod investment in
the present period, the interest rate--denoted by d--used
to discount the costs becomes a crucial parameter and
must be considered along with P, W, and V as part of the
individual's economic environment.

Therefore, investments--except for the multi-
period nature of their benefits and costs--are not dif-
ferent from other types of consumptions. For this
reason, they are also included in the individual's pro-
duction set G and are routinely considered by the
individual for consumption. That is, the individual's
demand correspondence associating equilibrium consumptions
with given parameter values is obtained, in the inter-
termporal case, by varying the price vector, the wage
rate, non-earned income, and the interest rate and noting

the individual's choice response to changes in these

17

parameters. In this way the present period's equilibrium

consumption (ZO) is obtained as

o_o o_
z — [21,..., zm] — ©(P,w,v,dl{f}) (8)

"Free" Goods

 

The last objectives of this chapter are to incor-
porate so called "free" goods into the analytical frame-
work and to explicitly obtain a demand correspondence
for such "free" goods. As explained in the introduction,

a "free" good is a good or service (i.e., an xi) that is
offered by a dispensing agency at an explicit money price
of zero. The good may have a single or a multiperiod
nature. The individual combines these "free" inputs with
his time to produce consumptions which maximize his utility.
Some "free" goods, such as food and welfare payments, are
used up in the immediate market period. Other "free"
goods, such as human capital-enhancing services, are

inputs in investment activities that alter the individual's
set of production functions and thereby change the
individual's future production possibilities. The benefits
of such "free" goods have a multiperiod nature.

From (2) it is known that the individual combines
market inputs and a time input to produce a consumed
commodity. In the production process, the individual

demands the market inputs used and supplies the time

18

input. The demand relationship for a market input (an
X?) is obtained from relation (8) as
i

= i
xj ¢j(P,w,V,dl{f}). (9)

Likewise, the supply relationship of time by the
individual is obtained from relation (8) as

r1 = ¢1(P.W.v,dl{f}). (10)

When an input or set of inputs is "free," the

individual's crucial consideration becomes the Opportunity
cost of time allocated to consume the "free" good. The
quantity demanded, therefore, of a "free" good will vary
with the time cost of consuming it. In order to discuss
the demand curve of a "free" good x; in the price-quantity

plane, we define the price or time cost as

W(—T) (11)

where Ti/X§ is the time required per unit of x; consumed,
and W is the individual's Opportunity wage rate. An
agency may initiate a change in the time cost by changing
the rate at which an individual may consume x; per Ti.
The individual's demand response to such a change
d(Ti/X§) < 0 will be characterized by the usual income

and substitution effects. The substitution effect will

19

be negative as usual. Since the time cost per unit of x;
consumed has fallen, x; will be relatively less expensive
to consume and the individual will substitute into it
dx§2>0. The income effect will be indeterminate because

for a given T, an individual can consume more of everything,

1
so dx.

-:—0.

Agencies supplying "free" goods view the process
differently. In supplying physical input "free" to an
individual, they demand time from him for production of
the consumption. The individual's time supply response
to an increase d(X§/Ti) >0 in the price per unit of time
supplied can also be analyzed in terms of the implied
income and substitution effects. Since this price per
unit of time supplied is really a negative price or a
benefit--like a wage rate--the substitution effect will
be positive dTi >0. The income effect will be indeter-
minate because the larger price may induce him to supply

more time, supply the same amount of time, or supply less

time.

CHAPTER III

THE EMPIRICAL APPLICATION

In the previous chapter a theoretical model of
static utility maximizing consumer equilibrium under
certainty, including "free" goods (zero priced market
inputs used by individuals in the production of certain
commodities), was developed. This model was constructed
from the individual's vieWpoint. It was in essence a
model describing how the individual allocates his time to
produce consumptions so as to maximize his utility.

Since the model considered the individual to be a pro-
ducer of the consumed commodities (by combining the
physical and time inputs required for their production),
the individual demands the market inputs and supplies the
time input needed for the production activity. The
chapter concluded with the establishment of demand rela-
tionships for the physical inputs used and of a supply
relationship of time.

The purpose for developing the consumer equilibrium
model was to derive the individual's demand relationship
for a "free" market input. This is the relationship that
is of paramount importance to the suppliers of such "free"

goods. The primary purpose of this chapter is to

20

21

establish explicitly the individual's demand relationship
for "free" vocational rehabilitation services. That is,
the model developed in the previous chapter will now be
used to obtain a demand function for a particular "free"
good which then will be estimated. Before deriving the
particular demand function, it will be necessary to
examine the Operation of the vocational rehabilitation
program in Michigan and the individual's vocational
rehabilitation decision.

The Supply of Vocational Rehabilitation
in Michigan

 

 

The vocational rehabilitation program in Michigan
is organized for the purpose of rehabilitating any
individual who for some physical or mental reason is
unable to participate in the activities of life to the
same extent as other individuals in his age and sex group.

The State of Michigan is divided into five regional
Offices which are further subdivided into 25 district
offices. The vocational rehabilitation program operates
as follows. First, a potential recipient arranges an
interview at one of the district Offices. He may have
decided to do this as a first step on his own initiative

or he may have been referred there by some other agency

 

1Michigan Department of Education, A State Plan
for Vocational Rehabilitation Services, Volume 2 Technical
Appendices (Lansing: Department of Education, 1970),
p. 55.

 

 

 

22

which he first contacted, such as the Department of Social
Services or the Department Of Health.

During this initial interview, the client is
assigned a particular counselor who records a series of
data about the individual, including the disability that
the client feels he is suffering from.

Next, the client is sent to a medical doctor for a
general examination and to investigate the reported dis—
ability to see if, in fact, such a disability exists, and,
if it exists, to determine exactly how severe it is. The
medical examiner then sends the counselor his findings.

On the basis of this information a decision is made about
accepting the client. If the client is accepted, another
interview is arranged with the counselor and another series
of data items are recorded that describe the client's
socio-economic situation before vocational rehabilitation
services are begun.

If the client is not accepted, it may be due to
any of a number of reasons, including the following:
unable to contact client, client moved, client died,
client failed to cooperate, client found to have no
disabling condition, client found to have no vocational
handicap, and client refused services. The primary
reason why a client would be refused services is that the

district does not have enough funds allocated from the

23

state to handle all the cases of this type of disability
that appear for treatment. The percentage of potential
recipients having a particular disability that are refused
services because of a lack of funds depends on the
particular disability they have. This is the case
because the state allocates vocational rehabilitation
funds on the basis of disability type, not in lump sums.
The district office then allocates its funds for each
specific disability type on a first-come, first-served
basis. When the allocation for a particular disability
type in any district becomes fully committed for the year,
then that district must refuse to accept any more clients
with that disability type for the remainder of that

fiscal year.

If the client is accepted for vocational rehabilita-
tion services, the counselor and the client then together
formulate the rehabilitation plan necessary for the
solution of the client's problems. Services available
include diagnostic procedures (medical, psychological,
social, and vocational), physical restoration services
(surgery, prosthetic appliances, and hospital care), and
training (remedial, literacy, vocational, academic, on-
the-job). Through a bargaining process, the client
attempts to procure his equilibrium set of vocational

rehabilitation services. After some negotiations, a

24

mutually agreeable program of vocational rehabilitation is
decided upon. Then, the plan is implemented.

Once services are begun, they continue until the
client is rehabilitated and a new job is found or until
he drOps out of the program. The rehabilitation process
involves two stages. First, medical treatment is used to
correct the client's disability. Second, the client is
then given some vocational training which will enable him
to pursue a new career. When the client either becomes
rehabilitated and finds a new job or else drops out of
the program, his case is closed. Closure of the client's
case includes the collection of a final set of data. A
total dollar value of services supplied is recorded.
Unfortunately, dollar quantities of each specific service
rendered are not recorded. However, available records do
indicate the broad categories (diagnostic, restoration,
training) from which the client's equilibrium set of
services come.

The Individual's Demand for Vocational
RehabilitaEIOn Services

 

 

Vocational rehabilitation is a "free" good whose
consumption alters the individual's production functions
and expands the set of activities available to the
individual. For example, acquisition of an artificial leg

by an amputee enables him to change the manner in which he

25

produces many commodities by reducing the need for outside
help. Also, it Opens up a whole new range of activities
which were previously denied him such as walking or
driving an automobile.

Since alterations Of the production set result from
consumption of vocational rehabilitation services, such
consumption is an investment activity. When the individual
surveys his production possibilities set G in the present
period (t = 0), one of the commodities he contemplates is
vocational rehabilitation. The benefits in the form of an
altered production set that would accrue to the individual
from consumption of vocational rehabilitation depend on
the severity of the disability and therefore on how
restricted his production function set presently is. If
he is not disabled, consumption of vocational rehabilita-
tion will allow only minimal alterations of his production
functions and therefore only a marginal expansion of his
production possibilities. Conversely, if the individual
is greatly disabled, many alterations in his production
functions are possible and therefore a large expansion of
his production possibilities is feasible.

Also, since vocational rehabilitation is a human
capital-enhancing commodity, its consumption will
facilitate not only an expansion of the production set G,

but, if labor skills are improved, the individual's

26

obtainable wage rate will rise and this will enable him
in the future to consume larger commodity bundles for any
given work effort (Tw). That is, since his wage rate
rises, for any given work effort, his budget constraint
will be expanded. This implies that his equilibrium

t
m),t>H

consumptions in future periods Zt = (ZE,..., Z
(where H is the final period in which the investment com-
modity vocational rehabilitation must be consumed) can be
greater. Thus, the benefits to be derived from consumption
of vocational rehabilitation are in the form of an expanded
production possibility set (in proportion to how badly
disabled the individual is) and of expanded equilibrium
consumptions in future periods due to an increased wage
rate facilitating an enlarged budget constraint.

Since consumption of most vocational rehabilita-
tion services requires more than one market period, the
costs, as well as the benefits, have a multiperiod nature.
To arrive at a price in the present period for a multi-
period vocational rehabilitation investment, it is
necessary to discount the multiperiod costs to the present
using the interest rate d. Thus, it is assumed that the
individual must, in the present period, set aside enough
money to pay for both the inputs in the present period
and, with interest collected on the remainder that is put

aside, the costs for the entire investment period when

they come due. However, since vocational rehabilitation

27

services are "free,' the individual does not have to
allocate income to pay for such inputs. Nevertheless,

the individual is still faced with a two component cost

of consuming vocational rehabilitation.

The first component consists of the time input
required for a vocational rehabilitation investment.

Since time is not transferable through time, it is
impossible for the individual to set aside time today
which can draw interest and be used in future periods

when it is needed. Rather, the individual can assess

the time cost by assuming that any amount of time spent

in consuming vocational rehabilitation could alternatively
be spent working and earning income at his present wage
rate. Thus, a magnitude for the time cost can be arrived
at by discounting, using d, the future time inputs valued
at the individual's present market wage rate.

The second component of the cost of consuming
vocational rehabilitation stems from the institutional
arrangements whereby agencies dispensing disability and
public assistance payments reduce the recipient's welfare
payments upon his completion of a vocational rehabilitation
program. The individual can place a present value on the
fall of such non-earned income by discounting to the
present, using d, the fall experienced in such benefits.

Therefore, combining the time lost component and the

28

forfeited public assistance component gives a cost for

consumption of vocational rehabilitation by the individual.

Formulation of the Hypotheses

 

In the preceeding section it was shown that the
benefits to be derived from consumption of vocational
rehabilitation are in the form of an expanded production
possibility set and of expanded equilibrium consumptions
in future periods due to an increased wage facilitating
an enlarged budget constraint. The magnitude of this second
benefit can be estimated by discounting to the present the
yearly increase in earnings attributable to the vocational
rehabilitation investment for the remainder of the
individual's working life.

It was also shown in the preceeding section that
the costs incurred in consuming vocational rehabilitation
consist of time that could be allocated to alternative uses
and disability payments that are forfeited upon completion
of the program. The time cost can be determined by obtain-
ing the product of the individual's monthly earnings and
an index of time supplied (in months). The forfeited dis-
ability payment cost can be determined by discounting to
the present the yearly fall in disability payments for the
remainder of the individual's life.

In contemplating the relative desirability of con-

suming vocational rehabilitation, the individual will weigh

29

the costs and benefits, and his demand for services and
his supply of time will depend on them. This leads to
the empirical hypothesis: "An individual's demand for
'free' vocational rehabilitation services is a function
of the costs and of the benefits generated from such a
demand." This also leads to the empirical hypothesis:
"An individual's supply of time to the vocational
rehabilitation process is a function of the costs and of

the benefits generated from such a supply."

Choice of Sample and Variables

 

Data collected at each of the different stages of
the vocational rehabilitation process for all clients in
Michigan has been recorded on computer tapes which are
kept in Lansing. It is this supply process that generates
the data on vocational rehabilitation that will be used to
estimate the demand function for services and the supply
function of time. Recently closed cases (third quarter of
fiscal year 1970) will be used for estimating the model
because the recorded responses from these cases are more
accurate and more observations are complete than on cases
from earlier years.

Data recorded on services consumed are observations
of individual's equilibrium consumptions. They observe
where supply equals demand for consumption of vocational

rehabilitation services. That is, the vector of vocational

3O

rehabilitation services that is observed and recorded as

being consumed is the vector agreed upon by the counselor
and the client. It is the equilibrium output determined

by the client's demand and the agency's supply.

In the present analysis it is possible to abstract
from simultaneous equation problems concerning the indi-
vidual's equilibrium consumption of vocational rehabilita-
tion services by choosing for examination that type of
disability for which there are minimal supply limitations,
i.e., no rationing is necessary. A direct regression of
the amount Of vocational rehabilitation services demanded
onto the independent variables is justified if, in fact,
the values of all the independent variables used in the
regressions are generated independently of the dependent
variable.

However, if supply limitations exist, then the
dispensing agency must ration the limited services accord-
ing to some criterion. If the agency has control over
the magnitude of some parameter, such as cost, that appears
in the individual's demand equation, then by varying its
magnitude, it can influence the quantity of services
demanded by the client. Hence, the magnitude of this
parameter would not be generated independently of the
dependent variable and it would not be truly exogenous.

This would call for a simultaneous equation model. The

31

observed (equilibrium) values of the dependent variable
and the agency determined parameter should be considered
as being determined simultaneously.

By choosing for examination that disability type
which has minimal rejections due to supply limitations,
it is possible to abstract from the simultaneous equation
problem. As was explained earlier, the state allocates
vocational rehabilitation funds to its district Offices on
the basis of disability type, not in lump sums. The
district office then allocates its funds for each specific
disability type on a first-come, first-served basis until
the allocation for that particular disability type in that
district becomes fully committed for the year. Then that
district must refuse to accept any more clients with that
disability type for the remainder of that fiscal year.
The size of the allocation each district receives for
each disability type depends on how the state allocates
its limited budget. The state's vocational rehabilitation
case priorities are revealed by the relative sizes of its
allocations for different disability types. Examinations
of the ratios of acceptances to rejections for each dis-
ability type reveals which disability types are well
funded (not subject to budget or supply limitations) and
which are not. Choosing that disability type having the

highest acceptance to rejection ratio minimizes the

32

possibility of simultaneous equation problems. Analysis
of the more than 6,000 cases closed in the third quarter
of fiscal year 1970 shows that the type of disability in
which individuals have lost one or both major lower
extremities (feet or legs) has the highest acceptance to
rejection ratio (9.7 to 1). This 55 case subset will be
the sample used.

Construction of empirically testable formulations
of the hypotheses requires observations On all the
parameters in the hypotheses. The hypotheses state that
both the services demanded and the time supplied depend
on the costs and the benefits generated by the demanding
and the supplying.

On the demand for services side, the benefits will
be calculated as a return per unit of service received.
The basic unit of service will be a dollar's expenditure
by the agency. The return will be the discounted present
value of the yearly increase in earnings (B) attributed
to the vocational rehabilitation investment for the
remainder of the individual's working life. The return
per unit of service is then computed as B/Q where Q is
the dollar amount of services the agency supplies. The
client is made aware of the approximate magnitude of B/Q
at the time he and his counselor negotiate his vocational

rehabilitation program.

33

Since data has been kept on the individual's
weekly earnings both before vocational rehabilitation
services were begun (wb) and after they were concluded
(we), B can readily be determined as

65 (we - wb)52

B = Z a (12)
a=n (1-+d)

 

where w is the number of months the client is in the
program divided by 12. Use of n is neceSsary because the
individual's higher wage rate does not begin until he
completes the program.

Though 17 individuals in the sample had positive
weekly earnings both before and after making the vocational
rehabilitation investment, 38 individuals did not have a
job before making the investment, 11 did not have a job
after making the investment, and 11 did not have a job
either before or after making the investment. Several
possibilities exist to explain why an individual did not
have a job before making the vocational rehabilitation
investment. First, he could have been so disabled that he
could not obtain a job. Second, he may be a young man just
coming onto the labor market but who has not yet found his
first job. Third, he may be able to get a job, but the
pay is so low that he values higher the domestic non-market

work that he can do by staying at home.

34

Since estimation of the significance of the
increased earnings benefit from vocational rehabilitation
requires data on earnings both before and after vocational
rehabilitation, it will be necessary to estimate the
Opportunity earnings of those individuals who did not have
earnings before, after, or both before and after consuming
vocational rehabilitation. This can easily be accom—
plished using a prediction equation. Since data has been
collected on most of the items likely to affect earnings,
it is possible to regress weekly earnings before vocational
rehabilitation, for all individuals who had such earnings,
on the set of explanatory variables. Similarly, it is
possible to regress weekly earnings after vocational
rehabilitation, for all individuals who had such earnings,
on the set of explanatory variables. Using the coeffi~
cients that result from this procedure, it is then possible
to construct two prediction equations-~one for weekly
earnings before vocational rehabilitation and one for
weekly earnings after vocational rehabilitation. These
two prediction equations will generate values for weekly
earnings before and after vocational rehabilitation for
those individuals who do not have them.

The explanatory variables to be included in the
prediction equation for weekly earnings and the reasons
for including them will now be examined. First, a

continuous variable indicating the number of years of

35

school completed by the individual will be used. Numerous
studies have shown the significance of education in
explaining earnings differentials between individuals.
Next, binary variables for race and sex will be
used. Studies by Thurow, Weiss, and Welch show that
education has lower financial value to blacks than to
whites, even if differences in the quality of schooling
are accounted for.2 Thurow's work suggests that blacks
fail to receive on—the—job training which is necessary,
in addition to education, to raise black incomes.
Studies by Holtman, Bayer, r, and White revealed
that employers were paying women less than men for the
same work in a wide range of occupations.4 A study by

Sanborn found discrimination against women by customers

 

2Lester C. Thurow, "The Occupational Distribution
of the Returns to Education and Experience for Whites and
Negroes," American Statistical Association, Proceedings
of the Social StatiStics Section 1967, X (1967), 233-44;
R. D. Weiss, “The Effect of Education on the Earnings of
Blacks and Whites," Review of Economics and Statistics,
LII (May, 1970), 150-59; and Finis Welch, "Labor—Market
Discrimination: An Interpretation of Income Differences
in the Rural South," Journal of Political Economy, LXXV
(June, 1967), 225-40.

 

 

 

3Thurow, Op. cit., p. 241.

4A. G. Holtman and A. E. Bayer, "Determinants of
Professional Income Among Recent Recipients of Natural
Science Doctorates," Journal of Business, XLIII (October,
1970), 410-18; Donald J. McNulty, "Differences in Pay
Between Men and Women Workers," Monthly Labor Review, XC
(December, 1967), 40-43; James White, "Women in the Law,"
Michigan Law Review, LXV (April, 1967), 1051-1122.

 

 

 

36

and fellow employees.5 These empirical studies support
the contention that individuals who are members of racial
minorities or who are women can expect lower earnings for

the same quantity and quality of work, ceteris paribus.

 

A continuous variable indicating the client's age
will also be included in quadratic form, the assumption
being that the individual's earnings will increase with
age, as he acquires more experience, but an age will
eventually come after which decreased physical productivity
will be reflected in falling earnings.

So, the weekly wage prediction equation for clients,
all of whom had the same disability category and were not
employed at the time they entered the vocational rehabili-

tation program, becomes

A _ 2
wb — Y0 + 71E + YZR + Y3S + Y4A + YSA + e (13)

where the y's are determined from the data on individuals
who were employed at the time of entrance into the voca-
tional rehabilitation program.

The weekly wage prediction equation for clients
who did not take a job at the end of their vocational

rehabilitation program becomes

 

5Henry Sanborn, "Pay Differences Between Men and
Women," Industrial and Labor Relations Review, XVII
(July, 1964), 534-50.

 

37

A _ 2
we — 60 + 61E + 62R + 538 + 64A + 65A + u (14)

where the 6's are determined from the data on individuals
who acquired employment at the conclusion of their voca-
tional rehabilitation program.

In the demand for services formulation, costs will
be calculated as a time price per unit of service demanded
and as forfeited disability payments per unit of service
demanded. The time price will be calculated by dividing
the monetary value of the time supplied C1 by Q. The
monetary value of the time supplied by the individual must
be estimated from the available data. An estimated time
input will be transformed into months of time supplied and
then multiplied by the individual's monthly earnings before
vocational rehabilitation services were begun to get a
dollar value for his time input. In estimating amount of
time supplied by the individual, data is available on
the type of services received (diagnostic, restoration,
on-the-job training, formal schooling such as vocational
or college, and maintenance) and for how many months these
services were received. By estimating how time intensive
reception of these different services was, it is possible
to construct an index of time inputs whose value depends
on the services received and the length of time in the
program. Once this quantity of time supplied in terms of

months is determined, it can be multiplied by the

38

individual's prevocational rehabilitation monthly earnings
to get a dollar value of time supplied. This opportunity

time cost Cl can be expressed as

(E months of time supplied) monthly earned income

C1 = for each service x before vocational
all services rehabilitation (l5)
requiring a time
input

Only those services that required use of the
individual's time were included in the index. Those
services included and their assigned weights are as
follows. First, all clients received diagnostic services
in the form of a thorough medical check-up. The time
assigned for diagnostic services is one day. Next,
restoration services were provided. Since the particular
type of disability selected for examination consiSts of
individuals who have lost one or both major lower
extremities (feet or legs), restoration services included
provision of and adaptation to artificial limbs. The time
assigned for these services is two weeks plus one day per
month for the number of months the individual stayed in
the program. These additional days represent medical
check-ups.

Academic training services, such as vocational
school or college, were the only other services provided
that were considered time consuming. They were weighted

as requiring full usage of the individual's time. [Other

 

39

services provided such as maintenance payments do not
require use of the client's time and are therefore not
included in the time index.

The forfeited disability payments per unit of
service demanded price is calculated by dividing the
discounted present value of forfeited yearly disability

income payments C by Q. Symbolically

2
c = Em (av-12)
2 at” "('1"""'+ (1)5 (16)

represents the forfeited disability income cost where AV
is the fall in monthly disability income payments and
E(A) is the expected number of years of life the individual
has remaining, as determined by an expected life table.
On the basis of this information, the demand

relation to be estimated is:

C1 B C2
= —— — —— +
Q a0+al(Q)+a2(Q)+a3(Q) 2; (17)
where
Q = the dollar amount of vocational rehabilitation
expenditures demanded by (supplied to) the
client
Cl/Q = the time price per unit of service demanded
B/Q = the discounted present value of the return
per unit of service demanded
CZ/Q = the discounted present value of forfeited

disability payments per unit of service
demanded.

40

It is expected that a will be positive in all

0
cases because it is expected that most individuals would
demand some of the services to increase their information
about the program. Also, available data do not record
the benefits generated by an expanded production possi-
bility set. This source of benefits is likely to be
important and will lead to positive do's. It is expected

that d1 will be negative because d1 is the coefficient

of the time price of the services. At higher prices,

less will be demanded. It is expected that d2 will be
positive because the larger the return per unit of service,

the more services will be desired. The sign of d in (17)

3
is expected to be negative because if an individual's
disability benefits decrease, he will be expected to
demand less services. Of the 55 cases in the sample, only
four had changes in disability benefits. Consequently, in
estimating (17), d3 will be restricted to zero because of
the lack of explanatory power of the four Observations in
this category.

On the supply side, time supplied will be measured
by the total number of calendar months M the individual
continues in his agreed upon vocational rehabilitation
program, supplying whatever amount of time per month

that his program requires. M will be regressed on a time

price per month in the program, on a services provided per

41

month supplied price, and on a measure of financial
benefits generated per unit of time supplied. The time
price is Cl divided by M. This is a measure of the time
cost for being in the program one month. It will be less
than the individual's potential monthly earnings unless
his rehabilitation program requires the full use of his
time. The services provided price per month in the program
is Q divided by M. This price, or benefit rate, tells how
intensively the agency is supplying services per unit of
time supplied by the client. The financial benefit per
month of time spent in the program will be measured as B
divided by M. This is a measure of the discounted present
value of the increased earnings generated per month spent
in the rehabilitation program.

This formulation of the supply of time relation

then becomes

C
_ _l 9 13.
M — 80 + 81(D4) + 82(M) + 83(M) + w (18)
where
M = the total number of calendar months the
individual supplies to the rehabilitation
program
Cl/M = the time price per month supplied
Q/M = the services provided per month supplied
price
B/M = the discounted present value of the increased

earnings per month of time supplied.

42

It is expected that 80 will be positive because
individuals are likely to be willing to supply some time
to gain more information about the program. Also, 80
will be positive because it will reflect the non-pecuniary
benefit of the expanded production possibility set. Since
the magnitude of this benefit is not observable, its
effect will appear in the constant term.

The sign of 81 is unpredictable because Cl is the
product of the client's time index and his monthly oppor—
tunity earnings. If we have two clients, alike in all
respects except that the second individual has a larger
monthly opportunity earnings, we might expect the second
individual to supply less time since his price of time per
month would be higher or we might expect him to supply more
time because he could acquire his needed consumptions by
working fewer hours and therefore have more time remaining
to supply to a rehabilitation program. Either possibility
follows from the model in Chapter II. If the substitution
effect generated by larger Cl/M dominates, he will supply
less time and 81 will be negative. If the income effect
dominates, he will supply more time and 81 will be positive.
Which effect dominates is an empirical question.

The sign of 82 is unpredictable. As shown in
Chapter II, the substitution effect generated by increased

Q/M will be positive since Q/M is actually a benefit rate.

Which effect dominates is an empirical question.

43

It is expected that 83 will be positive because
the larger the financial benefit per month spent in the
program, the more time the individual would be likely to
supply to the program.

It should be noted that implicit in the above
demand and supply formulations is the assumption that
marginal cost is equal to average cost and marginal
benefit is equal to average benefit over the domain of
C and B.

These relations will be estimated with several
variations. First, two different interest rates—-the
triple A bond rate and the bank savings rate--will be
used. In 1968, the time at which these vocational
rehabilitation decisions were made, the triple A bond
rate was 6.18%6 and the bank savings rate was 4%. Then,
because there is some evidence that individuals inten-
tionally understate their earnings during the initial
interview in order to establish greater need,7 the
relations will be estimated using before and after
earnings figures which are the maximum of reported
earnings and predicted earnings as given by the prediction

equations.

 

6Moody's Industrial Manual 1971, p. a18.

7U.S. Department of Health, Education, and Welfare,
Human Investment Programs: Vocational Rehabilitation
(Washington: Government Printing OffiCe, 1967), p. 24.

 

44

Empirical Results

 

The Wage Prediction Equations

 

Prevocational rehabilitation.--The prevocational

 

rehabilitation wage prediction equation (13) estimated for
the 17 individuals who had jobs before beginning voca-

tional rehabilitation is as follows:

Q = 71.147 + 1.584E + 38.483R + 55.7455 - 2.178A + .018A2

(232.86) (8.05) (46.96) (64.23) (9.05) (0.09)

(19)

Standard errors are given in parentheses below the coef-
ficients. R2 was .2114 and Hz was -.1471. There were 11
degrees of freedom. F(5,ll) was .5897 and its signifi-
cance was .708. The signs Of E, R, and S were all positive
as predicted. However, the signs of A and A2 were (—) and
(+), the opposite of what was expected. None of the co-
efficients were significantly different from zero.

These less than encouraging results prompted
several experiments which failed. First, a race-sex inter-
action term was introduced on the assumption that the
income differential between a male minority worker and a
male non—minority worker is greater than the differential
between a female minority worker and a female non-minority
worker. This effort failed because there was no black

female with a wage before vocational rehabilitation.

45

This made the matrix of independent variables (the X X
matrix) singular because the RS term was linearly
dependent on the constant term, R, and S.

Second, a rural-urban index was introduced to
reflect wage differentials between urban and rural areas,
the assumption being that if the cost of living is higher

in urban areas, ceteris paribus, then urban wage rates

 

will be higher than rural wage rates. Since each client's
county Of residence was known, census data was used to
separate all Michigan counties into either rural or urban
classifications and a rural-urban binary explanatory
variable was introduced. The problem that arose here was

that all 17 observations were from urban counties.

Postvocational rehabilitation.--The postvocational

 

rehabilitation wage prediction equation (14) estimated for
the 44 individuals who had jobs after concluding their
vocational rehabilitation services is as follows:

A 2
w = —76.l32 + 5.178E + 18.938R + 64.3208 + 4.430A - 0.056A
(53.14) (3.66) (19.21) (19.09) (2.15) (0.03)

(20)

Standard errors are given in parentheses below the
coefficients. R2 was .3851 and R2 was .3042. There were

38 degrees of freedom. was 4.7600 with signifi-

F(5,38)

cance .002. The signs of E, R, S, and A are all positive

46

as predicted while the sign of A2 is negative as pre-
dicted. The estimated coefficients of S, A, and A2 were
highly significant.

Again, experiments of including first a race-sex
interaction term, then a rural—urban index, and then both
together were performed. In all three cases, observations
were sufficient to permit execution of the regressions
(the matrix of independent variables was non-singular).
The regressions run with a race-sex interaction term, a
rural-urban index, and both all showed a slight increase

in R2, but R2

fell considerably. The significance levels
of the added regressors was low in all cases. Also, in
all three cases, the F statistic fell, but remained
highly significant.

Since the regression results obtained in (20) gave
a highly significant F test, I am led to conclude that
the approach used was satisfactory and that the poor
results Obtained in (19) were due to the limited number
of observations available. In estimating (13), only 17
observations were available while there were 44 observa-
tions available for estimating (14). So, (19) and (20)
will be used as the prediction equations for weekly
earnings for individuals who did not have them before

receiving vocational rehabilitation services and after

receiving them.

47

The Demand for Vocational
Rehabilitation Services

 

 

The estimation results for the demand for services
relation using both measures of before and after earnings
figures--actual reported earnings when a job was held

(wb and we) and predicted values (w and we) when not and

b

maximum of (wb, wb) and (we, we)--and both interest rates
are given in Table 1. Results are based on 54 observations
in each case.

In all four regressions d is positive (as

0

expected), large, and significant at the .005 level.

a is negative (as expected) in all cases, but not signifi-

1

cant at the .10 level. d2 is negative (not expected) in

all cases, but significant at the .05 level only in the

A . A
formulations u81ng maX1mum (w wb) and max1mum (we, w ).

b’ e

I attribute the negative value of a to the poor pre-

2
vocational rehabilitation wage prediction equation because
a poor estimate here will give a poor estimate for B and
therefore for B/C. Overall, significant F tests resulted
only when opportunity wages were estimated using maximum
b’ , we). The triple A bond rate

2

gives slightly more significant F statistics and R 's

(w wb) and maximum (w

e

than the bank savings rate, but the improvements are

marginal.

48

 

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. AAm.me . . . . . . n .n
Nope m u m mmeo o eemo 0 8mm on nee o- Hem can 0 e pee A a 3v essexes and
w e w
eeo.o ”.mwm Ammm.ov Aeee.oe Aeme.omv A a 3e essexme
. AAm.mV . . . . . . n .6
emem m u m Hmeo 0 400A 0 owe on mas o- mom NH» ma m can A a 3v easexme 05H
m m
one.p ".mem AAeH.ov Amee.ov Ammo.omv o x 3 mmmacs a
. Am.me . . . . . . n n
omen o u A m mmoo 0. Home 0 00A 0. AAA 0. AAA flee o e pee o x 3 mmmees a and
w 0
844.0 “.mem AemH.ov Aflee.ov AomA.omc o x 3 mmmaen e
. Aam.me . . . . . . e n
mmam o u m meoo o- flame o mefl 0. men 0. Ame New wed 8 new 0 x 3 mmmAcs e meH
m m m m A o

W! m a o e p 6063 coflmmmumwm

 

.muasmmn coflmmmummu coflumuflaflnmcwn HOCOflumoo> now OcmEmQII.H mqmde

49

The Supply of Time

 

The estimation results for the supply of time
relation using both ways to measure before and after
earnings--actual reported earnings when a job was held
(wb and we) and predicted values (wb and we) when not and
maximum of (wb, wb) and (we, we)--and both interest rates
are given in Table 2. Results are based on 55 observations
in each case.

In all four regressions 80 is positive (as expected)
and significant at the .0005 level. The sign of 81 is
positive and significant at the .01 level in all four
regressions. Apparently the income effect of larger
opportunity earnings dominates the substitution effect and

the client supplies more time. B is negative and signifi-

2
cant at the .10 level in all four regressions, especially
the two where the wages used are actual reported earnings
when a job was held (wb and we) and predicted values

(wb and we) when not. Apparently the income effect is
negative and large enough to swamp the positive substitu-
tion effect because less time is supplied as Q/M increases.
This implies a backward bending supply curve of time to
the vocational rehabilitation process. 83 is negative

(not expected) in all four regressions but significant at

the .10 level only in those formulations where wages are

taken as the maximum of (wb, wb) and (we, we). I attribute

50

 

 

 

 

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. AHm.mv -. . . . . . . n .n
0000 m u e Hook 0 meem 0 0A0 0- 0m0 0- eeo 0 400 mm 0 e 0:0 A e 3V seawxme 00H
m T
000.0 ".mem A0H0.0c AmN0.0V A0A0.0V A00m.mc A e . 3V Essexms
o AHm~MV o o o o o o a Q QQ
0000 m u 0 mmma 0 0040 0 000 0- 000 0: 040 0 000 em 0H 0 000 A e 3v gnawxme 00A
w w
0A0.0 ".mwm A0H0.0e Ammo.0e AeA0.0v Aemm.mv 0 x 3 mmmfics e
. Aam.mc . . . . . . . n a
00mm 0 u m 000A 0 Home 0 000 0- 040 0- 000 0 ewe em 0 e pee 0 x 3 mmmaes a sea
m m.
ma0.0 “.mem A0A0.0v Ammo.0v AeH0.0v Ammm.mc 0 x 3 mmwfies e
. Afim.mc . . . . . . . n n
Ammo m u m mmmfl 0 nema 0 000 0- 040 0- eeo 0 000 00 wmA 0 000 0 x 3 mmmfics a 00A
m WW mm mm mm Hm Om O mmmz coflmmmumom

 

.muasmmu coflmmwummu coflumuflafinmzmu Hmcofiumoo> on mEHu mo >Hmmsmnl.m mamme

51

the negative values of B3 to the poor prevocational
rehabilitation wage prediction equation because a poor
estimate here will give a poor estimate for B and there-
fore for B/M. Overall, all four F ratios were very
significant. The regressions using maximum (wb, 0b) and
maximum (we, we) for opportunity wages have larger R2's

and more significant F statistics. The results obtained

from both interest rates are not significantly different.

CHAPTER IV

SUMMARY AND CONCLUSIONS

The empirical results support the stated hypotheses.
In the demand for "free" services regressions, the constant
term (do) is always positive (as expected) and significant.
It reflectstfimabenefitcflfexpanded production possibilities.
The coefficient of the time price (0.1) for services is negative
(as expected) in all cases, but never more significant than
the . 30 level. The coefficient of the benefit per unit of ser-
vice received (d2)ifisnegative (not expected)ix1a11 cases
and significant in half the cases (the half using maximum

(w wb) and maximum (we, we) for Opportunity wages). These

b’
unexpected results are attributable to the poor prevocational
rehabilitation wage prediction equation because apoor esti-
mate here will give a poor estimate for B and therefore for 8/: .
Overall, significant F statistics and higher R2 " 5 result for the

formulations using maximum (w wb) and maximum (we, w ) for

b’ e
opportunity wages. The triple A bond rate gives slightly more
significant F statistics and Rz's than the bank savings
rate, but the improvements are marginal.

In the supply of time regressions, the constant
term (80) is positive (as expected) and highly significant

in all cases. 80 reflects the clients' willingness to

supply time to gain more information about the program
52

53

and the non—pecuniary expanded production possibilities
benefit. The coefficient (81) of the time cost for being
in the program is positive and highly significant in all
four regressions. Apparently the income effect of larger
opportunity earnings dominates the substitution effect and
the client supplies more time if his time price for being
in the program is larger. The coefficient (82) of the
services provided price per month supplied is negative and
significant in all four regressions, especially the two
where the wages used are actual reported earnings when a
job was held (wb and we) and predicted values (wb and we)
when not. Apparently the income effect is negative and
large enough to swamp the positive substitution effect
because less time is supplied as Q/M increases. This
implies a backward bending supply curve of time to the
vocational rehabilitation process. Though our observations
give a backward bending supply curve of time, it would not
be advisable to generalize that the entire supply curve of
time is negatively sloped. Since a client will consume
vocational rehabilitation services until he has consumed
that quantity (a target amount) required to rehabilitate
him and since there is some variability in proportions
available for combining services and time, the observed
backward bending supply curve Of time probably reflects
the fact that if the client can receive his target amount

of services more intensively (larger Q/M), then he will

54

not have to supply as much time. The technology of
rehabilitation would exclude the possibility that the
client could become rehabilitated by supplying a very
small M if Q/M were very large. The coefficient (83) of
the increased earnings benefit per month spent in the
program is negative (not expected) in all four regressions
but significant only in those formulations where wages are
taken as the maximum of (wb, wb) and (we, we). I
attribute the negative values of B3 to the poor pre-
vocational rehabilitation wage prediction equation

because a poor estimate here will give a poor estimate

for B and therefore for B/M. Overall, all four F ratios
are highly significant. The regressions using maximum
(wb, wb) and maximum (we, we) for opportunity wages have
larger R2's and more significant F statistics. The results
obtained from both interest rates are not significantly
different.

The implications of these empirical results is
that "free" vocational rehabilitation services are not
"free" to the recipient because of the time cost. As
expected, clients' demand functions are inversely related
to the time price. Using d1 (-.783) in 17c (the most
significant demand regression) and, from the transformed
data, average Q ($637.31) and average Cl/Q ($21.46), the
elasticity of demand for "free" services with respect to
their time price can be computed as .013. Using the

elasticity of demand formula, it can be calculated that if

55

the time price were zero, clients, on the average, would
demand $16.07 more of services, thus making the average
total demand $653.38. This would be a marginal increase
in the quantity of services demanded.

The implications of the empirical results for
supply of time considerations are that, since 81's are
positive, the individual will continue to stay in the
program additional months if his time cost for being in
the program is larger. Since all the clients in the sample
had similar disabilities and therefore similar prescribed
treatments, differences in time cost per month in the pro—
gram reflect differences in opportunity earnings. Those
individuals who have low opportunity earnings must spend
their time realizing those earnings to support themselves.
Those individuals who have larger opportunity earnings
can acquire the earnings needed to purchase required goods
and services by working a smaller number of hours than
individuals with low opportunity earnings and therefore
have more hours available to supply to rehabilitation
activities. That is, if an individual's opportunity
earnings should rise, it would make consumption of voca-
tional rehabilitation more expensive because of the more
costly time input. Thus, the individual would be motivated
to substitute out of rehabilitation. But, the income

effect generated by the larger opportunity earnings allows

56

the individual to acquire needed goods and services for
fewer hours of work and therefore release hours from
necessary work which may be used to consume vocational
rehabilitation.

Using 81 (.047) in 18d (the most significant
supply regression) and, from the transformed data, average
M (28.18) and average Cl/M ($87.92), the clients'
elasticity of supply of time with respect to its time
price can be computed as .079. Using the elasticity of
supply formula, it can be calculated that if the price of
supplying time were zero, clients, on the average, would
remain in the program 4.11 fewer months, thus making the
average total amount of time supplied 24.07 months. This
would be a marginal decrease in the number of months
supplied. The quite inelastic time supplied elasticity
suggests that attempts to keep clients in rehabilitation
programs additional months by attempting to increase their
opportunity earnings would have only marginal effects.

For example, a wage supplement program depending on hours
worked, would have limited success in increasing the time
period clients would stay with their rehabilitation pro-
grams.

Another implication of the empirical results for sup—
ply of time considerations is that the elasticity of time sup-
plied with respect to the services provided per month

supplied price is low. For example, computation of this

57

elasticity for regression 18d gives .031. Thus, if voca-
tional rehabilitation agencies desire to increase the
quantity of time clients remain in their programs, they
should lower Q/M. But such decreases will generate only
marginal increases in time supplied. It would be expected
that at lower levels of Q/M, the negative income effect
would cease to dominate the positive substitution effect
and supply of time would be positively sloped (not back—
ward bending).

In summary, the purpose of this dissertation was
to facilitate the inclusion of economic considerations in
"free" good demand forecasts by developing a comprehensive
model of consumer behavior incorporating "free" goods and
services. Such a model was developed and then used to
establish explicitly the individual's demand relationship
for "free" vocational rehabilitation services and his
supply relationship of time. The statistical results
generated by the application suggest that the variables

included in the model are important.

BIBLIOGRAPHY

58

BIBLIOGRAPHY

American Hospital Association. Cumulative Index of
Hospital Literature. Chicago: The Association,
annually.

 

 

Bayer, A. E., and A. G. Holtman. "Determinants of Pro-
fessional Income Among Recent Recipients of
Natural Science Doctorates." Journal of Business,
XLIII (October, 1970), 410-18.

 

Becker, Gary S. "A Theory of the Allocation of Time."
Economic Journal, LXXV (September, 1965), 493-517.

 

Conley, Ronald W. "A Benefit-Cost Analysis of the
Vocational Rehabilitation Program." The Journal
of Human Resources, IV (Spring, 1969), 226-52.

 

 

. The Economics of Vocational Rehabilitation.
Baltimore: Johns Hopkins Press, 1965.

 

The Cooperative Information Center for Hospital Management
Studies. Abstracts of HOSpital Management Studies.
Ann Arbor: University of Michigan Press.

Dano, Sven. Industrial Production Models. New York:
Springer Verlag New York, 1966.

 

Gornick, Marian. "Use of Medical Services as Demanded by
the Urban Poor." American Journal of Public
Health, LIX (August, 1969?) 1302-11.

 

Lepper, Mark. "Approaches to Meeting Health Needs of Large
Poverty Populations." American Journal of Public
Health, LVII (July, 1967), 1154-55.

 

McNulty, Donald J. "Differences in Pay Between Men and
Women Workers." Monthly Labor Review, XC
(December, 1967), 40-43.

 

Michigan Department of Education. A State Plan for Voca-
tional Rehabilitation Services. Vol. II, Technical

 

 

Appendices. Lansing: Department of EducatiOn,
1970.

 

59

60

Moody's Industrial Manual 1971. New York: Moody's
Investors ServiCe, 1971.

 

Samuelson, Paul A. Foundations of Economic Analysis.
Harvard Economic Series, Vol. LXXX. Cambridge:
Harvard University Press, 1947.

 

Sanborn, Henry. "Pay Differences Between Men and Women."
Industrial and Labor Relations Review, XVII
(July, 19647,,534-50.

 

Skolnik, Alfred M., and Sophie R. Dales. "Social Welfare
Expenditures, 1969-70." Social Security_Bulletin,
XXXIII (December, 1970), 8-11.

 

Thurow, Lester C. "The Occupational Distribution of the
Returns to Education and Experience for Whites and
Negroes." American Statistical Association,

 

Proceedings of the Social Statistics Section 1967,
X (1967), 233-44.

U.S. Department of Health, Education, and Welfare. Ap
Exploratory Cost-Benefits Analysis of Vocational
Rehabilitation. Washington: Government Printing
Office, 1967.

 

U.S. Department of Health, Education, and Welfare. Human
Investment Programs: Vocational Rehabilitation.
Washington: Government Printing Office, 1967.

U.S. Department of Labor. Manpower Administration.
Manpower Report of the President-—January 1969.
Washington: Government Printing Office, 1969.

Weiss, R. D. "The Effect of Education on the Earnings of
Blacks and Whites." Review of Economics and
Statistics, LII (May, 1970), 150-59.

 

Welch, Finis. "Labor-Market Discrimination: An Interpre-
tation of Income Differences in the Rural South."
Journal of Political Economy, LXXV (June, 1967),
225-40.

 

White, James. "Women in the Law." Michigan Law Review,
LXV (April, 1967), 1051-1122.

61

General References

 

Baumol, William J. "Activity Analysis in One Lesson."
The American Economic Review, XLVIII (December,
1958), 837—73.

Cowles Commission for Research in Economics. Activity
Analysis of Production and Allocation. Edited
by Tjalling C. Koopmans. Cowles Commission

1 Monographs, No. 13. New York: John Wiley and
' Sons, 1951.

Cramer, J. S. "A Dynamic Approach to the Theory of
Consumer Demand." The Review of Economic Studies,
XXIV (February, 1957), 73-86.

 

Dorfman, Robert; Paul A. Samuelson; and R. M. Solow.
Linear Programming and Economic Analysis. New
York: McGraw-Hill Book Co., 1958.

 

Ferguson, C. E. Microeconomic Theory. Homewood:
Richard D. Irwin, 1966.

 

Fishman, Leo, ed. Poverty and the Negro. New Haven:
Yale University Press, 1966.

 

Friedman, Milton. Price Theory: A Provisional Text.
Chicago: Aldine Publishing Co., 1962.

 

Gronau, Reuben. The Value of Time in Passenger Transporta-
tion: The Demand for Air Travel. National Bureau
of Economic Research OccasiOnal Paper 109. New
York: Columbia University Press, 1970.

 

 

Henderson, James M., and Richard E. Quandt. Microeconomic
Theory: A Mathematical Approach. New York:
McGraw-Hill Book Co., 1958.

 

 

Kmenta, Jan. Elements of Econometrics. Riverside:
Macmillan Co., 1971.

 

Koopmans, Tjalling C., ed. Activity Analysis of Production
and Allocation. Cowles Commission Monographs,
No. 13. New York: John Wiley and Sons, 1951.

 

 

Lancaster, Kelvin. Mathematical Economics. New York:
Macmillan Co., 1968.

 

62

Michigan Department of Education. A State Plan for
Vocational Rehabilitation Services. Vol. I,
Report and RecommendatiOns. Lansing: Department
of Education, 1970.

 

Pontney, Jack A., and Shih-Yen Wu. An Introduction to

 

 

Modern Demand Theory. New York: Random House,
1967.

Strotz, R. H. "Myopia and Inconsistency in Dynamic
Utility Maximization." The Review of Economic

 

Studies, XXIII (July, 1956), 165-80.

Tintner, G. "The Maximization of Utility Over Time."
Econometrica, VI (April, 1938), 154-58.

 

U.S. Bureau of the Census. 1960 Census of Population,
Michigan, Number of Inhabitants. Washington:
Government Printing Office, 1961.

 

U.S. Department of Commerce. Statistical Abstract of the
United States 1969. WaShington: Government
Printing Office, 1969.

 

APPENDIX

VOCATIONAL REHABILITATION DATA

63

VOCAT I ONAL REHAB IL I TAT ION DATA

In order to keep case materials confidential, the
Vocational Rehabilitation Division of the Michigan
Department of Education required me to sign a contract in
which I agreed to not disclose or reproduce the raw data
which they provided. 80, to give the reader some feel
for the data, some statistics for the raw data and for

the transformed data are given in Table 3.

64

65

TABLE 3.--Statistics on raw and transformed vocational rehabilitation

 

 

data.
Variable Mean Standard Minimum Maximum
Deviation Value Value
Education (E) 10.77a 2.29a 4.00a 16.00a
Race (R) 0.82a 0.39":1 0.00a 1.0051
a

Age (A) 36.23a 14.90a 0.00 67.00
Services Demanded (Q) $637.31 $643.84 $ 15.00 $3,980.00
Time Price of Services b b b b

(Cl/Q) $ 21.46 $114.69 $ 0.01 S 837.01
Service Benefit Rate b b b b

(B/Q) $122.37 $380.71 -$676.47 $1,632.90
Time Supplied (M) 28.18 22.79 1.00 99.00
Time Price Of Time 0 c c c

(Cl/M) $ 87.92 $166.17 5 0.56 $ 641.32
Services Price of Time

(Q/M) $ 47.48 $128.50 $ 0.00 $ 937.00
Time Benefit Rate c c c c

(B/M) $131.34 $361.14 -$449.41 $2,178.98

 

a . . .
These results are for the 44 indiViduals who acquired
employment after consuming vocational rehabilitation services.

These results are for regression 17c,

demand regression.

the most significant

C . . . .
These results are for regre551on 18d, the most Significant

supply regression.

MICHIGAN STATE UNIV. LIBRARIES
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