TABLE 5. Estimated transition probabilities of farms from 1978 to 1982.


 States S S S S S S 8 S S S S
 0 1 2 3 4 5 6 7 8 9 10
 S 1 0 0 0 0 0 0 0 0 0 0
 0
 S .176 .824 0 0 0 0 0 0 0 0 0
 1
 S .224 0 .776 0 0 0 0 0 0 0 0
 2
 S .352 0 0 .593 .055 0 0 0 0 0 0
 3
 S 0 0 0 0 .912 .088 0 0 0 0 0
 4
 S 0 0 0 0 0 .790 .210 0 0 0 0
 5
 8 0 0 0 0 0 0 1 0 0 0 0
 6
 8 .250 0 0 0 0 0 0 .625 .125 0 0
 7
 S .333 0 0 0 0 0 0 0 .556 .111 0
 8
 S 0 0 0 0 0 0 0 0 0 1 0
 9
 S .400 0 0 0 0 0 0 0 0 0 .600
 10



 step. The transition matrix, in addition to the initial starting
 state, completely defines the Markov process; i.e., with the
 foregoing information, it is possible to determine the outcome of
 the process at the n'h step.


 Estimation of the Transition Matrix
 The efficient estimation of transition probabilities often
 presents a major technical problem because such estimates de-
 pend on the quality of the data available. However, three alter-
 native approaches are possible: statistical estimation from micro-
 unit data, from aggregate data, and from conceptual considera-
 tions (Collins et al., 1974). This study, following Krenz (1964),
 adopts a conceptual approach to the estimation of transition pro-
 babilities. The Bureau of the Census does not provide data on in-
 dividual farm units in the quinquennial censuses, and only
 enumerates farms in one of the several categories given above. By
 making use of detailed information on the life of farms in the
 Virgin Islands, patterns of behavior are assumed and rules subse-
 quently adopted in order to determine the transition pro-
 babilities. In this regard, the following assumptions are made.

 First, it is presumed that most farm operators in the Virgin
Islands would want to expand their acreage if it is possible to do
so. Second, it is more likely that medium to average size farms
will expand because of financial resources available, and because
of economies of scale, than it is that small farm units will increase
their acreage. Third, any increase in farm size is likely to proceed
gradually by the acquisition of adjacent property. Such incremen-
tal aggregation is likely to be a function of the availability of
agriculturally zoned land, and of reasonable financial ar-
rangements for purchasing land. Fourth, individual farms are not
likely to decrease their unit size voluntarily, particularly because
of the problems of economies of scale. Rather, it is more probable
that a farm will go out of business than exist as a reduced entity.
It is on the basis of these assumptions that the following two rules
are adopted in determining the transition of farms from one state
to another.
 An increase in the number of farms in any state S;, from one
time period to the next, comes from the next smaller state, Si-. A
decrease in the number of farms in any state indicates a move-
ment to So, a state of demise. The application of these rules of
behavior to the data at hand produces the transition matrix that is
utilized below. One advantage of the transition matrix is that it
provides useful insights into the movements of farms that are not
readily available from other types of projection models.


26


TABLE 6. The fundamental matrix of mean five-year periods


The Application of Absorbing Chains
 In the movement of farms between states discussed above, it
was indicated that states So is that state in which all farms that go
out of business eventually end up, and the assumption was that
they remain there. Such a state in a Markov chain is defined as an
absorbing state if it is impossible to leave it. A chain is therefore
absorbing if it has at least one absorbing state and from every
state it is possible to go to an absorbing state (Kemeny et al.,
1962). With one absorbing state in this study, that of going out
of business, the model employed here is an absorbing Markov
chain. The application of the theory of absorbing Markov chains
thus permits one to generate very useful answers to a number of
questions (Kemeny et al., 1962; Bartholomew, 1982).
 First, what percentage of farms that are in a given state S, (or
size category) are likely to be amalgamated with farms in a larger
size category Sj after five years? This question can be answered by
examining the coefficients of the transition matrix. Second, for
any farm of a particular size, how long is it likely to stay within its
size classification before it amalgamates with another? Such a
question can be answered from the elements of a fundamental
matrix. Thus, the coefficients of this matrix give the mean
number of years (of five-year duration) in each transient state for
each possible nonabsorbing starting state.
 Third, on average, how long does it take before a farm in a
given size category is absorbed or goes out of business? The sum
of all the entries in a row of the fundamental matrix will give the
total length of (five-year) time periods that a farm is likely to sur-
vive before going out of business. Fourth, what, in absolute
numbers, is the distribution of farms, according to size
categories, likely to be in five years, in ten years, or in n years?
Or, how many farms in total will there be in five, ten, or n years?
Of the two alternative methods for projecting these numbers, use
is made of that which multiplies the distribution of farms in the
base year, 1978, by the canonical form of the transition matrix P
to generate the projected distribution for one period; then the
result is postmultiplied by P for the needed number of periods
(Krenz 1964). For complete details on the estimation of the fun-
damental matrix and related statistics, see Kemeny and Snell
(1976, pp. 43-50).

Empirical Results
 The frequencies used to estimate the transition matrix were
derived from the census data of 1978 and 1982, and by the ap-
plication of the rules stated previously. The particular quality of
the Virgin Islands census data dictated that only the data for 1978
and 1982 could be utilized in the generation of the transition


 PROCEEDINGS of the CARIBBEAN FOOD CROPS SOCIETY-VOL. XX