xt7gb56d3f7h https://exploreuk.uky.edu/dips/xt7gb56d3f7h/data/mets.xml   Agricultural Experiment Station, Department of Agricultural Economics, University of Kentucky 1969 journals kaes_research_rprts_01 English University of Kentucky Contact the Special Collections Research Center for information regarding rights and use of this collection. Kentucky Agricultural Experiment Station Research Report 1 : December 1969 text Research Report 1 : December 1969 1969 2014 true xt7gb56d3f7h section xt7gb56d3f7h   REPLACEMENT OF DAIRY COWS
A Multistoge Decision-Making Problem
John C. Redman and Lily P. H. Kuo
RESEARCH REPORT 1 : December 1969
University of Kentucky :: Agricultural Experiment Station
Department of Agricultural Economics
l :  
l .` N" `

THE PROBLEM .............................. l
PREVIOUS STUDIES ........................... 2
Objectives and Hypotheses ..................... 2
Assumptions Necessary for This Study ............... 2
THE METHODOLOGY ........................... 3
Background of Theoretical Development .............. 3
Delineation of the Method Used .................. 3
Variables Involved and Their Determination ............ 8
RESULTS ............................... I3
Solving the Equation for Dairy Cow Replacement Problem ...... 13
Interpretation of Results .................... I4
Comparison of Replacement Policies Followed by
Dairy Producers with the Optimal Policy ............ l7
SUMMARY AND CONCLUSIONS ....................... 18
REFERENCES .............................. 2l .
APPENDIX ............................... 23

1. Conditions at the End of the Enterprise ............. 13
2. Optimal Replacement Policies and Returns for Dairy Cows in
Production Level 2, Under Various Conditions ........... 16
l. Milk Production by Lactation and by Production Level ....... IO
2. Index of Milk Production and Feed Consumption ........... 10
3. Price Indexes of Dairy Cow, Feed, Beef and Milk ......... l2
4. Probability of Success of Dairy Cow by Lactation and Milk
Production Level ......................... 12
[ l

A Multistage Decision—Making Problem
John C. Redman and Lily P. H. Kuo
In a dairy enterprise, considerable quantities of time and other inputs are
allocated by the operator to raising or acquiring replacements for the producing
cows. The crux of the dairy cow replacement problem is basically concerned with
the optimum time at which a cow is to be replaced. Milk production as a bio-
logical process increases to a maximum point and then decreases, both within one
lactation and between lactations within a cow's life span. The problem has two
dimensions. First, a dairy producer must decide whether to replace a cow during
a given lactation, and second, if the decision is to replace her, when to do it.
However, in this study, the attention was concentrated on the problem of deter-
mining the best lactation number for replacement.
The replacement of dairy cows is a multistage decision—making problem since the
best decision depends upon subsequent and preceding decisions. The total plan—
ning problem for multiperiod production of the firm consists in determining the
value of all input and output variables for all time periods within a certain
planning span of time. The replacement problem is concerned only with determin-
ing value of durable input variables and only with cases where one durable asset .
is replacing another already in operation, instead of simply being added to the
total number of such assets. For example, if a farmer decides to add one dairy
cow to those already in production at a given time, this is not a replacement
situation but one of increasing the size of the dairy operation.
Further, it has been argued that a replacement situation exists only when the
level of production is not changed after the action of replacement. It would
be extremely difficult to maintain the distinction between replacement and ”
scale adjustment in an operational model. For instance, the farmer replaced
an old cow of low productivity with a young one with higher productivity. By
feeding less to the young cow, it might be possible to maintain milk production
at the same level as before; however, the economic result from such a policy
would not be a proper basis for studying the economic effect of the replacement.
For the purpose of this study, the term "replacement" will refer to any case
where one durable capital item is substituted for another one, whether the re-
placement is accompanied by a change in output or not. '

Very few economic studies have dealt explicitly with dairy cow replacement.
So far there are only two studies known by the authors. One, published by
Jenkins and Halter [14], must be considered a methodological study in the
field of dynamic programming. It uses the dairy cow replacement problem
as an illustration of given principles.
The second study was accomplished by Giaever [ll], using the method of dynamic I
programming with Markov processes as developed by Howard [I2].
Several research projects in the field of dairy science have dealt with current
replacement patterns and disposal causes. However, there is no known literature
in this field which attempts to determine the optimal replacement pattern for
dairy herds in terms of maximizing profit.
Objectives and Hypotheses
The principal objectives of this study were:
a. To test the hypothesis that by adopting the optimal replacement
policy of dairy cows, the dairy farm can increase net returns over
the life span of the enterprise.
b. To show how an optimal replacement policy can be obtained by the
dynamic programming model.
c. To determine the economic loss to dairy producers who follow re- .
placement policies which deviate from the optimal.
d. To study the sensitivity of optimal replacement policies to variations
in prices and in other parameters in the model.
e. To gain experience with practical problems which have to be faced when
applying this kind of models.
Assumptions Necessary for This Study
a. In any enterprise period, it was taken as certainty that replace-
ments in every specified lactation (i. e., from lst to 7th lactation)
were available.
b. The deviations of random replacements of cows on account of diseases, '
injury or death, were normally distributed. The probability of random
replacement was constant over years.
c. The deviations of production and feed consumption among cows were
normally distributed.

d. The motive of the dairy producer was profit maximization over the .
life span of the enterprise. c
e. The dairy producer followed a given “feeding system" which may have
varied over time as a result of price variations, but which, at any
given time, was the same for all cows in the herd.
f. All cows which had not been replaced before were replaced at the
end of the seventh lactation. ln actual practice few cows are kept
longer than to the end of the sixth lactation.
g. There was no significant difference between replacement with a pur-
chased animal and replacement with a self—raised heifer.
Background of Theoretical Development
As pointed out by Preinreich [15], the general theory of replacement is
simply the theory of maxima and minima. A solution to the optimization pro-
blem is attributed to the German forester Eaustman [9] as early as 1849.
Preinreich was the first to apply contemporary methods to the problem.
However, he discussed only the problem of replacing industrial equipment.
Gaffney [lO] has dealt with the optimal harvest age of timber. Faris [8]
discussed, more generally, replacement patterns for biological production
involving time. Winder and Trant [22] have discussed some of Earis' re-
placement rules and provided a more rigorous analysis of the principles.
Bellman [2] described the general principle of dynamic programming which
provides a convenient analytical and computational framework for dealing .
with the replacement problem. '
Delineation of the Method Used c
The replacement problems faced by dairy farmers are multistage decision
problems, i.e., time factor plays an important role in the replacement pro-
blems. Since the entire multistage decision process is considered in most ’
conventional farm management models as essentially one stage, it is difficult
to find the effective analytical solution of a problem with many dimensions,
i.e., a large number of equations to be solved simultaneously. Dynamic
programming is mainly designed for the multistage decision problems. The
advantage of this method is that it reduces one problem of many dimensions
to a problem of one dimension. This makes the problem analytically more
tractable and computationally vastly simpler.
Some additional terminologies need to be introduced, and a brief discussion
further of what is meant by a multistage decision process is presented below.
A Stage of the process is associated with an interval of time, and the time
interval is assumed constant from stage to stage.

The state of a system at any time is specified by a set of variables. In the f
course of time, this system is subject to changes of the variables describing
the system undergoing transformation.
A process may be assumed in which a choice of the transformations is available
to be applied to the system at any time. A process of this type is called
a decision process, with a decision being equivalent to a transformation. lf
a single decision is made the process is called a single-stage process; if a
sequence of decisions are to be made, then the term multistage decision process J
is used. I
ln a replacement decision process, a physical (or biological) system prevails
whose state at any time is determined by the values of a set of physical and
economic variables. At certain times, decisions must be made which affect
the state of the system and which are based on the prevailing state of the
physical and economic variables. The outcomes of past decisions are used to
guide the choice of future ones. This sequence of decisions is called
a policy. The purpose of the replacement decision process is to maximize
some function of the variables describing the final state. A policy which
is best according to some preassigned criterion will be called an optimal
In livestock enterprises, recurring decisions concerning the replacement
policy in a biological system are faced. If profit maximization is the goal,
the decisions are based on the expected net returns of the present animal on
hand compared with the expected net returns from the replacements, the cost
of replacing the present animal, the expected revenue from the sale of the
present animal, and the expected purchase price of a replacement. The
purpose of the decisions is to maximize the profit over the life span of the
An example of a process from a physical system is the determination of a
replacement policy for farm equipment. In this case, it is necessary to
consider the output of both old and new equipment, the maintenance cost for _
the old machine compared with that required by a new one, the purchase
price of the new machine, and the trade-in value of the old machine.
All members of the family of replacement models rest on the same set of »,
assumptions. (l) A "chain" of replacement is present, where one asset is
always replaced by another of identical or similar type. (2) Variation in
intensity of use of the asset is not considered. Thus, these models do
not consider the case where the life—span of one asset depends on how
heavily it is being used. (3) Income per time unit, outlays per time unit,
and salvage value of the old asset all are known mathematical functions of
age of the asset. (4) Capital is available at a given rate of interest.
(5) The objective criterion is maximization of the present value of I
future net returns.
The replacement decision process for the continually operating dairy enter-
prise used in this study can be represented by a recursive equation. The
approach by which the equation arises is to consider the set of all possible
sequences of decisions, i.e., the set of all feasible policies, compute the

returns from each such policy, and then maximize the return over the set of all V
policies. An optimal policy has the property that whatever is the initial
state and initial decision, the remaining decisions must constitute an optimal
policy with regard to the future state of the system.l Thus, the approach can
be expressed by a recursive equation as,
Hi,t = Vi,t + Max (Pj-NSj,t + Qj-NFj,t — Vj,t — 5j,t + l·Hj+l,t+l)
This equation determines the decisions that will maximize net returns within a I
certain planning time span, T, when the initial condition, Hj+1,0, are specified.
The sequence of decisions shows the most profitable age at which cows are to be
purchased in the beginning of the enterprise and subsequently, the most profit-
able replacements will be made in each enterprise period. In each enterprise
period, the farmer can either keep the cow for another time period or replace
her with a cow of a different age.
A description of the variables and a discussion of the relationships among
the variables may be made as,
t — planning time span, enterprise period (= 1,2,3, . . ., T)
i — lactation of cows on hand (= 1,2,3, . . ., I)
j — lactation of replacement cows (= 1,2,3, . . ., J)
The term, Hi,t, is the dependent variable for which the recursive function will
be solved and is the maximum net return, discounted to present value, for
time period t and subsequent time periods from a cow in lactation i.
The market value of the present animal, V1,t, and the market value of replace-
ments, Vj,t, are based on the value of beef plus the expected net returns of
dairy production from present and subsequent lactations. Each is a function of _
current production levels, the number of lactations remaining for the animal,
the price of milk, beef and feed, and the supply and demand for dairy cows. lt
is assumed that Viit = Vj,t when i = j. However, the difference, if any, between
Vj,t and Vj,t when i = j in the real market will be included in the transaction
cost (Sj,t).
The term, Pj·NSj,t + Qj·NFj t, expresses the expected net returns from a re-
placement cow in lactation j in time period t. There are four variables involved n
in determining the expected net returns. The probability of success of lac-
tation j, Pj, and the probability of failure of lactation j, Qj, are stochastic
factors in the equation. Failure of a dairy cow is defined as the removal of
the animal from the herd for sickness, injury or death. The variable NSj,t is
the net return from a cow of lactation j in time t if she completes lactation
j, i.e., the net returns from a successful lactation. The net returns
from failure, NFj,t, is the salvage value of a cow of lactation j, in time
period t if she fails to complete lactation j. ·
The transaction cost, Sj,t, includes the commission charge, cost of trans-
portation of the cows to and from the market, labor cost and any difference
1This is the basic principle of optimality [2, p. 83].

 6 T
between the selling price of the present animal and buying price of re-
placements when they are in the same lactation. There will be no
transaction cost if the cow is kept another time period, i.e., 5j,t = O when
The discount factor, A, is determined by interest rate. By using the discount
factor, outlays or revenues incurred during different time periods can be .
made comparable by discounting them to the same time period, the present
The term, Mj+J, t+J, is the maximum net return, discounted to present value,
of the decision process for a cow of age J + l in time period t + l. This
quantity represents the net returns to the enterprise if the optimal policy
is followed in future time periods. The subscript, J + l, is used instead
of the J because in the next time period, t + l, a cow must be one year
The meaning of the equation and the relationships between the variables can
be expressed with less symbolic notations.
At any time, t, the expected maximum net returns, discounted to present
value, for any cow i is measured by:
The market value of the cow, which includes the meat value of the cow
and her potential milk production for (t)th and the subsequent time
The maximum net returns that can be obtained by replacing the present
animal with cows of any of J different ages.
The maximum net returns of each replacement cow are measured by:
The expected net returns from the replacement in time period t, which
includes expected net returns from milk production in time period t and
meat value of the animal. W
The purchase price of the replacement cow, which includes her meat value
and potential milk production.
The cost of transaction.
The discounted net returns obtained in the future time periods when the
optimal replacement policy is followed.

The equation was designed to represent a recurring replacement decision process .
of a dairy production enterprise. The length of time between decisions may be ’
of any duration. However, given time duration, the other variables in the
equation will be specified by definite characteristics.
Since one of the most distinct features of the recursive model is that the
analysis of each time period is dependent on earlier time periods [5] the
initial state of the system, Hj+1,0, should be specified for the purpose of
solving the equation. The procedure is to begin with the last enterprise ;
period and proceed backward through time to the beginning of the enterprise. -
Because the only policy open at the end of the enterprise is to sell the
cows regardless of age, the selling price of the cows represents the maximum
returns of the replacement policy, i.e., the selling price is used initially
as the value for Hj+1,Q when deriving the replacement policy for the enter-
prise period next to the last. Without the possibility of carrying out this
procedure, there would be no means of giving a value to the Hj+1, {+1 term
which represents the net returns of the optimum policy in future enterprise
Another characteristic which should be mentioned here is that a single
(ixt)-dimensional problem has been reduced to a sequence of ixt one—dimension—
al problems. The utilization of structural properties of the equation and the
reduction in dimension combine to furnish computing techniques which greatly
reduce the time to solve the problem.
As discussed before, the equation will be solved by using the state of the
system at the end of the enterprise as the initial value of Hj+],O and
proceeding backward through time to a period t which equals some preassigned
value T. The enterprise periods are relabeled from the specified initial
position, i.e., instead of indexing the enterprise periods as t = 1,2,3,---,T,
they are now indexed as t = T,T—l, T—2,··-,3,2,1. The recursive equation can
now be written as,
Hiyt = Vi,t + Max (Pj‘NSj,t + Qj‘NFj,t — Vj,1; — 5j,t +*—‘Hj+1,t-1)
Once the value Hj+],Q is specified, the procedure is to add to it the expected
net returns of production from the possible replacements of j different lac-
tations and subtract the transaction cost when i ¢ j. Having then a vector
of j possible returns from replacements, the maximum is selected and added W
to the market value of the cows on hand of lactation i in time period one
to obtain Hi,t. This Hi,1 value is used for the Hj+],1 value Hj+l,t-1 in
calculating the returns from the j possible replacements in enterprise period
two; the maximum of these returns is added to the market value of the cow
On hand of age i, resulting in Hi);. Similarly, the Hi,2 value is used
for the Hj+l,2 value in determining the value Hi,3. The same type of iteration
is made for each enterprise period until t = T.
The solution of the equation was easily carried out with the use of an
electronic computer because the recursive equation is solved by successive
iterations, i.e., at each time t, the same procedure, computational—wise, is
repeated. This study was programmed in Fortran IV for the IBM 560 computer
which made the problem with its large number of dimensions, i.e., processes

involving a large number of possible replacements and a large number of planning ·
periods, relatively easy.
Variables Involved and Their Determination
Three hundred and fifty cows were selected at random from the Holstein D.H.I.A.
farms in Kentucky. Three hundred and ten observations were used, while the ·
remaining 40 cows were excluded from the sample because of incomplete in- ;
formation. ·
The 1965 milk production of the sample cows was collected from the D.H.I.A.
Lactation Report of the U.S.D.A. and University of Kentucky Cooperative
Extension Service.
To reduce variation of production owing to different quality of cows, the pro-
duction records were separated into three groups according to the production
level. Since total production of a cow varies with calving interval and age,
it is necessary to standardize the production record in order to decide to which
production level the cow should belong. The 305 days' yield at age 6 was
used to measure the productivity of a given cow. Three production levels
were defined: (a) the standardized production of less than 12,000 pounds,
(b) that between 12,000 and 15,000 pounds, and (c) that above 15,000 pounds.
One quadratic equation was fitted to each production level as follows:
Y1 = 5803.42 + 2162.37L - 2l8.llL2
Y2 = 9142.37 + 1782.90L — l66.0lL2
Y3 = 11241.04 + 2184.99L - l80.28L2
Where xi, Xg, is are estimated productions at three different levels,
(Appendix Table 2) L is the lactation number. These equations are plotted .
in Fig. 1. V
a z2s2 N — n . . -
Using the fgymulg df = -E··—-§—-3 the precision of estimates from these three
equations were calculated [24]. Where d= IXZYT, is the precision of the esti-
mate, Y`is the estimated value, K-is the true parameter in the population, z
is the reliability coefficient, s2 is the variance, N is the size of population,
n is the sample size. With reliability 95 percent, the precision of the esti-
mates is within 205 pounds, 153 pounds, 376 pounds for xi, Yé, xg, respectively. ”
The estimated milk production by lactations was used for the parameter of
milk production in 1965. The production of other years was obtained by multi-
plying the index of production by these estimates. The index of production was
calculated from the Dairy Herd Improvement Records (Appendix Table 3 and Fig.
Data on body weight of the sample cows were obtained from Kentucky Dairy Herd
Improvement Association Records Monthly Report. One arithmetic mean was found ’
for each lactation. Using average weight of first lactation as the base, the
index of body weight by lactation was calculated (Appendix Table 4).
Information on feed quantities actually given to individual cows at given

lactation was not available. Data on average yearly feed consumption for the ,
Holstein D.H.l.A. herds in Kentucky for years 1956-65 were obtained from the
Kentucky D.H.l.A. Yearly Herd Summary [16]. Feed consumption for cows at
different lactations was determined by modifying average herd consumption
by the index of body weight. The index of average herd consumption in years
1956-65 is shown in Figure 2 and Appendix Table 7. Number of cows failing
and total number of cows by lactation and production level were obtained by .
Jenkins and Halter from IBM cards of Pennsylvania D.H.l.A. program in 1960.
Data on market value of dairy cows by lactation were not available. Market `
value of an animal for purposes of this study was estimated using data ob-
tained from Blue Grass Stock Yards and Clay—Wachs Stock Yards in Lexington,
Ky. The market value of other years was adjusted by price index of dairy
cows for each year. The transaction cost involved in marketing was estimated
from data obtained from the same source.
Prices of milk, feed, beef and dairy cows were obtained from Agricultural
Statistics, U.S.D.A. Bureau of Agricultural Economics, Division of Agri-
cultural Price Statistics. The index of prices was calculated and is shown
in Appendix Table 6 and Fig. 3.
As mentioned before, the discount factor is determined by the interest rate.
ln this study, a decision—maker willing to accept a 6 percent interest rate
was assumed. Then, for the purpose of examining the sensitivity of interest
rate to the replacement model, interest rates of 8, 10, 15, and 20 percent
were used. ln all cases the interest rate was assumed constant over years.
ln the previous sections, the length of time between decisions has been
discussed. With different time duration, other variables in the equation
will be specified by different values. ln this study, the decision interval
was assumed to be equal to the enterprise period. An enterprise period is ,
defined as the time from the beginning of a lactation to the beginning of
the next, and corresponds to one year. The life span of the enterprise is
given as 10 years, 1956-65. As mentioned before, this is a backward ap-
proach, the starting point for solving the equation being the end of the
enterprise. The enterprise periods are indexed as:
t = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 U
+ +
End of Enterprise Period Beginning of Enterprise Period
(1965) (1956)
lt is assumed that all cows which have not been replaced before will be re-
placed at the end of the seventh lactation. The subscripts i, j appear as:
i=l,2,3, · · -,7; j=1,2,3, . . .,7. With time interval given as one year,
the other variables are specified. .
The term on the left hand side of the equation, Hi,t is the dependent vari-
able for which the equation will be solved.
The variables on the right hand side are all independent variables and will

Milk Production
(thousand pounds)
19 • • • Observed Values
-—— Predicted Values ·
16 `
Production Level 3 `
is °
12 Production Level 2
10 •
9 g Production Level 1
1 2 3 4 5 6 7
Fig. 1.—Milk Production by Lactation
and by Production Level
Index `
  Index of Milk Production
"'••• Index of Feed Consumption
100 /
/ ’ 
90 /·/
.# °
O ;
80 ·/
/1 • & ,.¢• ' 2
C - • V
70 ’
1956 1957 1958 1959 1960 1961 1962 1963 1964 1965
Fig. 2.——Index of Milk Production and Feed Consumption (1965 : 100)

be specified in the following discussion. The first variable is the market .
value of the present animal, Vi,t and is the salvage value of the animal
plus the expected net returns of dairy production from present and subse-
quent lactations. The market value of animals in 1965 was estimated
(Appendix Table 5). The market value in other years was adjusted by the
price index of dairy cows for each year.
The probability of failure of lactation j,Qj, was obtained by dividing the
number of age j cows that failed within one year by the total number of age j ;
cows in the same year. The probability of success was obtained by 1-Qj,
where O 5 P% 5 l, O 5 Qj 5 1. The probability of success is presented in
Appendix Tab e 1 and Fig. 4.
Net return from success, NSj,t was obtained by subtracting feed cost from
the market value of milk production. Costs and returns which are constant
to animals of all lactations, such as labor, barn, facility charges and
value of the calf are not considered here.
Net return from failure, NF·,t, is simply the salvage value of the animal.
This was obtained by multiplying the price of beef at time t by body
weight of the animal.
Market value of the replacement, Vj’t, was considered in the same manner
as the market value of the present animal, Vi t. The value of V· is
j t
assumed equal to the value of Vijt, when i = j. ,
The transaction cost, Sj,t, includes the commission charge, cost of trans-
portation of cows to and from the market, labor cost and any difference be-
tween the prices of Vi,t and Vj,t when i = j. The transportation cost of
dairy cows in Kentucky is around l2 cents per hundred weight per hundred
miles; average mileage per transaction is 100 miles. Allow $15 for com- .
mission charge, labor cost and the difference between buying and selling
prices for the cow at the same lactation. The transaction costs for each
lactation were estimated $27.71 for lactation 1, $28.45 for lactation 2, and
so forth. Since the replacement cows in the lactations other than first
lactation are not available in Kentucky, extra money should be added for find-
ing the replacements in other states. lt seems that the most convenient market
to buy a replacement is in Wisconsin. The extra transportation cost from Wis-
consin to Kentucky was estimated at $25 per head [17]. Therefore, $25 was 'I
added to the transaction costs for replacements other than first lactation.
Values of transaction costs by lactation are shown in Appendix Table 8.
1 . .
T§e discount factor, A, is determined by A = Tj;» T 1$lth€ 1¤t€T€$€ Tate- In
t is study 6 percent interest rate was used and A = = 0.9434. The
interest rate is assumed constant over years. 1+-06
The last variable, Hj+l’t_1, is the maximum return of the decision process
for a cow of age j+l in enterprise period t—l. The initial value for this
term, Hj+j,g, is the state of the system at the end of the enterprise,
December 31, 1965. The market value of dairy cows in 1965 was used as the
value for Hj+1,0,

Index A Index of Beef Price I-
· /\ /\
110 / V ’ `\ /
• • / \
/ . , ‘ \
• ( .
100 [ __ _ _ _ / \ ,
~~-....7:--·--· " "`·/—-;r""'{:·/
. / Index of Milk Price ~
é' \ Index of Feed Price
80 4
70 Index of Dairy Cow Price
1956 1957 1958 1959 1960 1961 1962 1963 1964 1965
Fig. 3.—Price Indexes of Dairy Cow, Feed, Beef and Milk (1965 : 100)
Probability of Success
1. 0
Production Level 1
0. 9
Production Level 2
A? » 
Production Level 3
O. S
1 2 3 4 5 6 7
Fig. 4. —Probability of Success of Dairy Cow by
Lactation and Milk Production Level

Solving the Eguation for Dairy Cow Replacement Problem
The initial conditions, Hj+l,O, at the end of the enterprise, 1965, is given
in Table 1.
DECEMBER 31, 1965
(Initial Value for Xj+j’t-j)
i+1 ~j+1,0
Lactation Dollars
2 317.66
3 338.83
4 358.83
5 348.62
6 345.98
7 340.68
8 337.60
Source: Refer to Appendix Table 5.
Once the value of jj+l’0 is specified, the procedure is to add to it the ex-
pected net returns of replacements of j different lactations during production
period one; subtract the market value of the replacements and subtract the _·
transaction cost when i # j. Having then a vector of j possible returns
from replacements, the maximum is selected and added to the market value of
the cows on hand, of lactation i in enterprise period one, to obtain Ii,1.
The computation can be demonstrated by considering two present animals of
lactation l and 3 in enterprise period one.
1.9457 x196.451.0543 x 141.87 - 281.49 — 0 4 .9434 x 317.661
1. 9245 x 233. 87 4 .0755 x 150. 39 - 317. 66 - 53.45 4 .9434 x 338. 85 1 ’ 
1.9063 x 253.284 .0937 K 161.74 · 338. 85 - 54.45 4 .9434 x 358. 83 1
11_1’1 281.49 4 I\1.1x1. 8804 v 271.13 4 .1196 x 165.99-- 358.83 - 54, 82 · .9434   348.621
. 8650 x 280. 36 4- .1350 x 167. 41 - 348. 62 · 54, 95 4 . 9434 x 345.98 1
.8443 x 271. 58 -1- . 1557 x 171. 67 - 345. 98 ·— 55. 32 4 .9434 x 340. 68
. 8424 x 249. 48 4 . 1576 x 175. 92 - 340. 68 — 55. 70 4- . 9434 x 337. 601
1.9457 x 196. 45 4- .. 0543 \ 141. 87 — 281.49 — 27.71+ .9434 x 317. 661
.9245 x 233. 87 4 .0755 x 150. 39- 317. 66 — 53.45 *-.9-.13-1 x 338. 85 1
.9063 N 253.28-1 .0937 x 161. 74 - 338. 85 — O 4 .9434 x 358. 83 1 .
113,] . 338. 851-h111N1. 8804 x 271.13 -1 .1196 x 165, 99 — 358. 83 - 54. 82 4 . 9434 x 348. 62
. 8650 x 280. 36 -1- . 1350 x 167. 41 — 348. 62 - 54. 95 4- . 9434 x 345. 98 1
1. 8443 x 271.58 -1 . 1557 x 171. 61 — 345.98 - 55.32 4- .9434 x 340. 68 1
1. 8424 x 249. 481 .1576 K 175. 92 — 340. 68 — 55. 70 4 .9434 x 337. 60 1
It is noticed that the only differences between the two sets of computations are
the market value of the animal on hand and the point at which the transaction cost

is O. The maximum returns from the decision process of cows at age 1-7 at the
end of enterprise period one, i.e., Hl,l, are given as follows:
Hl,l = 587.44. i
591.51 ;
577.42l .
These Hi 1 values are used for the Hj+1,l value (Kj+1_t_l) in calculating the
returns from the j possible replacements in enterprise period two; the maximum
of these returns is added to the market value of the cow on hand of lactation
i, resulting in Hi,2. The same type of iterati