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Unformatted text preview: Construction Management and Economics (2002) 20, 473—482 A theoretical framework for determining the
minimum number of bidders in construction bidding competitions STEPHEN C. NGAI', DEREK S. DREW”, H. P. L02 and MARTIN SKITMORE3 'Depamnent of Building and Real Estate, Hang Kong Polytechnic University, Hung Horn, Kowloon, Hong Kong
‘Depamnent of Management Science, City University of Hong Kong, Tat Ghee Avenue, Kowloon, Hong Kong
’Schaol of Commotion Management and HM, Queensland University of Technology, Gardens Point, Brisbane Q4001, Australia Received 28 February 2001 ; accepted 19 April 2002 A theoretical framework is proposed for determining the minimum number of bidders in competition for
projects in the construction industry. This is based on the neoclassical microeconomic theory for price
determination in construction and the assumption of random contractorselection. Empirical analysis of the
Hong Kong data set not only illustrates the applicability of the framework, but also supports the relevance
of the microeconomic model for construction price determination. The main implication for clients is that,
in order to obtain the most competitive bids for projects in the most costeﬁcit way, they should vary the
minimum number of bidders in competition according to market conditions. ngords: Construction price determination, tendering theory, commotion economics, number of bidders Introduction This paper proposes a theoretical framework for deter
mining the minimum number of contractors in com
petition for projects in the public contracting sector of
the Hong Kong commotion industry. The framework
aims to provide a more cost eﬂ'ective approach for the
Hong Kong Special Administrative Region (HKSAR)
Government to obtain competitive bids while contin
uing to maintain its public accountability. In order to
provide a theoretical foundation for the ﬁ'amework, the
linkage betwe market conditions and the degree of
competition is explored. Based on the neoclassical
microeconomic theory for construction price deter
mination, it is uggested that the number of potential
competitors in competition will depend on the market
conditions. A set of regression models is formulated to *Author for correspondence. email: [email protected] estimate the number of potential competitors in the
market. Following from this, the minimum number of
contractors to be included in competition is deter
mined. This is based on the asumption that the con
tractorselection process is random. However, the
framework does not quantify the cost eﬁ'ects of an addi
tional number of contractors in bidding competition
for projects. An empirical analysis ofa data set from
the HKSAR Governmt further conﬁrms the applic
ability of the framework. Since there is a large amount of literature on tender
ing theory, it is not realistic to expect a paper of this
format to provide a comprehensive literature review,
particularly, on the debate on the relevance of tender
ing theory and microeconomic theory for construction
price determination. See Runeson and Raftery (1998)
for a thorough literature review. Reference to the lit
erature is made whenever appropriate. The primary
purpose of this paper is to construct a framework for Construction Management and Economics
ISSN 0144—6193 print/ISSN 1466433X online 0 2002 Taylor & Francis Ltd
http://www.tand£co.ukliournala
DOI: 10.1080/01446190210151041 474 determining the minimum number of bidders in com
petition and to conduct an empirical analysis for testing
the applicability of the framework. Construction price determination The construction economics literature contains two
fundamentally different approaches to construction
price determination. The ﬁrst is the probabilistic
approach that originated from Friedman in 1956 and
has gained wide publicity. There is a large amount of
literature, that has become known as tendering theory,
on the analysis of how construction prices are deter
mined, e.g. Gates (1967, 1970, 1976), Rosenshine
(1972), Dixie (1974), Fuerst (1976), Weverbergh
(1978), Benjamin and Meador (1979) and Carr (1982,
1987). The second approach, by Hillebrandt (1974),
follows the neoclassical microeconomic theory of
price determination in construction. A more compre—
hensive literature review and evaluation on the rele
vance of neoclassical microeconomic theory for
construction price determination in the building indus
try was conducted by Runeson and Raftery (1998).
They concluded that the neoclassical microeconomic
theory is a more suitable analytical framework than
tendering theory, both in terms of its predictions and
in the conformity with empirical studies of the con
struction industry. It is outside the scope of this paper
to further evaluate further the appropriatenes of the
neoclassical microeconomic theory for construction
price determination. Instead, this line of thinking forms
the basis for the proposed empirical study. The basic assumption for the application of neo
classical microeconomic theory in construction is that
the building industry is very competitive and conforms
to the model of perfect competition. A perfectly com
petitive market is characterized by the stence of a
‘going market price' (i.e. perceived equilibrium price)
that all buyers pay and all sellers receive, and no one
player in the market can individually aﬁ'ect that price.
In other words, each buyer and each seller is much
toosmallapartoftheoverallmarketfortheiractions
to aﬁect the market price. Other standard descriptions
of such a market include homogeneity of the product,
perfect information and easy entry to and exit from
the market. Here the market can be considered as a
process of interaction between buyers and sellers of a
commodity for a mutually agreed price (Perman and
Scouller, 1999). A direct analogy in construction is
that the buyers areconstruction clients, either from the
public or private sectors, who are procurers of facili
ties. The sellers are construction companies (or con—
tractors) who construct facilities to customized designs
speciﬁed by the clients. Most contracts are awarded Ngai et a]. through competitive tendering processes where clients
and contractors reach a mutually satisfactory price.
Normally the contractors who submit lowest tender
prices obtain the contracts. In this competitive market, price determination is
based on the interaction of demand and supply. The
market price for a commodity is the equilibrium price
where the downwardsloping demand curve and the
upwardsloping supply curve intersect. The construc
tion industry responds to changes in demand in the
short run by changing the price of its product and in
the long run by a change in the capacity of the indus
try. On an a priori basis, it is assumed that ﬁrms in
the industry would tender only when they had, or anti
cipated they would have, excess capacity and, would
not tender when all capacity is being utilized. Consider a reduction in demand in the construction
industry; lower prices will result initially because of the
lower capacity utilization in the industry. The non
utilized capacity in the industry will lead to lower mar—
ginal costs. The lower the marginal costs, the higher
the opportunity costs of losing projects for individual
ﬁrms, and hence the lower the tender prices. As a
result, competitiveness increases. In the long run, the
industry will reduce the excess supply capacity because
of insuﬂicient proﬁt, and prices will be restored to their level. On the other hand, an increase in demand
in the construction market will result in higher capa
city utilization in the industry. The higher capacity uti
lization results in higher marginal costs and hence
lower opportunity costs of not winning projects, and
hence the higher the tender prices. In the long run,
the indusz supply capacity will be adjusted and prices
and proﬁts will return to their initial levels. Numberofpotendalcompeﬂtorsasa
measureofdegreeofcompetition Based on the above price determination model for the
construction industry, changes in dand and/or
supply will change the degree of competitiveness in the
industry initially, and result in movements in tender
price level. In the long run, however, the supply capac
ity will be adjusted and prices will be restored. In this
way, the degree of competition must be measured in
terms of capacity utilization rather than in terms of the
total level of output (Runeson and Bennett, 1983). In
addition, it is a reasonable assumption that the number
of potential competitors in the market is a reﬂection
of supply capacity utilization in the industry. In line
with this basic assumption, Runeson (1988) estimated
empirically that prices systematically changed by more
than :I: 20% over the economic cycle and that 85% of
these price changes could be explained by variables Minimum number of bidders for construction projects describing market conditions, such as changes in
demand and capacity utilization in industry. Therefore,
the degree of competition in the industry can be mea
sured in terms of the likely number of potential com
petitors for projects in the market, and the degree of
competition will depend on the market conditions.
There is much empirical evidence showing that
market conditions aﬁ'ect tendering behaviour (e.g. De
Neufville et al., 1977; Flanagan and Norman, 1985;
Runeson, 1990; Rawlinson and Raftery, 1997). The
market conditions aﬁ'ect at least the contractors’ bid
prices and number of competitors for a project. These
would seem obvious given the price determination
model just described. De Neufville et a1. (1977) showed
that in a boom period (which they referred to a ‘good’
years) wh there are more projects available in the
construction market, contractors generally bid for pro
jects at higher proﬁt margins, and competition for pro
jects is relatively less intense. In a slump period (i.e.
referred to as ‘bad’ years) with fewer projects avail
able, contractors bid lower than in the boom period,
and competition becomes more intense. They showed
also that market conditions aﬁ'ect the number of
competitors for a project. Interestingly enough, they
showed further that the market conditions aﬁ'ect con
tractors' bid prices independently of the competition
intensity (or the number of bidders) for a project. Measuring market conditions It seems that no deﬁnitive measure for market condi
tions in construction exists in the literature. It is sug
gested that, in these circumstances, the standard
approach is to identify a measurable quantity that can
be taken as an indicator, or proxy, for the variable
we are actually trying to measure (Flanagan and
Norman, 1983). These proxy variables’ values, and
changes in values, constitute an indirect measure of
the variable we are trying to measure. In this sense,
many possible proxy variables for market conditions
can be envisaged. McCaﬁ'er ct a1. (1983) ued the
ratio tender price index to construction cost index to
represent changes in prices due to market conditions.
Flanagan and Norman (1983) used number of bidders
received for particular projects as a manifestation of
market conditions. They further suggested that the rate
of change of a price index (such as tder price index)
would be a more appropriate proxy variable for market
conditions. Runeson (1990) derived an economic con
ditions index based on the average of all tenders’
markup, in an attempt to incorporate market conditions
into tendering models. Because of data limitations, as in our case, there is
no formal compilation of either a market condition 475 ind or building cost index in Hong Kong. For the
purpose of dis pirical study, we shall take the rate
of change of tender price index TPI (hereafter doted
as TPI,) as an indirect measure of market conditions.
In Hong Kong, generally the TPI is compiled by com
paring the prices of a proportion of the items within a
number of successful tenders during a given period
against the prices of similar items in a base schedule
of rates (Chau, 1998). It represents the cost a client
must pay for a building. It includes all input prices
and takes into account the prevailing market condi
tions. The movements of tender prices and input prices
are monitored by tender price and building cost
indices, respectively, and one of the major uses of the
TPI is for forecasting tender price level (I‘ysoe, 1981). Empirical analysis (I): minimum number of
bidders As stated above, we shall use TPIr (i.e. rate of change
of TPI) as an indirect measure of market conditions.
Additionally, we shall use l/N as a measure of the
degree of competition (since the number of potential
competitors also depends on type of project and geo
graphical location, the average number N of bidders
per project is used) and examine the relationship
between market conditions and the degree of compet
itivess for each project in a Hong Kong data set. In
order to develop a regression model for UN using the
TPIr as predictor, there are basically three steps
involved. The ﬁrst step is to use a polynomial to model the
time series of the TPI over a period of time. By using
time t as predictor in a polynomial regression analysis
of the TPI, we shall construct a best model of a poly nomial of degree n that provides a very good ﬁt to the
TPI: TPI = a0 + alt + trth + a3? + + alt" (1) Once the model of the time series of the TPI is con
structed, the second step is to diﬁ'erentiate the poly
nomial with respective to time t (see Eq. 2 below). By
substituting diﬁ'erent values of t into Eq. 2, TPIr values
can be obtained for diﬂ'erent values of t. TPIr = cu:l + 2a2t + 3113:2 + + notut'"l (2) When TPIr values are found, the third step is to work
out the ordered pairs of (UN, TPI,) for the time period.
Then a regression model for UN using TPIr as pre
dictor can be constructed, as shown in Eq. 3 below.
Thus, based on this approach, we can estimate the
average number of potential bidders N in the market
for further analysis. Since the TPI is a time series,
inferential timeseries regression or autoregressive 476 models can be constructed to forecast the TPI level in
the industry. While there is a key advantage in regres
sion analysis over other smoothing forecasting tech
niques (i.e. it provides a measure of reliability of each
forecast through prediction intervals), it is generally
risky for prediction outside the range of the observed
data that may make the model (i.e. Eq. 1) inappro
priate for predicting a future TPI level. Therefore, it
is suggested that the forecasting of a TPI level in the
industry generally be conﬁned to the short run. —1=b,,+b,x'n=1,+b,x N )1+... (3) One major problem facing construction clients, par
ticularly the HKSAR Government, is how to obtain
competitive bids for their projects in a cost eﬁ'ective
way and at the me time maintain its public account
ability. Traditionally, construction clits, at least for
the IlKSAR Government, encourage large numbers of
contractors to submit bids for each project. Drew and
Skitmore (1990, 1992) showed ﬁ'om their sample data
set taken from Hong Kong’s private and public sectors
that tendering competitions average 10 and 17 con
tractors, respectively. Empirical studies have shown
that a greater number of bidders in competition for
each project reduces the value of the lowest bid
(McCaﬁer, 1979; Skitmore, 2002). However, there has
been quite a body of literature concerning the issues
of limiting number of potential bidders and bid prepar
ation costs in competitive tendering (e.g. Engelbrecht—
Wiggans, 1980; Skinnore, 1981; Schweizer and
UngernSternberg, 1983; Flanagan and Norman,
1985; Samuelson, 1985; Wilson at al., 1987; Wilson
and Sharpe, 1988; De Neufville and King, 1991; Holt
er al., 1994; Remer and Buchanan, 2000). The key
idea is that a large number of contractors in tender—
ing competition will increase procurement costs. It is
a waste of limited resources when there are many com
petitors in tendering competition for projects in the
market, for instance, during a period of lower demand
level in the industry, when only the lowest bidder will
win the project. The high proportion of wasted
resources as a result of abortive tendering may oﬂ'set
any potential savings obtained from the lowest bid—win
tender. Therefore, policies of limiting the number of
bidders in competition would be beneﬁcial to the
industry as a whole. Some research ﬁndings recommend restricting com
petition to between four and eight contractors for each
project (Schweizer and UngernSternberg, 1983;
Flanagan and Norman, 1985; Wilson and Sharpe,
1988; De Neufville and King, 1991). The main argu
ment for this approach is that a higher number of con
tractors in competition has only marginal impact on
the value of the lowet bid received. Another approach Ngai et .11. suggests that there ts an optimum number of com
petitors for each construction project. This approach
is based on the assumptions that: (1) there is a quan
tiﬁable cost of tendering from the competitors associ
ated with every bid; (2) the total cost of tendering
increaes in proportion to the number of competitors;
and (3) potential savings diminish with increasing
numbers of competitors. The argument for this
approach is that ultimately this cost of tendering must
be recovered from clients in the long run. Which approach to adopt poses one fundamental
question: why do tenders vary? Only by answering this
question can a wellfounded theoretical basis for further
progressive thinking be formulated. For this, Runeson
and Raftery (1998) have given a comprehensive
account of assessing the variations between tenders.
They suggested that neeclassical microeconomic
theory provides an explanation of the variations in ten—
ders that is consistent with the available empirical evi
dence. If their argument is right, then the above
approaches fail to explain the fundamental question
properly, because they are based on the basic assump—
tions either implicitly or explicitly: (1) that tendering is
a random process; and (2) that there is a direct cost of
tendering. There are serious conceptual problems con
cerning thee approaches. First, based on neoclassical
microeconomic theory, more tenders would not neces
sarily guarantee a lower price, because price determin
ation is actually based on interaction of demand and
supply. Firms that are most desperate for jobs would
also be the ﬁrms most likely to tender, and thus prob
ably the number of bidders is unlikely have much effect
on the price. Second, even if it is assumed that more tenders result
in a lower price and there is a direct cost of tender
ing, the reduction in cost is for the individual project,
but the increase in cost of tendering is an industry wide
increase. A little reﬂection shows that two such differ
ent concepts cannot simply be combined and added
together. Moreover, if it is asumed that the cost of
tendering is a ﬁxed cost, then the arguments for these
approaches fail to stand as well. For instance, if it is
assumed that there are 10 ﬁrms each with an estimat
ing department set up to produce 20 estimates per
year, then there are 200 estimates per year for the
market. Ifit is assumed further that one year there are
100 new projects coming on the market, while the next
year there are only 20 new projects, the cost of ten—
dering has not changed for the industry or the ﬁrm
but the average number of estimates per project has
increased from two to 10. Instead of the above approaches as evaluated,
another approach will be suggested based on micro
economic theory for construction price determination
in rendering. This leads to variations in the tenders Minimum number of bidders for commotion projects received. The selection of contractors to submit tenders
can be assumed to be random for the purpose of public
accountability. If the random contractorselection assumption is
adopted, then the probl of predicting the lowest
tender is nondeterministic. Skiunore (1981) assumed
a random contractorselection process in tendering to
predict tender prices. He used an example in which
amples of six bidders to submit tenders were selected
from a population of 20 potential bidders. He then
worked out the ﬁ'equency distribution of the bidders’
success from all possible combinations of six bidders.
From this example he showed that random selection
of bidders reduces the predictability of the lowest
tender by reducing the chances of including potentially
low tenders, whereas increasing the number of bidders
in competition will increase predictability. In other
words, it is impossible to predict the lowest tender with
certainty. The best that can be achieved is to predict
a range of value where the lowest tender is expected
to land. Following Skitmore’s (1981) example, instead
of grouping a population of 20 potential bidders into
ascending order of potential values, N potential bids
will be arranged in ascending order of tender prices
and numbered X accordingly. After determining the average number of potential
bidders N in the market from the set of equations 1—3,
suppose that these N potential bidders in the market
are on the approved list and they would have estimated
their potential tender prices if they were asked to
submit tenders for this project. X= {1, 2, 3, . . ., N, where potential bid is
ranked Xth lowest in the group of N tenders. From these N potential contractors, k contractors will
be selected at random to submit their tenders. Then the total number of possible competitions for this
project can be calculated from Total number of possible competitions for (4)
this project = 1 The probability that the Xth lowest bid in N bids is
the lowest bid in competition is given by the proba
bility density function famed wh randomly select
ing k contractors from the group of N potential
contractors: [”_'
fav. M") = k
The simplest way to identify the lowest bid (i.e. X
= 1) from the group of N potential contractors is to where x=l, 2, ...N—k+l (5) 477 ask all of them to submit their tenders. However, as
mentioned before, this would not be cost eﬁ'ective.
Therefore, the most important question now becomes:
how can the chance of including the most competitive
bids be maximized, by randomly selecting k contrac
tors in competition for this project, once the average
number of potential contractors N competing for this
project in the market is estimated? Although there is
no theoretical deﬁnition for the meanings of maximiz
ing the chance and the most competitive bids, for sim
plicity and practicality, the chance will be
taken at the 95% conﬁdence level and the most com
petitive bids refer to one of the ﬁrst four lowest bids
among the N potential competitors. In other words,
there exists an ‘optimum’ value of k for each N such
that a 95% conﬁdence level of including one of the
ﬁrst four lowest bids in competition for this project
can be obtained. Based on this criterion, the minimum
number of contractors to submit tders for projects
will depend on the potential number of bidders N in
the market that in turn will depend on the market
conditions. Suppose it is estimated that the average number of
competitors N in the market for a particular project is
20, based on market conditions ﬁ'om the set of equa
tions 1—3. A probability density function for randomly
selecting the 1: bidders per project to submit bids can
be established. The probability and cumulative prob
ability distribution values of winning tenders for select—
ing k contractors ﬁ'om N = 20 potential competitors
in the market is shown in Table l. The cumulative
probability distributions as shown in Figure 1 indicate
that the k values aﬁ'ect the range of lowest bids
received. As It increases from 3 to 10, the range of
potential lowest bids in competitions reduces from the
lowest possible eighteenth bid in N bids received to
the lowest possible tenth bid in N bids received.
Therefore, it is desirable, at least from the client’s view
point, that those lowest potential bids in N bids will
have higher chances of being included in tendering
competitions, whereas those higher potential bids in N
bids will be excluded from the tendering process. The
winningtenderwillfallwithinarangeofvalues
depending on the choice of k. However, in order to
include one of the ﬁrst four lowest bidders in compe
tition for projects at the 95% conﬁdence level, the
minimum number of contractors to submit tenders is
when Ie = 10, giving the cumulative probability of
0.9567. The foregoing empirical analysis led to, the proposed
theoretical framework for determining the minimum
number of competitors to be included in construction
bidding competitions for projects. The following
empirical analysis on a HKSAR Government data set
sets out to test the applicability of the ﬁ'amework. 478 Ngar' et at.
Table 1 Probability and cumulative probability distribution values of tenders for N = 20
X fan, 3! (3‘) Va 3! (x) fao, 5) (x) 21129, 5) (1‘) fan, ma) Ef .20. m; (x)
1 0.1500 0.1500 0.2500 0.2500 0.5000 0.5000
2 0.1342 0.2842 0.1974 0.4474 0.2632 0.76.32
3 0.1193 0.4035 0.1535 0.6009 0.1316 0.8947
4 0.1053 0.5088 0.1174 0.7183 0.0619 0.9567
5 0.0921 0.6009 0.0880 0.8063 0.0271 0.9837
6 0.0798 0.6807 0.0646 0.8709 0.0108 0.9946
7 0.0684 0.7491 0.0461 0.9170 0.0039 0.9985
8 0.0579 0.8070 0.0319 0.9489 0.0012 0.9996
9 0.0482 0.8553 0.0213 0.9702 0.0003 0.9999
10 0.0395 0.8947 0.0135 0.9837 0.0001 1.0000
11 0.0316 0.9263 0.0081 0.9919 0.0000 1.0000
12 0.0246 0.9509 0.0045 0.9964 0.0000 1 .0000
13 0.0184 0.9693 0.0023 0.9986 0.0000 1.0000
14 0.0132 0.9825 0.0010 0.9996 0.0000 1.0000
15 0.0088 0.9912 0.0003 0.9999 0.0000 1.0000
16 0.0053 0.9965 0.0001 1.0000 0.0000 1 .0000
17 0.0026 0.9991 0.0000 1.0000 0.0000 1.0000
18 0.0009 1.0000 0.0000 1.0000 0.0000 1 .0000
19 0.0000 1 .0000 0.0000 1 .0000 0.0000 1 .0000
20 0.0000 1 .0000 0.0000 1 .0000 0.0000 1 .0000 12 3 4 5 G 7 I 91011121314151811181020
XMNI.) Figure 1 Cumulative probability distribution values of
winning tenders for N = 20 Enpiricalanalysisﬂ):ﬁongKongdntaset The following pirical analysis is based on a sample
of 229 projects with 3285 bids received over the period
from the fourth quarter of 1990 to the third quarter
of 1996. The sample was derived ﬁ'om HKSAR
Governmt Architectural Services Departmt (ASD)
data. Projects awarded through selective tendering
(where the number of competitors is an administrative
decision rather than a consequence of market condi
tions) have been excluded for homogeneity purposes.
Figure 2 shows the positively skewed frequency distri
bution of number of bidders per project for the data
set. On average, there are 14 contractors (betwe 3
and 33, with standard deviation about 7), competing
for each contract. Table 2 shows the variations in the TF1 level and
the average number N of contractors per project for
the data set. The TPI used is a quarterly index com
piled by ASD primarily as an aid to adjust building cost data for estimating purposes. It is prepared also
to provide an indication of the price level of tender
prices for new building works undertaken by ASD. The best model (i.e. using time t as a predictor in
a polynomial regression analysis of the TPI) is found
to be a polynomial of degree 3, as shown below. This
polynomial, as shown in Figure 3 (time series of the
TPI is shown as a solid line and predicted TPI as a
dotted line), provides a very good ﬁt to the TPI since
the corresponding R2 is 0.9773 and the residual plot
exhibits no special pattern for the violation of regres
sion assumptions. Manamarm Figure 2 Distribution of number of bidders per project for
ASD projects from 4th quarter of 1990 to the 3rd quarter
of 1996: (Std. Dev. = 7.14; Mean = 14 and number of pro
jects in sample = 229) Minimum number of bidders for construction projects 479 Table 2 Variations in the TPI and average number of bidders N (19901996) Year (muanur 'Tunet 111
1990 4 1 596
1991 l 2 608
2 3 592 3 4 573 4 5 515 1992 1 6 531
2 7 548 3 8 519 4 9 518 1993 1 10 527
2 11 527 3 12 541 4 13 563 1994 1 14 586
2 15 594 3 16 615 4 17 666 1995 1 18 708
2 19 712 3 20 733 4 21 747 1996 1 22 772
2 23 813 3 24 848 Ibud No. of Total number Av. no. of
projects of bidders bidders/project (N)
8 112 14
6 116 19
5 119 24
9 204 23
8 189 24
10 189 19
6 129 22
6 99 17
8 148 19
10 158 16
4 61 15
7 98 14
5 100 20
14 239 17
7 81 12
14 210 15
9 98 11
14 171 12
11 110 10
17 142 8
13 123 9
14 157 11
10 121 12
14 111 8
229 3285 Tmmlm
§§§§§§§§§§§ 412341234123412341234123
1H1 1 ms 1" 1“ 190s “II(mlﬂllll’lhm1m Figure 3 Time series ofthe TPI and polynomial regres
sion analysis of the TPI using time t as a predictor TPI = 695.6” — 37.62: + 2.893:2 — 0.04l9t3
(R2 = 0.9773) Hence, the rate of change TPIr at time t is given by
TPIT = —37.624 + 5.7862 — 0.1257:2 By substituting diﬂ'erent values of t into the above
equation, 24 ordered pairs of (UN, TPI.) can be found
as shown in Table 3, and the scatter plot of these 24
ordered pairs is produced as shown in Figure 4. The plot shows a nonlinear relationship between UN and
TPI,. A regression model can be esmblished to best ﬁt
the set of data with the following result (R2 = 0.7550;
F = 32.36; p < 0.0001): 1/N = 0.04843 + 0.0007134 x TPI,
+ 0.00003809 x TPII.2 Suppose the forecast of TPIr (i.e. the rate of change of
TPI) is 23, then from the above regression model, the
estimated potential number of competitors for a project
is N = 12. The probability density ﬁmction values for
randomly selecting the I: bidders per project to submit
bids are as shown in Table 4. At the 95% conﬁdoe
level, to include at least one of the ﬁrst four lowest bid
ders among 12 potential competitors in competition,
the minimum number of contractors to submit tenders
is when k = 6, giving the cumulative probability 0.9697. Therefore, by randomly selecting six contractors
from the approved list of qualiﬁed contractors to sub
mit tenders to compete for a project in this period, the
HKSAR Government can have a 95% conﬁdence that
at least one of the ﬁrst four lowest bids in the market
will be included in competition for projects. In this
way, not only will this approach be more cost eﬁ'ective
in terms of procurement costs, but also the Government
can still maintain its public accountability in the ten
dering competitions. 480 Ngat' at al.
Table 3 Variation of TPI, and UN (19901996)
Year Quarter Time I TPI TPIr Av. No. of bidders! LIN
Project (N) __
1990 4 1 596 —3l.9637 14 0.0714
1991 1 2 608 —26.5548 19 0.0517
2 3 592 —21.3973 24 0.0420
3 4 573 —l6.4912 23 0.0441
4 5 515 —11.8365 24 0.0423
1992 I 6 531  7.4332 19 0.0529
2 7 548 3.2813 22 0.0465
3 8 519 0.6192 17 0.0606
4 9 518 4.2683 19 0.0541
1993 1 10 527 7.6660 16 0.0633
2 11 527 10.8123 15 0.0656
3 12 541 13.7072 14 0.0714
4 13 563 16.3507 20 0.0500
1994 1 14 586 18.7428 17 0.0586
2 15 594 20.8835 12 0.0864
3 16 615 22.7728 15 0.0664
4 17 666 24.4107 11 0.0918
1995 1 18 708 25.7972 12 0.0819
2 19 712 26.9323 10 0.1000
3 20 733 27.8160 8 0.1197
4 21 747 28.4483 9 0.1057
1996 1 22 772 28.8292 1 1 0.0892
2 23 813 28.9587 12 0.0826
3 24 848 28.8368 8 0.1261 0.14 9
.a 1m
.0 .o .o
8 8 8 an 10 o 10 mundaneurn)
M4 Scatterplotoft:hangeofmarketconditionTPIr
with degree of competition llN b
8 Conclusions 'I'hispapersetsouttoexploreanddemomrrateame
oretical linhge between market conditions and the
number of potenn'al contractors in competition. Based
on neoclassical microeconomic theory for construc
tion price determination and the assumption of random
contractorselection in the bidding competitions, a the
oretical framework for determining the minimum
number of competitors in the tendering process has
been proposed. The framework comprises two basic parts. The ﬁrst part applies microeconomic theory in
linkingchangesindemandtochangesinpricesand
subseth changes in supply capacity to explain con
struction price determination and tender variations.
The rate of change of the TPI is used to measure
market conditions in the building industry, and the
degree of competition can be measured in terms of
number of potential competitors in the market. From
these, a set of regression models is formulated to esti
mate the number of potential contractors competing
inthemarketbytheforecastTTIlevel.1hesecond
part determines the minimum number of contractors
to submit tenders for the particular projects concerned
such that a 95% conﬁdence can be achieved that one
oftheﬁrstfourlowestbidsinﬂtemarketwillbe
included in the bidding The implication is that the HKSAR Governmt can
deriveamore costeﬁ'ectiveapproachinitsopentmder—
ingsystembyselectingtheminimumnumberkofcon—
tractors in bidding competitions based on the market
conditions (or the TPI, value) while its
public accountability for contractorselection in tender
ing. The limitations for this framework are: (l) gener
allythepredictionoftheTPIisconﬁnedtotheshort
termbecause ofreducedaccuracyoffmecastsfxther
intotheﬁiture,andhencemaymaketheTPIforecast
ing model inappropriate; and (2) the framework is Minimum number of bidders for construction projects 481 Table 4 Probability and cumulative probability distribution values of winning tenders for N = 12 __—____——_______———————— X faz, (3‘) Elm, (I) f 12, 5) (x)
1 0.3333 0.3333 0.4167
2 0.2424 0.5758 0.2652
3 0.1697 0.7455 0.1591
4 0.1 131 0.8586 0.0884
5 0.0707 0.9293 0.0442
6 0.0404 0.9697 0.0189
7 0.0202 0.9899 0.0063
8 0.0081 0.9980 0.0013
9 0.0020 1.0000 0.0000 10 0.0000 1.0000 0.0000 1 1 0.0000 1.0000 0.0000 12 0.0000 1.0000 0.0000 Zfaz, 5) (x) f 12, 9(3) 2’ (12, (x)
0.4167 0.5000 0.5000
0.6818 0.2727 0.7727
0.8409 0.1364 0.9091
0.9293 0.0606 0.9697
0.9735 0.0227 0.9924
0.9924 0.0065 0.9989
0.9987 0.001 1 1.0000
1.0000 0.0000 1.0000
1 .0000 0.0000 1 .0000
1 .0000 0.0000 1 .0000
1 .0000 0.0000 1.0000
1 .0000 0.0000 1 .0000 _____—__—_—_—————— based on the assumption that the contractorselection
process is random. It remains possible that the selec
tion of contractors in competition for particular projects
will not be simply a random choice that reduces the
range of lowest tenders. Further research is necessary
to quantify the total tendering cost eﬂ'ects of an addi
tional number of contractors in bidding competitions
for projects. Acknowledgement The research is supported by the Research Grants
Council, University Grants Committee of Hong Kong,
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 Economics, Ethics , The Land, TPI, minimum number, market conditions

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