This note explores the value of search capital in interfirm matches in the outsourcing trade, by extending Rauch and Casella's (2003) framework to a dynamic model of matching and searching. On provisional calculations, the sunken cost of this search imposes a similar order of magnitude trade barrier to most tariffs, and would be expected to affect both the intensive and extensive margins of trade.Trade, search, matching, capital

T.Huw Edwards

English

Research Papers in Economics

     
       
   
ISSN 1750-4171 
 
               
 
 
DEPARTMENT OF ECONOMICS DEPARTMENT OF ECONOMICS DEPARTMENT OF ECONOMICS DEPARTMENT OF ECONOMICS       
       
DISCUSSION PAPER SERIES DISCUSSION PAPER SERIES DISCUSSION PAPER SERIES DISCUSSION PAPER SERIES       
 
 
 
The capital value of the search for trading 
matches 
 
T.Huw Edwards 
 
 
WP 2009 - 08 
 
 
 
 
 
 
 
 
 
            
Dept Economics 
Loughborough University 
Loughborough  
LE11 3TU  United Kingdom 
Tel:  + 44 (0) 1509 222701 
Fax: + 44 (0) 1509 223910 
http://www.lboro.ac.uk/departments/ec The Capital Value of the Search for Trading Matches
T.Huw Edwards, Department of Economics, Loughborough University, LE11 3TU.
Tel: +44(0)1509-222717. Fax: +44 (0) 1509 223910. T.H.Edwards@lboro.ac.uk.
Abstract
This note explores the value of search capital in inter￿rm matches in the outsourcing trade, by extending
Rauch and Casella￿ s (2003) framework to a dynamic model of matching and searching. On provisional calcu-
lations, the sunken cost of this search imposes a similar order of magnitude trade barrier to most tari⁄s, and
would be expected to a⁄ect both the intensive and extensive margins of trade.
KEYWORDS: Trade, Search, Matching, Capital. JEL Classi￿cation: F00, F12, F13
11 Introduction
The trade literature emphasises that there are substantial unspeci￿ed barriers impeding the international ￿ ow
of goods and services. Tre￿ er (1995) shows that the factor content of international trade falls far short of that
predicted by the Heckscher-Ohlin-Vanek model. McCallum (1995) demonstrates the presence of large national
border e⁄ects.1 While there is no single cause for these discrepancies, two important factors have emerged: ￿rm-
level studies (Tybout, 2002) have found evidence of substantial, unspeci￿ed sunken costs of entering new markets
- a ￿nding which underlies much of the recent literature on ￿rm heterogeneity and trade (Bernard and Jensen,
1999, Ghironi and Melitz, 2005, Bernard et al, 2007). In addition, gravity studies indicate signi￿cant informational
barriers to trade (Rauch and Trindade, 2003). These two ideas are potentially interlinked2: if the search for
information on trade partners is costly, and that information yields a return to those searching, then it can be
viewed as a form of capital outlay. Of course, there are variants on this, depending upon the extent to which
information is private, excludable and sharable within networks (free, or alternatively in return for an fee3).
In this note, I develop the idea in more detail, concentrating especially on the capital formation process in
outsourcing relationships - such as those between Chinese garment producers and Western distribution ￿rms.
Typically, these are relationships between ￿ upstream￿and ￿ downstream￿trading ￿rms, which entail contracting
arrangements, even if they are relatively short-term.4 Such ties lead to a searching/matching process: if the
relationship between the ￿rms proves successful, it will be maintained in the long term, while otherwise the ￿rms
will move on.
1The di⁄erence in trade between Canadian provinces and that with neighbouring US states, when corrected for size and di⁄erence,
is a matter of 22-fold, though later studies (Anderson and van Wincoop, 2003) greatly reduce this discrepancy.
2Rauch (1996).
3As in Rauch and Watson (2004)
4Besedes and Prusa (2006) estimate the median trading relationship is around 1 year, and argue it supports a searching/matching
process.
2Time after start
of search
0
Expected
profit/loss
Expected profit of a successfully matched firm
Initial churn of partners as most firms
are still searching
Expected loss of a firm temporarily in a poor match
Figure 1: trend in average pro￿t of searching ￿rms over time, following start of search.
Following trade liberalisation5 there will be a churn of searching ￿rms, with many pairings making losses.
Eventually, pro￿table pairings will emerge as dominant. This process is shown in Figure 1., showing expected
pro￿t increasing over time. With free entry and exit and random match pro￿tability, then ￿rms entering the
industry will expect, ex ante, to make just su¢ cient long-run pro￿t to cover the interest costs of losses during the
initial search. In the absence of spillovers, we would expect the average pro￿t in long-run equilibrium to equal
the interest on the costs of informational capital formation. By modelling the dynamic process of searching for
matches, we can derive this long-run return, and hence estimate the implicit value of informational search capital.
2 A schematic model of match-searching in the outsourcing trade
I set up an illustrative model. Outsourcing requires the partnership of an upstream ￿rm, u, and a downstream
￿rm, d, which, for reasons of comparative advantage, will locate in di⁄erent countries when trade is unimpeded. u
sells a semi-￿nished good to d, who then completes the manufacture and sells it on to ￿nal consumers. The two
￿rms are of equal size and ex ante expected e¢ ciency: however, productivity varies depending on the goodness of
￿t of the match, ￿i. As in Rauch and Casella (2003), match quality, ￿; follows a uniform, rectangular distribution
5Or even, in some circumstances, in anticipation of trade liberalisation (see Edwards, 2006).
3between 0 and 1, and ￿rms do not know ￿i before entering a match i, though they know its overall distribution.
Search friction derives from the need for at least one ￿rm to make a relationship-speci￿c investment: in order
to avoid a potential hold-up problem,6 this generally requires a contractual relationship for at least some minimum
period, which I characterise by a ￿xed contract period, t, during which the two ￿rms have an exclusive relationship.
Firms employ just labour, both variable and ￿xed, the latter being renewable once per contract period.7 This
￿xed labour cost is normalised at unity, so that pro￿ts are
￿ui;￿di = ￿￿i before the ￿xed cost and (1)
￿ui;￿di = ￿￿i ￿ 1 excluding the ￿xed cost. (1a)
Since ￿i is uniformly distributed between zero and unity, expected initial match quality is ￿e
i = 1
2. Ex ante expected
pro￿tability from a random match is therefore
￿e
ui;￿e
di =
￿
2
; (2)
￿e
ui;￿e
di =
￿
2
￿ 1: (2a)
The annual discount rate, r, equates to a discount rate of ￿ per contract period, where
￿ = (1 + r)t ￿ 1: (3)
I concentrate on a case where the industry is small in comparison to the economy as a whole, and wages are
exogenous.
At the end of each contract period, a ￿rm which had still been searching in the previous period will assess
whether its latest match is worth sticking with (￿ ￿ ￿R, the reservation match quality,8 which will occur with
probability 1 ￿ ￿R) or whether it should again renew search (probability ￿R). If the quality of successive matches
is serially independent, then in period n + 1; a proportion 1 ￿ ￿n
R will still be searching:
6See Hart (1995).
7This means there are no long-run sunken costs, other than that of the search.
8The switchpoint of a search process (Kohn and Shavell, 1974).
4In period n+1; the expected pro￿t before ￿xed costs for those ￿rms which are still searching can be written as
￿￿
2 , where ￿ (=
(1￿(1+r)
￿t)
r ) is an adjustment for the length of the contract period. Expected pro￿t for ￿rms which
￿nd a satisfactory partner equals
￿￿
2 (1+ ￿R), while that for unsuccessful matches averages
￿￿￿R
2 . Expected pro￿t
over all ￿rms will therefore be
￿e
Sn+1 =
￿￿
2
(1 + ￿R ￿ ￿
n+1
R ): (4)
By contrast, if initial match quality is ￿i0, then if the ￿rm chooses from the beginning to stick with its initial
partner, its pro￿t in each period will be ￿￿￿i0. The net expected bene￿t, Be
Sn+1, in period n+1 of having started
by searching rather than not searching is
Be
Sn+1 =
￿￿
2
(1 + ￿R ￿ 2￿i0 ￿ ￿
n+1
R ): (5)
The expected net present value (to the beginning of the search process) of pro￿ts for a ￿rm which chooses to start
by searching, Ne
S; is a geometric progression, which can be summed to yield
Ne
S =
￿￿
2￿
(1 + ￿R ￿ 2￿i0) ￿
￿￿
2
￿R
(1 + ￿ ￿ ￿R)
: (6)
When ￿i0= ￿R, Ne
S= 0; which is satis￿ed by the roots of a quadratic equation, of which only the smaller is feasible:
￿R = 1 + ￿ ￿
p
￿(1 + ￿): (7)
￿R is decreasing in ￿ (for ￿ > 0), and hence in terms of r and t. For example, with t = 1 year and r = 5 per cent
per annum, ￿R will equal 0:82, and average long-run match quality will be
1+￿R
2 = 0:91.
53 Search information as capital
In a long-run equilibrium, where all ￿rms have found satisfactory partners, normalising the ￿xed labour cost of
each ￿rm to equal unity, the average annual pro￿t of a matched ￿rm after subtracting ￿xed costs is
￿f =
1 ￿ ￿R
2￿R
￿ 1 =
1 ￿ ￿R
2￿R
: (8)
This equals the interest cost on the average expected level of search capital, rSK.
4 Comparison to trade ￿ ows
To gauge the e⁄ect of trade search costs, I compare it to the value of trade ￿ ows. For this, it is necessary to develop
the model more explicitly. Assume that a large number ￿rm pairings sell their joint output in a Krugman-style
monopolistically competitive environment. Initially we will write a simpli￿ed demand function for the joint output
of ￿rm pair i as
Yu = Yd = Yi = (
Pi
A
)￿"; (9)
where the number of ￿rms is large and " approximates the Dixit-Stiglitz elasticity of substitution between varieties.
A will shift if rival ￿rm pairs enter or exit the market. The unit variable cost of each ￿rm is Ci(￿i), where Ci is
assumed equal for the two ￿rms and declining in terms of ￿i. Pro￿t-maximising output is
Y ￿
i = (
"
" ￿ 1
2Ci
A
)￿": (10)
Assuming equal Nash bargaining weights for u and d, pro￿t of each ￿rm before taking account of ￿xed costs,
￿u = ￿d =
A
2"
Y
￿
"￿1
"
i : (11)
Since we want ￿u and ￿d to be proportional to match quality, we set (11) equal to pro￿t in (1) and rearrange.
Where ￿i = ￿R, ￿u = ￿d = 1 and ￿ = 1
￿R; which gives ￿i =
￿i
￿R: Consequently, output,
Y
￿
"￿1
"
i =
2"
A
￿i
￿R
; (12)
6implying that the value of trade between the two ￿rms is also proportional to match quality:
V ￿
i =
P￿
i Y ￿
i
2
=
￿i
￿R
": (13)
The average match quality of successful matches is
1+￿R
2 ; so the value of trade ￿ ow between an average successful
pairing,
V =
1 + ￿R
2￿R
"; (14)
and hence the ratio of the interest cost on search capital to the trade ￿ ow between an average successful pairing is
rSK
V
=
1 ￿ ￿R
1 + ￿R
1
"
: (15)
As an illustration, where r = 5% and t = 1 year - in line with Besedes and Prusa (2006) - our model indicates
an annual interest cost of search capital of around 5 per cent of the total value of trade for a typical pairing facing
a demand elasticity of 2. This would drop to 2 per cent for a demand elasticity of 5. Longer contract periods or
higher interest rates can raise this cost quite considerably (see Figures 2a and 2b).
0.5 1 1.5 2 2.5
2.50%
5.00%
7.50%
10.00%
0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
7.00%
8.00%
9.00%
10.00%
Contract period t
Interest r
Figure 2a: interest cost of search as proportion of trade value, elasticity of demand=2.
70.5 1 1.5 2 2.5
2.50%
5.00%
7.50%
10.00%
0.00%
0.50%
1.00%
1.50%
2.00%
2.50%
3.00%
3.50%
4.00%
Contract period t
Interest r
Figure 2b: interest cost of search as proportion of trade value, elasticity of demand = 5.
Over the ranges investigated (2:5% 6 r 6 10%;0:5 6 t 6 2:5), the value of search capital stock varies from
48 to 118 per cent of trade value, when the typical elasticity of demand is 2, falling to 19 ￿ 87 per cent when the
elasticity is 5. For a pairing of given match quality, ￿i, the cost of search would reduce sales by between 5 and 17
per cent (in both elasticity cases). This represents the e⁄ect on ￿rms￿intensive margins: however, search costs can
also potentially a⁄ect extensive margins, since ￿rms will not enter into a search for partners at all, unless there is
a large enough comparative cost advantage (of proportion
1￿￿R
￿R ) for trading pairings.
5 Conclusions
By extending the one-o⁄ matching model (Rauch and Casella, 2003) into dynamic search by repeated matching,
it is possible to derive the scale of search capital costs and relate it to the value of trade . The functional form
is relatively simple, and no account is taken of the role of networks in information-sharing, which requires further
research. However, these rather rough calculations indicate provisionally that search costs may well be of a similar
order of magnitude to typical tari⁄ barriers.
The relationship to the recent literature on ￿rm heterogeneity and market entry (Ghironi and Melitz, 2005) is
interesting. Search costs are certainly a credible cause of the inferred sunk cost of entering new markets, and are
consistent both with the selection bias towards more e¢ cient ￿rms and with the fact that only the most e¢ cient
exporting ￿rms enter several markets (Bernard et al, 2007). In the long run, trading ￿rms may be more pro￿table,
8not just because the more productive ￿rms self-select to enter trade, but also because of the return on the short-run
search costs for partners, with trade search, in itself, producing further heterogeneity across ￿rms.
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10

The capital value of the search for trading matches.

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The capital value of the search for trading matches.

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