5.2 Case 2: Outsourcing dominates intermediate exports
When (27) holds there is a potential gain for the MNE from signing an outsourc-
ing agreement. Therefore, in principle all three modes are viable options for a
Y -firm to serve market M. However, by (15) and (26), it is readily shown that
ΠO > ΠE whenever (27) is satisfied. Thus, as before, only two modes are rele-
vant. These modes are X and O, which implies m = 0. Then (m,n) = (0,n*)
is an equilibrium distribution of firms between modes X and O whenever no
firm wants to change its mode. This happens for interior outcomes n ∈ (0, 1)
whenever:
∆χ,o(n) ≡ Πχ - ∏o = 4 A(0,n) τ -
', 2 h(1 - β) + τi2 = 0
(31)
and for corner outcomes n =1 (n =0) whenever ∆X,O (n) < 0 (> 0). In (31)
the market potential measure is obtained by solving (3) after substituting for
the equilibrium prices (6) and (25)
A(0,n) =
—8τ(1 — n) + 8ʌ/Lnβ2 (τ + 2(1 — β)) + τ2 (1 — n)2
2nβ2 (τ + 2(1 - β))
(32)
Solving (31) for n gives:
1r _|_9Г 1 <Λ]β2[τ + 2(1 - β)]2L - 8τ2
= 8[τ + 2(1 - β)∣------τ2 (2β - τ)------.
(33)
which can be shown to be a stable equilibrium since ∂∆X,O/∂n∣n=n* > 0. Given
(27), n* is increasing in L. It is constrained between 0 and 1 for L ∈ (LX ,LO)
with
8τ2
LX ≡ β2 [2(1 - β) + τ ]2 , LO ≡ 2(1 - β) + τ LX,
(34)
where LO >LX is granted once more by (27). Corner outcomes n =0and
n =1are attained for LX ≥ L and L ≥ LO respectively. It is readily verified
that LX is an increasing function of τ while LO is decreasing in τ whenever
τ > 4(1- β)
(35)
holds and increasing otherwise. Note that, for (35) to hold at some τ < 1,it
must be β > 3/4.
Thus, when β > τ/2 the chosen mode of supply depends on both the size
of the market and the level of trade costs. In particular, by simple inspection,
(33) shows that the share of firms choosing mode O always rises with market
size L (∂n* /∂L > 0). The impact of trade costs on n* is more complex. Indeed,
differentiating (33) with respect to τ yields:
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