∂n* _ β2[2(1 - β)+ τ]2[(3 - 2β)τ - 4β(1 - β)] L - 8τ3
(36)
(37)
∂τ 4τ3 (τ - 2β)2
which shows that ∂n*∕∂τ > 0 (< 0) as long as L > L (< L), with
8τ 3
β2[2(1 - β) + τ]2[(3 - 2β)τ - 4β(1 - β)].
Under (27) L is a decreasing function of τ that falls between LX and LO as long
as (35) holds. Thus, when L ∈ (Lχ, Lo) and (35) holds, we have ∂n*∕∂τ > 0
(∂n*∕∂τ < 0) for L > L (L < L). Moreover, since L is decreasing in τ, we can
have L>L (L<L) for large (small) τ and L>L for large τ. This happens
if L is large enough, namely, larger than the smallest possible value of L (i.e.,
L∣τ=1).
To sum up, we have:
Proposition 2 When β > τ/2 FDI plus intermediate exports (mode E) is
never chosen. The choice between final exports (mode X) and FDI plus ousourc-
ing (mode O) is affected by both market size and trade costs. In particular, the
share of firms choosing FDI increases with market size. It also increases with
trade costs if market size is small. On the contrary, the relation between the
share of firms chosing FDI and trade costs is U -shaped if market size is large
enough.
The source of this non-linearity lies in contractual incompleteness. This
makes the payoff from mode O depend on trade costs even though no trade
takes place under that mode. To understand why this happens, consider the
trade-offs a Y -firm faces when choosing its supply mode. First, it faces the
traditional proximity-vs-concentration trade-off: final exports incur trade costs
but save on the costs of distant assembly lines whereas the opposite is true for
FDI. Second, in the case of FDI, a Y -firm faces the trade-off between outsourcing
and intermediate exports. Outsourcing saves on trade costs but incurs the costs
of ex-post bargaining. FDI under intermediate exports incurs the former costs
but saves on the latter. This second trade-off is entirely due to contractual
incompleteness and arises only if the ex-post bargaining power of the Y -firm in
small enough with respect to the cost of shipping intermediates (β > τ /2).7
Crucially, the level of trade costs affects both trade-offs. When trade costs
are large, the first trade-off dominates and makes FDI more appealing than ex-
ports due to traditional proximity considerations. When trade costs are small,
the second trade-off dominates. It is still true that small trade costs make
exports more appealing than FDI due to proximity-vs-concentration considera-
tions. However, they also strengthen the outside option of FDI plus intermediate
7 This is formally shown in the next section, where the complete contract outcome is fully
characterized.
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