Uncertain Productivity Growth
3 THE OPTIMAL MARKET ENTRY MODE
The general functional form of the option values for both market entry strategies results as
(53)
Fiu (iï) = Aiiu Vβ1u + Ai2u Vβ2u
with
|
1 - 1 r - δu I β1u = 2 - 2+ V |
' r - δu _ 1 |
+ 22 > 1 (54) σ2 |
|
1 _ 1 r - δu S β2u = 2 - σ - V |
' r - δu _ 1 σ2 2 |
+ 2r2 < 0. (55) σ2 |
The optimal cut-offs iï*u and the two unknown Aiju can be determined by defining appropriate
boundary conditions. If the current productivity level iï approaches zero, the option value of an
uncertain investment should also tend to zero, as the probability of a sufficient increase in the
future is low. Therefore, the first boundary condition states
Fiu (0) = 0.
(56)
If the productivity level reaches the optimal cut-off level, the investor is indifferent between delay-
ing the uncertain investment (keeping the option alive) and executing the project by investing the
sunk costs Ii . As a consequence, the second condition is the matching condition which captures
the indifference at iï* with
iu
Fiu (iïi*u) = Vi(iïi*u) - Ii.
(57)
Finally, in order to find an optimal threshold value for iï the two functions need to be tangent in
the optimum. Tangency can be accounted for by imposing the smooth pasting condition with
∂F (iï*u ) = ∂V (iï*u )
diï diï
The first boundary condition necessitates that Ai2u = 0 as β2u is negative. Therefore, the option
functions for both market entry modes are reduced to
(58)
Fiu (iï) = Aiu iïeu
(59)
with βu = β1u and i ∈ {E, F}.
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