8
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To increase the number of goods it produces, a firm invests in innovative effort, i.e. spends
resources on R&D. A firm’s R&D choice is twodimensional. Each instant, it chooses both
the frequency of arrival of new innovations
λ
t
∈
[0
,
¯
λ
] and the quality of new innovations
θ
t
∈
[0
,
1]. The arrival intensity
λ
t
determines the Poisson rate at which innovations arrive.
Conditional on an innovation, the number of new product lines generated is given by
X
t
= min (
Y
t
,
¯
p

P
t

)
with
Y
t
∼
Bin
(
n, θ
)
,
where
n <
¯
p
is an exogenous upper bound on the number of new product lines that can be
developed following an innovation and
Bin
(
n, θ
) is the binomial distribution. This specifica
tion implies that the expected number of new product lines is approximately
nθ
. Therefore,
a higher quality
θ
leads to a higher expected number of new product lines. Bounding the
number of new product lines
X
t
from above by ¯
p

P
t

ensures that
P
t
never exceeds ¯
p
.
These assumptions imply that the total number of product lines the firm has developed up
to time
t
, denoted by
I
t
, evolves as
dI
t
=
X
t
dN
I
t
,
where
dN
I
t
is a Poisson process with intensity
λ
t
.
A firm’s existing product lines can become obsolete because some other firm innovates
on a good it is currently producing.
In this case, the incumbent producer loses the good
from its portfolio due to creative destruction. Since any firm is infinitesimal, we can ignore
the possibility that it innovates on a good it is currently producing.
Because of creative
destruction, each product becomes obsolete at an exponentially distributed time with inten
sity
f
. We call
f
the rate of creative destruction, that each firm takes as given. Subsection
C
embeds the singlefirm model into an industry equilibrium and endogenizes the rate
f
of
creative destruction. The total number
O
t
of product lines lost by the firm up to time
t
≥
0
9
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because of creative destruction evolves as
dO
t
=
dN
O
t
,
where
dN
O
t
is a Poisson process with intensity
fP
t

. The total number product lines in a
firm’s portfolio
P
t
is therefore given by
P
t
=
I
t

O
t
.
A firm with zero product lines exits the economy at time
τ
0
≡
inf
{
t >
0 :
P
t
= 0
}
.
A firm performing R&D with intensity and quality (
λ
t
, θ
t
) incurs flow costs
q
(
P
t
, λ
t
, θ
t
).
To make sure that shareholders are better off with more product lines, we impose that the
R&D cost function does not increase too fast in the number of product lines in that
q
(
p
+ 1
, λ, θ
)

q
(
p, λ, θ
)
<
1
.
(1)
An incumbent firm’s operating profit is the profit that comes from the operation of the
product lines minus the endogenous costs of performing R&D:
P
t

q
(
P
t
, λ
t
, θ
t
)
.
Profits are taxed at the constant rate
π >
0. As a result, firms have an incentive to issue
debt to reduce corporate taxes.
9
To stay in a simple timehomogeneous setting, we follow
the literature (e.g.
Leland
(
1994
),
Duffie and Lando
(
2001
), and
Manso
(
2008
)) and consider
debt contracts that are characterized by a perpetual flow of coupon payments
c
. The firm
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 Spring '17
 Enkhjin
 Finance, Corporate Finance, Debt, Test, The Natural, Kerr