Interaction product market vs. capital market
[Brander & Lewis]
Recap:
Conflict of interest owners / creditors
→ Owners invest in projects more risky than creditors
would like.
→ Argument against debt financing.
This disadvantage of debt financing may be turned into an
advantage in some cases of imperfect competition in the
product market.
Aggressive action by a firm on the product market may
result in the firm’s rivals behaving less aggressively.
Debt financing may be one way to convince rivals that one
is committed to competing aggressively.
If risky actions in this way pay off for the firm, then the
creditors may be interested in financing this aggressiveness,
and debt may gain an advantage over equity.
Tore Nilssen – Lecture 9: Corporate Finance III - 1
A duopoly model: Cournot with cost uncertainty
Two firms: 1 and 2
Inverse demand:
P = a – bQ,
where
Q = q 1 + q 2.
Uncertainty in costs of production:
Ci(qi, zi) = (ci - zi)qi,
i = 1, 2.
E(zi) = 0
z1 and z2 are independent stochastic variables,
distributed over the interval [z, z ],
density function f(zi).
Expected costs: ECi(qi) = ciqi.
A high zi is a positive shock for firm i.
Firm 1’s gross profit (before paying down any debt):
R1 = R(q1, q2, z1) = [a – b(q1 + q2) – c1 + z1]q1.
Expected gross profit for firm 1:
ER1 = [a – b(q1 + q2) – c1]q1;
similar for firm 2.
Tore Nilssen – Lecture 9: Corporate Finance III - 2
Nash equilibrium: Both firms simultaneously choose
quantities.
dER1
= a − b⎛⎜⎝ q1 + q2 ⎞⎟⎠ − c1 − bq1 = 0
dq1
dER 2
= a − b⎛⎜⎝ q1 + q2 ⎞⎟⎠ − c2 − bq2 = 0
dq2
⇒
a − 2c1 + c2
q1 =
3b
a − 2c2 + c1
q2 =
3b
q2
BR1(q2)
BR2(q1)
q1
Tore Nilssen – Lecture 9: Corporate Finance III - 3
Aggressive behaviour may be profitable.
How to become more aggressive?
→ lower marginal costs
- investing in new technology
- loan commitment: borrowing I costs
D = K + rI = r*I ⇒ r < r*.
Question:
How can debt financing make the firm more aggressive?
R1 = [a – b(q1 + q2) – c1 + z1]q1
R11 = ∂R1/∂q1 = a – 2bq1 – bq2 – c1 + z1
1 = – 2b < 0
R11
R21 = – bq1 < 0
R1z = q1 > 0
1
1 =–b<0
R12
(firm’s decisions are strategic substitutes)
R11z = 1 > 0
1
(an increase in the shock z1 is good both for gross profits
and for marginal gross profits)
Tore Nilssen – Lecture 9: Corporate Finance III - 4
Firm i’s debt: Di.
Profit is positive only if zi is such that Ri > Di.
Define ẑi according to: Ri ( qi , q j , zˆi ) ≡ Di.
Assumption: ẑi ∈ (z, z ).
Expected profit of firm i:
z
V i ( qi , q j , Di ) = ∫ ⎣⎡ R i qi , q j , zi − Di ⎦⎤ f ( zi )dzi
zˆi
(
)
Two stages:
Stage 1: Firms choose levels of debt financing, D1 and D2.
Stage 2: Firms choose quantities, q1 and q2.
Tore Nilssen – Lecture 9: Corporate Finance III - 5
At stage 2, firm i’s first-order condition is:
z i
dzˆ
i
Vi = ∫ Ri qi , q j , zi f ( zi )dzi − ⎡⎣ Ri qi , q j , zˆi − Di ⎤⎦ f ( zˆi ) i
dqi
zˆi
=0
z i
= ∫ Ri qi , q j , zi f (zi )dzi = 0
zˆi
Stage-2 equilibrium quantities, q1 and q2, solve the system
of equations V11 = 0, and V22 = 0.
(
)
(
(
)
)
Question: How do the equilibrium quantities qi and qj
respond to an increase in Di?
Total differentiation in the equations V11 = 0, and V22 = 0 :
Viii dqi + Viji dq j + ViDi dDi = 0
V jij dqi
+ V jjj dq j
i
j
+ V jD dDi
i
=0
But: V jDj = 0 . Rewrite:
i
⎡Viii Viji ⎤ ⎡ dqi ⎤ ⎡−ViDi dDi ⎤
i
⎢ j
⎥
⎥=⎢
j ⎥⎢
dq
V
V
0
j
jj ⎥
⎥⎦
⎢⎣ ji
⎦ ⎢⎣ ⎥⎦ ⎢⎣
Define B =
Viii
Viji
V jij V jjj
. Assumption: B > 0 (stability).
Tore Nilssen – Lecture 9: Corporate Finance III - 6
Solving the system, we find:
ViDi V jjj
dqi
=− i ;
dDi
B
i
j
dq j ViDi V ji
.
=
dDi
B
R jjj < 0 ⇒ V jjj < 0
(more generally: second-order conditions)
R jij < 0 ⇒ V jij < 0
(more generally: strategic substitutes)
i =Vi
ViD
izˆ
i
i
dzˆi
dDi
R i (qi , q j , zˆi )= Di ⇒ Rzi dzˆi = dDi ⇒
dzˆi
1
= i > 0.
dDi Rz
An increase in debt narrows the range of z over which
the firm earns a profit.
Tore Nilssen – Lecture 9: Corporate Finance III - 7
Viizˆ = − Rii (qi , q j , zˆi ) f (zˆi ) > 0
i
ERii = 0 & Rizi > 0 ⇒ Rii (⋅,⋅, zˆi )< 0
Rii
ẑ i
z
zi
When the critical value of z increases, the firm’s marginal
expected profit increases, i.e., the firm becomes more
aggressive.
In order for it to have anything left after paying its debt, an
increase in the debt level makes the firm more aggressive.
We summarise the analysis of stage 2:
ViDi V jjj
dqi
> 0;
=− i
dDi
B
i
j
dq j ViDi V ji
< 0.
=
dDi
B
Tore Nilssen – Lecture 9: Corporate Finance III - 8
Stage 1: The owners negotiate with the creditors the level
of debt that maximises the total expected value of the firm:
z i
= ∫ R qi , q j , zi f (zi )dzi
z
(
Yi
)
Stage-1 first-order condition for firm i:
YDi
i
⎡ zˆi
⎤ dq
i
= ⎢ ∫ Ri qi , q j , zi f (zi )dzi ⎥ i
⎢z
⎥ dDi
⎣
⎦
(
)
}<0
direct effect of
increased debt
⎡z
⎤ dq
i
+ ⎢ ∫ Ri qi , q j , zi f (zi )dzi ⎥ i
⎢ zˆ
⎥ dDi
⎣ i
⎦
} = 0 (FOC stage 2)
⎡z i
⎤ dq
+ ⎢ ∫ R j qi , q j , zi f (zi )dzi ⎥ j
⎢⎣ z
⎥⎦ dDi
} >0
(
(
)
)
strategic effect of
increased debt
=0
At Di = 0, the direct effect is zero, while the strategic effect
is positive. Therefore, some debt is better than no debt.
Tore Nilssen – Lecture 9: Corporate Finance III - 9