corporate finance: capital structure and pmcspiegel/teaching/corpfin/ppt-pmc.pdf · corporate...

Corporate Finance:Capital structure and PMC

Yossi Spiegel

Recanati School of Business

Brander and Lewis, AER 1986

“Oligopoly and Financial Structure: The Limited Liability Effect”

Corporate Finance 3

Cournot duopoly with differentiated products

� Two firms produce differentiated products for which the inverse demand functions are

� γ reflects the “degree of product differentiation”� γ = 1 the products are perfect substitutes� 0 < γ < 1 the products are imperfect substitutes

� γ = 0 the products are unrelated� γ < 0 the products are complements

� Both firms have a constant marginal cost k

� The profit functions:

122211 qqApqqAp γγ −−=−−=

( ) ( ) 222111 , qkpqkp −=−= ππ

Corporate Finance 4

The Nash equilibrium

� F.o.c for firm 1:

� The best-response function of firm 1

� The best response function of firm 2 is analogous

02 21

revenue Marginal

121

1

1 =−−−=−−−−=∂∂

kqqAkqqqAq

γγπ

44 344 21

2)( 2

21

qkAqBRq

γ−−==

Corporate Finance 5

Gaining strategic advantage with cost reduction from k to k’

2

12

qkAq

γ−−=

q1

q2

Direct effect Indirect effect

(A-k)/2

2

' 21

qkAq

γ−−=

2

21

qkAq

γ−−=

(A-k)/γ

(A-k)/(2+γ) (A-k)/γ

(A-k)/(2+γ)

(A-k)/2

Corporate Finance 6

The limited liability effect� Marginal cost is cH with prob. α or cL with prob. 1-α (oil prices fluctuate)

� The firm chooses quantity before knowing if cost is high or low (an airline determines its flight schedule in advance and buys fuel on the spot market)

� Expected profit: Eπ = α(p – cH)Q + (1-α)(p – cL)Q

� The firm issues debt with face value D that can be repaid in full only if cost is cL: D > (p–cH)Q

� If cost is cH, the firm goes bankrupt; due to limited liability equityholders get a payoff of 0

� The firm acts on behalf of equityholders and therefore maximizes:(1-α)(p – cL)Q

� The firm behaves as if its cost is cL rather than αcH + (1-α)cL� The firm gains a strategic advantage over rivals if strategies are strategic

substitutes� The strategy is costly: the firm goes bankrupt with probability α

Corporate Finance 7

The general model� The timing:

� Ri(qi,qj,z) has a unique maximizer qi

� Rij(qi,qj,z) < 0 – firm j has a negative externality on firm i

� Riz(qi,qj,z) > 0: z is a positive shock (either supply of demand

shock)

� z ~ [0, ∞) according to f(z) with CDF F(z)

Period 1 Period 2

Two rival firms issue debt with faces values D1 and D2

The two firms choose output levels q1 and q2

Profits are R1(q1,q2,z) and R2(q1,q2,z)

Corporate Finance 8

Examples

� Cost shocks:

� Ci = C(qi)/z

� Ci = (1-z)C(qi), z ~ [0, 1]

� Demand shocks� pi = zpi(qi), where pi = A-qi-γqj� pi = Az-qi-γqj

Corporate Finance 9

All-equity firm� Firm i chooses qi to maximize its expected profit

taking as given qj

� The best response of firm i against qj is defined by

� The NE is defined implicitly by the system:

( ) ( ) ( )∫∞

=0

,,,, zdFzqqRzqqVMax ji

i

ji

i

qi

( ) ( ) 0,,0

== ∫∞

zdFzqqRV ji

i

i

i

i

( ) ( ) 0,,0,, == zqqVzqqV ji

j

jji

i

i

Corporate Finance 10

Leveraged firm

� The firm is solvent iff:

( )iiji

i

i DzqqRzz =≥ *,,*

Ri(qi,qj,z)

z

Di

zi*

Comparativestatics:

• Di↑ ⇒ zi*↑


The problem of a leveraged firm

� Firm i’s problem:

� The best response of firm i against qj is defined by

� The NE is defined implicitly by the system:

( ) ( )( ) ( )∫∞

−=*

,,,,

i

iz

i

ji

i

ji

i

qzdFDzqqRzqqVMax

( ) ( )( ) ( ) ( ) ( )∫∞

=

=+∂∂

−−=*

0

0,,*

**,,,,

iz

ji

i

i

i

ii

i

iji

i

ji

i

i zdFzqqRq

zzfDzqqRzqqV

444 3444 21

( ) ( ) 0,,0,, == zqqVzqqV ji

j

jji

i

i


The effect of leverage on the firm’s strategy

� To find the effect of Di on qi, fully differentiate f.o.c.:

� The change in firm i’s best response function depends on the sign of ViiDi(qi,qj,z)

( ) ( ) ( )( )zqqV

zqqV

D

qDzqqVqzqqV

ji

i

ii

ji

i

iD

i

iiiji

i

iDiji

i

ii,,

,,0*,,,,

s.o.cby (-)−

=∂∂

⇒=∂+∂43421

( ) ( ) ( )

( ) ( )})(

*

***,,

,,,,

+

∞

∂∂

−=

∂∂

= ∫

i

iiiji

i

i

z

ji

i

i

i

ji

i

iD

D

zzfzqqR

zdFzqqRD

zqqV

i


The sign of Rii(qi,qj,zi*)

� Suppose that Riiz > 0 – the shock z boosts the marginal

profit

� Recall that total area between the horizontal axis and the graph of Ri

i(qi,qj,zi) sums up to 0

Rii(qi,qj,z)

z

0zi*

Rii(qi,qj,zi*) < 0


Effect of Di

� If Riiz > 0, then q

i ↑ with Di – firm i becomes more aggressive when it issues debt

� Why?� Due to limited liability, firm i ignores low states

of z in which it goes bankrupt and hence behaves as if z was on average high

� When Riiz > 0, the firm produces more when z is high

� In the Cournot model, when firm i is more aggressive, firm j is softer and firm i enjoys a strategic advantage vis-à-vis firm j


The choice of Di

� The value of debt:

� The value of equity:

� The total value of the firm:

( ) ∫∫∞

+=*

*

0

)()(*,*,)(z

i

z

ji

i

ii zdFDzdFzqqRDBi

( )∫∞

−=*

)()*,*,()(

iz

iji

i

ii zdFDzqqRDE

∫∞

=0

)()*,*,()( zdFzqqRDY ji

i

ii


The optimal choice of Di

44444 344444 21321444 3444 21

32144 344 21

444 3444 21

)(

0

)()(

*

0

)(

0)(

qfor F.O.Cby 0

*

*

0

0

*

*)()*,*,(

*

*)()*,*,(

*

*)()*,*,(

*

*)()*,*,(

*

*)()*,*,(

)(*

*)*,*,(

*

*)*,*,()('

i

+

∞

+−

−

∞

−

=

∞

∞

∫∫

∫

∫∫

∫

∂

∂+

∂∂

=

∂

∂+

∂∂

+∂∂

=

∂

∂+

∂∂

=

i

j

ji

i

j

i

i

z

ji

i

i

i

j

ji

i

j

i

i

z

ji

i

i

i

i

z

ji

i

i

i

j

ji

i

j

i

iji

i

iii

D

qzdFzqqR

D

qzdFzqqR

D

qzdFzqqR

D

qzdFzqqR

D

qzdFzqqR

zdFD

qzqqR

D

qzqqRDY

i

i

i


The optimal choice of Di

� The f.o.c for Di:

� The second term is the positive strat. effect

� The first term is the negative direct effect: debt distorts the firm’s choice of quantity

� As Di → 0 ⇒ zi* → 0 ⇒ the negative direct effect vanishes ⇒ Yi’(Di) > 0 ⇒ Di* > 0

0*

*)()*,*,(

*

*)()*,*,()('

)(

0

)()(

*

0

=∂

∂+

∂∂

=

+

∞

+−

∫∫44444 344444 21321444 3444 21 i

j

ji

i

j

i

i

z

ji

i

iiiD

qzdFzqqR

D

qzdFzqqRDY

i

Spiegel, JEMS 1996

“The Role of Debt in Procurement Contracts”


The model� The timing:

� Vk(k,z) > 0 > Vkk(k,z)

� Vz(k,z) > 0 and Vkz(k,z) ≥ 0

� z ~ U[0, 1]

A firm invests k which determines the net surplus from trading with a buyer, V(k,z)

z is realized, the firm and the buyer bargain over a price, D is due

Period 1 Period 2 Period 3

The firm issues debt with face value D

z↑

k

V(k,z)


The bargaining stage

� The buyer and the firm trade iff V(k,z) ≥ D, o/w the price the buyer pays is not enough to ensure solvency

� Let z* be such that V(k,z*) = D

� If V(k,z) ≥ D, the buyer makes a TIOLI with prob. γ and the firm makes an offer with prob. 1-γ

� The firm’s expected payoff when it issues debt is

( ) ( ) ( )( ) kDdzdzDzkVDkYzz

−+−−= ∫∫32144 344 21

Debt

1

*

Equity

1

*

,1, γ


The choice of debt

� The f.o.c for the firm’s value:

� At D = 0, YD(k,z) > 0 ⇒ D* > 0

( ) ( )( ) ( )

( ){

0*

*1

**1*11,

)(

=∂∂

−−=

∂∂

−−+−−−=

+

D

zDz

D

zDzzDkYD

γ

γ


Illustrating the model – all equity

z1

V(k.z)

(1−γ)V(k.z)


Illustrating the model

z1

V(k.z)

D

z*

(1−γ)V(k,z)


Investment

� The firm’s value as a function of k:

� The f.o.c for k:

( ) ( ) ( ) ( ){

( ) ( ) 01*

*,1

1*

*,1*

*,,

1

*

)(

1

*0

=−∂∂

−−=

−∂∂

−−+∂∂

=

∫

∫−

=

k

zDdzzkV

k

zDdzzkV

k

DDkYDkY

z

k

z

kDk

γ

γ43421

( ) ( ) ( )( ) kdzDdzDzkVDkYzz

−+−−= ∫∫43421444 3444 21

Debt

1

*

Equity

1

*

**,1, γ


The effect of debt on investment

� The marginal benefit of k:

� Since V(k,z*) = D:

� The effect of D:

( ) ( )k

zDdzzkV

z

k ∂∂

−− ∫*

*,1

1

*

γ

( ) ( )Dk

zD

k

z

k

zzkVk ∂∂

∂−

∂∂

−∂∂

−−*

***

*,12

γ

( )( )

0*,

*,*<−=

∂∂

zkV

zkV

k

z

z

k


The effect of debt on investment

� Suppose that

� The marginal benefit of k increases with D

� The firm invests more when it issues debt

0*2

<∂∂

∂Dk

z

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