Five models

Patent race with respect to process innovation Process innovation leads to reduction of

average cost: Incentives to innovate for

– benevolent dictator– monopolist – perfect competition – two symmetric firms – two asymmetric firms

cc

Benevolent dictator vs. monopolist

…for benevolent dictator

… for monopolist (according to Arrow)

c

c

BD dccD

cpDccpcpc MMMM ,

cpDcdc

dppcdc

cpcd MMM

M

MMMM

,

BDc

c

M dccpD

c

c

MMMA dc

dcd

cc

p

x

p

x

nondrasticinnovation

)c( Mp

)c( Mp

c

)c( Mp

)c( Mpc

c

)c(X

drastic innovation

Innovator enjoys blockaded monopoly, ccpM )(ccpM )(

)c(X

c

Perfect competition

c

c

nondrasticPC dccDcDcc ,

0, cMdrasticPC

BDPCA

Incentives to innovate– for nondrastic innovation

– for drastic innovation

Comparison: (Check this out yourself using the figures)

R1

R2

Innovation competition

p1

p2

1

2

Two symmetric firms

Patent race with respect to product innovation R&D-expenditure of firm 1: R&D-expenditure of firm 2: The measure of innovation difficulty :

Innovation probability of firm 1:

1R

2R

0R

210

11 RRR

Rw

Exercise (innovation probability)

Innovation probability of firm 1

R1 R2

½ R0 ½ R0

6R0

R0

34R0

R0

100R0

R0

R0

0

How does the innovation probability depend on R0 , R1 and R2.

1w

Equilibrium (symmetric case)

Profit function of firm 1

Reaction function of firm 1

Nash equilibrium ,

1021

1211 , R

RRRR

RR M

NN RR 21 ,

0021 8

41

41

21

21

RRRR MMMNN

00 21

0

2

0

1M

N

M

NNN RRwith

RR

RR

020221 )( RRRRRR MR

Nash equilibrium

)( 21 RRR

)( 12 RRR

04/ RM

0RM 1R

2R

0RM

04/ RM

Exercise (sequential symmetric case)

Consider the symmetric case with R0=0. Calculate an equilibrium in the sequential game:

1

2 1R 2R

One incumbent, one potential competitor

Monopolist (firm 1) with average cost , and potential competitor (firm 2)

Process innovations leads to reductionof average cost :

Profits– No firm innovates

– Established firm innovates

– Entrant innovates

cc

0)( 21 andcM

0)( 21 andcM

dd and 2211

c

Profit functions (asymmetric case)

Profit function of firm 1 (monopolist)

Profit function of firm 2

1021

0

1021

2

021

1211

,

RcRRR

R

RRRR

cRRR

RRR

M

dM

22021

2212 , R

RRRR

RR d

Incentives to innovate

No firm innovates.

Monopolistinnovates.

Entrant innovates.

GN1

GN2

A2

A1

Replacement effect (Arrow)

The Arrow terms are defined as the profit differences a firm enjoys by innovating rather than not innovating.

If the incumbent innovates, he replaces himself. If the entrant innovates, he achieves positive profits as compared to zero profits:

AdMMA cc 221

p

x

)c( Mp

p

xIncentive to innovate from replacement effect

for established firm

for potential competitor

)c( Mp

c

c

nondrasticinnovation

A1

A2

)c( Mp

pM (c)c

drasticinnovation

c

Efficiency effect (Gilbert, Newbery)

The Gilbert-Newbery terms are defined as the profit differences a firm enjoys if she herself rather than her competitor innovates.

The established firm´s incentive to remain a monopolist is greater then the entrant´s incentive to become a duopolist.

GNddMGN c 2211

Price competition– Nondrastic innovation

– Drastic innovation

or" " (1)in )(),( and 0 21 ccpcc Mdd

" " (1)in 0 and )(

" " (1)in )( and 0

21

21

c

cM

Mdd

(1) )(21 cMdd

or

Gilbert-Newbery effect

(deterrence)

(blockade)

"" 1in 0 and )( 21 cM

Gilbert-Newbery effect

Quantity competition– Nondrastic innovation: deterrence or

accommodation (exercise)– Drastic innovation: (see above)

(1) )(21 cMdd

Exercise (Gilbert-Newbery effect) Suppose demand curve p(x)=a-bx Marginal cost

– of firm 1 – of firm 2

Monopolist´s marginal cost Show that the monopolist´s profit is greater than the sum of the duopolists´ profit in Cournot-Nash equilibrium.

cc

c

Replacement versus efficiency effect

Replacement effect– entrant has a greater

incentive to innovate

Efficiency effect– established firm has a

greater incentive to innovate

GNGN21 AA

21

Equilibrium (asymmetric case)

Reaction function of firm 1

Reaction function of firm 2

Nash equilibrium: “forget it“

dMMMR cRccRRRRR 1200221 )(

GNA RRRR 121002

dR RRRRRR 2100112 )(

GNA RRRR 212001

Special case: efficiency effect only

Hypothesis: R0 = 0 i.e., it is certain that one of the two firms innovates

Reaction function of firm 1

Reaction function of firm 2

Nash equilibrium:

)()( 12221dMR cRRRR

cRRRR dR21112 )(

2

21

2122

21

211

)( ;

)()(

ddM

ddMd

ddM

ddMdM

c

c

c

cc

Special case: replacement effect only Hypothesis : a drastic innovation (the

innovating firm becomes a blocking monopolist and there is no GN effect)

The greater the monopoly´s profit without innovation, the less are the monopolist´s incentives to replace itself :

01 c

RM

R

Exercise (drastic or nondrastic innovation) Inverse-demand function p=a-Q All firms have identical unit costs , where

Only one firm reduces its unit cost to

Infer kind of innovation

cac 2

acc 2

c

Exercise (develop or don´t)

Consider 3 firms in the patent race, let R denote R&D-expenditure.

Probability that a firm discovers product w=1/2 Let V denote the monetary value of the patent

– If only one firm discovers the product, then its profit is V.– If only two firms discover, then each firm earns V/2.– If all three discover, then each firm has profit V/3.

Exercise (develop or don´t)

a) Calculate the minimal value of V that ensures that each firm will invest.

b) Suppose now only 2 firms, which have only one owner. Calculate the minimal value of V that would induce the owner to let both firms undertake separate research instead of joint research (1 firm).

Exercise (subsidy game)

Assume two firms in two countries.

a) The government of firm 2 gives a subsidy for developing. Calculate the minimal subsidy (sN) given to firm 2 that will ensure that it will develop the product.

-10 -10 50 0

0 50 0 0

Develop Don´t develop

Develop

Don´t developFirm 1

Firm 2

Exercise (subsidy game)

b) Suppose, the subsidy for firm 2 is 15 units. Which subsidy (sN) for firm 1 would guarantee that it will develop the product, too.