Technical Progress in the Relationship between

Competition and Investment

François Jeanjean 24 march 2017FSR Annual Scientific Seminar

2

Motivation

§ Research Question: – What is the impact of technical progress in the relationship between

Competition and Investment?

§ Methodology: – Theoretical framework crossing the literature of adoption of innovation

(Reinganum 1981) with literature on the relationship between competition and investment (Schmutzler, 2013).

– General framework illustrated with several examples

§ Results:– Technical progress understood as the size of innovation impacts the

relationship between competition and Investment. – The size of innovation tends to reduce the level of competition

(measured as the level of substitutability) which maximizes investment.

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Literature

§ Technological diffusion:

– Jennifer Reinganum (Reinganum 1981) shows that, provided the costof adoption is high enough, innovation diffusion is sequential on themarket, firm after firm. Firms choose when to adopt a new technology.They face a tradeoff between the cost of adoption and the benefitsfrom adopting which both decrease over time.

– If the cost of adoption is low enough, firms adopt immediately.

§ Competition-Investment relationship:

– Armin Schmutzler (Schmutzler, 2013) shows that the relationshipbetween competition and Investment may take any shape. The shapedepends on the market (the demand, the type of competition (à laCournot, à la Bertrand or à la Hotelling).

– Using the examples providen by schmutzler, I show that thedifferences may be explained by the size of innovation.

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The model§ A duopoly where firms are horizontally differentiated. The degree of

substitutability is 𝜃: 𝜃 = 0 means that firms offers are totally differentiatedand 𝜃 = 1means that they are perfect substitutes.

§ An (exogen) innovation which reduces marginal costs is available at time𝑡 = 0.

§ To adopt, firms have to pay a fixed cost 𝐹(𝑡), decreasing and convex.Firms adopt respectively at time 𝑇,and 𝑇-, with 𝑇- ≥ 𝑇, ≥ 0.

§ At time 𝑡 = 0, it is assumed that firms are symmetrical and face the samemarginal constant marginal cost 𝑐. They both earn a profit flow dependingon marginal cost and substitutability, 𝜋 𝜃, 𝑐̅, 𝑐̅ .

§ At time 𝑡 = 𝑇,, firm 1 adopts, reduces its marginal costs to 𝑐 < 𝑐, pays𝐹(𝑇,) and earns a higher profit flow 𝜋 𝜃, 𝑐, 𝑐 > 𝜋 𝜃, 𝑐̅, 𝑐̅ . Firm 2, for itspart, earns a lower profit flow 𝜋 𝜃, 𝑐, 𝑐 < 𝜋 𝜃, 𝑐, 𝑐

§ At time 𝑡 = 𝑇-, Firm 2 adopts in turn, pays 𝐹 𝑇- < 𝐹(𝑇,) and earns a profitflow 𝜋 𝜃, 𝑐, 𝑐 . Firm 1 earns then the same profit 𝜋 𝜃, 𝑐, 𝑐 .

§ Assumption 1:𝜋 𝜃, 𝑐, 𝑐 > 𝜋 𝜃, 𝑐, 𝑐 ≥ 𝜋 𝜃, 𝑐, 𝑐 > 𝜋 𝜃, 𝑐, 𝑐

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Action and reaction functions

§ I denote 𝑓 𝜃 = 𝜋 𝜃, 𝑐, 𝑐 − 𝜋 𝜃, 𝑐, 𝑐 the action function. This is thedifference between the profit flow before and after adoption for firm 1.

§ I denote g 𝜃 = 𝜋 𝜃, 𝑐, 𝑐 − 𝜋 𝜃, 𝑐̅, 𝑐 the reaction function, this is thedifference between profit flow before and after adoption by firm 2,reacting to the adoption of firm 1.

§ I denote ∆𝑐 = 𝑐 − 𝑐, the size of innovation.§ In the following, it is assumed that:

1) Assumption 2: 𝑓(𝜃) is positive and convex2) Assumption 3: 𝑓 𝜃 − 𝑔 𝜃 is positive, increasing and convex3) Assumption 4: 9:(;)

9∆<≥ 9= ;

9∆<≥ 0, ∆c impacts positively 𝑓 more than g.

4) Assumption 5: 𝑓A 𝜃 ≥ 0 ⇒ 9:C ;9∆<

≥ 0 and 𝑔A 𝜃 ≤ 0 ⇒ 9=C ;9∆<

≤ 05) Assumption 6: 𝑓 0 = 𝑔 0

§ (All assumptions are verified for all the examples)

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Choice of adoption date§ Firms choose their adoption date 𝑇, and 𝑇- to maximize the present value

of their profit flow. § For firm 1:

𝑉, = F 𝜋 𝜃, 𝑐, 𝑐 𝑒HIJ𝑑𝑡 + F 𝜋 𝜃, 𝑐, 𝑐 𝑒HIJ𝑑𝑡MN

MO

+ F 𝜋 𝜃, 𝑐, 𝑐 𝑒HIJ𝑑𝑡P

MN

MO

Q

− 𝐹 𝑇, 𝑒HIMO

§ For firm 2:

𝑉- = F 𝜋 𝜃, 𝑐, 𝑐 𝑒HIJ𝑑𝑡 + F 𝜋 𝜃, 𝑐, 𝑐 𝑒HIJ𝑑𝑡MN

MO

+ F 𝜋 𝜃, 𝑐, 𝑐 𝑒HIJ𝑑𝑡P

MN

MO

Q

− 𝐹 𝑇- 𝑒HIMN

𝑟 is the discount rate.§ Maximization of 𝑉, and 𝑉- leads to:

S𝑓 𝜃 = 𝑟𝐹 𝑇, − �̇� 𝑇,𝑔 𝜃 = 𝑟𝐹 𝑇- − �̇� 𝑇-

§ 𝑇, and 𝑇- decrease respectively with 𝑓 𝜃 and 𝑔 𝜃§ Investment writes: 𝐼 𝜃 = 𝐹 𝑇, 𝑒HIMO + 𝐹 𝑇- 𝑒HIMN

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Shape of 𝑓 𝜃 and 𝑔 𝜃

§ By assumption 2, 𝑓 𝜃 may be decreasing, increasing or U shaped, but not inverted-U shaped.

§ By assumption 3 and assumption 6:– If 𝑓 𝜃 is decreasing, 𝑔 𝜃 is also decreasing.– If 𝑓 𝜃 is U shaped, 𝑔 𝜃 may be decreasing or U shaped.– If 𝑓 𝜃 is increasing, 𝑔 𝜃 may be decreasing, increasing or inverted-

U shaped

𝑓(𝜃)

𝑔(𝜃)

𝜃𝜃

𝑓(𝜃)

𝑔(𝜃)

𝜃

𝑓(𝜃)

𝑔(𝜃)

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Relationship between 𝑓 𝜃 , 𝑔 𝜃 and Investment

§ 𝑇, and 𝑇- decrease respectively with 𝑓 𝜃 and 𝑔 𝜃§ Investment decreases with 𝑇, and 𝑇- and thus increases with 𝑓 𝜃

and 𝑔 𝜃§ For example, for 𝐹 𝑡 = 𝐹𝑒HVJ

𝑇, = W1𝛼 𝑙𝑛

𝑟 + 𝛼 𝐹𝑓 𝜃 𝑖𝑓𝑓 𝜃 < 𝑟 + 𝛼 𝐹

0𝑖𝑓𝑓 𝜃 ≥ 𝑟 + 𝛼 𝐹

𝑇- = W1𝛼 𝑙𝑛

𝑟 + 𝛼 𝐹𝑔 𝜃 𝑖𝑓𝑔 𝜃 < 𝑟 + 𝛼 𝐹

0𝑖𝑓𝑔 𝜃 ≥ 𝑟 + 𝛼 𝐹

§ However, 𝑇- ≥ 𝑇, ≥ 0.This means that when 𝑓 𝜃 ≥ 𝑟𝐹 0 − �̇� 0which means 𝑇, = 0, a growth of 𝑓 𝜃 does not increase Investment any more. (Idem for 𝑔 𝜃 )

§ When 𝑇- > 𝑇, > 0, Investment is impacted by both 𝑓 𝜃 and 𝑔 𝜃§ When 𝑇- > 𝑇, = 0, Investment 𝐼 𝜃 varies like 𝑔 𝜃 only.§ When 𝑇- = 𝑇, = 0, Investment remains constant 𝐼 𝜃 = 2𝐹 0

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Impact of the size of Innovation

§ By assumption 4, an increase in the size of innovation tends to increase 𝑓 𝜃 and g 𝜃 and thus reduces 𝑇,and 𝑇-.

§ When 𝑓 𝜃 is higher, the level of substitutability beyond which 𝑇, =0 is lower and, therefore, the level of substitutability which maximizes investment is also lower.

𝑟𝐹(0) − �̇� 0

𝜃

𝑓(𝜃)

𝑔(𝜃)

𝜃]

𝑓(𝜃)

𝑔(𝜃)

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The building of the inverted-U relationship

𝐼 𝜃 = 𝐹 𝑇, 𝑒HIMO + 𝐹 𝑇- 𝑒HIMN

𝑟𝐹(0) − �̇� 0

𝜃

𝑓 𝜃

𝑔 𝜃

𝑇, = 0

𝑇- = 0

𝐼 𝜃 = 𝐹 0 + 𝐹 𝑇- 𝑒HIMN

𝐼 𝜃 = 2𝐹 0

𝐼 𝜃

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Equilibrium with preemption

§ The payoffs of firm 1 is higher than the payoff of firm 2. It is therefore in the interest of both firms to adopt first. They are incited to preempt the leadership by adopting before 𝑇,. (Fudenberg and tirole 1985)

§ This incentive remains until the payoff of the leader adopting before 𝑇, equals the payoff of the follower adopting in 𝑇-.

§ Preemption leads the leader to adopt at 𝑇,^ ≤ 𝑇,where 𝑉,^ = 𝑉-§ This leads to:

𝐹 𝑇,^ 𝑒HIMO_ − 𝐹 𝑇- 𝑒HIMN = F 𝜙 𝜃 𝑒HIJ𝑑𝑡MN

MO_§ with 𝜙 𝜃 = 𝜋 𝜃, 𝑐, 𝑐 − 𝜋 𝜃, 𝑐, 𝑐 . The higher 𝜙 𝜃 , the sooner 𝑇,^.§ 𝜙 𝜃 ≥ 𝑓 𝜃 , 𝜙 0 = 𝑓 0 and ∆𝑐 increases 𝜙 𝜃 . Thus, 𝑇,^ plays

the same role as 𝑇,in the shape of I 𝜃 but reach the top for lower values of 𝜃.

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Ejection of the follower

§ For a large size of innovation and an important degree of substitutability (drastic innovation), it is possible that the follower is ousted from the market between 𝑇,or 𝑇,^and 𝑇-. In such case, the leader enjoys a higher profit from innovation during this time and 𝜋 𝜃, 𝑐, 𝑐 = 0. This tends to increase 𝑓 𝜃 and 𝜙 𝜃 and to decrease g 𝜃 . As a result, this tend to increase the probability that 𝑇, = 0 and I 𝜃 reaches its maximum.

𝑓(𝜃)

𝑔(𝜃)

𝑟𝐹(0) − �̇� 0

𝜃

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5 Examples (from Schmutzler 2013)

§ 3 types of competition (à la Bertrand, à la Cournot, à la Hotelling)§ 3 types of demand function :

– D1𝑝c 𝑞c, 𝑞e, 𝜃 = 1 − 𝑞c − 𝜃𝑞e (Shubik et Levitan)– D2 𝑝c 𝑞c, 𝑞e, 𝜃 = 1 − ,

,f;𝑞c −

;,f;

𝑞e ( Singh et Vives)

– D3 𝑞c 𝑝c, 𝑝e, 𝜃 = 𝑚𝑎𝑥 ,-+ ^jH^k ;

- ,H;; 0 (Hotelling)

Competition

Demand

Cournot Bertrand Hotelling

D1 Example1 Example2

D2 Example3 Example4

D3 Example5

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Example 1(Cournot D1)

F = 0.2α = 0.2r = 0.1

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Example 2 (Bertrand D1)

F = 0.2α = 0.2r = 0.1

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Example 3 (Cournot D2)

F = 0.2α = 0.2r = 0.1

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Example 4 (Bertrand D2)

F = 0.2α = 0.2r = 0.1

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Example 5 (Hotelling D3)

F = 0.7α = 0.2r = 0.1

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Consumer Surplus (with preemption)

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Consumer Surplus (relative growth)Dc growthfactor0,05 1,1720,1 1,5710,15 2,2090,2 3,1210,25 4,3430,3 5,9110,35 7,5630,4 9,000

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Welfare (with preemption)

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Conclusion

§ Technical progress increases Investment in cost reductiontechnology and reduces the degree of substitutability whichmaximizes Investment.

§ Technical progress improves Consumer surplus and reduces thedegree of substitutability which maximizes relative growth ofconsumer surplus.

§ The impact of Technical progress on the degree of substitutabilitythat maximizes absolute Consumer Surplus is ambiguous. Itdepends on the relative weight of static and dynamic effects.

§ The impact on Welfare is more ambiguous.

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Why an inverted-U relationship ?

§ In the examples, 𝑓 𝜃 and g 𝜃 are increasing quadratic functions of ∆𝑐, as a result, 𝐼, 𝜃

qrsq and 𝐼- 𝜃

qrsq are increasing quadratic functions of ∆𝑐

until ∆𝑐 is high enough to achieve 𝑓 𝜃 = 𝛼 + 𝑟 𝐹 or g 𝜃 = 𝛼 + 𝑟 𝐹

§ In frameworks where Investment is increasing and convex function of cost reduction, there are no limit, Investment can grow infinitely and thus Investment according to the degree of substitutability is shaped like 𝑓 𝜃 .However, technical progress sets a limit to investment according to a given cost reduction.

Example 4 (Cournot D2)

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Examples: Adoption dates and Investment

§ Cost of adoption:𝐹 𝑡 = 𝐹𝑒HVJ

§ Adoption dates are:

𝑇, = W1𝛼 𝑙𝑛

𝑟 + 𝛼 𝐹𝑓 𝜃 𝑖𝑓𝑓 𝜃 < 𝑟 + 𝛼 𝐹

0𝑖𝑓𝑓 𝜃 ≥ 𝑟 + 𝛼 𝐹

𝑇- = W1𝛼 𝑙𝑛

𝑟 + 𝛼 𝐹𝑔 𝜃 𝑖𝑓𝑔 𝜃 < 𝑟 + 𝛼 𝐹

0𝑖𝑓𝑔 𝜃 ≥ 𝑟 + 𝛼 𝐹§ Investment of the leader and the follower are:

𝐼, 𝜃 =𝑓 𝜃 ,fIV

𝛼 + 𝑟 ,fIV 𝐹IV𝑖𝑓𝑓 𝜃 < 𝑟 + 𝛼 𝐹

𝐹𝑖𝑓𝑓 𝜃 ≥ 𝑟 + 𝛼 𝐹

𝐼- 𝜃 =𝑔 𝜃 ,fIV

𝛼 + 𝑟 ,fIV 𝐹IV𝑖𝑓𝑔 𝜃 < 𝑟 + 𝛼 𝐹

𝐹𝑖𝑓𝑔 𝜃 ≥ 𝑟 + 𝛼 𝐹