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Jeanjean scientific seminar 2017
1. Technical Progress in the
Relationship between
Competition and Investment
François Jeanjean 24 march 2017
FSR Annual Scientific Seminar
2. 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.
3. 3
Literature
§ Technological diffusion:
– Jennifer Reinganum (Reinganum 1981) shows that, provided the cost
of adoption is high enough, innovation diffusion is sequential on the
market, firm after firm. Firms choose when to adopt a new technology.
They face a tradeoff between the cost of adoption and the benefits
from 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 relationship
between competition and Investment may take any shape. The shape
depends on the market (the demand, the type of competition (à la
Cournot, à la Bertrand or à la Hotelling).
– Using the examples providen by schmutzler, I show that the
differences may be explained by the size of innovation.
4. 4
The model
§ A duopoly where firms are horizontally differentiated. The degree of
substitutability is 𝜃: 𝜃 = 0 means that firms offers are totally differentiated
and 𝜃 = 1 means that they are perfect substitutes.
§ An (exogen) innovation which reduces marginal costs is available at time
𝑡 = 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 same
marginal constant marginal cost 𝑐. They both earn a profit flow depending
on marginal cost and substitutability, 𝜋 𝜃, 𝑐̅, 𝑐̅ .
§ At time 𝑡 = 𝑇,, firm 1 adopts, reduces its marginal costs to 𝑐 < 𝑐 , pays
𝐹(𝑇,) and earns a higher profit flow 𝜋 𝜃, 𝑐, 𝑐 > 𝜋 𝜃, 𝑐̅, 𝑐̅ . Firm 2, for its
part, 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 profit
flow 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
5. 5
Action and reaction functions
§ I denote 𝑓 𝜃 = 𝜋 𝜃, 𝑐, 𝑐 − 𝜋 𝜃, 𝑐, 𝑐 the action function. This is the
difference 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 the
difference 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 convex
2) Assumption 3: 𝑓 𝜃 − 𝑔 𝜃 is positive, increasing and convex
3) 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∆<
≤ 0
5) Assumption 6: 𝑓 0 = 𝑔 0
§ (All assumptions are verified for all the examples)
6. 6
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 𝑒HIJ
𝑑𝑡 + F 𝜋 𝜃, 𝑐, 𝑐 𝑒HIJ
𝑑𝑡
MN
MO
+ F 𝜋 𝜃, 𝑐, 𝑐 𝑒HIJ
𝑑𝑡
P
MN
MO
Q
− 𝐹 𝑇, 𝑒HIMO
§ For firm 2:
𝑉- = F 𝜋 𝜃, 𝑐, 𝑐 𝑒HIJ
𝑑𝑡 + F 𝜋 𝜃, 𝑐, 𝑐 𝑒HIJ
𝑑𝑡
MN
MO
+ F 𝜋 𝜃, 𝑐, 𝑐 𝑒HIJ
𝑑𝑡
P
MN
MO
Q
− 𝐹 𝑇- 𝑒HIMN
𝑟 is the discount rate.
§ Maximization of 𝑉, and 𝑉- leads to:
S
𝑓 𝜃 = 𝑟𝐹 𝑇, − 𝐹̇ 𝑇,
𝑔 𝜃 = 𝑟𝐹 𝑇- − 𝐹̇ 𝑇-
§ 𝑇, and 𝑇- decrease respectively with 𝑓 𝜃 and 𝑔 𝜃
§ Investment writes: 𝐼 𝜃 = 𝐹 𝑇, 𝑒HIMO + 𝐹 𝑇- 𝑒HIMN
7. 7
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)
8. 8
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, = W
1
𝛼
𝑙𝑛
𝑟 + 𝛼 𝐹
𝑓 𝜃
𝑖𝑓 𝑓 𝜃 < 𝑟 + 𝛼 𝐹
0 𝑖𝑓 𝑓 𝜃 ≥ 𝑟 + 𝛼 𝐹
𝑇- = W
1
𝛼
𝑙𝑛
𝑟 + 𝛼 𝐹
𝑔 𝜃
𝑖𝑓 𝑔 𝜃 < 𝑟 + 𝛼 𝐹
0 𝑖𝑓 𝑔 𝜃 ≥ 𝑟 + 𝛼 𝐹
§ However, 𝑇- ≥ 𝑇, ≥ 0. This means that when 𝑓 𝜃 ≥ 𝑟𝐹 0 − 𝐹̇ 0
which 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
9. 9
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) − 𝐹̇ 0
𝜃
𝑓(𝜃)
𝑔(𝜃)
𝜃]
𝑓(𝜃)
𝑔(𝜃)
11. 11
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 𝑒HIJ
𝑑𝑡
MN
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.
12. 12
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) − 𝐹̇ 0
𝜃
13. 13
5 Examples (from Schmutzler 2013)
§ 3 types of competition (à la Bertrand, à la Cournot, à la Hotelling)
§ 3 types of demand function :
– D1 𝑝c 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 Example 1 Example 2
D2 Example 3 Example 4
D3 Example 5
22. 22
Conclusion
§ Technical progress increases Investment in cost reduction
technology and reduces the degree of substitutability which
maximizes Investment.
§ Technical progress improves Consumer surplus and reduces the
degree of substitutability which maximizes relative growth of
consumer surplus.
§ The impact of Technical progress on the degree of substitutability
that maximizes absolute Consumer Surplus is ambiguous. It
depends on the relative weight of static and dynamic effects.
§ The impact on Welfare is more ambiguous.
23. 23
Why an inverted-U relationship ?
§ In the examples, 𝑓 𝜃 and g 𝜃 are increasing quadratic functions of ∆𝑐,
as a result, 𝐼, 𝜃
q
rsq and 𝐼- 𝜃
q
rsq 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)
24. 24
Examples: Adoption dates and Investment
§ Cost of adoption:
𝐹 𝑡 = 𝐹𝑒HVJ
§ Adoption dates are:
𝑇, = W
1
𝛼
𝑙𝑛
𝑟 + 𝛼 𝐹
𝑓 𝜃
𝑖𝑓 𝑓 𝜃 < 𝑟 + 𝛼 𝐹
0 𝑖𝑓 𝑓 𝜃 ≥ 𝑟 + 𝛼 𝐹
𝑇- = W
1
𝛼
𝑙𝑛
𝑟 + 𝛼 𝐹
𝑔 𝜃
𝑖𝑓 𝑔 𝜃 < 𝑟 + 𝛼 𝐹
0 𝑖𝑓 𝑔 𝜃 ≥ 𝑟 + 𝛼 𝐹
§ Investment of the leader and the follower are:
𝐼, 𝜃 =
𝑓 𝜃
,f
I
V
𝛼 + 𝑟
,f
I
V 𝐹
I
V
𝑖𝑓 𝑓 𝜃 < 𝑟 + 𝛼 𝐹
𝐹 𝑖𝑓 𝑓 𝜃 ≥ 𝑟 + 𝛼 𝐹
𝐼- 𝜃 =
𝑔 𝜃
,f
I
V
𝛼 + 𝑟
,f
I
V 𝐹
I
V
𝑖𝑓 𝑔 𝜃 < 𝑟 + 𝛼 𝐹
𝐹 𝑖𝑓 𝑔 𝜃 ≥ 𝑟 + 𝛼 𝐹
