STRATEGIC R&D IN COMPETITIVE RESOURCE...

STRATEGIC R&D IN COMPETITIVE RESOURCE MARKETS

MARK C. DUGGAN

Abstract. In today’s economic climate, energy is at the forefront of public attention. Renewable

energy is a field whose technology is constantly changing. Firms must adapt to the ever changing

market in order to keep up with their competitors’ rates of research and development (R&D). We

explore how different industry scenarios affect the optimal level of R&D for each firm. Some of

the properties that affect funds devoted to R&D include the number of firms, their relative costs,

competitive substitutable goods, and the method used to find the competitive equilibrium (Cournot

or Bertrand). Over time, each firm will make different decisions based upon each firm’s production

prices (as a result of R&D), thus we break a dynamic game into stages with respect to production

price. We explore how the different aspects of the industry hurt or promote R&D, using stochastic

processes, along with other tools of probability and game theory.

This competitive market problem in R&D can be modeled as a dynamic game, in particular, we

consider Counrot and Bertrand competition. In order to determine what decision each firm makes,

we consider different stages based on the level of technology of each firm. Technology improvement

is modeled as a Poisson process, where the amount of time for a firm to make an improvement is

a random variable. We observe that this process has the Markov property: the property that the

conditional probability distribution of future states of the process depends only upon the present

state. In order to understand what determines how much effort firms put into research, since, in

general, there are no closed form solutions, numerical experiments will be run using various software

(MATLAB in particular). The solutions are found considering a finite number of stages and using

the game theory concept of Nash equilibrium on each stage.

Date: June 7, 2013.Thanks go to the Materials Research Laboratory at UCSB for the support via the Reseach Internship in Science andEngineering, and to my research advisor Professor Michael Ludkovski of the Statistics Department at UCSB.

1

2 MARK C. DUGGAN

Contents

1. Introduction 3

2. Theoretical Background 3

2.1. Game Theory & Competition 3

2.2. Stochastic Processes 5

3. A Sequence of Increasingly Complicated Research Scenarios 6

3.1. A Simple Scenario: Cournot Monopoly 6

3.2. Cournot Duopoly: Single Firm Research 11

3.3. Cournot Duopoly: Dual Research 13

3.4. Cournot Substitution 18

4. When to Say “No” to R&D 18

4.1. Linear 18

4.2. Exponential 20

5. Figures 20

6. Further Research 23

References 23

STRATEGIC R&D IN COMPETITIVE RESOURCE MARKETS 3

1. Introduction

We build on the example by Ludkovski and Sircar [3]. This example describes competition

between a producer of an exhaustible resource (oil, natural gas, etc.) with a “green” producer that

has access to an inexhaustable resource, such as solar or wind. Rather than exploring the details

of exploration for more of an exhaustable resource, we focus on the details of R&D to reduce the

cost of producing some inexhaustable resource. We will consider a point process that counts the

number of (production) cost reductions, as a result of successful technology improvements, over

time. In order to determine the appropriate amount of effort (or funding) for research, we rely on

economic game theory, in particular the theories of Cournot and Bertrand.

Before getting into the details of jargon and concepts, we will discuss the ideas behind the process

in which the firm is able to reduce its cost. A firm chooses to devote some amount of resources

(effort) to devote to R&D, in hopes of reducing its production cost, in order to maximize its profits.

We suppose that research successes happen incrementally. More precisely, research successes are

represented by a point process, where a success corresponds to a cost reduction. We suppose that

this process is memoryless: that if a success has not occurred in the past t time, then the firm

faces the same scenario as it did t time ago. This means that under constant effort, the expected

time until the next success is the same for anywhere in (Ti, Ti+1), where Tj is the time of a success.

Furthermore, constant effort must be an equilibrium solution within one of these intervals, as the

game has not changed during the time between consecutive successes (interarrival time), and so

we suppose that effort is constant within each interval, because the equilibrium solutions are often

unique, forcing constant effort to optimize. From this we see that in (Ti, Ti+1) we have a point

process that is dependent upon the effort of a firm during that interarrival time.

So, costs can be lowered step by step by means of researching. Firms must decide between

putting in more effort now to reap increased profits later or making as much money as they can

now with minimal effort put into research. At each step, the optimum quantity each firm chooses

is found by using Cournot’s model of competition in order to find a Nash equilibrium, a solution

such that deviation from this solution by either firm would result in reduced profit for that firm.

2. Theoretical Background

This chapter introduces the concepts and definitions in economics and probability that are nec-

essary to understand the details of this paper.

2.1. Game Theory & Competition.

Definition. A game is described by mutually aware rational players, those players’ possible de-

cisions (strategies), and a set of payoffs for each player, conditioned on the decisions made by all

players. In the context of resource markets, players will be referred to as firms, and payoffs are

given in the form of a real number of dollars. A player is rational if his preferences define a total

ordering on R, i.e. a firm prefers more money to less money and has the ability to predict his

competitors’ decisions, because his competitors are rational as well.

4 MARK C. DUGGAN

Definition. A Nash equilibrium, often called an equilibrium, is a set of strategies for each firm,

such that no firm has incentive to change his strategy, i.e. were one firm to change his strategy, his

earnings would be less than in equilibrium.

Definition. In Competition, firms produce some quantity, q, of a good at some marginal cost, s,

and sell it to the public at a price, P , with profit π = q(P − s) − C, where C is the sum of costs

that are not dependent on the quantity produced, in this case the cost of research. We will also

use net gains less production cost ρ = q(P − s) = π + C.

Definition. The equilibrium price is the price that prevails in the market, the price at which

firms sell goods. Were some firm to charge more, he would sell nothing. Alternatively, were a firm

to charge less than the equilibrium price, his competitors would sell nothing (in the case where the

firm can supply enough to satisfy the entire demand, or clear the market); this lowering of prices

would necessarily cause him to make less money, by definition of equilibrium.

Definition. Goods are not always identical, so we can consider substitute goods: a pair of goods

such that an increase (decrease) in the price of one good increases (decreases) the demand for the

other good. This relationship can be described by the goods’ cross-price elasticity of demand. For

goods 1 and 2 with quantities q[1], q[2] and prices P [1], P [2], their elasticity is

δ1,2 =∂q[1]

∂P [2]

P [2]

q[1],

or more intuitively, the ratio: percent change of q[1] over percent change of P [2].

Definition. In Cournot’s Model of Competition, i = 0, 1, 2, . . . firms respond to a price

function by simultaneously choosing a quantity, qi ∈ [0, 1], to produce. The price function is

inversely proportional to quantity. For the duration of this paper, we will use the price function

P = 1 − Q, where Q =∑n

i=1 q[i]. It is supposed that consumers will purchase all goods supplied.

In general, having more firms drives the price down, as a larger quantiy of goods will be produced,

but each firm will produce fewer goods. This setup would encouragee collusion into a cartel, so

that the model becomes like that of a monopoly, because this is pareto efficient (the highest total

profit, summing all firms’ profits) for the firms; however, this is not possible, when we assume

that the game is competitive (allows no cooperation). The equilibrium price and quantities (Nash

equilibrium) can be found by solving the system of equations

∂

∂q[i]

(q[i](P (Q)− s[i])

)=

∂

∂q[i]

(q[i](1−Q− s[i])

)= (1−Q− q[i] − s[i]) = 0

=⇒ q[i] = max

{1− s[i] −

∑j 6=i q

[j]

2, 0

}∀i,

where s[i] is the cost of production for firm i. Notice how each firm’s optimum quantity is dependent

on the other’s. As mentioned, this solution is a Nash equilibrium, meaning that deviation by either

part would cause for reduced profits. For further discussion of equilibrium see [1]. Notice that the

strategy space is{

(q[i])∀i∣∣0 ≤ q[i] ≤ ∞ ∀i

}, but we can clearly limit the strategy space to quantity

being between 0 and 1, since producing more than 1 will always result in negative profit (since

production cost is always nonnegative).


Definition. In Bertrand’s Model of Competition, firms respond to consumer demand by

simultaneously choosing prices. Notice that when firms produce identical goods, the equilibrium

price is the marginal cost of production of the firm with the second lowest per-unit product cost

(marginal cost) of all firms. If firms have different marginal costs, then the firm with the lowest

marginal cost can set the price below all other firms’ marginal costs, blockading them from the

market, meaning they cannot produce, because they would lose money by doing so. This firm

wants to have the highest price possible that blockades all other firms from the market, so they set

their price to the second lowest marginal cost of all firms. All firms with the same marginal cost,

will compete and set the price at marginal cost, and every firm makes zero profit. This is because

if any firm were to raise its price, it would be undercut by another firm; alternatively, if a firm

lowers its price below marginal cost it would lose money.

Comparison between Bertrand and Cournot. The Bertrand model predicts that if there are

two or more firms, then price is driven down to the marginal cost of the firm with second lowest

marginal cost, whereas in the Cournot model, prices approach the same value (marginal cost), as

the number of firms approaches infinity.

Definition. Discounting is a method to determine the present value of future payoffs. We con-

sider the continuous time discount e−rt. For example, whether a firm prefers a dollar today (t = 0)

or two tomorrow (t = 1) can be determined by calculating whether 1e−r(0) or 2e−r(1) is larger, that

is, the firm prefers a dollar today, if r > − log 12 .

2.2. Stochastic Processes.

Definition. A stochastic process or random process is a family of random variables indexed

by some set T . We consider a family of random variables {N(t) : t ≥ 0}, where N(t) counts the

number of successes (these result in cost reductions) indexed by the set [0 = T0, T1, T2 . . . , TN ], the

times at which these successes occur. This example is a special case of stochastic process: a point

process.

Definition. A Markov chain is a collection of random variables such that given the present,

the future is conditionally independent of the past; this propert is known as the Markov property,

or memoryless property. We assume that our process satisfies the strong Markov property, which

states that for the Poisson processs X, with some stopping time T , (X(T + s) − X(T )|s ≥ 0) is

a Poisson process starting from time 0 and independent of stopping time T and independent of

history (X(r)|0 ≤ r ≤ T ).

6 MARK C. DUGGAN

Definition. A Poisson process is a Markov process which counts the number of events and the

occurance time of each event. The time between each pair of consecutive events is distributed

exponentially and independently of each other. Two occurrences have negligible probability of

happening simultaneously. A Poisson process is said to have intensity λ if

P(N(t) = j) =(λt)j

j!e−λt for j = 0, 1, 2, . . . ,

where N(t) is the cumulative number of occurances by time t [2]. That is, N(t) has Poisson

distribution with parameter λt, which we abbreviate as N(t) ∼Pois(λt). Notice that the expected

time between successes is the reciprocal of the intensity.

3. A Sequence of Increasingly Complicated Research Scenarios

In order to build up to a more complicated model of competition between two firms who can

each reduce production costs by researching, first we consider a sequence of models each adding a

new layer of complexity.

3.1. A Simple Scenario: Cournot Monopoly.

Model 1. A single firm produces some good that has production cost s, and price, P = 1 − q,where q is the quantity produced by that firm. In this scenario, for the firm to maximize profit, it

must optimize with respect to quantity.

∂

∂qq(P − s) = q(1− q − s) = 1− s− 2q = 0 =⇒ q∗ =

1− s2

.

This yields a profit of

q∗(P (q∗)− s) =1− s

2

((1− 1− s

2

)− s)

=

(1− s

2

)2

.

Definition. A firm chooses some amount of effort, a, or funding for research and development

(R&D), in order to reduce the cost of production; this effort has an associated cost C. We will

mostly use the cost of effort function C(a) = a2+ka, where k ≥ 0 is referred to as the effort weight.

In general, we will consider a Poisson process N with intensity λa that counts the number of cost

reductions (successes), and with success times [0 = T0, T1, T2, . . . TN ].

Proposition 1. Suppose T1 ∼Exp(λa), then

E[

1− e−rT1r

]=

1

λa+ r,

where E[X] is the expectation or expected value of the random variable X.

Proof. Notate FT1(t) = 1− e−λt, the cumulative distribution function of T1. Then

E[e−rT1

]=

∫ ∞0

e−rtFT1(t)dt =

∫ ∞0

e−rt(1− e−λt)dt =λa

λa+ r.

As a result we see that

E[

1− e−rT1r

]=

1− E[e−rT1

]r

=1− λa

λa+r

r=

1

λa+ r.


�

Building on Model 1: Effort. Now consider a firm that produces a good at initial cost s(t),

chooses some quantity to produce q(t) and can choose some amount of effort to put into research,

a(t), to reduce the production cost from s0 to s1 to . . . to sN . Define the discounted profit function

πN (s0) = E[∫ ∞

0e−rt (q(t)(P (q(t))− s(t))− C(a(t))) dt

∣∣∣∣s(0) = s0

],

which determines the discounted profit a firm receives from time zero onward, where N possible

arrivals are considered. We will drop the condition notation and remember that s(t = 0) = s0. To

maximize this, take the supremum at each time, t, with respect to quantity and effort; define

VN (s0) = supq(t)∈[0,1],a(t)∈[0,∞)|t∈R+

πN (s0) = supq,a

E[∫ ∞


].

If we consider the example above (Model 1), and suppose the firm decides to put in no effort,

then the value function simplifies to

E[∫ ∞

0e−rtq∗(P (q∗)− s)dt

]=q∗(P (q∗)− s)

r=

(1− s)2

4r.

Consider the simplest case that involves possible positive effort, N = 1; one chance for improve-

ment. Now let us consider V1(s0), which we can simplify by splitting the integral at the first success

time, T1, and noticing that it is optimum to put in no effort after T1, since there is no room for

cost reduction. Notice that the optimum effort is fixed between T0 = 0 (we use T0 to notate an

initial time, when no successes have occurred) and T1, since nothing changes during that time,

due to the memmoryless property. Thus between any two arrivals Ti, Ti+1 (when price is si), we

will call the optimum effort ai and the optimum quantity qi (maximizing πN (si)). To compactify

notation, define ρ(q(t)) = q(t)(P (q(t)) − s(t)), profit disregarding cost of effort, and its optimum

ρk = supq q(t)(P (q(t))− sk), and define profit,π(q(t), a(t)) = ρ(t)−C(a(t)), as well as its optimum

πk = ρk − C(ak).

V1(s0) = supq,a |t≤T1

E[∫ T1

0e−rt (ρ(q(t))− C(a(t))) dt

]+ supq,a |t≥0

[∫ ∞T1

e−rt (ρ(q(t))− C(a(t))) dt

]by the Markov property

= E[∫ T1

0e−rt (ρ0 − C(a0)) dt

]+ supq,a |t≥0

E[e−rT1

∫ ∞0


]= E

[1− e−rT1

r

](ρ0 − C(a0)) + E

[e−rT1

]V1(s1).

=1

λa0 + r(ρ0 − C(a0)) +

λa0λa0 + r

V1(s1),(3.1)

=π0

λa0 + r+

λa0λa0 + r

(ρ1r

),

since

8 MARK C. DUGGAN

V1(s1) =

∫ ∞0

e−rt (ρ1 − C(a1 = 0)) dt =

∫ ∞0

e−rtρ1dt =ρ1r.

For Pi = 1− qi, we previously found find the quantity that optimizes profit:

qi =1− si

2and ρi =

1

4(1− si)2.

Then to find a0, the optimizing effort value, find the supremum of Equation 3.1 with respect

to a0, remembering π0 is a function of a0. This one step process can be generalized to an N step

process. See figures 1 and 2 (on the following page) for an example of a solution to this process

with a linear cost scheme and an exponential cost scheme.


Figure 1. The figure shows the optimum effort (the effort that maximizes thevalue function, given production cost) of a firm with a monopoly at each productioncost for various step sizes N (for a linear cost scheme) with constants (k, λ, r) =(.5, 1, .01). This solution is found using Equation 3.1.

Figure 2. The figure shows the optimum effort of a firm with a monopoly at eachproduction cost for firm 1, sl = e−zl, l = 1, . . . N for the different z values givenwith constants (k, λ, r) = (.4, 1, .03). This solution is found using Equation 3.1.

10 MARK C. DUGGAN

Model 2. As the previous model alluded, we consider a firm that can make N improvements.

Consider a partition S of the interval [s0, sN ], {s0, s1, . . . , sN} , where s0 < s1 < . . . < sN . Then a

firm will choose optimum (qi, ai) that maximize VN (s0), while the price remains si. Ti ∼Exp(λai)

is the time of the ith price reduction for i ∈ N, and we will defineT0 = 0 (though there is no 0th

arrival). Over time, we have at time 0, cost is s0 until a research breakthrough at T1, at which

time the production cost is reduced to s1. The firm can continue putting in effort to achieve cost

reductions, until they decided to stop putting in effort at TN , where the production cost is sN .

As a result of the previous model, we see that

VN (sl) =πl + λalVN (sl+1)

λal + r, for l = 0, 1, . . . , N, where VN (sN+1) := 0, since

VN (sl) = supa,q |t≥0

E[∫ ∞


∣∣∣∣s(0) = sl

]= sup

a,q|0≤t≤T1E[∫ T1


]+ supa,q |t≥T1

E[∫ ∞

T1


]after splitting the integral, we use the strong Markov property for s,

= E[(ρl − C(al))

∫ T1

0e−rtdt

]+ supa,q |t≥T1

E[e−r(T1)

∫ ∞0


]= E

[1− e−rT1

r

](ρl − C(al)) + E

[e−rT1

]VN (s1).

then we use Proposition 1 to evaluate expectations,

=1

λal + r(ρl − C(al)) +

λalλal + r

VN (s1)

=πl + λalVN (sl+1)

λal + rfor l=0, 1,. . . N.

Notice that no matter how we define VN (sN+1), so long as it is finite,

VN (sN ) =πN+λaNVN (sN+1)

λaN+r = πNr , since aN = 0; so we will define VN (sN+1) = 0, for convenience.

In order to find VN (s0), we first solve for VN (sN ), then VN (sN−1), etc., as a result of the recursive

nature of VN (sl). Observe that while the production cost is si, a(t) is fixed at some level, al, because

it is an equilibrium solution; were the firm to make an adjustment during this time, it would be

less than optimal. The same holds for ql. We have already discussed how to find each ql, using the

methods of Cournot, and in Model 3, we will be more precise in how to find each al.

Lemma 1. Success & Decay Rates. For a single firm with the opportunity to reduce production

costs from s0 to s1 to . . . to sN , the amount of effort (at) that optimizes profit is the same in the

following scenarios:

1) The firm has success rate λ, decay rate r.

2) The firm has success rate λ′ = λb, and decay rate r′ = rb, for some constant b > 0.

Furthermore VN (sm) = bV ′N (sm) for all m ≥ 0, where V ′N (sm) is the value function for the firm

with success rate λ′, and decay rate r′.


Proof. First recall that for any m ≥ 0 (and less than N + 1),

VN (sm) =1

λam + rπm +

λamλam + r

V (sm+1)

where πm and am maximize VN (sm).

We will proceed by induction on production cost, sm. Begin with the base case, m = N :

bV ′N (sN ) = bπNr′

= bπN(br)

=πNr

= VN (sN ).

Now consider the general case,

bV ′N (sm) = b supa

(π

λ′a+ r′+

λ′a

λ′a+ r′V ′(sm+1)

)= sup

a

(b

π

λ′a+ r′+

λ′a

λ′a+ r′bV ′(sm+1)

)= sup

a

(b

π

(bλ)a+ (br)+

(bλ)a

(bλ)a+ (br)

(bV ′ (sm+1)

))= sup

a

(π

λa+ r+

λa

λa+ rV (sm+1)

)= VN (sm).

Notice that since we can factor b from both supremums, they are over the same quantity. Thus the

optimum effort is the same in both cases. The result follows by inductive reasoning. �

Because of this dependency, we will tend to let γ = λr , in order to compactify notation. It is

important to note that in the case of two firms, it may be reasonable to require that r[1] and r[2]

are the same, which would necessitate keeping λ and r separate.

3.2. Cournot Duopoly: Single Firm Research.

Model 3. Now we consider competition between two firms, where Firm 1 has the opportunity to

reduce its cost of production by means of R&D, but Firm 2 has fixed production cost s[2]. Suppose

Firm 1 has initial cost of production s[1]0 and N successes can reduce his cost of production to s

[1]N .

Then, consider a partition S of the interval [s[1]0 , s

[1]N ],

{s[1]0 , s

[1]1 , . . . , s

[1]N

}, where s

[1]0 < s

[1]1 < . . . <

s[1]N . This is an N + 1 stage game. The quantities each firm chooses is the same as in the classical

Cournot model: During [Ti, Ti+1), Firm 1 chooses quantity q[1]i = max

{1−s[1]i −q

[2]i

2 , 0

}and Firm 2

chooses quantity q[2]i = max

{1−s[2]−q[1]i

2 , 0

}(for more detail on how we arrive at these solutions see

the definition of Cournot’s model of competition). If one firm has production cost too high to enter

the market (1 + s[k] − 2s[j]i < 0), it produces no goods and the other firm chooses the quantity as

in the monopoly case (q[k] = 1−s[k]2 ). Supposing q

[j]i > 0, ∀j(

q[1]i , q

[2]i

)=

(1 + s[2] − 2s

[1]i

3,1 + s

[1]i − 2s[2]

3

).

12 MARK C. DUGGAN

From this we conclude that Pi =1+s[2]+s

[1]i

3 and that

ρi = qi(Pi − s[1]i ) =1 + s[2] − 2s

[1]i

3

(1 + s[2] + s

[1]i

3− s[1]i

)=

(1 + s[2] − 2s

[1]N

)29

= q2i .

Now we want to find ai. We start by finding VN (s[1]N ) = ρi

r , then solving (by finding aN−1 that

maximizes VN (s[1]N−1))

VN (s[1]N−1) =

1

λaN−1 + r(ρN−1 − C(aN−1)) +

λaN−1aN−1 + r

ρNr,

and so on, using

(3.2) VN (s[1]l ) =

1

λal + r(ρl − C(al)) +

λalal + r

VN (s[1]l+1),

in order to find the optimum amounts of effort at each stage. This notion is extends naturally

to Cournot competition with m firms, where Firm 1 is the only firm with the choice to put effort

into research.

In Equation 3.2, we can find al by differentiating VN (s[1]l ) with respect to al in order to find

critical points (we will use the cost funtion C(a) = a2 + ka) :

0 =∂

∂alVN (s

[1]l )

=∂

∂al

(1

λal + r

(ρl − a2l − kal

)+

λalal + r

VN

(s[1]l+1

))=

(λal + r)(−2al − k)− λ(ρl − a2l − kal)(λal + r)2

+λ(al + r)− λal

(al + r)2VN

(s[1]l+1

)= (λal + r)(−2al − k)− λ(ρl − a2l − kaN−1) + λrVN

(s[1]l+1

)= λa2l + 2ral −

(rk + λ

(rVN

(s[1]l+1

)− ρl

))

=⇒ al =

−r +

√r2 + λ

(rk + λ

(rVN

(s[1]l+1

)− ρl

))λ

.


3.3. Cournot Duopoly: Dual Research.

Lemma 2. For two firms, with success rate λ[1] and λ[2] respectively, the probability that Firm 1’s

first arrival occurs before Firm 2’s first arrival is

P(T[1]1 < T

[2]1

)=

a[1]0,0λ

[1]

a[1]0,0λ

[1] + a[2]0,0λ

[2],

where a[h]0,0 is the optimum amount of effort for Firm h between the initial time, t = T

[1]0 = T

[2]0 = 0

and t = min{T1, T2}, for h = 1, 2.

Proof. T[1]1 − T

[2]1 is the difference of two exponential distributions with parameters a

[1]0,0λ

[1] and

a[2]0,0λ

[2], so it has PDF

f(x) =

a[1]0,0λ

[1]a[2]0,0λ

[2]

a[1]0,0λ

[1]+a[2]0,0λ

[2]e−a

[2]0,0λ

[2]x for x ≥ 0

a[1]0,0λ

[1]a[2]0,0λ

[2]

a[1]0,0λ

[1]+a[2]0,0λ

[2]e−a

[1]0,0λ

[1]x for x < 0,

And thus

P(T[1]1 > T

[2]1

∣∣∣∣T [1]0 = T

[2]0 = 0

)= P

(T[1]1 − T

[2]1 > 0

∣∣∣∣T [1]0 = T

[2]0 = 0

)=

∫ ∞0

f(x)dx

=a[1]0,0λ

[1]

a[1]0,0λ

[1] + a[2]0,0λ

[2].

�

Model 4. Now we consider competition between two firms, where both firms have the opportunity

to reduce their costs of production by means of R&D. Suppose firm h has initial cost of production

s[h]0 , where Nh successes can reduce his cost of production to s

[h]Nh

. Then, consider a partition S of

the interval[s[h]0 , s

[h]Nh

],{s[h]0 , s

[h]1 , . . . , s

[h]Nh

}, where s

[h]0 < s

[h]1 < . . . < s

[h]Nh

. During [Ti, Ti+1), Firm 1

chooses quantity q[1]i,j = max

{1−s[1]i −q

[2]i,j

2 , 0

}and Firm 2 chooses quantity q

[2]i,j = max

{1−s[2]j −q

[1]i,j

2 , 0

}.

Notice that each firm need not be at the same step, that is for production prices s[1]i , s

[2]j , j and i

need not be the same. Supposing q[h]i > 0,

(q[1]i , q

[2]j ) =

(1 + s

[2]j − 2s

[1]i

3,1 + s

[1]i − 2s

[2]j

3

)if(

1 + s[2]j − 2s

[1]i > 0 and 1 + s

[1]i − 2s

[2]j > 0

).

Otherwise the firm with the lesser production cost treats the market as in the monopoly case,

and the other firm refrains from producing, so q =(1−s2 , 0

). From this we conclude that P

[1]i =

1+s[2]j +s

[1]i

3

(or P

[1]i =

1+s[1]i

2

)and that

ρ[1]i = q

[1]i

(P

[1]i − s

[1]i

)=

1 + s[2]j − 2s

[1]i

3

(1 + s

[2]j + s

[1]i

3− s[1]i

)=

(1 + s[2]j − 2s

[1]N )2

9=(q[1]i

)2.

14 MARK C. DUGGAN

Now we want to find ai. We start by finding V [1](s[1]N1, s

[2]N2

) =ρ[1]N1,N2r . We will consider when the

firms are at their ith and jth cost reductions, but by convention we will notate this as s[1]0 and s

[2]0 ,

because of the memmoryless property. Also note, when we consider P(T[1]1 < T

[2]1 ), we suppose that

the previous arrival time for each firm was T[1]0 and T

[2]0 respectively, so that we are cosidering the

next arrival for each firm.

First we notice that min{T[1]1 , T

[2]1

}is the minimum of two exponential distributions with pa-

rameters λ[1] and λ[2], so min{T[1]1 , T

[2]1

}∼Exp

(λ[1] + λ[2]

). In order to find the maximum amout

of effort for the firm to choose, consider the value function (we will prove that this is the correct

one soon)

V [1](s[1]0 , s

[2]0 ) =E

[π[1]0,0

∫ min{T [1]1 ,T

[2]1 }

0e−rtdt

]+ P

(T[1]1 < T

[2]1

)E[e−rT

[1]1 V [1]

(s[1]1 , s

[2]0

)]+

+ P(T[1]1 > T

[2]1

)E[e−rT

[2]1 V [1]

(s[1]0 , s

[2]1

)]=

π[1]0,0

λ[1]a[1]0,0 + λ[2]a

[2]0,0 + r

+

(a[1]0,0λ

[1]

a[1]0,0λ

[1] + a[2]0,0λ

[2]

)λ[1]a

[1]0,0

λ[1]a[1]0,0 + r

V [1](s[1]1 , s

[2]0

)

+

(a[2]0,0λ

[2]

a[1]0,0λ

[1] + a[2]0,0λ

[2]

)λ[2]a

[2]0,0

λ[2]a[2]0,0 + r

V [1](s[1]0 , s

[2]1

).

Notice that this recursion depends on a grid of possible production costs in [s[1]N1, s

[1]0 ]× [s

[2]N2, s

[2]0 ]

(usually [0, 1]× [0, 1] ), rather than just a line, as in previous examples.

Lemma 3. For two firms with the opportunity to research to reduce their production costs from s[1]0

to s[1]1 to . . . s

[1]N and s

[2]0 to s

[2]1 to . . . s

[2]N , respectively, we can find the optimum amount of effort

a[1]i,j and a

[2]i,j for each firm to put into research by maximizing the value function

V [1](s[1]0 , s

[2]0 ) =

π[1]0,0 + a

[1]0,0λ

[1]V [1](s[1]1 , s

[2]0

)+ a

[2]0,0λ

[2]V [1](s[1]0 , s

[2]1

)a[1]0,0λ

[1] + a[2]0,0λ

[2] + r,

with respect to effort and quantity, where λ[k] is the success rate (intensity) for firm k (that is

T ki − T ki−1 ∼ Exp(λa), where T[k]i is the time of the ith success for firm k), and r is the decay

factor.

Proof. We proceed as in in the first model, splitting up the integral by arrival times–because pro-

duction cost, s(t); quantity, q(t); price, P (q(t)); effort, a(t); and cost of research C(a(t)) are all

fixed between arrivals–but this time, arrivals for either firm must be considered. In the following

equations, we will drop the superscript indicating to which firm the variable is assigned, since

they are all variables for firm 1 (until we add superscripts back in). P(T[j]1 > T

[k]1

)is evalu-

ated by Lemma 2. E[e−rT

[j]1

]is evaluated as in Proposition 1, as is E

[e−rmin

{T

[1]1 ,T

[2]1

}], since


min{T[1]1 , T

[2]1

}∼Exp

(λ[1] + λ[2]

).

V [1](s[1]0 , s

[2]0 ) = sup

q,a|t∈R+

E[∫ ∞


∣∣∣∣s[1](t = 0) = s[1]0 , s

[2](t = 0) = s[2]0

]

= supq,a|0≤t≤min{T [1]

1 ,T[2]1 }

E

[∫ min{T [1]1 ,T

[2]1 }

0e−rt

(q(t)(P (q(t))− s[1]0,0)− C(a(t))

)dt

]

+ supq,a|t≥min{T [1]

1 ,T[2]1 }

E

[∫ ∞min{T [1]

1 ,T[2]1 }

e−rt (q(t)(P (q(t))− s(t))− C(a(t))) dt

]by the law of total probability,

= E

[π[1]0,0

∫ min{T [1]1 ,T

[2]1 }

0e−rtdt

]

+ P(T[1]1 < T

[2]1

)sup

q,a|t≥min{T [1]1 ,T

[2]1 }

E

[∫ ∞T

[1]1


∣∣∣∣∣T [1]1 < T

[2]1

]

+ P(T[1]1 > T

[2]1

)sup


[2]1 }

E

[∫ ∞T

[2]1


∣∣∣∣∣T [1]1 > T

[2]1

]

= E

[π[1]0,0

∫ min{T [1]1 ,T

[2]1 }

0e−rtdt

]+ P

(T[1]1 < T

[2]1

)sup

q,a|0≤t≤min{T [1]1 ,T

[2]1 }

E[e−rT

[1]1 E

[∫ ∞0

e−rt(q(t+ T

[1]1 )(P (q(t+ T

[1]1 ))− s(t+ T

[1]1 ))− C(a(t+ T

[1]1 ))

)∣∣∣∣s(T [1]1 )dt

]∣∣∣∣T [1]1 < T

[2]1

]+ P

(T[1]1 > T

[2]1

)sup


[2]1 }

E[e−rT

[2]1 E

[∫ ∞0

e−rt(q(t+ T

[2]1 )(P (q(t+ T

[2]1 ))− s(t+ T

[2]1 ))− C(a(t+ T

[2]1 ))

)∣∣∣∣s(T [2]1 )dt

]∣∣∣∣T [1]1 < T

[2]1

]by the Markov property and the tower rule,

= E

[π[1]0,0

∫ min{T [1]1 ,T

[2]1 }

0e−rtdt

]

+ P(T[1]1 < T

[2]1

)supq,a|t≥0

E[e−rT

[1]1

∫ ∞0


∣∣∣∣T [1]1 < T

[2]1

]+ P

(T[1]1 > T

[2]1

)supq,a|t≥0

E[e−rT

[2]1

∫ ∞0


∣∣∣∣T [1]1 < T

[2]1

]

= π[1]0,0E

1− e−rmin{T

[1]1 ,T

[2]1

}r

+ P(T[1]1 < T

[2]1

)E[e−rT

[1]1

∣∣∣T [1]1 < T

[1]1

]V [1]

(s[1]1 , s

[2]0

)+

+ P(T[1]1 > T

[2]1

)E[e−rT

[2]1

∣∣∣T [1]1 < T

[1]1

]V [1]

(s[1]0 , s

[2]1

)=

π[1]0,0

λ[1]a[1]0,0 + λ[2]a

[2]0,0 + r

+

(a[1]0,0λ

[1]

a[1]0,0λ

[1] + a[2]0,0λ

[2]

)λ[1]a

[1]0,0 + λ[2]a

[2]0,0

λ[1]a[1]0,0 + λ[2]a

[2]0,0 + r

V [1](s[1]1 , s

[2]0

)

+

(a[2]0,0λ

[2]

a[1]0,0λ

[1] + a[2]0,0λ

[2]

)λ[1]a

[1]0,0 + λ[2]a

[2]0,0

λ[1]a[1]0,0 + λ[2]a

[2]0,0 + r

V [1](s[1]0 , s

[2]1

).

16 MARK C. DUGGAN

V [1](s[1]0 , s

[2]0 ) =

π[1]0,0 + a

[1]0,0λ

[1]V [1](s[1]1 , s

[2]0

)+ a

[2]0,0λ

[2]V [1](s[1]0 , s

[2]1

)a[1]0,0λ

[1] + a[2]0,0λ

[2] + r.

�

Let us consider a simple case, where two firms each have the opportunity to make one research

breakthrough, reducing their production cost from s[1]0 = s

[2]0 > 1/2 to s

[1]1 = s

[2]1 = 0. Thus when

one firm makes a breakthrough, they have a monopoly, blockading the other from production. We

detail the firms value functions at each step below.

For firm k,

V [k](s[1]1 , s

[2]1 ) = V [k](0, 0) =

π[k]1,1

r=ρ[k]1,1

r=

1

9r,

where Pi,j = 1− q[1]i,j − q[2]i,j .

V [1](s[1]0 , s

[2]1 ) =

π[1]0,1 + a

[1]0,1λ

[1]V [1](s[1]1 , s

[2]1

)a[1]0,1λ

[1] + r

=ρ[1]0,1 −

(a[1]0,1

)2− ka[1]0,1 + a

[1]0,1λ

[1] ρ[1]1,1

r

a[1]0,1λ

[1] + r

=

ρ[1]0,1 + a

[1]0,1

(λ[1]

ρ[1]1,1

r − k)−(a[1]0,1

)2a[1]0,1λ

[1] + r

This is maximized when

0 =∂V [1](s

[1]0 , s

[2]1 )

∂a[1]=

(λ[1]a

[1]0,1 + r

)(λ[1]

ρ[1]1,1

r − k − 2a[1]0,1

)− λ[1]

(ρ[1]0,1 + a

[1]0,1

(λ[1]

ρ[1]1,1

r − k)−(a[1]0,1

)2)(λ[1]a

[1]0,1 + r)2

=

(λ[1]ρ

[1]1,1 − kr +

((λ[1])2 ρ[1]1,1

r− kλ[1] − 2r

)a[1]0,1 − 2λ[1]

(a[1]0,1

)2)

− λ[1](ρ[1]0,1 + a

[1]0,1

(λ[1]

ρ[1]1,1

r− k

)−(a[1]0,1

)2)

= λ[1](ρ[1]1,1 − ρ

[1]0,1

)− kr − 2ra

[1]0,1 − λ

[1](a[1]0,1

)2= λ[1]

(a[1]0,1

)2+ 2ra

[1]0,1 + λ[1]

(ρ[1]0,1 − ρ

[1]1,1

)+ kr

=⇒ a[1]0,1 =

−r+

√r2+(λ[1])

2(ρ[1]1,1−ρ

[1]0,1

)−λ[1]kr

λ[1]if(λ[1])2 (

ρ[1]1,1 − ρ

[1]0,1

)> λ[1]kr

0 if(λ[1])2 (

ρ[1]1,1 − ρ

[1]0,1

)≤ λ[1]kr

.


V [1](s[1]1 , s

[2]0 ) =

π[1]1,0 + +a

[2]1,0λ

[2]V [1](s[1]1 , s

[2]1

)a[2]1,0λ

[2] + r

=ρ[1]1,0 + λ[2]a

[2]1,0

ρ[1]1,1

r

λ[2]a[2]1,0 + r

,

which is maximized when

a[2]1,0 =

−r+

√r2+(λ[2])

2(ρ[2]1,1−ρ

[2]1,0

)−λ[2]kr

λ[2]if(λ[2])2 (

ρ[2]1,1 − ρ

[2]1,0

)> λ[2]kr

0 if(λ[2])2 (

ρ[2]1,1 − ρ

[2]1,0

)≤ λ[2]kr

,

since the problem is symmetric.

V [1](s[1]0 , s

[2]0 ) =

π[1]0,0 + a

[1]0,0λ

[1]V [1](s[1]1 , s

[2]0

)+ a

[2]0,0λ

[2]V [1](s[1]0 , s

[2]1

)a[1]0,0λ

[1] + a[2]0,0λ

[2] + r

=

π[1]0,0 + a

[1]0,0λ

[1]

ρ[1]1,0+λ

[2]a[2]1,0

ρ[1]1,1r

λ[2]a[2]1,0+r

+ a[2]0,0λ

[2]

ρ[1]0,1+a

[1]0,1

(λ[1]

ρ[1]1,1r−k)−(a[1]0,1

)2a[1]0,1λ

[1]+r

a[1]0,0λ

[1] + a[2]0,0λ

[2] + r.

Differentiating gives

0 =∂V [1](s

[1]0 , s

[2]0 )

∂a[1]0,0

=(a[1]0,0

)2+(k + 2V [1]

(s[1]1 , s

[2]0

))a[1]0,0 + (k − r)V [1]

(s[1]1 , s

[2]0

)+ a

[2]0,0λ

[2](V [1]

(s[1]0 , s

[2]1

)− V [1]

(s[1]1 , s

[2]0

))− ρ[1]0,0

=⇒ a[1]0,0 =

−12k − V

[1](s[1]1 , s

[2]0

)+

+

√(k+2V [1]

(s[1]1 ,s

[2]0

))2+4((r−k)V [1]

(s[1]1 ,s

[2]0

)+a

[2]0,0λ

[2](V [1]

(s[1]1 ,s

[2]0

)−V [1]

(s[1]0 ,s

[2]1

))+ρ

[1]0,0

)2

if (k − r)V [1](s[1]1 , s

[2]0

)< a

[2]0,0λ

[2](V [1]

(s[1]1 , s

[2]0

)− V [1]

(s[1]0 , s

[2]1

))+ ρ

[1]0,0

0 if (k − r)V [1](s[1]1 , s

[2]0

)≥ a[2]0,0λ

[2](V [1]

(s[1]1 , s

[2]0

)− V [1]

(s[1]0 , s

[2]1

))+ ρ

[1]0,0

,

(3.3)

and a symmetric equation holds for a[2]0,0. Then we solve for effort, a

[l]0,0, for each firm, l = 1, 2,

using these two equations. This is a Nash equilibrium as well. Recall that the quantiy we find

at each stage is a Nash equilibrium, which we find at each step seperately, making each step a

simple Cournot game. When each firm is in a given situation they choose a quantity produced, as

previously prescribed, and they also choose an effort to put into research, but this effort depends

18 MARK C. DUGGAN

on their opponents’ effort; thus a Nash equilibrium must be found by solving the pair of equations

given by Equation 3.3 and its counterpart.

3.4. Cournot Substitution.

Model 5. Previously, we assumed goods were identical, or perfect substitutes; however, this is not

always the case, so we consider firms selling goods that are substitutes, i.e. P [i] = 1−q[i]−∑

j 6=i δq[j],

where δ is the cross-price elasticity of demand, as defined in §2.1, between any two different firms

(often denoted δi,j for elasticity between good i and good j, but here we assume δi,j = δk,l for all

i 6= j, k 6= l). Consider just two firms now, and let us use the notation of the previous model. If

one firm has production costs too high to enter the market (2− δ+ δs[j]− 2s[k]i > 0), they produce

no goods and the other firm chooses the quantity as in the monopoly case (q = 1−s2 ). Supposing

q[h]i > 0,

(q[1]i , q

[2]j ) =

(2− δ + δs

[2]j − 2s

[1]i

4− δ2,2− δ + δs

[1]i − 2s

[2]j

4− δ2

).

From this we conclude that P[1]i =

2−δ+(2−δ2)s[1]i +s[2]j

4−δ2 and that ρ[1]i =

(q[1]i

)2. Notice that this

changes the ρl = πl − C(al) function, but no other part of VN (sl) =πl+λalVN (sl+1)

λal+r, so the general

process remains the same.

4. When to Say “No” to R&D

4.1. Linear. We explore under what conditions it is not worthwhile to continue putting effort into

research. Suppose a firm is deciding whether to start researching, in the monopoly case. Then the

firm will choose not to research if for all a ≥ 0,


V (sn)|(a = 0) > V (sn)|(a ≥ 0)

⇐⇒ ρ

r>

ρ− Cλa+ r

+λa

λa+ rV (sn+1)

⇐⇒ ρ

r>

ρ− Cλa+ r

+λa

λa+ r

ρn+1

r

⇐⇒ (λa+ r)ρ > rρ− rC + λaρn+1

⇐⇒ a(λρ− λρn+1) > −rC

⇐⇒ a <r

λ

C

ρn+1 − ρIf we let C = a2 + ka, then

λ

r(ρn+1 − ρ)− k < a.

Notice that if the condition must be satisfied for all a ≥ 0,

then it will suffice for it to be satisfied for a = 0

⇐⇒ γ(ρn+1 − ρ) < k.

⇐⇒ γ

((1− sn+1)

2

4− (1− sn)2

4

)< k,

where we define γ = λ/r. Notice that if k = 0, this condition is always satisfied. Otherwise, we can

consider a few different cases. For the simplest case, the linear case, sn+1 = sn− 1N (sn = s0− n

N ),

gives us the condition

γ

((1− sn+1)

2

4− (1− sn)2

4

)< k.

⇐⇒ γ

((1− sn + 1

N )2

4− (1− sn)2

4

)< k.

⇐⇒ 4k

γ>

(1

N2+

2

N− 2sn

N

)⇐⇒ sn >

1

2N+ 1− 2kN

γ

⇐⇒ s0 −n

N>

1

2N+ 1− 2kN

γ

⇐⇒ n <1

2+N(s0 + 1)− 2kN2

γ

In conclusion, the firm will not research when n < 12 + N(s0 + 1) − 2kN2

γ . Notice that if

N < 12 + N(s0 + 1) − 2kN2

γ ⇐⇒ 2kN2

γ − Ns0 − 12 < 0 ⇐⇒ N < −s0 +

√s20 + 4k

γ the firm

will not research at all.

20 MARK C. DUGGAN

4.2. Exponential. For the exponential case, e−z(n+1) = e−zne−z = sn+1 = sne−z for some z > 0,

gives us the condition:

γ

((1− sn+1)

2

4− (1− sn)2

4

)> k.

⇐⇒ γ

((1− sne−z)2

4− (1− sn)2

4

)> k.

⇐⇒ γ

(s2n(e−2z − 1) + 2sn(1− e−z)

4

)− k > 0.

⇐⇒ γ

(e−2zn(e−2z − 1) + 2e−zn(1− e−z)

4

)− k > 0.

⇐⇒ γ

(e−2zn−2z − e−2zn + 2e−zn − 2e−zn−z

4

)− k > 0.

⇐⇒ e−2zn−2z − e−2zn + 2e−zn − 2e−zn−z >4k

γ.(4.1)

Notice that we get equality when

e−zn = sn =

γ2 (e−z − 1)±√γ

√γ4 (1− e−z)2 + (e−2z − 1)k

γr (e−2z − 1)

,

or equivalently

n = −z log

γ2 (e−z − 1)±√γ

√γ4 (1− e−z)2 + (e−2z − 1)k

λ2r (e−2z − 1)

.

Thus the condition is satisfied in between the two critical values (plugging in extreme values of

1 and taking the limit as n→∞ makes the left hand side of the Inequality 4.1 very small).

5. Figures

Cournot Duopoly, Single Firm Research, Linear Cost


Figure 3. Shows optimum effort choices for Firm 1, given different production costs

for Firm 2 and constants(s[1]0 , N, k, λ, r

)= (1, 25, .5, 1, 0.01). Notice that total effort

is maximized when Firm 2’s production costs are around 12 , and minimized when

Firm 2’s production costs are 0. Notice the dip in the red curve around s = 0.2and in the green curve around s = 0.6. These dips are a result of changing from aduopoly to a monopoly; Firm 2 is blockaded from competing. To emphasize thesedips, see Figure 2.

22 MARK C. DUGGAN

Cournot Duopoly, Single Firm Research, Linear Cost

Figure 4. Shows optimum effort choices for Firm 1, given different step sizes (recallcosts improve linearly in this case), 1

N cost reduction for each success and constants(s[1]0 , s

[2], k, λ, r)

= (1, 0.6, .5, 1, 0.01).

Figure 5. The figure shows the optimum effort of a firm with who competes witha firm with fixed production cost 0.6 at each production cost for firm 1, e−zl, l =1, . . . N with constants (k, λ, r) = (.4, 1, .03).


6. Further Research

A few things that could be more closely observed are:

(1) Critical R&D points

When does a firm decide never to research? When does a firm decide to stop researching?

Those these questions are mentioned, a complete answer is not yet clear.

(2) How partitioniing of production costs affects optimum effort

Can we choose an effort a and find a production cost partition {s0, s1, . . . , sN} such that

the optimum effort is a?

(3) Refinements of cost partitions

If a firm has production cost partition S, and a second firm’s cost partition is a refinement

of S, then the expected profit of the second firm is greater than or equal to the expected

profit of the first firm.

References

[1] A. Cournot, Researches on the Mathematical Principles of the Theory of Wealth, (1838), 79-84.

[2] G. Grimmett and D. Stirzaker, Probability and Random Processes, Oxford University Press, 2001.

[3] M. Ludkovski and R. Sircar, Exploration and Exhaustibility in Dynamic Cournot Games, European Journal on

Applied Mathematics 23(3) (2012), 343-372.

Department of Mathematics, University of California, Santa Barbara, CA 93106.

E-mail address: [email protected]

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