The model of duopoly under uncertainty is based in Dixit & Pindyck book (1994, chapter 9, last section), but with some practical adaptation:

- Specifying as exponential the demand curve, instead a generic one;
- Allowing non-unitary production;
- Allowing leader and follower producing different quantities; and
- Allowing cost differentiation for the players.

This is an ** option-game** application. See other related
applications in the option-game webpage.

For a software related to this theory, go to the ** Duopoly
under Uncertainty Spreadsheet** webpage, from the real options
software section.

OBS: **This webpage is under revision**, in order to consider many additional aspects and a more rigorous discussion on equilibrium strategies. The revision is coming soon!

The market of a generic product has (inverse) demand curve given by P = f(q), where P is
the price and q is the demanded quantity. Consider an exponential demand
curve with parameter e which is a kind of *elasticity*.

This type of inverse demand curve has the some advantages over other more popular inverse demand curves. For example, the linear inverse demand curve can lead to negative prices for high output production.

A good alternative is the ** iso-elastic inverse demand curve**, which in the simplest format is given by:

One solution is analyzed in the interesting paper of Agliari & Puu (2002), where they suggest the modified iso-elastic inverse demand curve:

Suppose that the factor Y is uncertain, so the inverse exponential demand curves can be in different places along the time. The following figure shows two demand curves for different values of Y.

Modelling the factor Y uncertainty with a lognormal diffusion process, let Y follow a geometric Brownian motion (GBM):

The current demand function is known but the future demand function P =
Y D(q) is uncertain. The uncertainty is modeled with a GBM for the
oscillations in the demand factor Y. Its current value is known (Y = Y_{0}).

Let us examine a leader-follower duopoly under uncertainty, extending the Smets' model presented in the Dixit & Pindyck book.

For example, suppose there is a duopoly in a telecommunications market.

Consider one small city with telecom investment opportunity. Both firms,
leader and follower, have a perpetual option to invest in that market with
stochastic demand.

Firms have rational expectations and maximize value under demand
uncertainty, expressed in the factor Y.

The current demand function is known but the future demand function P = Y
D(q) is uncertain and modeled with a GBM (see item 2 above).

Let we allow different unitary investment costs for firms, so the leader
has a cost I_{1} per unity of product and the follower has a cost
I_{2} per unity of product. So, I extend the model to permit cost
differentiation between the players.

The valuation is performed backwards, first you should first calculate
the follower value, supposing that the leader already entered.

Given that follower behavior, the leader value is calculated. This means
that the leader value will be affected by the expected entry of the
follower (when the demand rise until the threshold Y_{2}).

The follower has a perpetual option to entry in the market, producing q_{2},
and waiting the demand increase until reach a threshold level Y_{2}
where is optimal the immediate investment of the follower (threshold for
the follower).

The demand curve is for the industry, so consider the sum of the both firms production, that is,

**P = Y e ^{- e q} = Y e^{-
e (q1 + q2)}**

Suppose that the price P is net of operational expenses, so that P can
be interpreted as a margin (profit).

To simplify, imagine that the present value of the operating project V is
the result of a perpetual flow from this profit P: **V = q _{2} . P/d**, where
d is the dividend yield or the rate of cash
flow distribution given by:

With the presented premisses we can calculate the threshold Y_{2}
and the follower value F_{2}, for a perpetual option:

Where:

If Y > Y_{2} the follower value is:

**F _{2} = q_{2} (Y e ^{- e
(q1 + q2)})/d - q_{2}
I_{2}**

If Y < Y_{2}, the follower will wait until the demand factor
reach for the first time the threshold level Y_{2}. The expected
time to occur that is called ** first hitting time** T* of the
stochastic process of Y.

When this happen, the follower will get the same anterior value for F

The expected value for the discount factor E[e^{-
r T*}] is given below:

**E[e ^{- r T*}] = (Y/Y_{2})
^{b1}**

See the Appendix A, Proof for the First Expectation for the demonstration and details.

So, the follower value is:

A typical call option value chart represents the follower value.

The leader will, for a certain time (until T*), remain acting alone in the market, without the follower. So, the demand curve is:

**P = Y e ^{- e q} = Y e^{-
e (q1)}**

After the time T* the leader profit will be shared with the follower
that, entering at this instant, will change the demand curve to: **P = Y e ^{- e q} = Y e^{-
e (q1 + q2)}**

The value of the expected cash flow, discounted by the rate
r until the moment that the follower enters,
added to the expected present value after the follower entry, gives the
leader value F_{1}:

If Y > Y_{2} : ** F _{1} = q_{1} Y e^{-
e (q1 + q2)})/ d
- q_{1} I_{1}**

If Y < Y_{2}, the leader knows that the follower will wait
until the demand shock Y reach the threshold Y_{2}. The value of
the leader is given by the present value of the expected cash-flow
producing alone in the market until T* plus the expected cash-flow after
the follower entry at T*, less the investment cost to become a leader.
This is given by the expression below:

Is necessary the first hitting time concept to calculate the
expectations (for the expectation of the integral, see the
Appendix B "Proof for the Second
Expectation" and for the other expectation, see
Appendix A).

By substituting the results from the appendixes (and remembering
d = r - a) we get:

**Leader Threshold Y _{1}**

If the firms are equal (same cost) and if becoming leader doesn't mean a
larger market share (same q), we have the case analyzed in Dixit &
Pindyck chapter 9.

In this case the threshold value Y_{1} is given by the point that
both firms are indifferent to assume the leader or follower role. This
happen when **F _{1}(Y_{1}) = F_{2}(Y_{1})**
provided that Y

Equaling these two equations, F

The following picture shows the leader and follower values and the entry thresholds:

The above chart is built with homogeneous firms in the sense of same
unitary cost I and producing the same quantity (the firms have the same
market share of 50%).

This is very similar to the chart presented in Dixit & Pindyck (1994,
p.311).

In case of *first move advantage* in the sense that the leader
obtains a **larger market share**, the threshold Y_{1} is
given by the point that each firm is indifferent between the leader and
follower roles, but this time at Y_{1} the value of leader
producing q_{1} > q_{2} is equal to the value of the
follower that (probably) will produce in the future a lower quantity q_{2}.

That is, **F _{1}(Y_{1}, q_{1}) = F_{2}(Y_{1},
q_{2})**.

The chart below presents this case of different quantities or *different
market shares*. In this case the leader gets a higher market share and
produces q_{1} = 15,000 units, whereas the follower produces q_{2}
= 10,000 units.

The leader and follower points are as before, but when the follower
enters the leader value is higher because his/her higher market share.

Note also that, with both in the market, the inclination of value line of
leader is higher than the follower. This means that when the demand rises,
the leader value grows more than the follower value. But when the demand
value decreases, the leader valuer drops faster than the follower.

Note also that in order to produce a larger quantity the leader
investment is larger because the larger capacity demands more investment
(remember that the leader investment = I_{1} q_{1}).
Hence the threshold Y_{1} can be larger than the previous case.

Here is not allowed incremental investment in capacity. If we allow an
* option to expand* production, the leader could invest in a
small capacity at lower threshold Y

Let us examine the case of **different costs** between the players
but with the *same market share*. Suppose I_{1} = 1500,
lower than I_{2} = 2000, so that there is a competitive advantage
for the first firm.

In this case the leader will be the firm with the lower cost, because
the point of indifference between leader and follower roles for the firm
with lower cost Y_{1}^{(1)} is lower than the point of
indifference for the firm with higher cost Y_{1}^{(2)}.
In this case, the firm with lower cost doesn't need to invest at Y_{1}^{(1)}
because the threat of the other firm entering the market is believable
only at the level Y_{1}^{(2)}.

The entry threshold Y* for the firm with lower cost will be some point in
the interval: Y_{1}^{(1)} < Y* < Y_{1}^{(2)}.

The chart below present this case of different costs and same market shares.

In the above chart the firms produce the same quantity q_{1} =
q_{2} = 10,000 units.

Let us allow both different market shares and different costs. The chart below shows this case.

The final question is to set the optimal quantities q_{1} and q_{2}
for the leader and follower.

For the **Stackelberg equilibrium**, the leader *commit* a production level q_{1} and the follower reacts according firm 2 reaction curve function q*_{2}(q_{1}).

There is a discussion if the Stackelberg equilibrium is the most likely outcome or if it is the **Cournot-Nash equilibrium**.

Stackelberg outcome is possible if we assume a *immutable capacity commitment* by the leader. However, if the game continues it is not Nash equilibrium because there exists an incentive for the leader to reduce its production in order to maximize profit. After this, the follower will react in the same way, until reaching the Cournot-Nash equilibrium (see the Appendix of the asymmetric duopoly webpage for a chart and additional issues).

Fudenberg & Tirole textbook (*Game Theory*, pp.74-76) points out that there is a problem of "**time consistency**" with the Stackelberg equilibrium, because the leader quantity in Stackelberg is not a best response for the follower production. In short, also in timing games with one firm entering before, the Cournot outcome is the most likely equilibrium.

So, let us assume that the most likely equilibrium is the Cournot-Nash outcome. The problem beconmes:

The follower maximize the profit given q_{1}; and

The leader maximize profit given q_{2} = f(q_{1}).

The follower profit is:

**p _{2} = (q_{2} (Y e^{-
e(q1 + q2)})/ d)
- I_{2} q_{2}**

By taking the partial derivative of the follower profit in relation to q_{2},
and equaling to zero in order to get the optimal production q_{2},
gives:

**(e ^{- e(q1 + q2)}
Y (1 - e q_{2})/d)
- I_{2} = 0**

The value of q_{2} is given for the above non-linear equation.
The solution can be found by an iterative process.

Note that the value of q_{2} is a function of the stochastic
factor Y.

Now is necessary to calculate the leader optimal quantity q_{1}.
This is done by looking the reaction curve q_{2} = f(q_{1}).

**p _{1} = (q_{1} Y e^{-
e(q1 + q2)})/ d)
- I_{1} q_{1}**

Where q_{2} is given for the non-linear equation presented
before, that is function of q_{1}.

The solution is to find the maximum profit p_{1}
by choosing q_{1}.

One solution method is: in the calculus of optimal, for each q_{1}
the value of q_{2} is calculated by the non-linear equation. With
these values is calculated a profit value p_{1}.
By calculating several profits for each q_{1}, is easy to choose
the higher one.

In this way are calculated the leader and follower values under
Cournot-Nash equilibrium for a given level of demand Y.

Things become more complicated if we invest in excess capacity in order to get the option to adjust the production output q_{1} to the oscillations in the demand level Y. This interesting complication is left for a future update.....

In this appendix I prove the expectation **E[e ^{-
r T*}] = (Y/Y_{2}) ^{b1}**
by following the appendix in the chapter 9 of Dixit & Pindyck
textbook, but including some intermediate steps not showed in that book.

To make more general the notation, let the threshold

Suppose an expected discount factor in continuous time, with a generic (exogenous) discount rate r:

**f(Y) = E[e ^{- r t}]**

Denoting the first hitting time as **T*** (first time that Y is
equal or larger than Y*), here representing when the option to invest will
be optimally exercised, the expected discounted payoff from T* to current
date is exactly the current value of the option to invest. Assuming that
the current Y < Y* and by choosing an interval dt sufficiently small
that hitting the threshold Y* in the next short time interval dt is an
unlike event, the problem restarts from a new level (Y + dY). Therefore we
have the dynamic programming-like recursion expression:

f (Y) = e^{- r dt}
E [ f (Y + dY) | Y] = e^{- r dt} {
f(Y) + E [ df(Y) ] }

By noting that:

(a) Y follows a geometric Brownian motion with drift a
and volatility s; and

(b) Using the Itô's Lemma for expanding df (Y), and using the
subscripts to denote derivatives we have:

df = f_{Y} (a Y dt +
s Y dz) + 0.5 f_{YY} (s^{2}
Y^{2} dt) = f_{Y} a Y dt + f_{Y}
s Y dz + 0.5 f_{YY}
s^{2} Y^{2} dt

Note that we are supposing the infinite time horizon (perpetual option)
case so that the variable time is not included in the Itô's Lemma.

By substituting df into the previous equation and by noting that E[dz] = 0,
and letting e^{- r dt} = 1
- rdt for a very small dt, we get:

f (Y) = (1 - rdt) { f + f_{Y}
a Y dt + 0.5 f_{YY}
s^{2} Y^{2} dt }

With a few algebra (remember dt^{2} is zero) the reader can
find out the following differential equation:

**0.5 s ^{2} Y^{2}
f_{YY} + a Y f_{Y}
- r f = 0 **

Where the subscripts denote derivatives. The general solution is:

f (Y) = A_{1} Y^{b1}
+ A_{2} Y^{b2}

Where b_{1} and b_{2}
are respectively the positive and the negative roots of the standard
quadratic characteristic equation from the differential equation (see an
instructive discussion of the characteristic equation in chapter 5,
section 2.A, of Dixit & Pindyck book).

Applying two boundary conditions: as Y approximates to the threshold Y*,
T* is probable to be small and the discount factor f(Y) close to 1, so
f(Y*) = 1. When Y is close to zero, T* is likely to be large and so the
discounted factor close to zero, therefore f(0) = 0 (note: alternatively,
I think is possible to see Y = 0 as an absorbing barrier, so when Y tends
to zero, T* tends to infinite or there is no finite time for Y to reach
the threshold from Y = 0).

With these results, we can see that A_{1} = (1/Y*)^{b1}
and A_{2} = 0.

Therefore the solution for the expected discount factor is:

Where Y is the initial value of the stochastic variable (that is at t =
0).

The parameter b_{1} is given by the
positive root of the fundamental quadratic equation (see Dixit &
Pindyck):

We want to prove the following result of the expectation below (defined by the function g(Y)), being T* the first time that the stochastic process (GBM) of Y reaches the threshold level Y*.

In the expectation of the integral, the main difference - when comparing with the first expectation - is that in the interval between 0 and dt there exists a dividend like profit p given by the value of the integral in this time interval:

Calculating the integral, we get:

Assuming that currently Y < Y* and by choosing an interval dt sufficiently small that hitting the threshold Y* in the next short time interval dt is an unlike event, the problem restarts from a new level (Y + dY). Therefore we have the dynamic programming-like recursion expression - this time including the profit or dividend-like term p(Y):

**g(Y) = p(Y) + e ^{-
r dt} E [ g (Y + dY) | Y]**

The term p(Y) is calculated above and the
term e^{- r dt} E [ g (Y + dY) | Y]
is calculated using the Itô's Lemma in the same way as calculated in
the first expectation (see Appendix A above). Hence,

g (Y) = Y dt + (1 - rdt) { g
+ g_{Y} a Y dt + 0.5 g_{YY}
s^{2} Y^{2} dt }

With a few algebra it is easy to get the ordinary differential equation (ODE) showed in the chapter 9 appendix from Dixit & Pindyck book:

**0.5 s ^{2} Y^{2}
g_{YY} + a Y g_{Y}
- r g + Y = 0 **

Where the subscripts denote derivatives. This ODE has a homogeneous part (equal to ODE from Appendix A) and a nonhomogeneous part (the last term of the left side in the equation above). The solution of this ODE is the sum of the homogeneous general solution with the particular solution:

g(Y) = B_{1} Y^{b1}
+ B_{2} Y^{b2} +
Y / (r - a)

As before, b_{1} and
b_{2} are respectively the positive
and the negative roots of the standard quadratic characteristic equation

The value of the constants are calculated by using the boundary
conditions for Y = 0 and for Y = Y*. Look the definition of g(Y) for the
following reasoning.

For g(0) = 0 is necessary that B_{2} = 0;

For g(Y*) = 0 (because T* = 0) we get B_{1} = -
(Y*)^{1 - b1} / (r
- a).

Substituting the constants into the ODE solution, we complete the proof: