1. Introduction

Bertrand's model is one of the classical models of differentiated duopoly and is a forerunner of the Nash equilibrium. In this model demand for each firm increases in the competitor's price while decreasing in its own price [1-3]. In this classical model production costs are usually assumed to be linear. As such is often not the case, (e.g. [4, 5]), we explore duopoly and tri-opoly [6] scenarios with increasing marginal production costs. We also consider a two-period model where a firm's demand in the second period depends on its and its competitors’ price in the first period ("reference price"), as well on the second period prices [7, 8]. While such consumer behavior may not be entirely rational, there is ample empirical evidence in the marketing and psychology literature for its prevalence.

For completeness, we first sketch the well-known Bertrand's linear duopoly model of differentiated products [2, 9] and then do so for a tri-opoly (cf. [6, 10]) and we include comparative statics. We show conditions under which a tri-opolist would charge a higher price than a duopolist. We then generalize them to non-linear production costs, where we focus on the case of quadratic costs for which we obtain an explicit solution. Finally, we consider a two-period duopoly with reference prices.

2. The Basic Model

The basic Bertrand duopoly model of differentiated products is

where and where and are, respectively, the prices set by duopolists i and j, and the resulting demand of duopolist i (e.g. [2, 9]).

is the cross effect of on .

is the quantity demanded when prices are zero. Thus this quantity is between and . are assumed to be identical for both duopolists, so the model is symmetric.

Now, is positive if and or if and .

Note that profit is

where is the unit production cost, and .

, so is concave in .

Here the Nash-equilibrium is the solution of

The symmetric solution is thus , which increases in and and decreases in .

(Note that since for a monopolist with this parametrization , therefore

, which increases in , decreases in and increases in if

, which increases in and decreases in and ; all are intuitive.

For example, if

and

3. Three Firms

Here the assumption is that

so the effect of other products’ prices on the demand of a product is symmetric (cf. [6]). So

Denoting by , we have

The optimality condition implies that

So the symmetric solution is

and thus

which increases in , .

which increases in and decreases in

Here are some relevant comparative statics.

4. Non-Linear Costs Duopoly

Here are the possibly, non-linear costs of duopolist producing units, with and

thus

To obtain an explicit solution we need a specific (family of) function(s) C.

Example

so

Thus

so

i.e.,

For the symmetric solution

so we have:

Proposition 1

For quadratic costs

, which increases in and . if or .

Under this condition

, which increases in and decreases in . It decreases in if . Under this condition

i.e.,

It is possible to demonstrate that if then .

5. With Reference Price(s)

See Güler et al. [8] and Popescu and Wu [7] for suggestions for incorporating reference prices in supply chain models, motivated by Marketing and Psychological studies (e.g. [12, 13] that found it to be prevalent in (consumer) behavior.

Let be the prices and quantities in period 1, and similarly for period 2.

2^nd period:

The demand model for the second period^(*) is

where is the effect of own price change, and , the effect of competitor's price change. We shall assume that .

Note that if and the contribution of the first reference-price difference is negative, and of the second positive. If and , the signs of these contributions reverse, but the expressions remain the same.

Also, [noise has no effect]. Thus

Proposition 2

In the symmetric case

. Note that the denominator is positive since and .

As for comparative statics, is linear in . It is increasing in v and c.

Or, in different form, if

increases in and decreases in .

Now, moving to the first period,

and

As the denominator is independent of , and positive, we shall focus on the numerator.

It can be shown that

Thus

where means “has the same sign as”.

So

sufficient conditions for are . These conditions hold with some restrictions on the parameters.

6. Concluding Remarks

We generalized the basic Bertrand model of differentiated duopoly in three ways: 1. Extension from two firms to three (tri-opoly). 2. Extension from linear to non-linear (convex) production costs (with two and three firms). 3. Extension to a two-period model where demand in the second depends also on the price in the first (reference price). Note that the term "reference effects" is used in some literature when the price of a product influences one's attitude to the price of another (e.g., [14]) which is a totally different meaning than our temporal "reference effect".

The tri-opolists are shown to charge, under certain conditions, higher price than the duopolists (who charges a higher price than a monopolist). For non-linear costs, we used a quadratic function. We provide complete comparative statics.

Our reference price model assumes that demand in the second period depends on one's own price difference and competitor's price difference between periods one and two. We find the optimal price and profit in the second period (as a function of the price in the first), then add to it the profit in the first period (which is also a function of the price in that period) and then optimize the sum over that price.

One possible extensions is to embed non-linear production costs in the reference price model.

One other possible generalization of the Bertrand model and its extensions is to assume a non-linear demand model, like

(e.g. [15] and references therein).

Another direction is the extension of the reference price model to more than two periods (for a continuous time model where demand always also depends on the price at time zero, see Fibich et al. [16]).

Note

^(*) For the first period, we still have as in the basic model.