Bertrand’s Paradox, its assumptions and models

Bertrand’s Paradox:

1) Assumptions:

* Each firm is assuming that the prices of other firms are constant in deciding the quantities.
* Subsequently in this model we can think about the motive to set a lower price than any other firm.
* Price-undercutting as well as finally up to most competitive level.

2) Model:

Unlike Cournot model, firms will determine the price. And quantity is decided in accordance with the price.

* Market Demand: P = a − Q or Q = a − P (a > 0)

Suppose there are two flash memory (USB) companies TDK and Maxell. Assuming that the USBs are entirely identical in every aspect, then consumers try to purchase a cheaper one.

If PT < PM , QT = a − PT = Q and QM = 0

If PT = PM , QT = 1/2 (a – PT) = QM

If PT > PM , QT = 0 and QM = a − PM = Q , where Q = QM + QT .

The unique (Nash) equilibrium is accomplished if PT = PM = c, and at this level

πT = πM = (c − c) ×1/2 (a − c) = 0.

* The rationale is as follows;

At PT = PM = c, no firm can set a price lower than c. If PT > c, then Maxell can dominate the market by setting PM < PT even slightly.

If PT > PM , πT = (PT − c) × 0 = 0

If PT = PM , πT = (PT − c) × 1/2 (a − PT)

If PT < PM , πT = (PT − c) × (a − PT)

* For example, if TDK sets PT = PM − ε (ε > 0 ), then its profit will be

(PM − ε − c)(a − PM +ε ). Simply, as ε approaches to zero, its profit will go to

(PM − c)(a − PM). If TDK sets PT = PM , profit is 1/2(PM − c)(a − PM) which is about half of (PM − ε − c)(a − PM + ε) .

Example: Suppose a = 2000 , c = 800 , PM = 1000 , and PT = 999, ε = 1 .

Maxell earns 0, and TDK earns (999 − 800) × (2000 − 999) = 199,199 .

If PT = PM = 1000, πT = 1/2(1000-800)(2000-1000) = 100000

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