# How to Calculate Bertrand Equilibrium

by Jennifer Fleming; Updated September 26, 2017Price elasticity generally assesses how well consumers respond to changes in price; price duplicity between two competitors demonstrates the concept of zero profit. Vigorous competition amongst two competitors in an unrestricted open market can lead to a folklore known as zero profit theory or Bertrand equilibrium. According to economic analysts and authors Michael Baye and John Morgan, the concept dates back to the 19th century and was founded by Joseph Bertrand who proposed that in a market where two proprietors are repeatedly undercutting each other’s prices to attract buyers there is no reasonable hope for equilibrium in which case both lose to zero profit.

Identify the formula. Bertrand equilibrium is best demonstrated in an example, as the atomic probability distribution is lengthy and unwieldy. Consider the components of demand, profit and quantity.

Determine the components. Given a situation where the demand curve is Q equal 6 minus price, where P is the lower of the two prices. If proprietor A charges $2 and B charges $1, then all sales go to B. If they charge the same price, then they split the market demand (6 - P/2). Assuming the cost of production is 0, and each proprietor can only price in whole dollars $0 through $6, compute for zero profit slope.

Compute the zero profit. Invert the demand curve to P equals 6 - Q, then multiply by quantity to get total revenue or profit since the cost per unit produced is zero. TR equals PQ equals 6Q minus Q squared. Maximum profit will occur where the profit function has a slope of zero, such as DTR divided by dQ equals 6 - 2Q equals 0, thus the quantity at which the slope is zero is when Q equals 3 and P equals 3. If each proprietor repeatedly undercuts his opponent in an open market when the competitive pricing descends to any value below $3 dollars and the activity persists this means they have reached Bertrand equilibrium and battle is meaningless to continue.

#### References

- Temple University; "Bertrand Price Competition and Iterative Elimination of Dominated Strategies"; Michael Baye and John Morgan
- Journal of Economic Theory; Duopoly Information Equilibrium:Cournot and Bertrand; JET; August 1983

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