Monopoly Adjustments to Shifts in Demand - Duncan, Acheson J. (Econometrica, 1942)
A PROBLEM COMMONLY TREATED in monopoly theory is the effect of a shift in demand on monopoly price and output. In most instances attention centers upon the positive or negative character of the shift, and on occasion the accompanying change in elasticity is considered. The direction of shift in demand is usually ignored, however. This would appear to be a serious omission. The following analysis indicates that the direction of shift in demand may have significant bearing on the results and that it is advisable in all cases to give it explicit consideration. 1 That the addition of a constant increment to the quantities that will be taken at various prices (a horizontal shift in demand) is not in general the mathematical equivalent of the addition of a constant increment to the prices that will be paid for various quantities (a vertical shift in demand) is readily demonstrated. Geometrically one need only shift a demand curve to the right by a constant amount at all levels and then shift the same curve upward by a constant amount for all abscissa points to note that the two new demand curves are not the same. Algebraically it may be shown as follows: Let x = F(p) be the equation of the demand curve in which the quantity is expressed as a function of the price and let p =f(x) be the inverse relationship. Then if a small constant amount is added to the quantity that will be taken at any price, the first equation becomes x' = F(p) +Ax and if Ax is small, the inverse becomes approximately p'=f(x) -f'(x)Ax. Hence the amount that would have to be added to the price at each quantity level to yield the practical equivalent of adding a constant amount to the quantity that will be taken at each price depends upon the slope of the demand curve f'(x) at each point. A linear (constant slope) demand curve is thus the only instance in which the addition of a constant amount horizontally is the equivalent of adding a constant amount vertically. The frequent use of linear demand curves in graphic analysis is possibly one reason why the direction of shift in demand has not been given more consideration.