The second most commonly specified demand relationship is the multiplicative form, which gives constant elasticity. The form is found in power functions.
An example is:
Q = aPbAcYd
where P = price, A advertising, and Y = income.
Differentiating equation (1) with respect to price (P), we obtain
Therefore, e p = abPb-1AcYd .(P/Q)
Substituting (1) for Q in (3) gives:
e p = abPb-1AcYd .(P/ aPbAcYd)
Combining terms and canceling where possible in equation (4), we obtain
Thus, the price elasticity of demand is equal to b(whose usual sign is negative), the exponent of the price variable in the multiplicative demand function given as in equation (1). Therefore, the elasticity is not a function of the price/quantity (P/Q) ratio and hence is constant. In a similar fashion, we can prove that (1) the income elasticity is c, and (2) the advertising elasticity is d.
The power function form can be transformed into a log-linear form using logarithm and then estimated by the least-squares method. Equation (1) is equivalent to
log Q = log a + b log P + c log A + d log Y
The property of constant elasticity is useful, since it means that changes in one of the independent variables, such as income, will result in proportionate changes in quantity demanded. Note that the elasticity of a linear function changes over the entire range of the demand curve.
Given the demand function in multiplicative form:
Q = 2077P-.144PX.097A.314
The price elasticity is always -.144, which means that a 1% reduction (increase) in price leads to .14% increase (decrease) in demand.
OPTIMAL PRICING POLICY AND PRICE ELASTICITY
We note, without a mathematical derivation, the relationship between the point price elasticity of demand, marginal revenue, and price:
(Note that ep is a negative number.)
This follows directly from the mathematical definition of marginal revenue (MR).
This formula is useful in setting a firm`s pricing policy. From the profit-maximizing condition MR = MC, we can derive the formula for determining the profit-maximizing price level, which is shown below.
MC = MR
Solving for the optimal or profit-maximizing P*, yields
Suppose the manager of a toy store notes a 2.5% increase in weekly sales following a 1% price discount on the Bamboo doll. The store`s wholesale cost per doll plus display and marketing expenses total $30 per unit. The price elasticity of demand is:
The profit-maximizing price is then:
P = $30/(1 + 1/-2.5) = $50
Suppose the manager can reduce, through quantity buying, marginal costs per unit by $3 to $27. Then the new optimal price is:
P = $27/(1 + 1/-2.5) = $45
Thus, the profit-maximizing price would fall by $5 following a $3 reduction in the store`s marginal costs.