To illustrate the economic application of differentials, let us consider the notion of the elasticity of a function. Given a demand function Q = f{P), for instance, its elasticity is defined as (AQfQ)f(APfP). Using the idea of approximation explained in Fig. 8.1, we can replace the independent change A P and the dependent change A Q with the differentials dP and dQ, respectively, to get an approximation elasticity measure known as the point elasticity of demand and denoted by £(i (the Greek letter epsilon, for ^elasticity");*
dQ/Q_JQ/dP
Observe that on the extreme right of the expression we have rearranged the differentials dQ and dP into a ratio dQfdP> which can be construed as the derivative, or the marginal function, of the demand function Q = f{P). Since we can interpret similarly the ratio QfP in the denominator as the average function of the demand function, the point elasticity of demand ej in (8.5) is seen to be the ratio of the marginal function to the average function of the demand function.
TThe point-elasticity measure can alternatively be interpreted as the limit of - as
AP ^ 0, which gives the same result as (8.5). A ' ^
Indeed, this last-described relationship is valid riot only for the demand function but also for any other function, because for any given total function y = f(x) we can write the formula for the point elasticity of>' with respect to x as dy/dx marginal function eyx =
y/x average function
As a matter of convention, the absolute value of the elasticity measure is used in deciding whether the function is elastic at a particular point. In the case of a demand function, for instance, we stipulate:
The demand is elastic of unit elasticity inelastic at a point when \ej\ ^ 1
Example 1
Example 2
Find ej if the demand function is Q = 100 - IP. The marginal function and the average function of the given demand are
so their ratio will give us
As written, the elasticity is shown as a function of P. As soon as a specific price is chosen, however, the point elasticity will be determinate in magnitude. When P = 25, for instance, we have e^ = -1, or = 1, so that the demand elasticity is unitary at that point. When P = 30, in contrast, we have = 1.5; hence, demand is elastic at that price. More generally, it may be verified that we have \ed\ > 1 for 25 < P < 50 and < 1 for 0 < P < 25 in the present example, (Can a price P > 50 be considered meaningful here?)
Find the point elasticity of supply zs from the supply function Q- P2 + IP, and determine whether the supply is elastic at P -2. Since the marginal and average functions are, respectively, dQ dP
their ratio gives us the elasticity of supply and j = P + 7
When P = 2, this elasticity has the value 11 /9 > 1; thus the supply is elastic at P = 2,
At the risk of digressing a trifle, it may also be added here that the interpretation of the ratio of two differentials as a derivative—and the consequent transformation of the elasticity formula of a function into a ratio of its marginal to its average—makes possible a quick way of determining the point elasticity graphically. The two diagrams in Fig. 8.2 illustrate the cases, respectively, of a negatively sloped curve and a positively sloped curve. In each case, the value of the marginal function at point A on the curve» or at x — in the domain, is measured by the slope of the tangent line AB. The value of the average function, on the
Chapter4 8 Comparative-Srafic A tw lysis of General-Function Models 183
FIGURE 8.2
Chapter4 8 Comparative-Srafic A tw lysis of General-Function Models 183
FIGURE 8.2
FIGURE 8.3
FIGURE 8.3
other hand, is in each case measured by the slope of line OA (the line joining the point of origin with the given points on the curve, like a radius vector), because at point A we have v = xo A and x — Qxo> so that the average is y/x = x^AjQxo = slope of OA. The elasticity at point A can thus be readily ascertained by comparing the numerical values of the two slopes involved: if AB is steeper than OA, the function is elastic at point A; in the opposite case, it is inelastic at Accordingly, the function pictured in Fig. &.2a is inelastic atyl (or at:* = xo), whereas the one in Fig. 8.2h is elastic at A,
Moreover, the two slopes under comparison are directly dependent on the respective sizes of the two angles 9m and 0a (Greek letter theta; the subscripts m and a indicate marginal and average, respectively). Thus we may, alternatively compare these two angles instead of the two corresponding slopes. Referring to Fig. 8.2 again, you can see that 0m < B(t at points in diagram a, indicating that the marginal falls short of the average in numerical value; thus the function is inelastic at point A. The exact opposite is true in Fig. %2h.
Sometimes, we are interested in locating a point of unitary elasticity on a given curve. This can now be done easily. If the curve is negatively sloped, as in Fig. 8.3a, we should find a point C such that the line OC and the tangent BC will make the same-sized angle with the x axis, though in the opposite direction. In the case of a positively sloped curve, as in Fig. 8.36, one has only to find a point C such that the tangent line at C when properly extended, passes through the point of origin.
We must warn you that the graphical method just described is based on the assumption (hat the function y = / (.v) is plotted with the dependent variable y on the vertical axis. In particular, in applying the method to a demand curve, we should make sure that Q is on the vertical axis, (Now suppose that Q is actually plotted on the horizontal axis. Mow should our method of reading the point elasticity be modified?)
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