purchasers make for the commodity. Such a demand may therefore be called a constant outlay demand. The curve which represents it, or constant outlay curve, as it may be called, is a rectangular hyperbola with Ox and Oy as asymptotes." This passage, it appears to me, contains a definite mathematical error, due to a failure to distinguish between point elasticity and arc elasticity For a constant outlay curve, the point elasticity of demand is always equal to one, but the arc elasticity is never equal to one. The arc elasticity for such a curve is always greater than one for a fall in price and an increase in demand, and is always less than one for a rise in price and a decrease in demand; and the arc elasticity differs more greatly from one in either direction, the greater is the length of the arc. Again, for a constant outlay curve, it is uot true that “any (finite) fall in price will cause a proportionate increase in the amount bought." This is only true where the arc elasticity is equal to one, which, as we have seen, it can never be for a constant outlay curve. §3. The argument may be made clear by a simple numerical illustration, followed by a mathematical proof. Suppose a constant outlay curve, such that, whatever the price of a certain commodity, £1,000 worth of it is demanded per year in a given market. Then if the price per unit is £10, the amount demanded is 100 units. The amount demanded will fall 10%, to 90 units, if the price rises to £11, a rise of 11%. Here the arc elasticity of demand If, on the other hand, the price = ΙΟ 9 falls 50%, to £5, the amount demanded will rise 100% to 200 units, and arc elasticity of demand = ΙΟΟ 50 = 2. In neither case does a given percentage movement in price cause an equal percentage movement in the amount demanded. More generally, let x units be demanded at a price y per unit, and (x + h) units at a price (y + k) per unit, either hor k being negative. Then the arc elasticity of demand = hy (When h and k are very small, we have the special case of point elasticity ydx = xdy In the case of a constant outlay curve, xy is constant. is always greater than one when h is positive, that is to say when price falls, and less than one when h is negative, that is to say when price rises. It should also be noticed that for any demand curve the arc elasticity for a given arc is different according to which end of the arc is taken as the base, from which elasticity is measured. If the number of units demanded changes from x to (x + h), when the price changes from y to (y+k), the arc elasticity, measured from the base while measured from the base (x + h, (x, y) is hy y+k) it is kx h (y + k) k (x + h) But the difference between these two elasticities will not generally be large. §4. The distinction between point elasticity and arc elasticity is of no practical importance in qualitative statements, e.g., that a certain result is the more likely, the greater some elasticity of demand. But it is of practical importance in quantitative statements, e.g., that a certain result will follow, if some elasticity of demand is greater than a certain definite amount. An illustration is provided by Professor Pigou's formula, noted in the previous chapter, to the effect that an increase in the supply of a factor of production, its demand curve remaining unchanged, will increase its absolute share of the product, provided that its elasticity of demand is greater than one. Elasticity here is point elasticity, and the formula is strictly correct only when the proportionate increase in supply is so small as to be negligible. But this is obviously a case of no practical interest. When, as will be the case in reality, the increase of supply is of moderate amount, the relevant elasticity will be arc elasticity, and, as argued in the preceding chapter and in §3 of this Note, the formula should be amended, so as to read that the absolute share of the factor, whose supply increases, will be increased, provided that its elasticity of demand is greater than one plus the proportionate increase in supply. Similarly, when the supply of a factor decreases, its absolute share increases, provided that its elasticity of demand is less than one minus the proportionate decrease in supply. §5. Marshall's geometrical construction for point elasticity of demand is easily modified for arc elasticity of demand as follows M Let any straight line tT cut a demand curve DD' in two points P and P'. Draw PM, P'M' perpendicular to OX and P'Q perpendicular to PM. Then, for the arc PP', the elasticity of demand (with base P) P'Q PM OM PQ The argument of this Note applies to elasticity of supply no less than to elasticity of demand. CHAPTER VI THE DIVISION OF INCOME BETWEEN WORKERS AND PROPERTY OWNERS. §1. The argument of the last chapter will now be applied to the broad problem of the division of the total product, or income, between workers on the one hand and property owners on the other.1 It will be necessary, to begin with, to think in terms of a number of homogeneous units of work on the one side, and of a number of homogeneous units of property on the other. Then changes in the division of income between workers on the one hand and property owners on the other depend upon the elasticities of demand and supply for work and for property, and upon the extent to which work and property are complementary or rival factors of production. In a given state of knowledge and of consumers' demand, it seems clear that work and property are on the whole complementary and not rival. Whatever may happen to particular sub-groups within each category, there seems no reasonable doubt that in all modern communities an increase in the total supply of property increases the total demand for work, and that an increase in the total supply of work increases the total demand for property.2 Again, modern developments tend to increase the 1 Compare with the argument of this chapter Cannan, Economic Outlook, pp. 232-238, and Wealth, pp. 173-181. Compare Marshall, Principles, pp. 522-4 and 540-4, and Pigou, Wealth and Weljare, p. 84. |