Elasticity Using Calculus

Elasticity Using Calculus

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Definition : General

The elasticity of Y with respect to X is defined as : the percentage by which Y changes, when X changes by 1 percent. In this definition, X and Y do not need to be economic variables. Elasticity just describes the relation between the proportional change in the dependent variable (Y ) in response to a proportional change in the independentvariable (X). If X is the quantity (in kilogrammes) of sheep manure a person spreads in his garden, and Y is the quantity (in kilogrammes) of tomatoes the garden yields, then the elasticity of Y with respect to X describes the percentage change in tomato yield in response to a given percentage increase in the quantity used of sheep manure. In general, the formula for the elasticity of Y with respectto X is dY X (1) dX Y dY where Y X is short for “the elasticity of Y with respect to X”, and dX is the derivative of Y with respect to X. One nice feature of elasticity is that the value of the elasticity does not depend on the units being used to measure the variable. That is, if we calculated the elasiticity of demand for shirts with respect to a person’s income, we would get the same answerwhether we measured income in Canadian dollars or in euros.
YX

≡

Warning : The elasticity defined by the general formula (1) might be negative, and it might be positive, depending on whether dY /dX was negative or positive. It depends on whether increases in X tend to increase Y , or decrease it.

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When we look at the most important specific example of an elasticity — the own–priceelasticity of demand — we just drop the negative sign, because we already know the direction of the effect : we know increases in price lead to decreases in quantity demanded, so we just take the absolute value of the elasticity of quantity demanded with respect to the price.

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Definition : Own–Price Elasticity of Demand

Start with a demand curve for some good. If p is the price of the good,then the quantity demanded might be written Q = D(p) as a function of the price. Warning : Notice that here the quantity demanded of the good is being written as a function of the price. That is how we think consumers behave : they find out the price of a good, and then decide what quantity they wish to buy. But (following the old tradition in economics), the price is written on the vertical axis, andthe quantity on the horizontal. So the derivative D (p) of this demand function is not the slope of the demand curve. It’s one over the slope of the demand curve, since price is on the vertical, quantity is on the horizontal, and quantity is the dependent variable. The derivative of quantity demanded with respect to the good’s price will be written dQ/dp ≡ D (p). [The derivative is not theelasticity : but we can use the derivative to calculate the elasticity.] If D(p) is the quantity demanded by consumers when the price is p, then the own–price elasticity of demand is defined by η ≡ −D (p) p D(p) (2)

(The Greek letter η (“eta”) is often used in economics to represent an elasticity.) So the own–price elasticity of demand is the derivative of quantity demanded with respect to the price —times the price, and divided by the quantity, and with a minus sign stuck in front of it. The minus sign ensures that the own–price elasticity comes out as a positive number : since D (p) < 0, therefore −D (p) > 0. 2

If η > 1, then demand is elastic ; if 0 < η < 1, then demand is inelastic ; and if η = 1 then demand is unit elastic.

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Straight–Line Demand Curves
D(p) = a − bp

If thedemand curve were a straight line, then its equation would be (3)

where a and b are positive numbers. In this case, the rules of calculus say that D (p) = −b, so that equation (2) implies that η=b p a − bp (4)

Equation (4) shows that the own–price elasticity of demand is not constant if the demand curve is a straight line. In expression (4) b is a constant. p p But a−bp is not : as the...