Machina Math Handout

So we get that the elasticity of y with respect to x

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Unformatted text preview: respect to x is given by: ( Δy y ) ( Δx x ) ⎛ Δy ⎞ ⎛ x ⎞ ⎛ dy ⎞ ⎛ x ⎞ ⎛x⎞ =⎜ ⎟ ⋅ ⎜ ⎟ ≈ ⎜ ⎟ ⋅ ⎜ ⎟ = f ′( x ) ⋅ ⎜ ⎟ ⎝ Δx ⎠ ⎝ y ⎠ ⎝ dx ⎠ ⎝ y ⎠ ⎝ y⎠ Note that if ƒ(x) is an increasing function the elasticity will be positive, and if ƒ(x) is a decreasing function, it will be negative. E y ,x = Recall that since the percentage change in a variable is simply 100 times its proportional change, elasticity is also as the ratio of the percentage change in y to the percentage change in x: E y ,x = (Δy y ) ( Δx x ) = ( y) 100 ⋅ Δy ( 100 ⋅ Δx x ) = % change in y % change in x A useful intuitive interpretation: Since we can rearrange the above equation as % change in y = Ey, x ⋅ % change in x we see that Ey, x serves as the “conversion factor” between the percentage change in x and the percentage change in y. Although elasticity and slope are both measures of how responsive y is to changes in x, they are different measures. In other words, elasticity is not the same as slope. For example, if y = 7⋅x, the slope of this curve is obviously 7, but its elasticity is 1: E y ,x = dy x ⋅ dx y = 7⋅ x 7x ≡1 That is, if y is exactly proportional to x, the elasticity of y with respect to x will always be one, regardless of the coefficient of proportionality. Econ 100A 3 Mathematical Handout Constant Slope Functions versus Constant Elasticity Functions Another way to see that slope and elasticity are different measures is to consider the simple function ƒ(x) = 3 + 4x. Although ƒ(⋅) has a constant slope, it does not have a constant elasticity: Eƒ( ⋅),x = d ƒ( x ) x ⋅ dx ƒ( x ) = 4⋅ x 3 + 4x = 4x 3 + 4x which is obviously not constant as x changes. However, some functions do have a constant elasticity for all values of x, namely functions of the form ƒ(x) ≡ c⋅ xβ, for any constants c > 0 and β 0. Since it involves taking x to a fixed power β, this function can be called a power function. Deriving its elasticity gives: Eƒ( x ), x d ƒ(x ) x = ⋅ dx ƒ( x ) = β ⋅ c ⋅ x ( β −1) ⋅ x c ⋅ xβ ≡β x Conversely, if a function ƒ(⋅) has a constant elasticity, it must be a power function. In summary: ƒ( x ) ≡ α + β ⋅ x ⇔ d ƒ (x ) ≡β dx x ƒ( x ) ≡ c ⋅ x β linear functions all have a constant slope: ⇔ Eƒ( x ), x ≡ β x power functions all have a constant elasticity: x x C. LEVEL CURVES OF FUNCTIONS If ƒ(x1, x2) is a function of the two variables x1 and x2, a level curve of ƒ(x1, x2) is just a locus of points in the (x1, x2) plane along which ƒ(x1, x2) takes on some constant value, say the value k. The equation of this level curve is therefore simply ƒ(x1, x2) = k, where we may or may not want to solve for x2. For example, the level curves of a consumer’s utility function are just his or her indifference curves (defined by the equation U(x1, x2) = u0), and the level curves of a firm’s production function are just the isoqua...
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This document was uploaded on 09/18/2013.

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