Machina Math Handout

A strange and wonderful result if we had

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Unformatted text preview: wonderful result: if we had differentiated in the opposite order, that is, first with respect to x2 and then with respect to x1, we would have gotten the same result. In other words, we have ƒ12(x1, x2) ≡ ƒ21(x1, x2) or equivalently ∂2ƒ(x1, x2)/∂x1∂x2 ≡ ∂2ƒ(x1, x2)/∂x2∂x1. Approximation Formulas for Small Changes in Functions (Total Differentials) If ƒ(x) is differentiable, we can approximate the effect of a small change in x by: Δƒ = ƒ(x+Δx) – ƒ(x) ≈ ƒ (x)⋅Δx where Δx is the change in x. From calculus, we know that as Δx becomes smaller and smaller, this approximation becomes extremely good. We sometimes write this general idea more formally by expressing the total differential of ƒ(x), namely: dƒ = ƒ (x)⋅dx but it is still just shorthand for saying “We can approximate the change in ƒ(x) by the formula Δƒ ≈ ƒ (x)⋅Δx, and this approximation becomes extremely good for very small values of Δx.” When ƒ(⋅) is a “function of a function,” i.e., it takes the form ƒ(x) ≡ g(h(x)), the chain rule lets us write the above approximation formula and above total differential formula as Δ g ( h( x )) ≈ dg ( h( x ))) ⋅Δx = g ′(h( x )) ⋅ h ′( x ) ⋅Δx dx so dg ( h( x )) = g ′( h( x )) ⋅ h ′( x ) ⋅ dx For a function ƒ(x1,..., xn) that depends upon several variables, the approximation formula is: ∂ ƒ( x1 ,..., xn ) ∂ ƒ( x1 ,..., xn ) ⋅ Δx1 + ... + ⋅ Δ xn ∂x1 ∂xn Once again, this approximation formula becomes extremely good for very small values of Δx1,…,Δxn. As before, we sometimes write this idea more formally (and succinctly) by expressing the total differential of ƒ(x), namely: Δ ƒ = ƒ( x1 +Δx1 ,..., xn +Δxn ) − ƒ( x1 ,..., xn ) = dƒ = ∂ ƒ( x1 ,..., xn ) ∂ ƒ( x1 ,..., xn ) ⋅ dx1 + ... + ⋅ dxn ∂x1 ∂xn or in equivalent notation: dƒ = ƒ1(x1,..., xn)⋅dx1 + ⋅⋅⋅ + ƒn(x1,..., xn)⋅dxn Econ 100A 2 Mathematical Handout B. ELASTICITY Let the variable y depend upon the variable x according to some function, i.e.: y = ƒ(x) How responsive is y to changes in x? One measure of responsiveness would be to plot the function ƒ(⋅) and look at its slope. If we did this, our measure of responsiveness would be: absolute change in y absolute change in x slope of ƒ(x ) = Δy Δx = dy dx ≈ = ƒ′( x ) Elasticity is a different measure of responsiveness than slope. Rather than looking at the ratio of the absolute change in y to the absolute change in x, elasticity is a measure of the proportionate (or percentage) change in y to the proportionate (or percentage) change in x. Formally, if y = ƒ(x), then the elasticity of y with respect to x, written Ey, x , is given by: E y ,x = proportionate change in y proportionate change in x = ( Δy y ) ( Δx x ) ⎛ Δy ⎞ ⎛ x ⎞ =⎜ ⎟⋅ ⎝ Δx ⎠ ⎜ y ⎟ ⎝⎠ If we consider very small changes in x (and hence in y), Δy/Δx becomes dy/dx = ƒ (x), so we get that the elasticity of y with...
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