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elasticity of substitutionrevised.tex

elasticity of substitutionrevised.tex - Lecture Notes on...

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Lecture Notes on Elasticity of Substitution Ted Bergstrom, UCSB Economics 210A March 3, 2011 Today’s featured guest is “the elasticity of substitution.” Elasticity of a function of a single variable Before we meet this guest, let us spend a bit of time with a slightly simpler notion, the elasticity of a a function of a single variable. Where f is a differentiable real-valued function of a single variable, we define the elasticity of f ( x ) with respect to x (at the point x ) to be η ( x ) = xf 0 ( x ) f ( x ) . (1) Another way of writing the same expression 1 is η ( x ) = x df ( x ) dx f ( x ) = df ( x ) f ( x ) dx x . (2) From Expression 2, we see that the elasticity of of f ( x ) with respect to x is the ratio of the percent change in f ( x ) to the corresponding percent change in x . Measuring the responsiveness of a dependent variable to an independent variable in percentage terms rather than simply as the derivative of the func- tion has the attractive feature that this measure is invariant to the units in which the independent and the dependent variable are measured. For exam- ple, economists typically express responsiveness of demand for a good to its price by an elasticity. 1 In this case, the percentage change in quantity is the 1 Some economists find it tiresome to talk about negative elasticities and choose to define the price-elasticity as the absolute value of the percentage responsiveness of quantity to price. 1
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same whether quantity is measured in tons or in ounces and the percentage change in price is the same whether price is measured in dollars, Euros, or farthings. Thus the price elasticity is a “unit-free” measure. For similar rea- sons, engineers measure the stretchability of a material by an “elasticity” of the length of the material with respect to the force exerted on it. The elasticity of the function f at a point of x can also be thought of as the slope of a graph that plots ln x on the horizontal axis and ln f ( x ) on the vertical axis. That is, suppose that we make the change of variables u = ln x and v = ln y and we rewrite the equation y = f ( x ) as e v = f ( e u ). Taking derivatives of both sides of this equation with respect to u and applying the chain rule, we have e v dv du = e u f 0 ( e u ) (3) and hence dv du = e u f 0 ( e u ) e v = xf 0 ( x ) f ( x ) = η ( x ) , (4) where the second equality in Expression 4 is true because e u = x and e v = f ( x ). Thus dv du is the derivative of ln f ( x ) with respect to ln x . We sometimes express this by saying that η ( x ) = d ln f ( x ) d ln x . (5) It is interesting to consider the special case where the elasticity of f ( x ) with respect to x is a constant, η that does not dependent on x . In this case, integrating both sides of Equation 5, we have ln f ( x ) = η ln x + a (6) for some constant a . Exponentiating both sides of Equation 6, we have f ( x ) = cx η (7) where c = e a . Thus we see that f has constant elasticity η if and only if f is a “power function” of the form 7.
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