elasticity of substitutionrevised.tex

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

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Lecture Notes on the Elasticity of Substitution Ted Bergstrom UCSB Econ 210A 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 differen- tiable 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 . There is yet another way to think of elasticities. 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 example, 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 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 reasons, engineers measure the stretchability of a material by an “elasticity” of the length of the material with respect to the force exerted on it. Suppose that we were to draw the graph of the function f on a log-log paper. That is, let us define z = ln x , so that x = e z . Let us also define g ( z ) = ln f ( e z ). Then applying the chain rule, g 0 ( z ) = e z f 0 ( e z ) f ( e z ) = xf 0 ( x ) f ( x ) = η ( x ) . (3) Since g ( z ) = ln f ( z ) and z = ln x , we can also express the result of equation 2 as η ( x ) = d ln f ( x ) d ln x . (4) 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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In general, the elasticity of f with respect to x depends on the value of x . For example if f ( x ) = a - bx , then η ( x ) = - bx a - bx . In this case, as x ranges from 0 to a/b , η ( x ) ranges from 0 to -∞ . 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 3 with respect to z , we have g ( z ) = ηz + b for some constants a and b . Since g ( z ) = ln f ( z ) and z = ln x , this equation is equivalent to ln f ( x ) = η ln x + b. (5) Exponentiating both sides of Equation 5, we have f ( x ) = cx η (6) where c = e b . Thus we see that f has constant elasticity η if and only if f is a “power function” of the form 6.
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