DIFFERENTIATION. MICROECONOMICS Principles and Analysis Frank Cowell. July 2017 Frank Cowell: Differentiation

Transcription

1 DIFFERENTIATION MICROECONOMICS Principles and Analysis Frank Cowell 1

2 Overview... Differentiation Basics Basic definitions Chain rule Elasticities l Hôpital s rule 2

3 Definition (1) Take the univariate case first Let f be a function from R to R as usual R denotes the set of real numbers the real line Let x and x be real numbers The derivative of f at x is where the limit exists This simple definition is often useful in practice 3

4 Examples Take an example where the limit is easy to evaluate Let f(x) = x 2 So f(x + x) = x 2 + 2x x + [ x] 2 Therefore [f(x + x) f(x)] / x = 2x + x Take the limit of this as x 0 clearly we have df(x)/dx = 2x Some other examples 4

5 Definition (2) Now the multivariate case Let f be a function from R nn to R: y = f(x 1, x 2,, x n ) = f(x) a variation in the ith component: f(x 1, x 2,, x i 1, x i + x i, x i+1,, x n ) The derivative of f with respect to x i is If all the components of x are allowed to vary then we find the total variation in y thus: if f is differentiable in each of its n arguments the total differential is 5

6 Overview... Differentiation Basics Differentiation involving a function of a function Chain rule Elasticities l Hôpital s rule 6

7 Chain rule (1) Also known as function-of-a-function rule when one differentiable function has as its argument something that is itself a differentiable function of something else how to carry out differentiation in this composition arrangement? Let f and ϕ be functions from R to R and, for some x R : y = f(x), z = ϕ(y) y = f(x+ x) f(x), z = ϕ(y+ y) ϕ(y) Clearly z can be written as a function g of x such that z = g(x) = ϕ(f(x)) the function g is a composition of f and ϕ We also have z = g(x+ x) g(x) = ϕ(f(x+ x)) ϕ(f(x)) How do we proceed to the differentials? 7

8 Chain rule (2) For x 0 and y 0 we could write z/ x = ( z/ y) ( y/ x) follows from simple rearrangement of expressions on previous page Now assume that f and ϕ are differentiable as x 0, we have y 0 and [f(x+ x) f(x)]/ x becomes f '(x), [g(x+ x) g(x)]/ x becomes g'(x) as y 0, [ϕ(y+ y) ϕ(y)]/ y becomes ϕ '(y) Drawing these results together in the limit we have the differential of g(x) = ϕ(f(x)): g'(x) = ϕ'(y) f '(x) This chain rule can be extended indefinitely to the function of a function of a function of a 8

9 Chain rule (3) Extend function-of-a-function approach to multivariate case now f is a function from R nn to R ϕ is (again) a function from R to R y = f(x), z = ϕ(y) so that z = g(x) = ϕ(f(x)) Partial derivative of g with respect to x i is: Now suppose ϕ is from R mm to R so that z = ϕ (y 1,, y m ) with differentiable f 1,,f m such that then the partial derivative is 9

10 Overview... Differentiation Basics Practical application Chain rule Elasticities l Hôpital s rule 10

11 Elasticity (1) Let s look at the intuition and basic definition Let f be a function from R to R let y = f(x) and y = f(x+ x) f(x) for some x > 0 ratio of y/y to x/x is (approximately) the elasticity η of f at x the proportionate change in y for a given proportionate change in x If f is differentiable, η is the limit of this ratio as x 0: Now for an alternative, equivalent form define u := log x and v := log y so that v = log ( f (e u )) applying the chain rule: dv/du = f '(e u ) e u / f (e u ) = f '(x) x/y the right-hand side is just η so the elasticity can be expressed as 11

12 Elasticity (2) Consider a natural multivariate extension Suppose we have differentiable f 1,,f m such that Then the elasticity of y j with respect to x i is An equivalent way of writing this elasticity is 12

13 Overview... Differentiation Basics A useful result Chain rule Elasticities l Hôpital s rule 13

14 l Hôpital s rule Let I = (x 0, x 1 ) be an interval in R For two functions f and g and some ξ in I suppose that: f and g are everywhere differentiable on I\{ξ} (with derivatives f', g' ) g'(x) 0 everywhere on I\{ξ} lim x ξ f(x) = 0; lim x ξ g(x) = 0 lim x ξ f'(x) / g'(x) exists (call this limit z) Then also Example. What happens to [a x 1] / x as x 0? let f(x) = a x 1 and g(x) = x so that f'(x) = a x log a and g'(x) = 1 we have lim x 0 f'(x) / g'(x) = log a therefore lim x 0 [a x 1] / x = log a 14