Leibniz integral Rule - Aniket

Leibniz integral Rule

Dr. Kumar Aniket University of Cambridge

1. Integrals

1.1. Leibniz integral Rule. ? Differentiation under the integral sign with constant limits.

d y1

y1

f (x, y) dy =

f (x, y) dy

dx y0

y0 x

for

x

(x0, x1)

provided

that

f

and

f x

are

continuous

over

a

region

in

the

form

[x0, x1] ?

[y0, y1].

? Differentiation under the integral sign with variable limits that are a function of the variable

used for differentiation.

d b()

b()

f (x, ) dx = b () ? f (b(), ) - a () ? f (a(), ) +

f (x, ) dx

d a()

a() x

1.1.1. Application to Consumer Demand.

? Consumer demand for product X varied and determined by each consumer's t [0, 1] ? Consumer Demand:

0

p > P (t)

h(t, p) p < P (t)

P (t) Reservation price: given t the price has to be low enough for the consumer to buy. h(t, P (t)) > 0, ht(t, p) > 0, hp(t, p) < 0, P (t) > 0

T (p) is the inverse function of P (t): for a given p the taste t has to be high enough for the consumer to buy

1

Distribution of t: f (t) is the distribution of t with the following properties: i. Number of people in [a, b] where 0 < a < b < 1 is given by

b

f (t)dt

a

ii.

1

f (t)dt = N

0

iii. For a small , there would be f (t) consumers1 with t (t, t + )

For a given price p, consumers with positive demand are t (T (p), 1)

Consumer demand (t (T (p), 1)) :

1

x = f (t)h(t, p)dt

T (p)

Using Leibinz formula

dx

1

h(t, p)

= -f T (p) h T (p), p ? T (p) + f (t)

dt

dp

T (p)

p

- Both terms on the RHS are negative since hP (t, p) < 0

1.1.2. Integrating over a Stochastic Distribution.

This is what I always get stuck on:

dG(w) = g(w)

dw

w

G(w) = g(w)dw

0

1

t+

f (t)dt f (t)

t

2

?

w random variable g(w) probability distribution G(w) cumulative distribution

3

?

 ................
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