Exercise 9

Exercise Session 9, Nov 22nd ; 2006 Mathematics for Economics and Finance Prof: Norman Schürho¤ TAs: Zhihua (Cissy) Chen, Natalia Guseva Exercise R 1 xEvaluate the following integrals: (a) pxe dx: 1+ex 1 R x2 p (b) Show e 2 dx = 2 : 0

Exercise 2 Suppose that a small business earns a net pro…t stream y(t) for t 2 [0; T ]. At time s 2 [0; T ]; the discounted value (DV) of future pro…ts is V (s; r) =

ZT

y(t)e

r(t s)

dt:

s

Where r is the constant rate of discount. Compute Vs0 (s; r). Exercise 3 Suppose that the random variable X has an exponential distribution with pdf fX (x) = exp( x); x > 0 and 0 elsewhere. (a) Find the pdf for Y = 1=X; (b) Y = ln(X); (c) Y = 1 FX (X): Exercise 4 Suppose the direct utility function is CES function: U (x) = [

1 x1

+

1= 2 x2 ]

;

where x = (x1 ; x2 )0 are the quantities of the two goods, p = (p1 ; p2 )0 are the prices of the goods, and 1 ; 2 ; and are given constants, with 1 ; 2 positive, and < 1, with budget constraint: px0 = I: Notes: (a)-(c) from Exercise Session 6, Q5 (a) Show that when = 1, indi¤ erence curves become linear. (b) Show that as ! 0; this utility function comes to represent the same preferences as the (generalized) Cobb-Douglas utility function u(x) = x1 1 x2 2 : (c) Show that as ! 1; indi¤ erence curves become “right angles”; that is, this utility function has in the limit the indi¤ erence map of the Leontief utility function u(x1 ; x2 ) = M in fx1 ; x2 g : (d) To make our life easy, let 1 = 2 = 1:Consider the consumer is trying to minimize the expenditure required for attaining a given target utility level. Show that the expenditure function is of the form E(p; u) = [pr1 + pr2 ]1=r u; where u is the utility level, and r can be expressed in terms of . (e) Using envelop property, explain why compensated demand functions (costminimizing commodity choices for a given utility level) can be expressed as Ep0 (p; u):

1