Exercise 6

Exercise Session 6, Nov 10th ; 2006 Mathematics for Economics and Finance Prof: Norman Schürho¤ TAs: Zhihua (Cissy) Chen, Natalia Guseva

Exercise 1 Consider a competitive …rm that produces a single output y using two inputs x1 and x2 . The …rm’s production technology is described by a CobbDouglas function y(x) = f (x1 ; x2 ) = x1 x2 ; where + < 1, > 0 and > 0. Taking as given the output price p and the input prices w1 and w2 , the …rm maximizes its pro…ts given by (x) = py(x)

wT x:

a) Does the production function exhibit decreasing, constant, or increasing returns to scale? Show. b) Write the …rst-order conditions and check whether su¢ cient conditions for a maximum are satis…ed. c) Solve for the …rm’s factor demands, giving the optimal input levels xi as functions of input and output prices. Exercise 2 De…ne the transformation f from A = (x1 ; x2 ) : x21 + x22 R2 into R2 by f (x1 ; x2 ) = (x21 x22 ; x1 x2 ):

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(a) Compute the determinant of the Jacobian of f and show that it is = 6 0 in the whole of A:(b) What does f do to the points (1; 1) and ( 1; 1). (c) Comment on the results in (a) and (b). Exercise 3 The function f (x1 ; x2 ; x3 ) = x21 + x22 + 3x23 x1 x2 + 2x1 x3 + x2 x3 de…ned on R3 has only one stationary point. Show that is a local minimum point. Exercise 4 Assume the consumer’s utility function is Cobb-Douglas utility function: u(x1 ; x2 ) = kx1 x12 for some 2 (0; 1) and k > 0;and his budget constraint follows: p1 x1 + p2 x2 = w: Find the consumer’s optimal consumption bundle. Exercise 5 Suppose that in a two-commodity world, the consumer’s utility func1 tion takes the form u(x) = [ 1 x1 + 2 x2 ] :(CES) (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 : Exercise 6 Exercise4. Q5.

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