Download Ec511 Problem Set 1: Differential Calculus. Question 1. Using

Ec511 Problem Set 1: Di¤erential Calculus.
Question 1. Using the Implicit Function Theorem, …nd the slope of an indifference curve for the following utility functions
(a) U = x + y; (b) U = xy; (c) U = x y (d) U = [ax + by ] =
where a; b; ; are positive constants.
Question 2. Preferences are homothetic if the slope of an indi¤erence curve
is constant along a ray out of the origin; i.e. Ux =Uy is a function only of y=x:
Which of the utility functions in question 1 imply homothetic preferences? What
implications do homothetic preferences have for income e¤ects?
Question 3. Consider a utility function U = U (x; y) and a second utility
function V = V (x; y) where V = R(U (x; y)) and R is a strictly increasing,
di¤erentiable function. By …nding the slopes of their corresponding indi¤erence curves, establish the marginal rates of substitution implied by these utility
functions are the same.
Question 4. For the utility function U (x; y) = x2 + y 2 :
(a) …nd @U=@x and @U=@y and express dU in terms of dx and dy;
(b) …nd dU=dx where y = y(x) satis…es px + qy = M ;
(c) …nd the values of x; y where dU=dx = 0;
(d) noting that
@ dU dy
@ dU
d2 U
[
]+
[
]
=
dx2
@x dx
@y dx dx
use your answer in (b) to …nd d2 U=dx2 : What is d2 U=dx2 at (x,y) given in part
(c).
(e) does your answer in (c) describe a maximum or a minimum? Draw an
indi¤erence curve diagram which depicts your answer.
Question 5. Repeat question 4 for the utility function U (x; y) = xy:
Question 6 [More di¢ cult] Repeat question 4 for the utility function U (x; y) =
x y :
Question 7. [Really di¢ cult]. Consider an indi¤erence curve U (x; y) = U0
where U is strictly increasing in x; y: The slope of the indi¤erence curve is
dy
=
dx
Ux
:
Uy
Find d2 y=dx2 : Show d2 y=dx2 > 0 if and only if
0
Ux
Uy
Ux
Uxx
Uxy
Uy
Uxy > 0:
Uyy
Question 8 [Extremely di¢ cult] A function U is homogenous of degree k if
and only if
U (tx; ty) = tk U (x; y)
for any t. Show a homogenous utility function implies homothetic preferences.
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