xy
d) U :Ux, y a)
1 2
U 1 x 2
1
U 1
x 2 y ; and
b)
U 1 3 x 3
c)
U
2
x x 3y
12 3
U x
x yy
1
1
1
y3
xy xy
x 2 y2 2
U 1 y 3 1
3 ; and
d)
1
.
x2 y2 2
y
1
x ; and
x
xy
e) V :V x , y
xy
U
y
x y 2 y
2
2 3
y
1
x3y
1
2 3
; and
2
U y
2 xx y
xy xy
x yx
e) Begin by writing V(x, y) = (x-2 + y-2)-0.5. Then V
X9.7
3
x2
x
x
V y
x2 x2 y2
y
2 1.5
x3
x3
2
1.5
y2 x2 y2
x2 y2
; and by the same argument,
1.5
Confirm that for the functions in X9.6c–e, we can rely on Expressions 9.4 and 9.5 to calculate the marginal utilities.
X9.6c) MUX(x, y) =
2
1a
1
X9.6d) MUX(x, y) =
MUY(x, y) = y x X9.6e) MUX(x, y) =
1
y
3
2
3
1
3
3
x
2
1
1
2
1
y
2 1 1
x
2
= x x
1
y1
2
2y
x
3 2 0.5
x
3
= x x
2
y2
; and similarly,
2 1.5
y
a
U x ,y
a
, if ax 11a ; and MU x , y 1 U x , y Y, if a 1
MU X x , y x a 1 x a 1 y a a ; now we rewrite
x 1 ya
MU X x , y
MUY x , y
a
a
x
1
x 1y
1
a
1a
aa
a
y1 a
1
Ux ,y x1 a
a
1
y
1
1 as a
1a
a
1
a
, and then:
a
; and likewise, we obtain a
Ux , y y1 a
Use Expression 9.6 to confirm that along any line passing through the origin that has the equation y = kx, the marginal rate of substitution, MRS(x, y) = –k1 – a. MRS =
X9.10
1.5
Confirm that for the general form of CES utility functions in Expression 9.1:
MUX x, y
X9.9
; and similarly,
12
1a U U x ,y x x
MUY(x, y) = y x
= x 3 x 3 y 3 ; and similarly,
3
x 2
1
x
1a U U x ,y x
2
y
2
1
MUY(x, y) = y 3 x 3 y 3
x
U x ,y x
U x
X9.8
1.5
x2y2
1 1.5 1 x y2
1
y 1 a kx 1 a x
= -kx 1 – a.
For the following utility functions, obtain the marginal utilities of x and y and calculate the marginal rate of substitution: