Chapter 2: Preferences and Utility Functions

Question 1

Monotone Preferences

Answer 1

if for any two bundles x\geq y then x\succeq y.

Question 2

Strictly Monotone Preferences

Answer 2

if x\geq y and x\neq y, then x\succ y.

Question 3

If u represents preferences, preferences are monotone

Answer 3

iff u is nondecreasing.

Question 4

If u represents preferences, preferences are strictly monotone

Answer 4

iff u is strictly increasing.

Question 5

Strict monotonicity for strict increases

Answer 5

if x>y then x\succ y.

Question 6

Preferences are globally insatiable if

Answer 6

for every x\in X, y\succ x for some bundle y\in X.

Question 7

Preferences are locally insatiable if

Answer 7

for every x\in X and \forall \epsilon>0, there exists y\in X that is no more distant from x such that y\succ x. (Note that this is stronger than globally insatiable.

Question 8

Preferences are convex if

Answer 8

for every x, y\in X, the bundle ax+(1-a)y\geq min{x,y}.

Question 9

Preferences are strictly convex if

Answer 9

for every x and y, x\neq y, x\succeq y, then ax+(1-a)y\succ min{x,y}.

Question 10

Preferences are semi-strictly convex if

Answer 10

they are convex and if for every pair x and y, x\succ y, ax+(1-a)y\succ y. (what is the difference from strictly?)

Question 11

Preferences are convex iff

Answer 11

for every point x, the set NWT(x) is convex.

Question 12

If preferences are convex, c(A) is

Answer 12

convex. What happens if we have strictly convex?

Question 13

A function is concave if,

Answer 13

f(ax+(1-a)y)\geq af(x)+(1-a)f(y) (what about strict?)

Question 14

A function is quasi concave if

Answer 14

f(ax + (1-a)y)\geq min{f(x), f(y)} (what about strict?)

Question 15

A function is semi-strictly quasi concave if

Answer 15

for all values x,y\in A, f(ax+(1-a)y)>f(y). How is this different from strict concavity?

Question 16

Preferences represented by a concave function u are

Answer 16

convex. (what about strictly concave function?)

Question 17

The utility function u is quasi concave if and only if the preferences

Answer 17

are convex.

Question 18

The utility function is strictly quasi concave if and only if preferences

Answer 18

are strictly convex.

Question 19

The utility function is semi-strictly quasi concave if and only if preferences

Answer 19

are semi-strictly convex. (why?)

Question 20

How do strong separability and weak separability differ?

Answer 20

Any arbitrary set of sub-bundles is also separable. Equivalently, strong separability is given when the aggregator v is the sum, rather than something that increases in only the relevant sub-bundles.

Question 21

Preferences have a quasi-linear representation iff (hint: three properties)

Answer 21

1) (x,m)\succeq (x,m’) iff m\geq m’; 2) (x,m)\succeq (x’,m’) iff (x,m+m’’)\succeq (x,m’+m’’) (no interaction among money and util); 3) For all x,x’, (x,m)~(x’,m’) for some m’.

Question 22

Preferences are homothetic if

Answer 22

x\succeq y implies \lambda x\succeq \lambda y for all positive lambda.

Question 23

Preferences are continuous and homothetic iff

Answer 23

they can be represented by a continuous and homogeneous utility function (why? apply debreu and direct application of homogeneity)