Chapter 2: Preferences and Utility Functions Flashcards
Monotone Preferences
if for any two bundles x\geq y then x\succeq y.
Strictly Monotone Preferences
if x\geq y and x\neq y, then x\succ y.
If u represents preferences, preferences are monotone
iff u is nondecreasing.
If u represents preferences, preferences are strictly monotone
iff u is strictly increasing.
Strict monotonicity for strict increases
if x>y then x\succ y.
Preferences are globally insatiable if
for every x\in X, y\succ x for some bundle y\in X.
Preferences are locally insatiable if
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.
Preferences are convex if
for every x, y\in X, the bundle ax+(1-a)y\geq min{x,y}.
Preferences are strictly convex if
for every x and y, x\neq y, x\succeq y, then ax+(1-a)y\succ min{x,y}.
Preferences are semi-strictly convex if
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?)
Preferences are convex iff
for every point x, the set NWT(x) is convex.
If preferences are convex, c(A) is
convex. What happens if we have strictly convex?
A function is concave if,
f(ax+(1-a)y)\geq af(x)+(1-a)f(y) (what about strict?)
A function is quasi concave if
f(ax + (1-a)y)\geq min{f(x), f(y)} (what about strict?)
A function is semi-strictly quasi concave if
for all values x,y\in A, f(ax+(1-a)y)>f(y). How is this different from strict concavity?
Preferences represented by a concave function u are
convex. (what about strictly concave function?)
The utility function u is quasi concave if and only if the preferences
are convex.
The utility function is strictly quasi concave if and only if preferences
are strictly convex.
The utility function is semi-strictly quasi concave if and only if preferences
are semi-strictly convex. (why?)
How do strong separability and weak separability differ?
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.
Preferences have a quasi-linear representation iff (hint: three properties)
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’.
Preferences are homothetic if
x\succeq y implies \lambda x\succeq \lambda y for all positive lambda.
Preferences are continuous and homothetic iff
they can be represented by a continuous and homogeneous utility function (why? apply debreu and direct application of homogeneity)
