Mod 3 Flashcards
The preference relation is said to be non-decreasing if?
x is as least as good ay y whenever x is greater than or equal to y
The preference relation is increasing if?
it is non-decreasing and x is strictly better than y whenever x is bigger than y in every dimension.
The preference relation is strictly increasing if ?
it at least equal to y in every dimension and strictly greater than in one dimension
In other words increasing preference relation is a way of expressing the idea?
that more of a good thing is always better, and that if you like something, you will always like more of it.
Non-decreasing -
it never hurts to have more
Increasing
If you have strictly more of everything than you’re better off
Strictly increasing
if you have at least as much of everything and m ore of at least one the things.
Locally non-satiated preference
For any given bundle of x and any positive integer e, there exists another bundle of goods y that is arbitrarily close to x “ ie the distance between y and x is less that e (||y-x||<e)”
that the consumer prefers to x (i.e. y is weakly preferred to x)
u(x,y) = 2x + y represents preferences that are?
locally non-satiated but not increasing.
u(x,y) = 2x +y explain how it locally non-satiated but not increasing.
To see why, let’s consider two bundles (1, 1) and (1, 2) where y2 > y1. Since the utility function is u(x,y) = 2x + y, we have:
u(x2, y2) = 2x2 + 2 = 6
u(x1, y1) = 2x1 + 1 = 3
Since y2 > y1, it follows that u(x2, y2) > u(x1, y1). Now, let’s consider a bundle (x3, y3) such that x3 > x2. Since the utility function is locally non-satiated, we know that if we add a sufficiently small positive amount δ to x2, we can make the new bundle (x2 + δ, y2) preferred to (x2, y2). Similarly, if we add a sufficiently small positive amount δ’ to y2, we can make the new bundle (x2, y2 + δ’) preferred to (x2, y2).
However, we cannot say that the preferences are increasing because there may exist bundles (x4, y4) such that x4 < x2 and y4 > y2, but u(x4, y4) > u(x2, y2). In other words, preferences are not always determined by the level of utility, so the preferences are not increasing.
u(x1, x2) = x1 + x2
locally non-satiated and strictly increasing
u(x, y) = x^2 + y^2
locally non-satiated and increasing
u(x, y) = x + y^2
locally non-satiated and non-decreasing
increasing x alone does not guarantee an increase in utility; an increase in y is also necessary.
convex preferences
the average of the two consumption bundles is preferred to the less preferred to the less preferred of the two original bundles.
means that the consumer is indifferent between combinations of goods that lie on a straight line connecting two points on the curve. In other words, the consumer is willing to trade off some of one good for more of the other good, but only up to a certain point.
Suppose that the preference relation is convex. Suppose
that y and z are in U ≿(x) so y ≿ x and z ≿ x. Now since ≿ is complete either y ≿ z or z ≿ y. Let us, if necessary, relabel the consumption plans y and z so that z ≿ y. Now, since the preferences are convex for any 0 < α < 1 we have?
we have αz + (1 − α)y ≿ y and since y ≿ x, by the
transitivity of ≿, we have αz + (1 − α)y ≿ x, that is αz + (1 − α)y is
in U ≿(x).
We say that the preference relation is strictly convex if?
Strictly convex preferences refer to preferences that satisfy the following condition: for any two bundles of goods x and y, and any α between 0 and 1, the bundle αx + (1-α)y is strictly preferred to any bundle that is a convex combination of x and y and lies strictly between them.
In other words, if preferences are strictly convex, the consumer is strictly averse to “averages” or “compromises” between two bundles, and always prefers one bundle that is closer to x or y. This implies that the consumer has a strong preference for variety and is willing to trade-off some quantity of one good for a different good in a non-linear way.
u(x,y) = xy convex or strictly convex
This utility function exhibits strict convexity because for any two bundles (x1, y1) and (x2, y2) where x1<y1 and x2<y2, and any α between 0 and 1, the bundle α(x1, y1) + (1-α)(x2, y2) lies strictly between (x1, y1) and (x2, y2), and the utility of the convex combination is strictly preferred to the utility of (x1, y1) and (x2, y2).
I(x) = {y ∈ X | u(y) = u(x)}
The indifference set of the consumption bundle
A non-decreasing preference relation on x=RL+ is homothetic if whenever
x is indifferent to y the any integer greater than 0 will still mean they are indifferent
A preference relation is said to be homothetic if the shape of the indifference curves does not change with changes in the level of income or wealth. In other words, a preference relation is homothetic if the same relative quantities of goods are preferred regardless of the level of income or wealth.
Quasilinear
A utility function is quasilinear if it can be expressed as the sum of a linear function of one good and a function of the other goods.
More formally, a utility function u(x) defined over a consumption bundle x = (x1, x2, …, xn) is quasilinear if it can be written as:
u(x) = v(x1) + f(x2, x3, …, xn)
continuity
In economics, continuity is often used to describe the behavior of economic variables. For example, a demand function is said to be continuous if small changes in the price of a good result in small changes in the quantity demanded
Increasing preferences = what type of indifference curve
downward sloping indifference curve
lexicographic ordering R2+
x is as least as good or better than y in the the first dimension or its equal in the first dimension but no smaller in the second dimension.
this preference is transitive and complete.
It is strictly monotonic and strictly convex.
Let X ⊂ Rn be a convex set. We say that a function f : X → R is
concave if,
for all x, y in X and all 0 < α < 1,
f (αx + (1 − α)y ) ≥ αf (x) + (1 − α)f (y ),
Let X ⊂ Rn be a convex set. strictly concave if,
for all x, y in X with x can’t equal y and all 0 < α < 1,
f (αx + (1 − α)y ) > αf (x) + (1 − α)f (y ),
quasi-concave if, for all
x, y in X and all 0 < α < 1,
f (αx + (1 − α)y ) ≥ min{f (x), f (y )}
strictly quasi-concave
strictly quasi-concave if, for all x, y in X with x 6 = y and f (x) ≥ f (y ),
and all 0 < α < 1, f (αx + (1 − α)y ) > f (y ).
convex if,
for all x, y in X and all 0 < α < 1,
f (αx + (1 − α)y ) ≤ αf (x) + (1 − α)f (y )
strictly convex if,
for all x, y in X with x 6 = y and all 0 < α < 1,
f (αx + (1 − α)y ) < αf (x) + (1 − α)f (y ),
quasi-convex if,
for all x, y in X and all 0 < α < 1,
f (αx + (1 − α)y ) ≤ max{f (x), f (y )}
strictly quasi-convex if,
for all x, y in X with x 6 = y and f (x) ≤ f (y ),
and all 0 < α < 1, f (αx + (1 − α)y ) < f (y )
Suppose that the rational preference relation % is represented by the utility
function u : X → R. Then
If the utility function u is concave then the preference relation is ______.
If the utility function u is strictly concave then the preference relation is _____.
The utility function u is quasi-concave if and only if the preference relation is ____.
The utility function u is strictly quasi-concave if and only if the
preference relation is ___.
J
convex ,strictly convex ,convex ,strictly convex
Homogenous utility function
Means the same share of income will be spent on any given good no matter how the consumers income may change over time.
