Utility Flashcards
What is a utility function?
A function which ordinally captures preferences
How would you prove the theorem that if a preference relation over a finite set A of alternatives is complete and transitive, then it has a utility representation, i.e. there exists a function u: A –> R such that a ≽ b <==> u(a) ≥ u(b)?
The function u(a) = number of elements in {b ∈ A | a ≽ b} satisfies the theorem
What form do the utility functions of perfect substitutes and complements take?
Substitutes: u(x, y) = ax + by
Complements: u(x, y) = min(ax, by)
What form does the utility function for preferences with a bliss point take?
u(x, y) = -(x - a)2 - (y - b)2
Negative everywhere except at bliss point (a, b)
How is the indifference curve passing through (x, y) denoted?
IC(x, y) = {(a, b) | u(x, y) = u(a, b)}
What is an example of a utility function representing non-convex preferences?
u(x, y) = x2 + y2
What is the Marginal Rate of Substitution and how can it be calculated?
The slope of the indifference curve (Δy/Δx)
This captures the relative valuation of the two goods at some bundle for the consumer
MRS1, 2(x, y) denotes the MRS of y for x
What does it mean graphically if the MRS does not depend on one or both of the inputs?
If the MRS only depends on one of the two inputs, the ICs will be shifts of each other along the axis which does not affect the MRS
If the MRS has no dependence on inputs, the slope is constant (perfect substitutes)
How does MRS relate to the utility function?
|MRS| is the ratio of the partial derivatives of the utility function (marginal utilities)
MRS1, 2(x, y) = - u1(x, y) / u2(x, y)
u1 also denoted MU1
This shows that MRS, the relative value of importance of two goods, is also the ratio of the importance of each good for utility
What is the relationship between MRS and convexity for monotonic preferences?
Convex preferences have |MRS| non increasing, strictly convex preferences have |MRS| strictly decreasing
What form does a Cobb-Douglas utility function take and what are some properties?
u(x, y) = xy
Represents well-behaved preferences
Here MRS1, 2(x, y) = -y/x
More generally there can be more inputs and each can have different exponents which can always be transformed to sum to 1
What is a quasilinear utility function?
u(x, y) = x + v(y) is a quasilinear function in x where v is a concave function
ICs will be shifts of each other along the axis of the linear term
How can utility functions be transformed?
Since utility functions only need to produce ordinal rankings, a monotonic transformation of u represents the same preferences
If u: A –> R represents preferences and if t: R –> R is a strictly increasing function, then t ∘ u also represents the preferences
Why might MRS be more useful than MU?
The magnitude of MU depends on the specific function u but MRS is always the same for the same preferences (unchanged by monotone transformations)
One consequence of this is that a utility function with diminishing MU can be transformed to one that doesn’t have diminishing MU, but MRS will be diminishing irrespective of transformations
Which of assumptions for well-behaved preferences are required for there to be a continuous utility function?
If preferences are complete, transitive, and continuous, then they can be represented by u: A –> R
