Lec 2: Preferences and Utility

What is an indifference curve?

An indifference curve represents all combinations of two goods that yield the same level of utility for an individual. Each bundle on the curve provides the same utility.

What are the four components of the consistent choice model?

An individuals consumption set, preference ordering, constraint set, and the “individual welfare maximisation assumption”.

What is the “individual welfare maximisation” assumption?

the individual chooses the object in the constraint set that has the highest rand according to the preference ordering.

What is the Consumption Set equation for the set of all non-negative bundles of two commodities?

C = {(q1, q2) € R2: q1 ≥ 0, q2 ≥ 0} = R2+

What is the Rationality Assumption?

We can use weak preference relations that are “rational” to construct a consistent ranking of every consumption bundle in the consumption set.

What axioms satisfy the Rationality Assumption?

Direct comparison (completeness) and indirect comparison

What is the direct comparison axiom?

We can use the weak preference relation to determine the relative ranking of any two bundles in the consumption set. x≥y or y≤x or both.

What is the indirect comparability axiom?

Allows us to use a sequence of direct binary comparisons between consumption bundles to make indirect binary comparisons. If x≥ y and y≥z then it must be the case that x≥z.

it also rules our the possibility of preference cycles of the form x≥y≥z≥x when it is not the case that x~y~z as we have a theory of “consistent” choice.

If a binary relation ≥ is a weak preference relation that is rational then it must be both…

Strongly complete (direct comparability) and transitive (indirect comparability)

What is the equation for an indifference curve?

U(q1,q2)=k, where k is a constant utility level

What happens when you vary q1 and q2 on an indifference curve?

The utility remains constant, so there is no change in total utility, i.e., dU=0

What is the first-order differential approximation to the change in utility?

dU ≈ MU1dq1+MU2dq2, where MU1 and MU2 are marginal utilities

What is the equation for the slope of an indifference curve?

The slope of the indifference curve is dq2dq1 ≈ −MU1MU2 = −MRS1,2

What does the marginal rate of substitution (MRS) represent?

The MRS represents the rate at which an individual is willing to give up one good in exchange for an additional unit of another good, while keeping utility constant.

What is the relationship between the marginal rate of substitution and marginal utilities?

MRS1,2 = −MU1 MU2, which is the ratio of the marginal utilities of the two goods.

What is the significance of the ordinal nature of utility functions?

Utility functions are ordinal, meaning they rank preferences but do not provide exact levels of satisfaction. This affects how marginal utility is interpreted across different utility function representations.

How does the marginal utility change when applying a strictly increasing transformation to a utility function?

The marginal utility values change, but the marginal rate of substitution (MRS) remains the same, as it is invariant under strictly increasing transformations.

What does the marginal benefit (MB) of a commodity represent?

The marginal benefit represents the maximum amount an individual is willing to pay for an additional unit of a commodity, holding other factors constant.

How is marginal benefit related to the marginal rate of substitution (MRS)?

The marginal benefit of a commodity is a special case of the MRS, specifically the rate at which the individual is willing to trade the commodity for money income.

What is the formula for the marginal benefit of a commodity?

The marginal benefit of commodity i is MBi=MRSi,y = MUi MUy, where MUi is the marginal utility of the commodity and MUy is the marginal utility of income.

What is the Desirability Assumption?

More is preferred to less

What are the three most common Desirability Assumptions?

Local non-satiation, monotonicity and strong monotonicity

Define Local non-satiation

Given a certain amount of a good, a consumer will always prefer to have a little more rather than less, as long as they are within a nearby range of consumption

Define Monotonicity

A function is monotonic if it is either entirely non-increasing or non-decreasing throughout its domain. It is non-decreasing, whenever x1<x2 has the output f(x1)≤f(x2)

Define Strong Monotonicity

A function is said to be strongly monotonic if it is either strictly increasing or strictly decreasing throughout its domain. It is strictly decreasing whenever x1<x2 has the output f(x1)<f(x2)

What is the relationship between the three most common desirability assumptions

SM => M => LNS

What will an agent that had locally no -satiated preferences choose to do

Exhaust their budget

What is the situation if every agent in the economy exhaust their budgets?

The sum of the values of excess demand is equal to zero - Walras law

What are the Curvature assumptions

The Curvature assumptions capture the trade-offs that individuals are prepared to make between different commodities. These often take the form of diminishing marginal rates of substitution and a taste for diversification (aggregates are preferred to extremes).

What are the two alternative curvature assumptions that are commonly employed?

Convexity and strict convexity