L2: Utility

Part B Intermediate Microeconomics

General Overview

Course Code: ECB002
Lecture Focus: Consumer Theory - Preferences and Utility
Lecturer: Valeria Unali
Slides: Included in the lecture notes

Key Objectives of Consumer Theory

Purpose: To build a model that helps us understand and predict how markets behave, which is essential for making smart policy decisions. This theory helps explain consumer choices.
Core Question: How do consumers choose to spend their money on different goods?
Today’s Focus: Understanding what consumers like (preferences) and how satisfied they are (utility) when considering different combinations of goods.
Next Steps: We'll introduce the idea of a 'budget constraint' (how much money a consumer has) to figure out the best choices a consumer can make.
Definition of a bundle: A specific collection of different goods a consumer might consider, like a specific grocery list. For instance, (X, Y) could represent a bundle with 'X' number of good X and 'Y' number of good Y; (2 bananas, 1 apple) is a bundle (2,1).

Theoretical Fundamentals

Consumer's Optimal Point: We can imagine a consumer's ideal choice on a graph where the axes show quantities of various goods, like X and Y.
Aim: Our goal is to precisely describe this ideal choice mathematically so we can predict how consumers will react if prices change, their income changes, or new taxes and policies are introduced.

Lecture Outline

Axioms of Consumer Preferences
Indifference Maps
Properties of Indifference Curves (Private Study)
Common Properties of ‘Standard’ Indifference Curves
Marginal Rate of Substitution (MRS)
Utility Functions (Private Study)
Mathematical Description of Indifference Curves
Special Forms of Indifference Curves

Detailed Axioms of Consumer Preferences

These fundamental assumptions help us understand how consumers make choices, forming the basis of consumer theory.

Axiom 1: Completeness

Consumers can always compare any two bundles of goods and decide if they prefer one over the other or if they are equally happy with both. They never say 'I don't know'.
For any two bundles A = (XA, YA) and B = (XB, YB): - Either A is preferred over B, B over A, or the consumer is indifferent between the two.

Axiom 2: Transitivity

Consumer preferences are consistent and logical. If a consumer prefers bundle A to bundle B, and also prefers bundle B to bundle C, then they must logically prefer bundle A to bundle C.
Similarly, if they are equally happy with A and B, and also equally happy with B and C, then they will also be equally happy with A and C.

Axiom 3: Non-Satiation (Monotonicity)

Generally, consumers always prefer more of a good to less of it; they are never fully satisfied. This idea simplifies our analysis by assuming that getting even a little bit more of something good will increase their overall satisfaction or 'utility'.

Realism and Limitations of Assumptions

While these basic assumptions might seem too simple or unrealistic, we consider them for several practical reasons:
1. Upon closer examination, many assumptions are less irrational than they seem.
2. The theory doesn't imply individuals explicitly calculate decisions this way, but rather that their behavior acts as if they do, allowing us to focus on predictable outcomes.
3. The theory serves as a useful approximation—we aren't aiming for perfect prediction, but a good general understanding.
4. For situations where traditional theory falls short, newer fields like behavioral economics offer alternative models.

Indifference Maps

Most of our analysis simplifies complex choices down to just two goods, but the principles can be expanded to situations with many goods.
Important Concept: An indifference curve (IC) is a line on a graph that connects all the bundles of goods that provide a consumer with the exact same level of satisfaction (utility). Different curves represent different levels of utility (e.g., U1, U2).

Properties of Indifference Curves (Private Study)

Utility and Distance: Generally, bundles located farther away from the origin (0,0) on the graph offer higher utility, meaning more satisfaction. For example, a bundle in area A provides more of at least one good than bundle 'e', making it preferred. - Conversely, a bundle in area B represents less utility than 'e'.

Indifference Curve Characteristics:

Passes Through All Bundles: Because of the completeness axiom, every single possible combination of goods must lie on some indifference curve, showing its utility level.
Cannot Cross: Indifference curves can never intersect. If they did, it would create a logical contradiction, violating the transitivity axiom and making it impossible to consistently rank bundles.
Cannot Have Positive Gradient: Indifference curves must slope downwards. If they sloped upwards, it would mean a consumer is equally happy with less of both goods, which contradicts the non-satiation axiom (that 'more is better').
Not ‘Thick’: An indifference curve is a thin line. It cannot encompass areas, as a thick curve would imply that different bundles within that 'thickness' provide both the same and different utility, leading to contradictions.

Common Structure of Indifference Curves

Generally, indifference curves are 'bowed inward' towards the origin, which we call being convex to the origin.
Convexity Meaning: This shape implies that consumers prefer a mix of goods rather than having a lot of just one good. Mathematically, it means that if you draw a straight line between any two points on or above an indifference curve (a 'preferable' region), the entire line will stay within that preferred region.

Marginal Rate of Substitution

Definition: The MRS tells us how much of good Y a consumer is willing to give up to get one additional unit of good X, while maintaining the same level of satisfaction. It is measured by the slope of the indifference curve at any given point.
Example: An MRS of $-2$ means the consumer is willing to give up 2 units of Y for one more unit of X to stay equally happy.
- Typical consumer behavior shows a diminishing MRS, meaning that as a consumer gets more and more of good X, they are willing to give up less and less of good Y to get even more of X. The more of good Y they have, the less valuable an additional unit of good X becomes to them.

Utility Functions

Consumers don't actually calculate their satisfaction using explicit mathematical utility functions in their heads. However, their observed preferences can often be represented by these underlying, unobserved utility functions.
- For any bundle A = (XA, YA) compared to bundle B = (XB, YB):
- Preferred: If bundle A provides more utility than B, we write it as $U(XA, YA) > U(XB, YB)$.
- Indifferent: If bundles A and B provide the same utility, meaning the consumer is equally happy with both, we write $U(XA, YA) = U(XB, YB)$.
Critical Note: Only the ranking or order of utility matters, not the specific numerical value (this is called ordinality).
- Example: If $U(A) = 10 > U(B) = 1$ and also $U(A) = 150 > U(B) = 133$, both examples simply tell us that bundle A is preferred to bundle B. The actual numbers (10 vs 150, or 1 vs 133) don't change the preference ranking.

Utility Function Examples

Different Functions Providing Same Preference Rankings: It's possible for two entirely different mathematical functions, like $U(X,Y)=3X+5Y$ and $V(X,Y)=8+6X+10Y$ , to represent the same exact preferences for a consumer, meaning they rank bundles in the same order.
Positive Monotonic Transformation: A utility function can be altered or 'transformed' into another function while still preserving the original ranking of preferences. This happens if the new function is derived by multiplying the old function by a positive number and/or adding a constant (e.g., $V(X,Y) = a + bU(X,Y)$ where b > 0).
- Various methods exist for transformation, including logarithmic and square root forms.
- General Form: If bundle $(X,Y)$ is preferred to $(X',Y')$ according to utility function $U(.)$ , it will remain preferred according to $V(.)$ if $V(.)$ is a positive monotonic transformation of $U(.)$ .

Mathematical Framework for Indifference Curves

For a given utility function $U(X,Y)$ and a specific level of satisfaction $\bar{U}$ (a constant utility level), all the bundles that provide this equal utility can be grouped together.
Formulation: The associated Indifference Curve (IC) is defined as the set of all $(X,Y)$ combinations for which $U(X,Y) = \bar{U}$ .
Example: If a consumer's utility is described by $U(X,Y) = X^{0.5} + Y^{0.5}$ , we can plot different indifference curves by setting $U(X,Y)$ to various constant utility levels (e.g., $\bar{U}=2, \bar{U}=3$ ).

Marginal Utility

Marginal utility measures the additional satisfaction a consumer gets from consuming one more unit of a specific good, while keeping the amount of other goods constant.
For Good X: The marginal utility of X ( $MU_X$ ) tells us how much utility changes when we increase X by a tiny amount: $\frac{\partial U}{\partial X}$ .
For Good Y: Similarly, the marginal utility of Y ( $MU_Y$ ) describes the change in utility from a tiny increase in Y: $\frac{\partial U}{\partial Y}$ .
Note: The previous formula for marginal utility was incorrect. The correct definitions are partial derivatives as shown here.

Exercises and Worked Examples

Tasks include deriving the MRS and marginal utilities for various utility functions, such as those provided in the lecture.
For example, you might derive these for functions like $U(X,Y) = aX^{1/3} + bY^{2/3}$ .

Special Forms of Indifference Curves

Here we look at specific types of indifference curves that represent particular kinds of preferences:

Perfect Substitutes: - This occurs when a consumer views two goods as completely interchangeable (e.g., different brands of generic aspirin). The utility function is linear, like $U = aX + bY$ , which results in a constant MRS, meaning the consumer is always willing to swap one for the other at a fixed rate.
Perfect Complements: - These are goods that are consumed together in fixed proportions (e.g., left shoes and right shoes). The utility function is $U = \text{min}(X,Y)$ , meaning utility is determined by the smaller quantity of the pair. This is illustrated by L-shaped (right-angle) indifference curves.
Quasi-Linear Preferences: - With this type of preference, utility increases linearly with one good, but at a decreasing rate with the other (e.g., $U = u(X) + Y$ ). The unique feature is that these indifference curves are typically parallel and shift vertically as the utility level changes, implying that the marginal utility for the second good (Y) remains constant, regardless of how much of X is consumed.

Summary of Learning Outcomes

By the conclusion of the session, students should be able to:
- Explain and apply the foundational assumptions (axioms) of consumer preference to understand why indifference curves have certain properties.
- Analyze the characteristics of different utility functions and what they tell us about consumer choices.
- Derive and interpret the Marginal Rate of Substitution, and understand other mathematical aspects that describe indifference curves.
- Articulate the nature and importance of special forms of indifference curves, like perfect substitutes and complements, and what they reveal about consumer behavior.