Consumer Theory

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1 Lecture 1: Introduction to Microeconomics

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2 Lecture 2: Preferences and Utility Functions

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3 Lecture 3: Budget Constraints

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4 Exercises

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Lecture 1: Introduction to Microeconomics

1.1.1 What is microeconomics?

Microeconomics is the study of how individuals and firms maximize their well-being in a world of scarcity. The core of microeconomics is the study of constrained optimization and assessing tradeoffs. The key concept behind tradeoffs is opportunity cost: every action or inaction has a cost in terms of what could have been done instead.

💡 핵심 개념

Study of constrained optimization
Focus on scarcity and tradeoffs
Opportunity cost is central

1.1.2 Modeling in microeconomics

A model is any description of the relationship between two or more economic variables. Economic models are simplified representations of relationships between variables.

💡 핵심 개념

Models simplify economic relationships
Supply and Demand Model is fundamental

Supply and Demand Model

Demand curve is downward-sloping. It shows the relationship between price and quantity demanded. It measures the willingness of consumers to buy a certain good. Supply curve is upward-slowing. It shows the relationship between price and quantity supplied. It measures the willingness of producers to sell. The intersection of supply and demand curve is the market equilibrium. Each point on the demand curve shows how much consumers will demand at a given price. Each point on the supply curve shows how much producers will supply at a given price. At the equilibrium price, suppliers are willing to supply as much as demanders will demand.

💡 핵심 개념

Demand curve: downward-sloping
Supply curve: upward-sloping
Market equilibrium at intersection

1.1.3 Positive vs. normative economics

Positive analysis is the study of the way things are (e.g. eBay auctions). Normative analysis is the study of the way that things should be (e.g. should organ sales be legal?).

💡 핵심 개념

Positive: what is
Normative: what should be

1.1.4 Market economy

Capitalistic economy: individuals and firms decide what to produce and consume, subject to limited restrictions by the government (similar to laissez-faire). Command economy: government in control with production and allocation (e.g. inefficiency and corruption of Soviet Union). Invisible hand: Adam Smith's concept, self-regulating nature of markets and self-interest

💡 핵심 개념

Capitalistic economy: individual choice
Command economy: government control
Invisible hand: self-regulation

1.1.5 TO KNOW - Conceptual Understanding

Microeconomics studies how individuals and firms make optimal choices under scarcity, focusing on trade-offs and opportunity cost. Models are simplified tools to understand behavior; positive economics explains what is, while normative economics debates what should be

💡 핵심 개념

Understand scarcity and opportunity cost
Know difference between positive and normative analysis

Lecture 2: Preferences and Utility Functions

1.2.1 Consumer preferences

Consumer choices are based on preferences and budget constraints. To model consumer preferences, there are three assumptions: Completeness (when comparing two bundles of goods, you either prefer one, prefer the other, or are indifferent), Transitivity (If consumer prefers bundle x to bundle y, and bundle y to bundle z, then must prefer bundle x to bundle z), Non-Satiation (More of a good is always better, consumers never get satiated)

💡 핵심 개념

Completeness
Transitivity
Non-Satiation

1.2.2 Indifference curves

We use indifference curves as the basic graphical tool of consumer theory. There are four important properties of indifference curves: Consumers prefer higher indifference curves, Indifference curves are downward-sloping, Indifference curves never cross, There is one indifference curve through each possible consumption bundle

💡 핵심 개념

Higher curves preferred
Downward-sloping
Never cross
One curve per bundle

1.2.3 Utility

Utility is a way of mapping preferences. We use utility to get ordinal ranking, not cardinal ranking. Utility function translates consumer utility from different consumption bundles into units, that can then be compared. Marginal utility is the derivative of utility with respect to good. It measures how utility changes as consumers consume more of a good. The important principle of diminishing marginal utility states that consumers receive less utility from each unit of a good they consume.

MRS = -\frac{MU_x}{MU_y} = -\frac{\delta U/\delta x}{\delta U/\delta y}

💡 핵심 개념

Utility provides ordinal ranking
Marginal utility measures change
Diminishing marginal utility principle
MRS is ratio of marginal utilities

Marginal rate of substitution (MRS)

The slope of the indifference curve is called the marginal rate of substitution (MRS). Marginal rate of substitution (MRS) = rate at which consumers are willing to trade Y axis for X axis. MRS is the ratio of marginal utilities. MRS is diminishing as you move along the indifference curve

💡 핵심 개념

MRS = slope of indifference curve
MRS diminishes along curve

1.2.4 TO KNOW - Graphical and Math Understanding

Know required skills for graphical and mathematical understanding

💡 핵심 개념

Prove indifference curves never cross using a figure
Prove indifference curves are downward sloping using a figure
Draw indifference curves for perfect complements and perfect substitutes
Sketch indifference curve given verbal description
Calculate marginal utilities given a utility function
Calculate marginal rate of substitution given a utility function

Lecture 3: Budget Constraints

1.3.1 Budget constraint

Consumers have limited resources: their budget constraint. One simplifying assumption is that budget is equal to income (I). Budget over two goods X and Y is defined to be

\[I = p_X X + p_Y Y\]

MRT = -\frac{p_X}{p_Y}

💡 핵심 개념

Budget = Income assumption
Budget line equation: $I = p_X X + p_Y Y$
MRT is slope of budget constraint
Price/income changes shift/rotate budget line

The slope of budget constraint

The slope of budget constraint is defined as marginal rate of transformation (MRT): rate at which you can transform one good into the other in the marketplace. Intuitively, with a fixed budget, by choosing one thing you are by definition reducing the money you have to spend on other things. Shifts in price and income alter the position and slope of the budget constraint. For example, if the price of good X increases, the budget constraint flattens. If the income decreases, the budget constraint shifts inwards.

💡 핵심 개념

MRT = marginal rate of transformation
MRT = slope of budget line
Price changes rotate the line
Income changes shift the line

1.3.2 Constrained optimization

The goal of constrained choice is to maximize utility subject to the budget constraint. Preferences are represented by indifference curves. The optimal bundle that a consumer can choose is defined by the point where indifference curve is tangent to the budget constraint

MRS = -\frac{MU_X}{MU_Y} = -\frac{\delta U/\delta X}{\delta U/\delta Y} = -\frac{p_X}{p_Y} = MRT

💡 핵심 개념

Maximize utility subject to budget
Optimal point: indifference curve tangent to budget line
At optimum: MRS = MRT
Interior solutions vs corner solutions

Optimal bundle condition

At this point, slope of indifference curve = slope of budget constraint. This is equivalent to equating the marginal cost and benefit of consuming each good. The above equation defines an interior solution (in which the consumer consumes some of each good); if indifference curves are fat, there can also be corner solutions in which the consumer only consumes one good

💡 핵심 개념

Slope equality at optimal point
Interior vs corner solutions

1.3.3 TO KNOW - Graphical and Math Understanding

Required skills for understanding budget constraints

💡 핵심 개념

Know how to write down a budget constraint given prices and income
Show graphically how to find the bundle that maximizes the consumer's utility subject to the budget constraint

Exercises

🎯 연습문제

1. Prove that indifference curves never cross using a figure

귀류법으로 증명:
1. 두 무차별곡선이 점 A에서 교차한다고 가정
2. 곡선1 위의 점 B와 곡선2 위의 점 C를 선택
3. A~B (곡선1), A~C (곡선2)이므로 이행성에 의해 B~C
4. 그러나 B와 C는 서로 다른 무차별곡선 위에 있어 모순

∴ 무차별곡선은 교차할 수 없다

🎯 연습문제

2. Prove that indifference curves are downward sloping using a figure

단조성(monotonicity) 가정 사용:
1. 소비자는 더 많은 것을 선호 (more is better)
2. 점 A(x₁, y₁)에서 오른쪽 위로 이동하면 더 나은 상태
3. 같은 효용을 유지하려면 y를 감소시켜야 함

$\frac{dy}{dx} < 0$ (우하향)

🎯 연습문제

3. Draw indifference curves corresponding to perfect complements and perfect substitutes

완전보완재: L자형 곡선
예: 왼쪽 신발과 오른쪽 신발
$U = \min(x, y)$

완전대체재: 직선
예: 1달러 동전과 100센트
$U = x + y$

🎯 연습문제

4. Know how to sketch an indifference curve given a verbal description of a consumer's preferences

주요 패턴:
1. 일반재: 볼록한 곡선 (원점 방향)
2. 완전대체재: 직선
3. 완전보완재: L자형
4. 중립재: 수직선 또는 수평선
5. 나쁜 재화: 우상향 곡선

🎯 연습문제

5. Calculate marginal utilities given a utility function

효용함수 $U(x,y)$가 주어졌을 때:

$MU_x = \frac{\partial U}{\partial x}$

$MU_y = \frac{\partial U}{\partial y}$

예: $U = x^{0.5}y^{0.5}$
$MU_x = 0.5x^{-0.5}y^{0.5}$

🎯 연습문제

6. Calculate marginal rate of substitution given a utility function

$MRS = -\frac{MU_x}{MU_y} = -\frac{\partial U/\partial x}{\partial U/\partial y}$

예: $U = x^{0.5}y^{0.5}$
$MRS = -\frac{0.5x^{-0.5}y^{0.5}}{0.5x^{0.5}y^{-0.5}} = -\frac{y}{x}$

🎯 연습문제

7. Know how to write down a budget constraint given prices and income

예산제약식:

$p_x \cdot x + p_y \cdot y = I$

또는 y에 대해 정리:
$y = \frac{I}{p_y} - \frac{p_x}{p_y}x$

기울기: $-\frac{p_x}{p_y}$
y절편: $\frac{I}{p_y}$

🎯 연습문제

8. Show graphically how to find the bundle that maximizes the consumer's utility subject to the budget constraint

최적 조건 (내부해):
1. 무차별곡선과 예산선이 접하는 점
2. $MRS = -\frac{p_x}{p_y}$
3. $\frac{MU_x}{p_x} = \frac{MU_y}{p_y}$ (한계효용 균등화)

그래프: 예산선과 가장 높은 무차별곡선의 접점