Lecture 3. Consumer Theory (consumer theory)

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1. Overview
2. Utility Function
3. Marshallian Demand Function
4. Hicksian Demand Function
5. Demand Curve

1. Overview

1) Goods : denoted by x, y
2) Price : denoted by p
3) Budget : denoted by I

2. Utility Function (utility function)

1) Consumption bundle : a set of goods considered for consumption
2) Utility : the satisfaction consumers expect to obtain when purchasing goods or services
3) Preference : microeconomics assumes consumers consider utility over all feasible choices and then choose the best option
4) Utility function (utility function)

(1) Definition

Total utility obtained by consuming a particular bundle among various bundles

A function that provides a way to rank consumption bundles based on utility levels

There are few restrictions on the choice of a utility function: we care only about which option is preferred, not by how much

If you compose a utility function with a strictly increasing function, the resulting function can also serve as a utility function

(Note) A representative example of a utility measure is monetary value converted under a currency system

(2) Cardinal utility vs. ordinal utility

Cardinal utility: a utility function where concrete absolute values can be defined

Ordinal utility: a utility function that cannot assign meaningful absolute values but can rank (compare) bundles

Using ordinal utility is usually not a problem: the marginal rate of substitution does not require the utility function to be unique

(3) Example 1. Cobb–Douglas utility function

(4) Marginal utility (marginal utility): related to uₓ and uᵧ

Definition: the additional utility gained from consuming one more unit of a particular good

(5) Average utility (average utility)

5) Indifference curve : related to y(x) (because it can be separated in an implicit-function form)

(1) Definition

A curve representing combinations that yield the same utility for an individual (also called an equal-utility curve)

An indifference curve can be viewed as a contour line of “height” given by utility

(2) Properties

Property 1: indifference curves representing different utility levels cannot intersect

Property 2: indifference curves slope downward (negative slope)

Reason: even if consumption of x decreases, increasing y appropriately can keep utility the same

This reflects that consumers can substitute one good for another to maintain a fixed level of satisfaction

(3) Marginal rate of substitution (MRS) (marginal rate of substitution): related to -y′(x)

Definition: the rate at which the consumer is willing to substitute one good for another while maintaining the same satisfaction

Relationship to marginal utilities: the negative sign is used so that MRS is defined as a positive number

Over an infinitesimal interval, if x decreases and y increases by MRS, utility remains constant

Tip: MRS is proportional to dy/dx; since uₓ is inversely related to dx and uᵧ is inversely related to dy, the numerator/denominator take that form

Law of diminishing MRS: as you move to the right along an indifference curve, MRS gradually decreases

6) Convexity : related to y″(x) (convex to the origin; also described as “convex downward”)

(1) Intuitive definition

An “average” choice yields higher utility than an extreme choice

(2) Mathematical definition

y = y(x) is convex downward. In fact, a “convex function” commonly refers to a function that is convex downward in this convention

If MRS is a monotonically decreasing function, the indifference curve is convex (because MRS = -y′(x))

Example 1: u(x, y) = max{x, y} is not convex

Example 2: u(x, y) = min{x, y} is not convex

(3) Law of diminishing marginal utility (law of diminishing marginal utility) or diminishing marginal rate of substitution

Definition: as consumption of a specific good increases, its marginal utility decreases (i.e., marginal utility is a decreasing function)

In other words, as consumption of one good increases, the amount of the other good the consumer is willing to give up for one more unit decreases

Reason: consumers prefer more balanced consumption bundles

Example: after eating the first 100 g of beef, the additional satisfaction from eating another 100 g is relatively small

3. Marshallian Demand Function (Marshallian demand function)

1) Assumption: x and y are types of goods; pₓ and pᵧ are their respective prices
2) Budget constraint (budget constraint)

(1) Definition

Consumers cannot spend beyond their budget (income level)

It is straightforward to see that when pₓx + pᵧy = I, utility u(x, y) is maximized at the optimal choice

3) Utility maximization problem (utility maximization problem)

(1) Solution method

Solve using the Lagrange multiplier method

(2) Example 1

Choose goods in quantities inversely proportional to their prices

(3) Example 2

There is no interior maximum; when MRS is larger than pₓ / pᵧ, it is advantageous to increase x

Figure 1. Indifference curves for Example 2

⒜: tangent line to the utility function; the magnitude of the slope is MRS

⒝: set of (x, y) that keeps the budget constant; the magnitude of the slope is pₓ / pᵧ

Interpretation 1: At ⒜, increasing x leaves unused budget, so it is advantageous

Interpretation 2: At ⒝, increasing x moves to a higher indifference curve, so it is advantageous

(Note) Since pₓ is in the numerator and pᵧ in the denominator, this corresponds to a situation where x is relatively cheap and y is relatively expensive

(4) (Note) Corner solution

Solve the utility maximization problem under x ≥ 0, y ≥ 0

(5) (Note) Interior solution

Solve the utility maximization problem under x > 0, y > 0

4) Budget line (budget line)

(1) Definition

A graph of consumption bundles that can be chosen given a particular budget

If the utility function is convex, utility is maximized at the point where the line pₓx + pᵧy = I is tangent to an indifference curve

5) Marshallian demand function : refers to the solutions x(pₓ, pᵧ, I) and y(pₓ, pᵧ, I) to the utility maximization problem

(Note) In general, “demand function” refers to the Marshallian demand function

Gross substitutes: when the price of another good rises, this good sells relatively better

Gross complements: when the price of another good rises, the complementary good sells relatively worse

6) Indirect utility function : the utility value evaluated at x(pₓ, pᵧ, I) and y(pₓ, pᵧ, I), i.e., the maximum utility under given prices and budget

4. Hicksian Demand Function (Hicksian demand function, compensated demand function)

1) Expenditure minimization problem (expenditure minimization problem)
2) Hicksian demand function : refers to the solutions xᶜ(pₓ, pᵧ, k) and yᶜ(pₓ, pᵧ, k) to the expenditure minimization problem
3) Expenditure function (expenditure function) : the minimum required budget to achieve a given preference (utility) level

5. Demand Curve (demand curve)

1) Demand curve

(1) If we fix pᵧ and I, we can find a function x = x(pₓ)

Figure 2. Price-consumption curve and demand curve

(2) The graph of x = x(pₓ) is called the demand curve
(3) In general, x = x(pₓ) has an inverse function, so we can also find pₓ = pₓ(x)
(4) Change in quantity demanded: movement along the demand curve (a change in quantity)
(5) Change in demand: a change in consumption caused by a shift of the demand curve itself
(6) The original demand curve is typically downward sloping and convex to the origin, but there are exceptions

Figure 3. Demand curves for perfect substitutes, perfect complements, and when one is a bad

2) Changes in consumer choice

(1) Price effect (price effect)

Definition: the change in consumption when a good’s price changes

Price effect = substitution effect + income effect

(2) Substitution effect (substitution effect)

Definition: the change in consumption needed to maintain the same satisfaction as before a price change, while achieving that satisfaction at minimum cost after the price change

The difference in demands between points on the same indifference curve

(3) Income effect (income effect)

Also called the “real income effect”

Definition: the change in consumption induced by the change in real income caused by the price change

The difference in demands between points on different indifference curves when the price ratio pₓ/pᵧ is held constant

(4) Formalization

Figure 4. Diagram of substitution effect and income effect

Price effect: ● → ●

Substitution effect: ● → ●

Income effect: ● → ●

(5) Special case 1: Giffen good (e.g., potatoes)

A special case of an inferior good: an inferior good for which the income effect is larger than the substitution effect

The demand curve slopes upward

(6) Special case 2: Network effect (network effect)

An individual’s demand is influenced by other people’s demand

With network effects, the market demand curve cannot be obtained by simply summing individual demand curves horizontally

(7) Special case 3: Bandwagon effect (bandwagon effect)

Purchasing something because it is fashionable

(8) Special case 4: Snob effect (snob effect)

Demand decreases as the number of people consuming the good increases

Example: luxury-brand consumption motivated by exclusivity

3) Elasticity of demand

(1) Price elasticity of demand (price elasticity of demand): related to the price effect

Definition: a measure of how much the quantity demanded changes in response to a change in the good’s price

A negative sign is attached

Tip: among several demand curves that intersect at a point, the steepest one is the most inelastic

Magnitude of price elasticity

Elastic ( ε > 1): when price falls, P × Qd rises (necessary and sufficient condition)

Inelastic ( ε < 1): when price falls, P × Qd falls (necessary and sufficient condition)

Unit elastic ( ε = 1): total revenue is maximized; when price falls, P × Qd stays constant (necessary and sufficient condition)

(Note) “Elastic” means consumers can more easily forgo purchasing when the price rises

Determinants of price elasticity

Luxury vs. necessity: luxuries are easier to give up → elasticity ↑

Availability of substitutes: more substitutes → elasticity ↑

Budget share: higher share → elasticity ↑

Market definition: narrower markets transmit information faster → elasticity ↑ (e.g., Coke vs. Pepsi)

Time horizon: in the long run, adaptation raises elasticity → elasticity ↑

Example: illegal drugs in the U.S.

Demand for illegal drugs is inelastic

Supply suppression raises prices and thus increases suppliers’ revenues, which is undesirable

Demand-reduction policies (e.g., anti-drug campaigns) are more desirable

(Note) If quantity demanded increases when price rises, price elasticity can be negative

Example 1: conspicuous consumption (luxury goods)

Example 2: Giffen goods

(2) Income elasticity of demand (income elasticity of demand): related to the income effect

Definition: a measure of how much quantity demanded changes in response to a change in income

No negative sign is attached

Normal vs. inferior goods

Engel curve: relationship between income and quantity demanded

Normal good: income elasticity > 0

Inferior good: income elasticity < 0 (e.g., staple foods)

Necessity: 0 < income elasticity < 1

Luxury: income elasticity > 1

(3) Cross elasticity of demand (cross elasticity of demand): related to the substitution effect

Definition: a measure of how much the quantity demanded of one good changes when the price of another good changes

No negative sign is attached

Substitute: cross elasticity is positive

Complement: cross elasticity is negative

4) Consumer surplus (consumer surplus)

(1) Definition

The integral of the difference between the demand curve and the price line pₓ = p₀, representing the value gained from obtaining good x

In other words, it indicates the satisfaction consumers obtain when they purchase a good at a lower price

(2) Note

It reflects “I almost paid more, but I bought it cheaper” (because the demand curve is decreasing: when q is small, consumers are willing to pay more)

(3) Formula

(Note) Typically, it is preferred to compute using the pₓ–x curve

Figure 5. Diagram of consumer surplus

(4) Note

In Producer Theory, there is a similar concept called producer surplus

Together, the two relate to marginal revenue

Input: 2020.03.16 10:26

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