Intermediate Microeconomics

DEMANDBEN VAN KAMMEN, PHDPURDUE UNIVERSITY

IntroductionWe have already discussed how consumers maximize utility subject to a budget constraint.◦ They get on the “highest” indifference curve possible and

consume a bundle of goods where their Marginal Rate of Substitution equals the Price Ratio of the two goods.

Deriving the consumer’s demand function for a good is only a small step beyond this principle.◦ How quantity demanded depends on prices, income,

and preferences.

Outline of objectives1. Derive a demand function from the consumer’s

maximization problem.

2. Evaluate assumptions about consumers that are embedded in a model and its solution.

3. Break down price-quantity changes into income and substitution effects.

4. Solve for market demand by aggregating individual demand functions.

Upshot of Consumer Choice: the Demand FunctionHow quantity demanded depends on prices, income, and preferences.

𝑄𝑄 = 𝑄𝑄 𝑃𝑃𝑗𝑗 ,𝑀𝑀 ,

where 𝑃𝑃𝑗𝑗 are the relevant prices and 𝑀𝑀 is income.

The consumer’s problemA consumer maximizes a utility function, such as:

𝑈𝑈 𝑋𝑋,𝑌𝑌 = 𝑋𝑋𝛼𝛼𝑌𝑌1−𝛼𝛼;𝛼𝛼 ∈ 0,1 .subject to a budget constraint such as:

𝑀𝑀 = 𝑋𝑋𝑃𝑃𝑋𝑋 + 𝑌𝑌𝑃𝑃𝑌𝑌.The marginal utilities of each good are:

𝑀𝑀𝑈𝑈𝑋𝑋 = 𝛼𝛼𝑋𝑋𝛼𝛼−1𝑌𝑌1−𝛼𝛼 and𝑀𝑀𝑈𝑈𝑌𝑌 = 1 − 𝛼𝛼 𝑋𝑋𝛼𝛼𝑌𝑌−𝛼𝛼 .

The consumer’s problemSo his MRS is:

𝑀𝑀𝑈𝑈𝑋𝑋𝑀𝑀𝑈𝑈𝑌𝑌

=𝛼𝛼

1 − 𝛼𝛼∗ 𝑋𝑋𝛼𝛼−1𝑌𝑌1−𝛼𝛼 ÷ 𝑋𝑋𝛼𝛼𝑌𝑌−𝛼𝛼

Or,𝑀𝑀𝑈𝑈𝑋𝑋𝑀𝑀𝑈𝑈𝑌𝑌

=𝛼𝛼

1 − 𝛼𝛼∗𝑋𝑋𝛼𝛼−1𝑌𝑌1−𝛼𝛼

𝑋𝑋𝛼𝛼𝑌𝑌−𝛼𝛼=

𝛼𝛼1 − 𝛼𝛼

∗𝑋𝑋−1𝑌𝑌1

1

=𝛼𝛼

1 − 𝛼𝛼∗𝑌𝑌𝑋𝑋

.

MRS = price ratioSetting 𝑀𝑀𝑈𝑈𝑋𝑋

𝑀𝑀𝑈𝑈𝑌𝑌= 𝑃𝑃𝑋𝑋

𝑃𝑃𝑌𝑌:𝛼𝛼

1 − 𝛼𝛼∗𝑌𝑌𝑋𝑋

=𝑃𝑃𝑋𝑋𝑃𝑃𝑌𝑌

The optimal quantity of YThis tells you the utility-maximizing level of Y is:

𝑌𝑌 =1 − 𝛼𝛼 𝑋𝑋𝑃𝑃𝑋𝑋𝛼𝛼𝑃𝑃𝑌𝑌

.

Constraint with optimal YCombining the previous slide with the budget we get:

𝑀𝑀 = 𝑋𝑋𝑃𝑃𝑋𝑋 +1 − 𝛼𝛼 𝑋𝑋𝑃𝑃𝑋𝑋𝛼𝛼𝑃𝑃𝑌𝑌

𝑃𝑃𝑌𝑌.

Or,

𝑀𝑀 = 𝑋𝑋𝑃𝑃𝑋𝑋 1 +1 − 𝛼𝛼𝛼𝛼

= 𝑋𝑋𝑃𝑃𝑋𝑋1𝛼𝛼

.

Demand functionSo,

𝑋𝑋 =𝛼𝛼𝑀𝑀𝑃𝑃𝑋𝑋

,

the Marshallian Demand Function for good X.The Law of Demand holds:

𝜕𝜕𝑋𝑋𝜕𝜕𝑃𝑃𝑋𝑋

= −𝛼𝛼𝑀𝑀𝑃𝑃𝑋𝑋2 < 0.

A typical demand curve

What we are used to . . . but shouldn’t the axes be reversed?

Alfred MarshallIf you want to graph demand on Marshall’s axes, you have to take its inverse.

For example instead of 𝑋𝑋 = 𝛼𝛼𝑀𝑀𝑃𝑃X

, you would re-write

it as:

𝑃𝑃𝑋𝑋 =𝛼𝛼𝑀𝑀𝑋𝑋

.

Outline of objectives1. Derive a demand function from the consumer’s

maximization problem.

2. Evaluate assumptions about consumers that are embedded in a model and its solution.

3. Break down price-quantity changes into income and substitution effects.

4. Solve for market demand by aggregating individual demand functions.

Introduction 1.6Graphically the demand curve is depicted beginning with the indifference curve map with which we are already familiar (see next slide).

For a consumer to maximize his utility, he finds a consumption bundle where the indifference curve is tangent to the budget constraint.

We want to analyze the effects of a price change beginning from this state.

Significance of αMarshallian Demand can be written:

𝑋𝑋𝑃𝑃𝑋𝑋 = 𝛼𝛼𝑀𝑀.α is the proportion of income that the consumer spends on good X.◦A fixed proportion of income on good X.

Demand and utility relationshipThe (“Cobb-Douglas” form) utility function,

𝑈𝑈 = 𝑋𝑋𝛼𝛼𝑌𝑌1−𝛼𝛼 ,

produced the demand function,

𝑋𝑋 =𝛼𝛼𝑀𝑀𝑃𝑃𝑋𝑋

.

Example of the Constant Elasticity of Substitution (CES) utility functions.

Elasticity of substitutionElasticity of substitution, denoted 𝜎𝜎, is

𝜎𝜎 =%Δ 𝑋𝑋

𝑌𝑌%Δ𝑀𝑀𝑀𝑀𝑀𝑀

=𝛿𝛿 𝑋𝑋

𝑌𝑌𝛿𝛿𝑀𝑀𝑀𝑀𝑀𝑀

𝑀𝑀𝑀𝑀𝑀𝑀𝑋𝑋𝑌𝑌

.

Higher 𝜎𝜎 means MRS doesn’t diminish, and the goods are more substitutable.

“Constant elasticity of substitution”

𝑀𝑀𝑀𝑀𝑀𝑀 =𝛼𝛼

1 − 𝛼𝛼𝑌𝑌𝑋𝑋

.

So the Cobb-Douglas elasticity of substitution is:

1 − 𝛼𝛼𝛼𝛼

∗𝛼𝛼

1 − 𝛼𝛼𝑌𝑌𝑋𝑋

𝑌𝑌𝑋𝑋

= 1.

Perfect complements (𝜎𝜎 = 0), and

Perfect substitutes (𝜎𝜎 → ∞) are also constant CES.

CES utility and price elasticityPrice elasticity of demand is also constant:Δ𝑋𝑋Δ𝑃𝑃𝑋𝑋

= −𝛼𝛼𝑀𝑀𝑃𝑃𝑋𝑋2 . . . times 𝑃𝑃𝑋𝑋

𝑋𝑋is:

𝜀𝜀𝑃𝑃 = −𝛼𝛼𝑀𝑀𝑋𝑋𝑃𝑃𝑋𝑋

.

𝜀𝜀𝑃𝑃 = −1 (substituting 𝑋𝑋 = 𝛼𝛼𝑀𝑀𝑃𝑃𝑋𝑋

from the demand function), so the price elasticity is constant (-1).

Outline of objectives1. Derive a demand function from the consumer’s

maximization problem.

2. Evaluate assumptions about consumers that are embedded in a model and its solution.

3. Break down price-quantity changes into income and substitution effects.

4. Solve for market demand by aggregating individual demand functions.

Optimal consumption

Budget constraints with different prices

Optimal consumption bundles

The budget constraint rotates outward as 𝑃𝑃X decreases.

Substitution effect

Income effect

Substitution and income effects together

Normal and inferior GoodsNormal Good: A good that is bought in greater quantities as income increases.

Inferior Good: A good that is bought in smaller quantities as income increases.

Income elasticity of demand (𝜀𝜀𝑀𝑀):◦𝜀𝜀𝑀𝑀 > 0 indicates a normal good;◦𝜀𝜀𝑀𝑀 < 0 indicates an inferior good.

Substitutes and complementsSubstitutes: Two goods such that, if the price of one increases, the quantity demanded of the other good rises.

Complements: Two goods such that, if the price of one increases, the quantity demanded of the other good decreases.

Complementarity and Substitutability are measured by the cross price elasticity of demand (𝜀𝜀𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑜𝑜𝑜𝑜 𝑜𝑜𝑜𝑜𝑜𝑃𝑃𝑃𝑃 𝐺𝐺𝑜𝑜𝑜𝑜𝐺𝐺). ◦If it is positive, the two goods are substitutes. If it is negative, the two goods are complements.

Consumer’s surplus

Outline of objectives1. Derive a demand function from the consumer’s

maximization problem.

2. Evaluate assumptions about consumers that are embedded in a model and its solution.

3. Break down price-quantity changes into income and substitution effects.

4. Solve for market demand by aggregating individual demand functions.

Market demandThe horizontal sum of individual demand functions.

3 consumers with demand functions:1. 𝑋𝑋1 = 3

𝑃𝑃𝑋𝑋

2. 𝑋𝑋2 = 2𝑃𝑃𝑋𝑋

3. 𝑋𝑋3 = 4𝑃𝑃𝑋𝑋

The market demand (X) is X1 + X2 + X3.

𝑋𝑋 =3 + 2 + 4

𝑃𝑃𝑋𝑋=

9𝑃𝑃𝑋𝑋

.

SummaryUtility maximization: the origin of the demand function.

Demand function maps prices, income to the quantity of the good consumed.

Price elasticity of demand: measures the responsiveness of quantity demanded to a price change.◦ Can be decomposed into a substitution and income effects.

Welfare measured by consumer’s surplus—the area below the demand curve and above the equilibrium price.Market demand is the sum of the individual consumers’ quantities at each given price.

Applying utility and pricesThe next topic is the simplest conceivable economy:◦ 2 Goods, 2 Consumers◦ The “Desert Island” exchange economy.

Equilibrium and gains from trade.