Review of
Consumer Theory

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 15

What's On The Test?

  • Budget line / budget set
  • Utility maximization
  • Demand curve
  • Price offer curve
  • Income offer curve
  • Hicks decomposition
  • Cost minimization

Graphs / Representations

  • Complements
  • Substitutes
  • Normal Goods
  • Inferior Goods
  • Income Effect
  • Substitution Effect
  • Compensating Variation
  • Equivalent Variation

Concepts

  • Utility-maximizing bundle given a budget constraint
  • Marshallian demand function
  • Cost-minimizing bundle given a utility constraint
  • Hicksian demand function

Derivations

You should be able to perform this kind of analysis for any utility function,
including ones you haven't seen before (or seen before in this context).

What's On The Test?

  • Budget line / budget set
  • Utility maximization

Graphs / Representations

Concepts

  • Utility-maximizing bundle given a budget constraint

Derivations

You should be able to perform this kind of analysis for any utility function,
including ones you haven't seen before (or seen before in this context).

m = \text{money income}
p_1 = \text{price of good 1}
p_2 = \text{price of good 2}
\text{horizontal intercept} = \frac{m}{p_1}
\text{vertical intercept} = \frac{m}{p_2}
\text{slope of budget line} = -\frac{p_1}{p_2}

Gravitational Pull Argument

move to the right along the budget line

move to the left along the budget line

MRS > {p_1 \over p_2}
MRS < {p_1 \over p_2}

IF...

THEN...

The consumer's preferences are "well behaved"
-- smooth, strictly convex, and strictly monotonic

\(MRS=0\) along the horizontal axis (\(x_2 = 0\))

The budget line is a simple straight line

The optimal consumption bundle will be characterized by two equations:

MRS = \frac{p_1}{p_2}
p_1x_1 + p_2x_2 = m

More generally: the optimal bundle may be found using the Lagrange method

\(MRS \rightarrow \infty\) along the vertical axis (\(x_1 \rightarrow 0\))

How do you tell if a preferences are "well behaved"?

Strictly monotonic

Strictly convex

Smooth

\(MU_1 > 0\) and \(MU_2 > 0\) for any \(x_1,x_2\)

\(\frac{\partial MRS}{\partial x_1} \le 0\) and \(\frac{\partial MRS}{ \partial x_2} \ge 0\), with at least one strict

MRS has no "jumps" (not defined piecewise)

Continuous

Utility function has no "jumps" (not defined piecewise)

(i.e., indifference curves get flatter as you move down and to the right)

The Tangency Condition

What happens when the price of a good increases or decreases?

The Tangency Condition when Lagrange sometimes works

What happens when income decreases?

What's On The Test?

  • Budget line / budget set
  • Utility maximization
  • Demand curve
  • Price offer curve
  • Income offer curve
  • Hicks decomposition
  • Cost minimization

Graphs / Representations

  • Complements
  • Substitutes
  • Normal Goods
  • Inferior Goods
  • Income Effect
  • Substitution Effect
  • Compensating Variation
  • Equivalent Variation

Concepts

  • Utility-maximizing bundle given a budget constraint
  • Marshallian demand function
  • Cost-minimizing bundle given a utility constraint
  • Hicksian demand function

Derivations

You should be able to perform this kind of analysis for any utility function,
including ones you haven't seen before (or seen before in this context).

✅

✅

✅

What's On The Test?

  • Budget line / budget set
  • Utility maximization
  • Demand curve

Graphs / Representations

Concepts

  • Utility-maximizing bundle given a budget constraint
  • Marshallian demand function

Derivations

You should be able to perform this kind of analysis for any utility function,
including ones you haven't seen before (or seen before in this context).

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✅

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Specific Prices & Income

General Prices & Income

\text{Constraint: }2 x_1 + x_2 = 12
\text{Constraint: }p_1x_1 + p_2x_2 = m
\text{Objective function: } x_1^{1 \over 2}x_2^{1 \over 2}
MRS(x_1,x_2) = {x_2 \over x_1}
x_1^* = 3
x_1^*(p_1,p_2,m) = {m \over 2p_1}
x_2^* = 2x_1^* = 6
x_2^*(p_1,p_2,m) = {m \over 2p_2}

OPTIMAL BUNDLE

DEMAND FUNCTIONS

(optimization)

(comparative statics)

What's On The Test?

  • Budget line / budget set
  • Utility maximization
  • Demand curve
  • Price offer curve
  • Income offer curve
  • Hicks decomposition
  • Cost minimization

Graphs / Representations

  • Complements
  • Substitutes
  • Normal Goods
  • Inferior Goods
  • Income Effect
  • Substitution Effect
  • Compensating Variation
  • Equivalent Variation

Concepts

  • Utility-maximizing bundle given a budget constraint
  • Marshallian demand function
  • Cost-minimizing bundle given a utility constraint
  • Hicksian demand function

Derivations

You should be able to perform this kind of analysis for any utility function,
including ones you haven't seen before (or seen before in this context).

✅

✅

✅

✅

✅

What's On The Test?

  • Budget line / budget set
  • Utility maximization
  • Demand curve
  • Price offer curve
  • Income offer curve

Graphs / Representations

  • Complements
  • Substitutes
  • Normal Goods
  • Inferior Goods

Concepts

  • Utility-maximizing bundle given a budget constraint
  • Marshallian demand function

Derivations

You should be able to perform this kind of analysis for any utility function,
including ones you haven't seen before (or seen before in this context).

✅

✅

✅

✅

✅

Remember what you learned about demand and demand curves in Econ 1 / high school:

  • The demand curve shows the quantity demanded of a good at different prices
  • A change in the price of a good results in a movement along its demand curve
  • A change in income or the price of other goods results in a shift of the demand curve
    • If two goods are substitutes, an increase in the price of one will increase the demand for the other (shift the demand curve to the right).
    • If two goods are complements, an increase in the price of one will decrease the demand for the other (shift the demand curve to the left).
    • If a good is a normal good, an increase in income will increase demand for the good
    • If a good is an inferior good, an increase in income will decrease demand the good
x_1^*(p_1,p_2,m)\ \

Three Relationships

...its own price changes?

Movement along the demand curve

...the price of another good changes?

Complements

Substitutes

Independent Goods

How does the quantity demanded of a good change when...

...income changes?

Normal goods

Inferior goods

Giffen goods

(possible) shift of the demand curve

x_1^*(p_1,p_2,m)\ \

Three Relationships

...its own price changes?

Movement along the demand curve

How does the quantity demanded of a good change when...

The demand curve for a good

shows the quantity demanded of that good

as a function of its own price

holding all other factors constant

(ceteris paribus)

The price offer curve shows how the optimal bundle changes in good 1-good 2 space as the price of one good changes.

x_1
x_1
x_2
p_1

DEMAND CURVE FOR GOOD 1

BL_{p_1 = 2}
BL_{p_1 = 3}
BL_{p_1 = 4}
2
3
4

"Good 1 - Good 2 Space"

"Quantity-Price Space for Good 1"

BL

PRICE OFFER CURVE

x_1^*(p_1,p_2,m)\ \

Three Relationships

...the price of another good changes?

How does the quantity demanded of a good change when...

Substitutes

Complements

When the price of one good goes up, demand for the other increases.

When the price of one good goes up, demand for the other decreases.

Independent

Demand not related

x_1
x_2
x_1
x_2

Complements:    \(p_2 \uparrow \Rightarrow x_1^* \downarrow\)

What happens to the quantity of good 1 demanded when the price of good 2 increases?

Substitutes:    \(p_2 \uparrow \Rightarrow x_1^* \uparrow\)

COMPLEMENTS:

UPWARD-SLOPING

PRICE OFFER CURVE

SUBSTITUTES:

DOWNWARD-SLOPING

PRICE OFFER CURVE

x_1^*(p_1,p_2,m)\ \

Three Relationships

How does the quantity demanded of a good change when...

...income changes?

Normal Goods

Inferior Goods

When your income goes up,
demand for the good increases.

When your income goes up,
demand for the good decreases.

The income offer curve shows how the optimal bundle changes in good 1-good 2 space as income changes.

x_1
x_2
x_1
x_2

Good 1 normal:    \(m \uparrow \Rightarrow x_1^* \uparrow\)

What happens to the quantity of good 1 demanded when the income increases?

Good 1 inferior:    \(m \uparrow \Rightarrow x_1^* \downarrow\)

BOTH NORMAL GOODS:

UPWARD-SLOPING

INCOME OFFER CURVE

ONE GOOD INFERIOR:

DOWNWARD-SLOPING

PRICE OFFER CURVE

CES Utility

= \begin{cases}\infty & \text{ if } x_1 < x_2 \\ 0 & \text{ if } x_1 > x_2 \end{cases}
-\infty
1
0
MRS = \left(x_2 \over x_1\right)^\infty
MRS = {x_2 \over x_1}
MRS = 1
r
u(x_1,x_2) = \min\{x_1,x_2\}
u(x_1,x_2) = x_1x_2
u(x_1,x_2) = x_1 + x_2

PERFECT
SUBSTITUTES

PERFECT
COMPLEMENTS

INDEPENDENT

PERFECT
SUBSTITUTES

u(x_1,x_2) = (x_1^r+x_2^r)^{1 \over r}

Constant Elasticity of Substitution (CES) Utility

MRS = \left(x_2 \over x_1\right)^{1-r}
  • A change in the price of a good results in a movement along its demand curve
  • A change in income or the price of other goods results in a shift of the demand curve

What's On The Test?

  • Budget line / budget set
  • Utility maximization
  • Demand curve
  • Price offer curve
  • Income offer curve
  • Hicks decomposition
  • Cost minimization

Graphs / Representations

  • Complements
  • Substitutes
  • Normal Goods
  • Inferior Goods
  • Income Effect
  • Substitution Effect
  • Compensating Variation
  • Equivalent Variation

Concepts

  • Utility-maximizing bundle given a budget constraint
  • Marshallian demand function
  • Cost-minimizing bundle given a utility constraint
  • Hicksian demand function

Derivations

You should be able to perform this kind of analysis for any utility function,
including ones you haven't seen before (or seen before in this context).

✅

✅

✅

✅

✅

✅

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Two Effects

Substitution Effect

Effect of change in relative prices, holding utility constant.

Effect of change in real income,
holding relative prices constant.

Income Effect

Decomposition Bundle

Suppose that, after a price change,
we compensated the consumer
just enough to afford some bundle
that would give the same utility
as they had before the price change?

The Hicks decomposition bundle
is the lowest-cost bundle
that satisfies this condition.

Approach

TOTAL EFFECT

A
C
B

INITIAL BUNDLE

FINAL BUNDLE

DECOMPOSITION BUNDLE

SUBSTITUTION EFFECT

INCOME  EFFECT

Utility Maximization

Cost Minimization

\max \ u(x_1,x_2)
\min \ p_1x_1 + p_2x_2
\text{s.t. }p_1x_1 + p_2x_2 = m
\text{s.t. }u(x_1,x_2) = U

Solution functions:
"Ordinary" Demand functions

x_1^*(p_1,p_2,m)
x_2^*(p_1,p_2,m)

Solution functions:
"Compensated" Demand functions

x_1^c(p_1,p_2,U)
x_2^c(p_1,p_2,U)

Hicks Decomposition Bundle

Suppose the price of good 1 increases from \(p_1\) to \(p_1^\prime\).

The price of good 2 (\(p_2\)) and income (\(m\)) remain unchanged.

Initial Bundle (A):
Solves
utility maximization
problem

Final Bundle (C):
Solves
utility maximization
problem

\max \ u(x_1,x_2)
\min \ p_1^\prime x_1 + p_2x_2
\text{s.t. }p_1x_1 + p_2x_2 = m
\text{s.t. }u(x_1,x_2) = U(A)

Decomposition Bundle (B):
Solves
cost minimization
problem

\max \ u(x_1,x_2)
\text{s.t. }p_1^\prime x_1 + p_2x_2 = m

Complements and Substitutes (one last time)

Complements

Substitutes

When the price of good 1 goes up...

Net effect: buy less of both goods

Net effect: buy less good 1 and more good 2

Substitution effect: buy less of good 1 and more of good 2

Income effect (if both goods normal): buy less of both goods

Substitution effect dominates

Income effect dominates

Compensating and Equivalent Variation

Compensating Variation

Equivalent
Variation

When the price of a good changes...

How much would your income need to change for you to be able to afford the initial utility at the new prices?

How much would your income need to change for you to be able to afford the new utility at the initial prices?

How to Calculate CV and EV

Compensating Variation

Equivalent Variation

How much would your income need to change for you to be able to afford the initial utility at the new prices?

How much would your income need to change for you to be able to afford the new utility at the initial prices?

  • Find lowest-cost way to achieve initial utility at the new prices (Hicks Decomposition bundle); this is at the intersection of \(U_1\) and \(IOC_2\)
  • Evaluate the cost of that bundle at the new prices
  • Compare that cost to your actual income
  • Find lowest-cost way to achieve new utility at the initial prices (this is actually the Hicks Decomposition bundle for the reverse price change); at the intersection of \(U_2\) and \(IOC_1\)
  • Evaluate the cost of that bundle at the initial prices
  • Compare that cost to your actual income

General Things To Think About

  • Read each question carefully. Answer that question.
    (A highlighter is a good idea!)
  • Watch out for corner solutions or other weirdness.
  • Clearly (but succinctly) write down your process!
    You can get lots of points even if you make a math error,
    but only if we know what you were trying to do.
  • People who "run out of time" generally spend too long
    in a rabbit hole, or write far too much in their answers.
    Get something down for each question before tackling algebra!

Good luck!