Microeconomic Theory

SAROO0OON

webpage_4.pdf

Home >Business & Finance homework help >Economics homework help >Microeconomic Theory

Consumer Choice

Intermediate Micro Theory ECN 312

From Last Class

• We characterized what bundles a consumer can afford.

• Previously, we looked at how people compare happiness from bundles (preferences).

• But, (out of the bundles that the consumer can afford) which one will they choose?

From Last Class

Preferences Budget (Income)

Choice (Individual Demand)

Population

Market Demand

From Last Class

Preferences Budget (Income)

Choice (Individual Demand)

Population

Market Demand

Quick Preview y

Which of these bundles makes the consumer

happiest?

Quick Preview y

Which of these bundles makes the consumer

happiest?

Which ones are

affordable? Which one does she choose?

Quick Preview y

BUT – can she do

better than B?

• What feasible choice will make me as happy as possible?

• In other words – how do I get to the “best” indifference curve without spending more money than I have?

• In other words – how do I pick x and y to make u(x,y) as large as possible while satisfying Pxx + Pyy ≤ m

Decision Problem

Decision Problem - Graphically Clothing

Food

PC = 3 PF = 2 m = 30

A B C

Decision Problem - Graphically Clothing

Food

PC = 3 PF = 2 m = 30

A B C

What is wrong with bundle A?

Decision Problem - Graphically Clothing

Food

PC = 3 PF = 2 m = 30

A B C

What is wrong with bundle B?

Decision Problem - Graphically Clothing

Food

PC = 3 PF = 2 m = 30

A B C

What is wrong with bundle C?

Decision Problem - Graphically Clothing

Food

PC = 3 PF = 2 m = 30

A B C

What is wrong with bundle D?

Decision Problem - Graphically Clothing

Food

PC = 3 PF = 2 m = 30

A B C

What is wrong with bundle E?

Decision Problem - Graphically Clothing

Food

PC = 3 PF = 2 m = 30

What is true about bundle E?

Decision Problem - Graphically Clothing

Food

PC = 3 PF = 2 m = 30

1-Consumer spends all of her money

2-Slope of the IC = slope of the BL

Decision Problem Analytically

• At the optimal bundle:

– The slope of the IC will be equal to the slope of the BL (tangency)

– All money will be spent

• The slope of the IC is given by: -MRS = -

• The slope of the BL is given by: -

• So, at the optimum: =

MUx MUyPx

MUx MUy

Px Py

Decision Problem Analytically

• Let’s look at our food and clothing example:

– The prices (PC = 3, PF = 2) and income (m=30) hold

– Preferences are simple Cobb-Douglas:

u(F,C) = FC

• The slope of the IC is given by:

-MRS = - = -

• The slope of the BL is given by:

- = -

• So, at the optimum: - = - C = F

MUF MUC

PF PC

C F

2 3

C F

2 3

Decision Problem Analytically

• We know that the consumer chooses C = F

• We know that the consumer spends all her income, i.e., we know that the optimal bundle will satisfy the constraint:

PC C + PF F = m

3 C + 2 F = 30

• Putting these together:

3( F) + 2 F = 30

2 F + 2 F = 30

4F = 30

F* = 7.5

2 3

Decision Problem Analytically

• We know that the consumer chooses C = F

• And we now know that F* = 7.5

• Putting these together:

C = F

C = (7.5)

C* = 5

2 3

Decision Problem Analytically

The formal way to solve constrained optimization problems is as a Lagrangian

Decision Problem Analytically

The formal way to solve constrained optimization problems is as a Lagrangian

L = (objective function) – λ(constraint function)

Decision Problem Analytically

The formal way to solve constrained optimization problems is with a Lagrangian

L = (objective function) – λ(constraint function)

L = U(x,y) – λ(Px x + Py y - m)

Decision Problem Analytically

L = U(x,y) – λ(Px x + Py y - m)

To solve, you three first-order conditions:

1. With respect to x: 𝜕ℒ

𝜕𝑥 = 0

2. With respect to y: 𝜕ℒ

𝜕𝑦 = 0

3. With respect to L: 𝜕ℒ

𝜕λ = 0

Decision Problem Analytically

L = U(x,y) – λ(Px x + Py y - m)

To solve, you three first-order conditions:

1. 𝜕ℒ

𝜕𝑥 = 𝜕𝑈

𝜕𝑥 − 𝜆𝑃𝑥 = 0  MUx = 𝜆𝑃𝑥

2. 𝜕ℒ

𝜕𝑦 = 𝜕𝑈

𝜕𝑦 − 𝜆𝑃𝑦 = 0  MUy = 𝜆𝑃𝑦

3. 𝜕ℒ

𝜕λ = 𝑃𝑥𝑥 + 𝑃𝑦𝑦 − 𝑚 = 0  𝑃𝑥𝑥 + 𝑃𝑦𝑦 = 𝑚

Decision Problem Analytically

L = U(x,y) – λ(Px x + Py y - m)

To solve, you three first-order conditions:

1. 𝜕ℒ

𝜕𝑥 = 𝜕𝑈

𝜕𝑥 − 𝜆𝑃𝑥 = 0  MUx = 𝜆𝑃𝑥

2. 𝜕ℒ

𝜕𝑦 = 𝜕𝑈

𝜕𝑦 − 𝜆𝑃𝑦 = 0  MUy = 𝜆𝑃𝑦

3. 𝜕ℒ

𝜕λ = 𝑃𝑥𝑥 + 𝑃𝑦𝑦 − 𝑚 = 0  𝑃𝑥𝑥 + 𝑃𝑦𝑦 = 𝑚

𝑀𝑈𝑥 𝑀𝑈𝑦

= 𝑃𝑥 𝑃𝑦

Decision Problem Analytically

L = U(x,y) – λ(Px x + Py y - m)

To solve, you three first-order conditions:

1. 𝜕ℒ

𝜕𝑥 = 𝜕𝑈

𝜕𝑥 − 𝜆𝑃𝑥 = 0  MUx = 𝜆𝑃𝑥

2. 𝜕ℒ

𝜕𝑦 = 𝜕𝑈

𝜕𝑦 − 𝜆𝑃𝑦 = 0  MUy = 𝜆𝑃𝑦

3. 𝜕ℒ

𝜕λ = 𝑃𝑥𝑥 + 𝑃𝑦𝑦 − 𝑚 = 0  𝑃𝑥𝑥 + 𝑃𝑦𝑦 = 𝑚

𝑀𝑈𝑥 𝑀𝑈𝑦

= 𝑃𝑥 𝑃𝑦

What about λ?

• The mathematical interpretation is how sensitive the optimization problem is to the constraint.

• Economists often call it the shadow price.

• In utility maximization, it tells us how much our utility will go up by with a relaxation of the constraint (or an increase in income). Can you think of another way to describe λ???

Decision Problem - Graphically Clothing

Food

PC = 3 PF = 2 m = 30

EC* = 5

F* = 7.5

Decision Problem

• In general, a consumer’s optimal bundle is found when two conditions are satisfied:

– The MRS is equal to the price ratio (PR)

– All income is spent

• Think of this as the “usual” case. It occurs when we have “nice” preferences and “nice” budget lines.

What do we mean by “nice”???

Decision Problem

• Why spend all income?

– If consumer has money left over, she could buy more goods and be better off.

– If consumer can be better off, she’s not at the optimum.

• Why satisfy tangency condition (MRS = PR)?

– Suppose not. Then (we’ll show this) the consumer can be made better off by re-arranging her bundle.

– If consumer can be better off, she’s not at the optimum.

Decision Problem y

Suppose that you choose a bundle A where MRS > PR

Decision Problem y

Now, suppose that you increase the amount of x by 1

unit….

Decision Problem y

Now, suppose that you increase the amount of x by 1

unit….

Decision Problem y

Now, suppose that you increase the amount of x by 1

unit….

How much y do you have to give up to

raise $1?

Decision Problem y

Now, suppose that you increase the amount of x by 1

unit….

How much y do you have to give up to

raise $Px?

Decision Problem y

Now, suppose that you increase the amount of x by 1

unit….

How much y do you have to give up to

raise $Px?

Px/Py

Decision Problem y

Now, suppose that you increase the amount of x by 1

unit….

How much y will you give up to stay on the same IC?

Decision Problem y

PR MRS

Progress so far…..

• A general rule for optimization:

- Choose the IC as far from the origin as possible, while not exceeding the budget.

• The usual rule for optimization:

– Look for a bundle that lies at the tangency of the IC and the budget line and exhausts all $$$.

• The usual case is when the MRS is diminishing and the ICs do not touch the axes.

Corner Solutions

• Sometimes (not a “usual” case), it is better to buy zero units of a product.

• (Almost always) happens with perfect substitutes (can you think of one time it wouldn’t?)

• Can happen with other preferences (as long as the ICs have intercepts)

Perfect Substitutes y

Perfect SubstitutesPaper Towels 10

Napkins

Paper towels and napkins with:

U = T + 2N

Price of T = 1

Price of N = 3

m = 10

3.33

Perfect SubstitutesPaper Towels 10

Napkins

Paper towels and napkins with:

U = T + 2N

Price of T = 1

Price of N = 3

m = 10

3.33

And if the price of

napkins fell to 10

cents???

Perfect Complements y

A B

Perfect Complements Jelly

Peanut Butter

U = min{PB,J}

Price of PB = 1

Price of J = 1

m = 1010

105

Perfect Complements Jelly

Peanut Butter

U = min{PB,J}

Price of PB = 1

Price of J = 1

m = 1010

105

What if price of J falls to 50

cents?

Perfect Complements Jelly

Peanut Butter

U = min{PB,J}

Price of PB = 1

Price of J = .5

m = 10

6.67

Intercept = 20

Example: Cobb-Douglas Utility

Consider xylophones and yo-yos with:

u(x,y) = 3x1/2y1/4

Px = 4, Py = 3, m = 12

• Cobb-Douglas  “nice” preferences

• Constant price ratio  “nice” budget

 use tangency condition to find optimum

Cobb-Douglas – Solve for MRS

Consider xylophones and yo-yos with:

u(x,y) = 3x1/2y1/4

Px = 4, Py = 3, m = 12

• MRSxy= MUx/MUy • MUx= (1/2)3 x

-1/2y1/4 = (3/2) x-1/2y1/4

• MUy= (1/4)3 x 1/2y-3/4 = (3/4) x1/2y-3/4

• MRSxy= MUx/MUy = (2y)/x

Cobb-Douglas – Set equal to price Ratio

Consider xylophones and yo-yos with:

u(x,y) = 3x1/2y1/4

Px = 4, Py = 3, m = 12

• The price ratio (negative slope of the BL)= 4/3

• Setting MRS = PR  (2y)/x = 4/3

 2y*3 = 4*x

 y = (2x)/3

Cobb-Douglas – Plug into budget

Consider xylophones and yo-yos with:

u(x,y) = 3x1/2y1/4

Px = 4, Py = 3, m = 12

• We know that at the optimum:

– MRSxy = PRxy  y = (2x)/3

– All money will be spent  4 x + 3 y = 12

• Putting this together:

4x + 3((2x)/3)) = 12  x* = 2 and y*=(4/3)

Example: Gas Taxes or Standards

• We have spent considerable energy learning how to describe optimal choices between two goods.

• What is the benefit??

– We can now explain attributes of consumer choices that are hard with simple demand analysis

• Example: Suppose we want to reduce gasoline consumption. Should we use mileage standards or gas taxes?

Example: Gas Taxes or Standards

• Gary Becker (Business Week) argues that gas taxes are the best way. Note this this is not an uncommon stance among economists….

• Suppose that consumers care about two things:

– The miles that they travel (T)

– All other consumption (C)

• Each product has a price and consumers have some (limited) budget

Example: Gas Taxes or Standards

• Questions:

– Why is the good “travel” instead of gasoline?

– What is the effect of a gas tax, holding car type fixed?

– What is the effect of better fuel efficiency, holding gas prices fixed?

Example: Gas Taxes or Standards C

What does a gas tax do?

Example: Gas Taxes or Standards C

What does a gas tax do?

Example: Gas Taxes or Standards C

What does a gas tax do?

What has happened to

gas consumption?

Example: Gas Taxes or Standards C

What does a mileage standard (increase fuel

efficiency) do?

Example: Gas Taxes or Standards C

What does a mileage standard (increase fuel

efficiency) do?

Example: Gas Taxes or Standards C

What does a mileage standard (increase fuel

efficiency) do?

What has happened to

gas consumption?

Example: Gas Taxes or Standards

• What did we learn?

– Without our analysis, it would have been hard to understand what we know about this policy issue and where additional evidence is needed.

– Simple models can illuminate important (and otherwise complicated) issues

– Much of microeconomic analysis applies to the idea that prices matter and shift behavior.

Consumer Choice: Main Ideas

• We have looked at how consumers make choices, conditional on prices and income.

– Choices depend on preferences

– Choices depend on affordability

• Consumer goal: maximize happiness given budget.

• “Usual” way to find optimum: set MRS = PR and spend all money.