# The Edgeworth Box Framework

Christopher Makler

Stanford University Department of Economics

Econ 51: Lecture 5

## Today's Agenda

Part 1: Efficiency

Part 2: Equity

From preferences to allocations

**Pareto improvements**

**Pareto efficiency** and the "contract curve"

Some applications

The Utility Possibilities Frontier

Social preferences

Altruism

Fairness

## From Preferences to Allocations

An **endowment** is a vector saying how much of different goods an agent has.

### Definition

### Example

Alison has 120 oreos and 20 twizzlers

Bob has 80 oreos and 80 twizzlers

## Constrained Optimization

## Canonical Constrained Optimization Problem

Suppose \(g(x_1,x_2)\) is monotonic (increasing in both \(x_1\) and \(x_2\)).

Then \(k - g(x_1,x_2)\) is **negative** if you're **outside** of the constraint,

**positive** if you're **inside** the constraint,

and **zero** if you're **along** the constraint.

OBJECTIVE

FUNCTION

CONSTRAINT

FIRST ORDER CONDITIONS

3 equations, 3 unknowns

Solve for \(x_1\), \(x_2\), and \(\lambda\)

## How does the Lagrange method work?

## It finds the point** along the constraint** where the

**level set of the objective function **passing through that point

is** tangent **to the constraint

FIRST ORDER CONDITIONS

TANGENCY

CONDITION

CONSTRAINT

## Example: Fence Problem

You have 40 feet of fence and want to enclose the maximum possible area.

## Example: Fence Problem

You have 40 feet of fence and want to enclose the maximum possible area.

OBJECTIVE

FUNCTION

CONSTRAINT

FIRST ORDER CONDITIONS

TANGENCY

CONDITION

CONSTRAINT

TANGENCY

CONDITION

CONSTRAINT

## Meaning of the Lagrange multiplier

Suppose you have \(F\) feet of fence instead of 40.

TANGENCY

CONDITION

CONSTRAINT

SOLUTIONS

## Meaning of the Lagrange multiplier

Suppose you have \(F\) feet of fence instead of 40.

SOLUTIONS

Maximum enclosable area as a function of F:

## Meaning of the Lagrange multiplier

Suppose you have \(F\) feet of fence instead of 40.

# Verbal Analysis: MRS, MRT, and the “Gravitational Pull" towards Optimality

## Fish vs. Coconuts

- Can spend your time catching
**fish**(good 1)

or collecting**coconuts**(good 2) - What is your optimal
**division of labor**

between the two? - Intuitively: if you're optimizing, you

**couldn't reallocate your time**in a way

that would make you better off. - The last hour devoted to fish must

bring you the**same amount of utility**

as the last hour devoted to coconuts

**Marginal Rate of Transformation** (MRT)

- The number of coconuts you
**need**to give up in order to get another fish - Opportunity cost of fish in terms of coconuts

**Marginal Rate of Substitution** (MRS)

- The number of coconuts you are
**willing**to give up in order to get another fish - Willingness to "pay" for fish in terms of coconuts

**Both of these are measured in
coconuts per fish**

(units of good 2/units of good 1)

**Marginal Rate of Transformation** (MRT)

- The number of coconuts you
**need**to give up in order to get another fish - Opportunity cost of fish in terms of coconuts

**Marginal Rate of Substitution** (MRS)

- The number of coconuts you are
**willing**to give up in order to get another fish - Willingness to "pay" for fish in terms of coconuts

Opportunity cost of marginal fish produced is **less** than the number of coconuts

you'd be willing to "pay" for a fish.

Opportunity cost of marginal fish produced is **more** than the number of coconuts

you'd be willing to "pay" for a fish.

Better to spend **less** time fishing

and **more** time making coconuts.

Better to spend **more** time fishing

and **less** time collecting coconuts.

Better to produce

**more good 1**

and **less good 2**.

## “Gravitational Pull" Towards Optimality

Better to produce

**more good 2**

and **less good 1**.

These forces are **always true**.

In **certain circumstances**, optimality occurs where MRS = MRT.

# Graphical Analysis:

PPFs and Indifference Curves

The story so far, in two graphs

Production Possibilities Frontier

Resources, Production Functions → Stuff

Indifference Curves

Stuff → Happiness (utility)

Both of these graphs are in the same "Good 1 - Good 2" space

Better to produce

**more good 1**

and **less good 2**.

Better to produce

**less good 1**

and **more good 2**.

# Mathematical Analysis:

Lagrange Multipliers

We've just seen that, at least under certain circumstances, the optimal bundle is

"the point along the PPF where MRS = MRT."

CONDITION 1:

CONSTRAINT CONDITION

CONDITION 2:

TANGENCY

CONDITION

This is just an application of the Lagrange method!

(see other deck for worked examples)

## Next Time

Examine cases where the optimal bundle is **not** characterized by a tangency condition.

New concepts:

**corner solutions** and **kinks.**

#### Econ 51 | 5 |The Edgeworth Box Framework

By Chris Makler

# Econ 51 | 5 |The Edgeworth Box Framework

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