# Constrained Optimization when Calculus Doesn't Work

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 9

Think about maximizing each of these functions subject to the constraint \(0 \le x \le 10\).

Plot the graph on that interval; then find and plot the derivative \(f'(x)\) on that same interval.

**Which function(s) reach their maximum in the domain [0, 10] at a point where \(f'(x) = 0\)?**

Recall: from Monday

Sufficient conditions for an interior optimum characterized by \(f'(x)=0\) with constraint \(x \in [0,10]\)

- \(f'(0) > 0\)
- \(f'(10) < 0\)
- \(f'(x)\) continuous and strictly decreasing on \([0,10]\)

# Last Time: MRS, MRT, and the “Gravitational Pull" towards Optimality

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.

# Corner Solutions

## Interior Solution:

## Corner Solution:

Optimal bundle contains

**strictly positive** quantities of both goods

Optimal bundle contains **zero** of one good

(spend all resources on the other)

If only consume good 1: \(MRS \ge MRT\) at optimum

If only consume good 2: \(MRS \le MRT\) at optimum

What would Lagrange find...?

What would Lagrange find...?

Remember: the Lagrange method finds the point along the PPF where the MRS equals the MRT.

Set MRS = MRT:

Plug into PPF:

What would Lagrange find...?

What would the Lagrange method find for the optimal quantity of fish, \(x_1^*\)?

pollev.com/chrismakler

You: Lagrange, I'd like you to find me a maximum please.

Lagrange: Here you go.

You: but that has a negative quantity of good 1! That's impossible!

Lagrange:

Lagrange will only find you

the point along the *mathematical description of the constraint* where the slope of the constraint is equal to the slope of the level set of the objective function passing through that point.

It doesn't care about your petty insistence on positivity.

# Kinks (Discontinuities)

## Kinked Indifference Curve:

## Kinked Constraint:

Discontinuities in the MRS

(e.g. Perfect Complements utility function)

Discontinuities in the MRT

(e.g. homework question with two factories)

What are some bundles that give the same utility as (4,8)?

What is the MRS at (4,8)? What about in the other place where the indifference curve intersects the PPF?

Here are some more indifference curves.

Where is the optimal bundle?

How can we solve for the optimal bundle?

# Monotonicity and Convexity

If preferences are **nonmonotonic**,

you might be satisfied consuming something within the interior of your feasible set.

FISH

COCONUTS

PPF

If preferences are **nonconvex**,

the tangency condition might find a **minimum** rather than a **maximum**.

FISH

COCONUTS

PPF

## Conditions for Calculus to Work

avoids a satiation point within the constraint

At the left corner of the constraint, \(MRS > MRT\)

avoids a corner solution when \(x_1 = 0\)

Monotonicity (more is better)

At the right corner of the constraint, \(MRS < MRT\)

avoids a corner solution when \(x_2 = 0\)

MRS and MRT are continuous as you move along the constraint

avoids a solution at a kink

ensures FOCs find a maximum, not a minimum

Convexity (variety is better)

## Next Steps

- Wednesday: apply all this to the situation of a consumer who is buying things and constrained by money, rather than someone who is making things and constrained by resources
- Section: Review old exam
- Friday: Midterm!!

#### Econ 50 | Lecture 09

By Chris Makler

# Econ 50 | Lecture 09

Constrained Optimization when Calculus Doesn't Work

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