# Constrained Optimization when Calculus Doesn't Work

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 5

pollev.com/chrismakler

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

Insight #1:

If preferences are monotonic, the optimal bundle must lie **along** the budget line.

If preferences are **nonmonotonic**,

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

Apples

Bananas

BL

BL'

...or it might not!

Better to buy

**more good 1**

and **less good 2**.

## “Gravitational Pull" Towards Optimality

Better to buy

**more good 2**

and **less good 1**.

These forces are **always true**.

In **certain circumstances**, optimality occurs where MRS = p1/p2.

Indifference curve is

**steeper** than the budget line

Indifference curve is

**flatter** than the budget line

Moving to the **right**

along the budget line

would increase utility.

Moving to the **left**

along the budget line

would increase utility.

**More** willing to give up good 2

than the market requires.

**Less** willing to give up good 2

than the market requires.

IF...

THEN...

The consumer's preferences are "well behaved"

- smooth
- strictly monotonic
- strictly convex

The indifference curves do not cross the axes

The budget line is a simple straight line

The optimal consumption bundle will be characterized by two equations:

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

# Constrained Optimization with One Variable

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

### Review: Section

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

**Which functions have a maximum at the point where \(f'(x) = 0\)? Why?**

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 \([0,10]\)

# Corner Solutions

How does the "Gravitational Pull" argument apply in each of these cases?

Is the solution characterized by \(MRS = p_1/p_2\)?

## Quasilinear

UTILITY FUNCTION

BUDGET CONSTRAINT

What is the MRS if you spend half your money on good 1?

What is the MRS if you spend *all* your money on good 1?

## Quasilinear

UTILITY FUNCTION

BUDGET CONSTRAINT

What would you get if you set the MRS equal to the price ratio?

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.

## Quasilinear

- Use Lagrange first
- If Lagrange finds solution in the first quadrant, you're done.
- If Lagrange finds a solution in the second or fourth quadrants, go to the nearest corner.
- "Gravitational force" argument applies still!

## Perfect Substitutes

UTILITY FUNCTION

BUDGET CONSTRAINT

What would you get if you tried to set the MRS equal to the price ratio?

What is the relationship between the MRS and the price ratio along the budget constraint?

## Perfect Substitutes

- MRS is constant; compare it to the price ratio ("gravitational pull")
- If \(MRS > p_1/p_2\), buy only good 1
- If \(MRS < p_1/p_2\), buy only good 2
- If \(MRS = p_1/p_2\), indifferent between all points along budget line

Concave Utility

Budget Line:

12

6

12

6

Plot some indifference curves.

What is the "gravitational pull" showing you?

Concave Utility

1. Plot utility along the budget line

2. Plot MRS vs. price ratio along the same budget line

(12,0)

(6,6)

utility

(12,0)

(6,6)

units of 2

units of 1

(0, 12)

(0, 12)

## Concave Utility

- Utility
*increases*as you move toward an axis - Tangency condition will find you the
*minimum**utility*point along the budget line - Solution method: check utility at corners. Which good should you spend all your income on?

# Solutions at Kinks

How does the "Gravitational Pull" argument apply in each of these cases?

Is the solution characterized by \(MRS = p_1/p_2\)?

## Perfect Complements

UTILITY FUNCTION

BUDGET CONSTRAINT

## Perfect Complements

If \(3x_1 < 2x_2\), you would get more utility from additional good 1, and not give up any utility by having less good 2; you should move to the right along your budget constraint.

If \(3x_1 > 2x_2\), you got no utility from the last unit of good 1, but you would get utility from additional good 2; you should move to the left along your budget constraint.

## Perfect Complements

- Set minimands equal to each other: e.g., if \(u(x_1,x_2) = \min\{2x_1,x_2\}\) , set \(x_2 = 2x_1\).
- Substitute into budget line to find point.
- Note that the "gravitational pull" argument holds: to the left of the ideal point, MRS is infinite; to the right, MRS is zero.

## Summary of Not-Well-Behaved Utility Functions

- Have to apply logic: Lagrange will not always work!
- Not monotonic (e.g., satiation point): check if optimum lies
*within*budget set - Not convex (e.g., concave or perfect substitutes)

or if convex but indifference curves cross the axes (quasilinear): check corners - Not smooth (e.g., perfect complements): use "gravitational pull" argument

# Kinked Constraints

## Kinked Budget Constraints

- Check MRS at the kink
- If it's less than the price ratio to the left, use equation for BL to the left
- If it's more than the price ratio to the right, use equation for BL to the right
- If it's between the two, kink is optimal
- If price ratio to the left is
*greater than*the price ratio to the right,

solve both ways and compare utility (kind of like concave)

## Conditions for Calculus to Work

avoids a satiation point within the constraint

At the left corner of the constraint, \(MRS > p_1/p_2\)

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

Monotonicity (more is better)

At the right corner of the constraint, \(MRS < p_1/p_2\)

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

MRS and price ratio 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)

# Conclusions and Next Steps

Bringing preferences and budget sets together, we find the most preferred bundle in a budget set.

Under **certain important conditions**

the optimal consumption bundle will be the point along the budget line

where the consumer's MRS is equal to the price ratio.

However, if those conditions are not met, we need to apply logic:

**always compare MRS and price ratio!!!**

Along a budget line, if the MRS is greater than the price ratio,

the consumer gets more "bang for their buck" from good 1 than good 2;

so they can be made better off by choosing more of good 1 and less of good 2; and vice versa.

Next time: examine what happens when prices and income change

#### Econ 50| 5 | Constrained Optimization without Calculus

By Chris Makler

# Econ 50| 5 | Constrained Optimization without Calculus

Constrained optimization when the solution can't be found by setting the MRS equal to the price ratio

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