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 \(f'(x) = 0\)?

f(x) = 5 + 4x - x^2
f(x) = 10 - |2-x|
f(x) = 9 - (x-11)^2
f(x) = 1 + \tfrac{1}{5}(x-5)^2
f(x) = 10 - x
f(x) = 3

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 \([0,10]\)
f(x)
f'(x)
x
x
10
0
10

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

Better to produce
more good 1
and less good 2.

MRS
>
MRT
MRS
<
MRT

“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.

MRS
>
MRT
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...?

MRS = {100 \over x_1}

What would Lagrange find...?

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

MRT = {1 \over 2}
PPF: x_1 + 2x_2 = 100
{100 \over x_1} = {1 \over 2}

Set MRS = MRT:

x_1^* = 200

Plug into PPF:

200 + 2x_2 = 100
2x_2 = -100
x_2^* = -50
MRS = {100 \over x_1}

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)

PPF: x_1^2 + x_2^2 = 80
u(x_1,x_2) = \min\left\{{x_1 \over 2}, x_2\right\}

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?

PPF: x_1^2 + x_2^2 = 80
u(x_1,x_2) = \min\left\{{x_1 \over 2}, x_2\right\}

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

  • Two old exams have been posted on Canvas, which cover material through today's class. (Due to scheduling differences between quarters, that's as far as we had gotten those quarters.) These are the only old exams that will be posted, so use them wisely. Solutions will be visible as of Sunday.
  • Monday: 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. This material is fair game for the exam! Some old exam questions will be part of the homework for Monday's class.
  • Wednesday: derive demand curves from the consumer's optimal behavior. This is not covered on this exam, but will be on Checkpoint 2.
  • Friday: Checkpoint 1!!