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.
MRS
>
p_1/p_2
MRS
<
“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.
p_1/p_2
MRS > \frac{p_1}{p_2}
MRS < \frac{p_1}{p_2}
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:
MRS = \frac{p_1}{p_2}
p_1x_1 + p_2x_2 = m
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?
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
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
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
u(x_1,x_2) = 100\ln x_1 + x_2
x_1 + 2x_2 = 100
\displaystyle{MRS = {MU_1 \over MU_2} = {100/x_1 \over 1} = {100 \over x_1}}
UTILITY FUNCTION
BUDGET CONSTRAINT
What is the MRS if you spend half your money on good 1?
\displaystyle{\text{Price ratio} = {p_1 \over p_2} = {1 \over 2}}
What is the MRS if you spend all your money on good 1?
\displaystyle{{100 \over 50} = 2}
\displaystyle{{100 \over 100} = 1}
\displaystyle{> {1 \over 2}}
\displaystyle{> {1 \over 2}}
Quasilinear
u(x_1,x_2) = 100\ln x_1 + x_2
x_1 + 2x_2 = 100
\displaystyle{MRS = {MU_1 \over MU_2} = {100/x_1 \over 1} = {100 \over x_1}}
UTILITY FUNCTION
BUDGET CONSTRAINT
\displaystyle{\text{Price ratio} = {p_1 \over p_2} = {1 \over 2}}
What would you get if you set the MRS equal to the price ratio?
\displaystyle{{100 \over x_1} = {1 \over 2}}
\displaystyle{x_1^* = 200}
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
u(x_1,x_2) = x_1 + 4x_2
x_1 + 2x_2 = 100
\displaystyle{MRS = {MU_1 \over MU_2} = {1 \over 4}}
UTILITY FUNCTION
BUDGET CONSTRAINT
\displaystyle{\text{Price ratio} = {p_1 \over p_2} = {1 \over 2}}
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
u(x_1,x_2) = x_1^2 + (2x_2)^2
u(x_1,x_2) = x_1^2 + x_2^2
Budget Line:
x_1 + x_2 = 12
12
6
12
6
x_1
x_2
Plot some indifference curves.
What is the "gravitational pull" showing you?
Concave Utility
u(x_1,x_2) = x_1^2 + (2x_2)^2
u(x_1,x_2) = x_1^2 + x_2^2
1. Plot utility along the budget line
2. Plot MRS vs. price ratio along the same budget line
x_1 + x_2 = 12
(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
u(x_1,x_2) = \min\{3x_1,2x_2\}
x_1 + x_2 = 100
UTILITY FUNCTION
BUDGET CONSTRAINT
= \begin{cases}3x_1 & \text{ if }3x_1 < 2x_2\\2x_2 & \text{ if }3x_1 > 2x_2\end{cases}
MU_1 = \begin{cases}3 & \text{ if }3x_1 < 2x_2\\0 & \text{ if }3x_1 > 2x_2\end{cases}
MU_2 = \begin{cases}0 & \text{ if }3x_1 < 2x_2\\2 & \text{ if }3x_1 > 2x_2\end{cases}
Perfect Complements
MRS = \begin{cases}\infty & \text{ if }3x_1 < 2x_2\\0 & \text{ if }3x_1 > 2x_2\end{cases}
MU_2 = \begin{cases}0 & \text{ if }3x_1 < 2x_2\\2 & \text{ if }3x_1 > 2x_2\end{cases}
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.
MU_1 = \begin{cases}3 & \text{ if }3x_1 < 2x_2\\0 & \text{ if }3x_1 > 2x_2\end{cases}
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
