Beyond Well-Behaved Preferences
Christopher Makler
Stanford University Department of Economics
Econ 50: Lecture 11Q
Desirable Properties of Preferences
We've asserted that all (rational) preferences are complete and transitive.
There are some additional properties which are true of some preferences:
- Monotonicity
- Convexity
- Continuity
- Smoothness
Monotonic Preferences: “More is Better"
\text{For any bundles }A=(a_1,a_2)\text{ and }B=(b_1,b_2)\text{, }A \succeq B \text{ if } a_1 \ge b_1 \text{ and } a_2 \ge b_2
\text{In other words, all marginal utilities are positive: }MU_1 \ge 0, MU_2 \ge 0
Convex Preferences: “Variety is Better"
Take any two bundles, \(A\) and \(B\), between which you are indifferent.
Would you rather have a convex combination of those two bundles,
than have either of those bundles themselves?
If you would always answer yes, your preferences are convex.
Concave Preferences: “Variety is Worse"
Take any two bundles, \(A\) and \(B\), between which you are indifferent.
Would you rather have a convex combination of those two bundles,
than have either of those bundles themselves?
If you would always answer no, your preferences are concave.
Some Examples of
Not-Well-Behaved Preferences
- Perfect Complements
- Concave
- Satiation Point
- "Curve Balls"
Preferences over Tea and Biscuits
Charlotte always has 2 biscuits with every cup of tea;
if she has a plate of biscuits and one cup of tea, she'll only eat two. (And if she has a whole pot of tea and 6 biscuits, she'll stop pouring after 3 cups of tea.)
Each (cup + 2 biscuits) gives her 10 utils of joy.
Which other bundles give her the same utility as
3 cups of tea and 4 biscuits (A)?
Which other bundles give her the same utility as
1 cup of tea and 4 biscuits (B)?
3
6
Any combination that has 4 biscuits
and 2 or more cups of tea
Any combination that has 1 cup of tea and
at 2 or more biscuits
4
5
1
2
3
6
4
5
1
2
A
B
Perfect Complements
Goods that you like to consume
in a constant ratio.
Left shoes and right shoes
Sugar and tea
u(x_1,x_2) = \min \left\{\frac{x_1}{a},\frac{x_2}{b}\right\}
Concave
The opposite of convex:
as you consume more of a good,
you become more willing to give up others
-
MRS is increasing in \(x_1\) and/or decreasing in \(x_2\)
-
Indifference curves are bowed away from the origin.
e.g., u(x_1,x_2) = ax_1^2 + bx_2^2
Satiation Point
There is some ideal bundle; utility falls off as you move away from that bundle
-
Not monotonic
-
Realistic, but often the satiation point is far out of reach.
e.g., u(x_1,x_2) = 100 - (x_1 - 10)^2 - (x_2 - 10)^2
Satiation Point
e.g., u(x_1,x_2) = 100 - (x_1 - 10)^2 - (x_2 - 10)^2
Curve Balls
u(x_1,x_2,x_3) = x_1 + \min\{x_2,x_3\}
Good 1: burritos
Good 2: burgers
Good 3: fries
Constrained Optimization with Not-Well-Behaved Preferences
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
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
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\}
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
Big Takeaways
- As long as preferences are monotonic, the "gravitational pull" argument always works!
- For most of these, you have to apply logic -- there isn't a simple formula.
Sufficient Conditions for Lagrange 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)
(i.e. to ensure the optimum is characterized by the unique point
along the PPF in the first quadrant where MRS = MRT)
For the Econ 50Q Exam
- Know how capacity constraints (Leonteif production functions) lead to a kinked PPF
- Know how to maximize perfect complements, satiation point, and concave preferences
- No "curve balls" :)
Econ 50 | Lecture 11Q
By Chris Makler
Econ 50 | Lecture 11Q
Constrained Optimization with Calculus
