pollev.com/chrismakler
Exam Locations for Monday
| Group | Location |
|---|---|
| OAE | Check your email! |
| Econ 50Q | 420-040 |
| Econ 50, Last name A-L | 420-040 |
| Econ 50, Last name M-Z | 320-105 (this room) |
Because the class is so large, we need to have multiple rooms for the exam on Monday.
I emailed OAE students this morning at around 10:40 with their location.
Please confirm via the poll that you know where to go!
Constrained Optimization when Calculus Doesn't Work
Christopher Makler
Stanford University Department of Economics
Econ 50: Lecture 11
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
CONSTRAINT
UTILITY FUNCTION
u(x_1,x_2) = 100 \ln x_1 + x_2
x_1 + 2x_2 = 100
\mathcal{L}(x_1,x_2,\lambda)=
100 \ln x_1 + x_2
100 - x_1 - 2x_2
+ \lambda\ (
)
\displaystyle{\partial \mathcal{L} \over \partial x_1} =
FIRST ORDER CONDITIONS
\displaystyle{\partial \mathcal{L} \over \partial x_2} =
\displaystyle{\partial \mathcal{L} \over \partial \lambda} =
\displaystyle{100 \over x_1}
1
100 - x_1 - 2x_2
=0
- \lambda
{1 \over 2}
- \lambda\ \times
=0
=0
\displaystyle{\Rightarrow \lambda\ =}
\displaystyle{\Rightarrow \lambda\ = }
2
\Rightarrow
Utility from last hour spent fishing
Utility from last hour spent collecting coconuts
Equation of PPF
\displaystyle{100 \over x_1}
x_1 + 2x_2 = 100
{1 \over 2}
\displaystyle{\lambda\ =}
\displaystyle{\lambda\ = }
Utility from last hour spent fishing
Utility from last hour spent collecting coconuts
Equation of PPF
\displaystyle{100 \over x_1}
x_1 + 2x_2 = 100
What would the Lagrange method find for the optimal quantity of fish, \(x_1^*\)?
What would the Lagrange method find for the optimal quantity of coconuts, \(x_2^*\)?
MRS = {100 \over x_1}
What would Lagrange find...?
MRS = {100 \over x_1}
What would Lagrange find...?
CONSTRAINT
UTILITY FUNCTION
u(x_1,x_2) = x_1 + 4x_2
x_1 + {x_2^2 \over 100} = 100
MRS =
MRT =
{1 \over 4}
{50 \over x_2}
What would the Lagrange method find for the optimal quantity of fish, \(x_1^*\)?
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 problem on specialization)
50Q only - Friday!
Next Steps
- A practice midterm has been posted on Canvas (separate for 50/50Q).
Solutions will be posted on Friday night at 8pm. - Section: old exam questions, including some pretty tricky ones!
- Friday: Econ 50Q lecture on optimization with some not-well-behaved utility functions (concave, satiation point, perfect complements, etc.)
- Saturday: SEA study party!
- Sunday: Review session, 3-5pm, Shriram 104
- Monday: exam! Look out for what room and what seat you will be assigned to!
Econ 50 | Lecture 11
By Chris Makler
Econ 50 | Lecture 11
Constrained Optimization when Calculus Doesn't Work
