Christopher Makler
Stanford University Department of Economics
Econ 51: Lecture 5
Part 1: Efficiency
Part 2: Equity
From preferences to allocations
Pareto improvements
Pareto efficiency and the "contract curve"
Some applications
The Utility Possibilities Frontier
Social preferences
Altruism
Fairness
An endowment is a vector saying how much of different goods an agent has.
Alison has 120 oreos and 20 twizzlers
Bob has 80 oreos and 80 twizzlers
Suppose \(g(x_1,x_2)\) is monotonic (increasing in both \(x_1\) and \(x_2\)).
Then \(k - g(x_1,x_2)\) is negative if you're outside of the constraint,
positive if you're inside the constraint,
and zero if you're along the constraint.
OBJECTIVE
FUNCTION
CONSTRAINT
FIRST ORDER CONDITIONS
3 equations, 3 unknowns
Solve for \(x_1\), \(x_2\), and \(\lambda\)
FIRST ORDER CONDITIONS
TANGENCY
CONDITION
CONSTRAINT
You have 40 feet of fence and want to enclose the maximum possible area.
You have 40 feet of fence and want to enclose the maximum possible area.
OBJECTIVE
FUNCTION
CONSTRAINT
FIRST ORDER CONDITIONS
TANGENCY
CONDITION
CONSTRAINT
TANGENCY
CONDITION
CONSTRAINT
Suppose you have \(F\) feet of fence instead of 40.
TANGENCY
CONDITION
CONSTRAINT
SOLUTIONS
Suppose you have \(F\) feet of fence instead of 40.
SOLUTIONS
Maximum enclosable area as a function of F:
Suppose you have \(F\) feet of fence instead of 40.
Marginal Rate of Transformation (MRT)
Marginal Rate of Substitution (MRS)
Both of these are measured in
coconuts per fish
(units of good 2/units of good 1)
Marginal Rate of Transformation (MRT)
Marginal Rate of Substitution (MRS)
Opportunity cost of marginal fish produced is less than the number of coconuts
you'd be willing to "pay" for a fish.
Opportunity cost of marginal fish produced is more than the number of coconuts
you'd be willing to "pay" for a fish.
Better to spend less time fishing
and more time making coconuts.
Better to spend more time fishing
and less time collecting coconuts.
Better to produce
more good 1
and less good 2.
Better to produce
more good 2
and less good 1.
These forces are always true.
In certain circumstances, optimality occurs where MRS = MRT.
The story so far, in two graphs
Production Possibilities Frontier
Resources, Production Functions → Stuff
Indifference Curves
Stuff → Happiness (utility)
Both of these graphs are in the same "Good 1 - Good 2" space
Better to produce
more good 1
and less good 2.
Better to produce
less good 1
and more good 2.
We've just seen that, at least under certain circumstances, the optimal bundle is
"the point along the PPF where MRS = MRT."
CONDITION 1:
CONSTRAINT CONDITION
CONDITION 2:
TANGENCY
CONDITION
This is just an application of the Lagrange method!
(see other deck for worked examples)
Examine cases where the optimal bundle is not characterized by a tangency condition.
New concepts:
corner solutions and kinks.