Christopher Makler
Stanford University Department of Economics
Econ 50: Lecture 14
Part 1: Resource Constraints and the PPF
Part 2: Optimization
The "Desert Island" Model
Resource constraints and the PPF
Deriving the equation of the short-run PPF
Shifts in the PPF
The Marginal Rate of Transformation
Relationship between MPL and MRT
The "Gravitational Pull" argument
Tangency when calculus works
Corners and kinks
Economics is the study of how
we use scarce resources
to satisfy our unlimited wants
Resources
Goods
Happiness
🌎
⌚️
🤓
Production functions
Utility functions
Demand
Supply
Equilibrium
🤩
🏪
⚖
Labor
Fish
🐟
Coconuts
🥥
[GOOD 1]
⏳
[GOOD 2]
Fish production function
Coconut production function
Resource Constraint
Fish production function
Coconut production function
Resource Constraint
Consider an economy with \(\overline L = 100\) units of labor and \(\overline K = 100\) units of capital.
In the short run, \(K_1 = 64\) and \(K_2 = 36\).
In the long run, capital can be reallocated in any combination between goods 1 and 2.
Max in SR
Max in LR
The New York Times, Oct. 29, 2013
Insider, July 23, 2020
Consider an economy with \(\overline L = 100\) units of labor and \(\overline K = 100\) units of capital.
In the short run, \(K_1 = 64\) and \(K_2 = 36\).
In the long run, capital can be reallocated in any combination between goods 1 and 2.
Max in SR
Max in LR
Note: we will generally treat this as a positive number
(the magnitude of the slope)
Suppose we're allocating 100 units of labor to fish (good 1),
and 50 of labor to coconuts (good 2).
Now suppose we shift
one unit of labor
from coconuts to fish.
How many fish do we gain?
100
98
300
303
How many coconuts do we lose?
Fish production function
Coconut production function
Resource Constraint
PPF
CHECK YOUR UNDERSTANDING
Chuck has \(\overline L = 8\) total hours of labor,
and the production functions
\(x_1 = 2 \sqrt{L_1}\) and \(x_2 = 4\sqrt{L_2}\).
What is his MRT if he spends
half his time producing each good?
CHECK YOUR UNDERSTANDING
Charlene has the PPF given by
\(2x_1^3 + 3x_2^4 = 1072\)
What is her MRT if she produces the output combination \((8,2)\)?
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.
What would Lagrange find...?
What would Lagrange find...?
What would Lagrange find...?
Discontinuities in the MRS
(e.g. Perfect Complements utility function)
Discontinuities in the MRT
(e.g. homework question with two factories)
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?