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Intro to Multivariate Calculus
Christopher Makler
Stanford University Department of Economics
Econ 50: Lecture 3
Today's Agenda
- Sketching a Function
- Solution Functions
- Multivariable Functions and Level Sets
- Unit Overview
Sketching a Function
Homework QuestionΒ
How can you sketch this without a calculator?
Solution Functions
\(N_C\) identical consumers, each of whom
has the demand function
\(N_F\) identical firms produce good 1, each of which
has the supply function
Solve for the equilibrium price and quantity if \(\alpha = \frac{1}{4}, m = 100, N_C = 64, w = 4, \overline K = 2, N_F = 16\)
Demand
Supply
Suppose that instead of 16 firms, we had only 9 firms.
Then, we would expect the equilibrium price to _____ and the equilibrium quantity to _____.
(Hint: think about what happens to the market demand and supply curves.)
\(N_C\) identical consumers, each of whom
has the demand function
\(N_F\) identical firms produce good 1, each of which
has the supply function
Solve for the solution functionsΒ for the equilibrium price and quantity.
Demand
Supply
Multivariable Functions
[INDEPENDENT VARIABLES]
[DEPENDENT VARIABLE]
Multivariable Chain Rule
Total Derivative Along a Path
Total Derivative Along a Path
The total change in the height of the function due to a small increase in \(x\)
The amount \(f\) changes due to the increase in \(x\)
[indirect effect through \(y\)]
The amount \(f\) changes due to an increase in \(y\)
The amount \(y\) changes due to an increase in \(x\)
[direct effect from \(x\)]
Unit I Overview
Unit I: Scarcity and Choice
Economics is the study of how
we use scarce resourcesΒ
to satisfy our unlimited wants
Resources
Goods
Happiness
π
βοΈ
π€
Monday: Modeling Production
with Multivariable Functions
Labor
Fish
π
Capital
Coconuts
π₯₯
[GOODS]
β³
β
[RESOURCES]
Unit I: Scarcity and Choice
Economics is the study of how
we use scarce resourcesΒ
to satisfy our unlimited wants
Resources
Goods
Happiness
π
βοΈ
π€
π
π₯₯
Production Possibilities Fronier
Feasible
Wednesday:
Resource Constraints
Unit I: Scarcity and Choice
Economics is the study of how
we use scarce resourcesΒ
to satisfy our unlimited wants
Resources
Goods
Happiness
π
βοΈ
π€
π
π₯₯
π
π
π
π’
π
Unit I: Scarcity and Choice
Economics is the study of how
we use scarce resourcesΒ
to satisfy our unlimited wants
Resources
Goods
Happiness
π
βοΈ
π€
Week 3:
Preferences and Utility
π
π₯₯
Optimal choice
π
π
π
π’
π
Unit I: Scarcity and Choice
Economics is the study of how
we use scarce resourcesΒ
to satisfy our unlimited wants
Resources
Goods
Happiness
π
βοΈ
π€
Week 4:
Constrained Optimization
Econ 50 | Lecture 03
By Chris Makler
Econ 50 | Lecture 03
Review of Univariate Calculus
- 206