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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 Question
f(x) = \ln(20x)
\text{Hint: }\ln(20) \approx 3
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
d(p_1|p_2,m) = \frac{\alpha m}{p_1}
\displaystyle s(p_1|w,r) = \frac{\overline K p_1}{2w}
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
d(p_1|p_2,m) = \frac{\alpha m}{p_1}
\displaystyle s(p_1|w,r) = \frac{\overline K p_1}{2w}
Solve for the solution functions for the equilibrium price and quantity.
Demand
Supply
Multivariable Functions
x
f()
z = f(x,y)
y
z
[INDEPENDENT VARIABLES]
[DEPENDENT VARIABLE]
\text{Level set for }z=\{(x,y)|f(x,y)=z\}
\displaystyle{{\partial f \over \partial x} = \lim_{\Delta x \rightarrow 0} {f(x + \Delta x, y) - f(x, y) \over \Delta x}}
\displaystyle{{\partial f \over \partial y} = \lim_{\Delta y \rightarrow 0} {f(x, y + \Delta y) - f(x, y) \over \Delta y}}
Multivariable Chain Rule
\text{Example: }h(x,y)=(3x+y)^2
h(x,y)=f(g(x,y))
{\partial h \over \partial x} = {df \over dg} \times {\partial g \over \partial x}
f(g)=g^2
g(x)=3x+y
Total Derivative Along a Path
\text{How does }f(x,y)\text{ change along a path?}
\Delta f \approx
\displaystyle{{\partial f \over \partial x} \times \Delta x}
\displaystyle{{\partial f \over \partial y} \times \Delta y}
+
\text{Suppose the path is defined by some function }y(x):
\Delta f \approx
\displaystyle{{\partial f \over \partial x} \times \Delta x}
\displaystyle{{\partial f \over \partial y}}
+
\displaystyle{\times {dy \over dx}}
\displaystyle{\times \Delta x}
\displaystyle{\Delta f \over \Delta x} \approx
\displaystyle{{\partial f \over \partial x}}
\displaystyle{{\partial f \over \partial y}}
+
\displaystyle{\times {dy \over dx}}
Total Derivative Along a Path
\displaystyle{\Delta f \over \Delta x}\ \ \ \ \ \ \approx
\displaystyle{{\partial f \over \partial x}}
\displaystyle{{\partial f \over \partial y}}
+
\displaystyle{\times\ \ \ \ \ \ {dy \over dx}}
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 \(x\)]
y(x)=4-0.4x
Unit I Overview
Unit I: Scarcity and Choice
Economics is the study of how
we use scarce resources
to satisfy our unlimited wants
Resources
Goods
Happiness
🌎
⌚️
🤓
Monday: Modeling Production
with Multivariable Functions
Labor
Fish
🐟
Capital
Coconuts
🥥
[GOODS]
⏳
⛏
[RESOURCES]
Unit I: Scarcity and Choice
Economics is the study of how
we use scarce resources
to satisfy our unlimited wants
Resources
Goods
Happiness
🌎
⌚️
🤓
🐟
🥥
Production Possibilities Fronier
Feasible
Wednesday:
Resource Constraints
Unit I: Scarcity and Choice
Economics is the study of how
we use scarce resources
to satisfy our unlimited wants
Resources
Goods
Happiness
🌎
⌚️
🤓
🐟
🥥
🙂
😀
😁
😢
🙁
Unit I: Scarcity and Choice
Economics is the study of how
we use scarce resources
to satisfy our unlimited wants
Resources
Goods
Happiness
🌎
⌚️
🤓
Week 3:
Preferences and Utility
🐟
🥥
Optimal choice
🙂
😀
😁
😢
🙁
Unit I: Scarcity and Choice
Economics is the study of how
we use scarce resources
to satisfy our unlimited wants
Resources
Goods
Happiness
🌎
⌚️
🤓
Week 4:
Constrained Optimization
Econ 50 | Lecture 03
By Chris Makler
Econ 50 | Lecture 03
Review of Univariate Calculus
