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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