# Econ 299 Chapter 04a.. - University of Alberta - Edmonton_ Alberta

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```					          4. Models with Multiple
Explanatory Variables
Up until this point, we have assumed that our
dependent variable (Y) is affected by only
ONE explanatory variable (X).

Sometimes this is the case. Example: The
amount of dollars in your bank account is
equal to the number of cents in your bank
account/100.

Usually, this is not the case. Example: your
mark on the midterm depends on how well
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you study and your innate intelligence.
4. Multi Variable Examples:
Demand = f( price of good, price of substitutes,
income, price of compliments)

Consumption = f( income, tastes, wages)

Graduation rates = f( tuition, school quality,
student quality)

Christmas present satisfaction = f (cost, timing,
knowledge of person, presence of card, age,
etc.)                                      2
4. The Partial Derivative
It is often impossible to analyze all various
movements of explanatory variables and
their impact on the dependent variable.

Instead, we analyze one variable’s impact,
assuming ALL OTHER VARIABLES
REMAIN CONSTANT

We do this through the partial derivative.
This chapter uses the partial derivative to
expand the topics introduced in chapter 2.
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4. Calculusand Applications
involving More than One Variable
4.1 Derivatives of Functions of More Than
One Variable
4.2 Applications Using Partial Derivatives
4.3 Partial and Total Derivatives
4.4 Unconstrained Optimization
4.5 Constrained Optimization

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4.1 Partial Derivatives
Consider the function z=f(x,y). As this function
takes into account 3 variables, it must be
graphed on a 3-dimensional graph.

A partial derivative calculates the slope of a
2-dimensional “slice” of this 3-dimensional
graph.

The partial derivative ∂z/∂x asks how x affects z
while y is held constant (ceteris paribus).

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4.1 Partial Derivatives
In taking the partial derivative, all other variables
are kept constant and hence treated as
constants (the derivative of a constant is 0).
There are a variety of ways to indicate the
partial derivative:
1) ∂y/∂x
2) ∂f(x,z)/∂x
3) fx(x,z)
Note: dy=dx is equivalent to ∂y/∂x if y=f(x); ie: if
y only has x as an explanatory variable.
Therefore often these are used interchangeably   6
in economic shorthand
4.1 Partial Derivatives
Let y = 2x2+3xz+8z2

∂y/ ∂x = 4x+3z+0
∂y/ ∂z = 0+3x+16z
(0’s are dropped)
Let y = xln(zx)
∂ y/ ∂ x = ln(zx) + zx/zx
= ln(zx) + 1
∂ y/ ∂ z = x(1/zx)x
=x/z
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4.1 Partial Derivatives
Let y = 3x2z+xz3-3z/x2

∂ y/ ∂ z=3x2+3xz2-3/x2

∂ y/ ∂ x=6xz+z3+6z/x3

Try these:

z=ln(2y+x3)
Expenses=sin(x2-xz)+cos(z2-xz)
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4.1.1 Higher Partial Derivatives
Higher order partial derivates are evaluated
exactly like normal higher order derivatives.
It is important, however, to note what variable to
differentiate with respect to:

From before:
Let y = 3x2z+xz3-3z/x2
∂ y/ ∂ z=3x2+3xz2-3/x2

∂ 2y/ ∂ z2=6xz
∂ 2y/ ∂ z ∂ x=6x+3z2+6/x3                     9
4.1.1 Higher Partial Derivatives
From before:
Let y = 3x2z+xz3-3z/x2
∂ y/ ∂ x=6xz+z3+6z/x3

∂ 2y/ ∂ x2=6-18z/x4
∂ 2y/ ∂ x ∂ z=6x+3z2+6/x3

Notice that d2y/dxdz=d2y/dzdx
This is reflected by YOUNG’S THEOREM, order
of differentiation doesn’t matter for higher
order partial derivatives
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4.2 Applications using Partial Derivatives
As many real-world situations involve many
variables, Partial Derivatives can be used to
analyze these situations, using tools
including:

 Interpreting coefficients
 Partial Elasticities
 Marginal Products

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4.2.1 Interpreting Coefficients
Given a function a=f(b,c,d), the dependent
variable a is determined by a variety of
explanatory variables b, c, and d.
If all dependent variables change at once, it is
hard to determine if one dependent variables
has a positive or negative effect on a.
A partial derivative, such as ∂ a/ ∂ c, asks how
one explanatory variable (c), affects the
dependent variable, a, HOLDING ALL
OTHER DEPENDENT VARIABLES
CONSTANT (ceteris paribus)
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4.2.1 Interpreting Coefficients
A second derivative with respect to the same
variable discusses curvature.
A second cross partial derivative asks how the
impact of one explanatory variable changes
as another explanatory variable changes.
Ie: If Happiness = f(food, tv),
∂ 2h/ ∂ f ∂tv asks how watching more tv affects
food’s effect on happiness (or how food
affects tv’s effect on happiness). For
example, watching TV may not increase
happiness if someone is starving.
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4.2.1 Corn Example
Consider the following formula for corn
production:

Corn = 500+100Rain-Rain2+50Scare*Fertilizer
Corn = bushels of corn
Rain = centimeters of rain
Scare=number of scarecrows
Fertilizer = tonnes of fertilizer

Explain this formula
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4.2.1 Corny Example
Intercept = 500
-if it doesn’t rain, there are no scarecrows
and no fertilizer, the farmer will harvest 500
bushels
dcorn/drain=100-2Rain
-positive until rain=50, then negative
-more rain increases the harvest at a
decreasing rate until rain hits 50cm, then
additional rain decreases the harvest at an
increasing rate
(∂ 2corn/ ∂ rain2=-2<0, concave)
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4.2.1 Corny Example
dcorn/dscare=50Fertilizer
-More scarecrows will increase the harvest 50
for every tonne of fertilizer
-if no fertilizer is used, scarecrows are useless
(∂ 2corn/ ∂ scare2=0, straight line, no curvature)
dcorn/dfertilizer=50Scare
-More fertilizer will increase the harvest 50 for
every scarecrow
-if no scarecrows are used, fertilizer is useless
(∂ 2corn/ ∂ fertilizer2=0, straight line, no curvature)
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4.2.1 Demand Example
Consider the demand formula:
Q = β1 + β2 Pown + β3 Psub + β4 INC
Where quantity demanded depends on a
product’s own price, price of substitutes, and
income.
Here ∂ Q/ ∂ Pown= β2 = the impact on quantity
when the product’s price changes
Here ∂ Q/ ∂ Psub= β3 = the impact on quantity
when the substitute’s price changes
Here ∂ Q/ ∂ INC= β4 = the impact on quantity
when income changes
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4.2.3 Partial Elasticities
Furthermore, partial elasticities can also be
calculated using partial derivatives:
Own-Price Elasticity = ∂ Q/ ∂ Pown(Pown/Q)
= β2(Pown/Q)
Cross-Price Elasticity = ∂ Q/ ∂ Psub(Psub/Q)
= β3(Psub/Q)
Income Elasticity = ∂ Q/ ∂ INC(INC/Q)
= β4(INC/Q)

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4.2.2 Cobb-Douglas Production Function
A favorite function of economists is the Cobb-
Douglas Production Function of the form
Q=aLbKcOf
Where L=labour, K=Capital, and O=Other
(education, technology, government, etc.)
This is an attractive function because if b+c+f=1,
the demand function is homogeneous of
degree 1. (Doubling all inputs doubles
outputs…a happy concept)

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4.2.2 Cobb-Douglas University
Consider a production function for university
degrees:

Q=aLbKcCf

Where L=Labour, K=Capital (excluding
computes), and C=Computers
More simply, labour can reflect professors,
capital can reflect classrooms, and
computers can reflect computers.
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4.2.2 Cobb-Douglas University
Finding partial derivatives:
∂ Q/ ∂ L =abLb-1KcCf
=b(aLbKcCf)/L
=b(Q/L)
=b* average product of labour
will contribute a fraction of the average
product of each current professor
-this partial elasticity gives us the MARGINAL
PRODUCT of labour                         21
4.2.2 Cobb-Douglas Professors
For example, if 20 professors are employed by
the department, and 500 students graduate
yearly, and b=0.5:

∂ Q/ ∂ L   =0.5(500/20)
=12.5

Ie: Hiring another professor will graduate 12.5
more students. The marginal product of
professors is 12.5
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4.2.2 Marginal Product
Consider the function Q=f(L,K,O)
The partial derivative reveals the MARGINAL
PRODUCT of a factor, or incremental effect
on output that a factor can have when all
other factors are held constant.
∂ Q/ ∂ L=Marginal Product of Labour (MPL)
∂ Q/ ∂ K=Marginal Product of Capital (MPK)
∂ Q/ ∂ L=Marginal Product of Other (MPO)

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4.2.2 Cobb-Douglas Elasticities
Since the “Professor Elasticity of Demand” (or
PED…an arbitrary name) is expressed:
PED = ∂ Q/ ∂ L(L/Q)

We can find that
PED =b(Q/L)(L/Q)
=b

The partial elasticity with respect to labour is b.
Find the remaining derivatives and
elasticities.                               24
4.2.2 Logs and Cobbs
We can highlight elasticities easier by using
logs:
Q=aLbKcCf
Ln(Q)=ln(a)+bln(L)+cln(k)+fln(C)

We now find that:
PED= ∂ ln(Q)/ ∂ ln(L)=b

Therefore using logs, elasticities more apparent.

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4.2.2 Logs and Demand

Reverting to a log-log demand example:
Ln(Qdx)=ln(β1) +β2 ln(Px)+ β3 ln(Py)+ β4 ln(I)

We now find that:
Own Price Elasticity = β2
Cross-Price Elasticity = β3
Income Elasticity = β4

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4.2.2 ilogs
Considering the demand for the ipad, assume:

We now find that:
Own Price Elasticity = -1, demand is unit elastic
Cross-Price Elasticity = β3, a 1% increase in the
price of netbooks causes a 4% increase in