4. Models with Multiple
Explanatory Variables
Chapter 2 assumed that the dependent variable
(Y) is affected by only ONE explanatory
variable (X).
Sometimes this is the case.
Example: Age = Days Alive/365.25
Usually, this is not the case.
Example: midterm mark depends on:
 how much you study
 how well you study
 intelligence, etc
1
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 analyze ONE variable’s
impact if ALL variables are changing.
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.
3
4. Calculus and 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
4
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).
5
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
7
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(a2-ab)+cos(b2-ab)
8
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 Young’s Theorem
From before:
Let y = 3x2z+xz3-3z/x2
∂ y/ ∂ x=6xz+z3+6z/x3
∂ 2y/ ∂ x2=6z-18z/x4
∂ 2y/ ∂ x ∂ z=6x+3z2+6/x3
Notice that ∂2y/∂x∂z=∂2y/∂z∂x
This is reflected by YOUNG’S THEOREM:
order of differentiation doesn’t matter for
higher order partial derivatives
10
4.2 Applications using Partial Derivatives
As many real-world situations involve many
variables, Partial Derivatives can be used to
analyze our world, using tools including:
 Interpreting coefficients
 Partial Elasticities
 Marginal Products
11
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)
12
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 hungry.
13
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
14
4.2.1 Corny Example
1) Intercept = 500
-if it doesn’t rain, there are no scarecrows and
no fertilizer, the farmer will harvest 500
bushels
2) ∂Corn/∂Rain=100-2Rain
-each additional cm of rain changes corn
production by 100-2Rain
-positive impact if rain < 50 cm
-negative impact if rain > 50 cm
15
4.2.1 Corny Example
3) ∂2Corn/∂Rain2=-2<0, (concave)
-More rain has a DECREASING impact on the
corn harvest
-More rain DECREASES rain’s impact on the
corn harvest by 2
4) ∂Corn/∂Scare=50Fertilizer
-More scarecrows will increase the harvest 50
for every tonne of fertilizer
-if no fertilizer is used, scarecrows are useless
16
4.2.1 Corny Example
5) ∂ 2Corn/∂Scare2=0 (straight line, no curvature)
-Additional scarecrows have a CONSTANT
impact on corn’s harvest
6) ∂ 2Corn/∂Scare∂Fertilizer=50
-Additional fertilizer increases scarecrow’s
impact on the corn harvest by 50
17
4.2.1 Corny Example
7) ∂Corn/∂Fertilizer=50Scare
-More fertilizer will increase the harvest 50 for
every scarecrow
-if no scarecrows are used, fertilizer is useless
8) ∂ 2Corn/∂Fertilizer2=0, (straight line)
-Additional fertilizer has a CONSTANT impact on
corn’s harvest
18
4.2.1 Corny Example
9) ∂ 2Corn/∂Fertilizer ∂Scare =50
-Additional scarecrows increase fertilizer’s
impact on the corn harvest by 50
19
4.2.1 Demand Example
Consider the demand formula:
Q = β1 + β2 Pown + β3 Psub + β4 INC
(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
20
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)
21
4.2.2 Cobb-Douglas Production Function
A favorite function of economists is the CobbDouglas 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)
22
4.2.2 Cobb-Douglas University
Consider a production function for university
degrees:
Q=aLbKcAf
Where
L=Labour (ie: professors),
K=Capital (ie: classrooms)
A=Administration
23
4.2.2 Average and Marginal Products
Finding partial derivatives:
∂ Q/ ∂ L =abLb-1KcAf
=b(aLbKcAf)/L
=b(Q/L)
=b* average product of labour
-in other words, adding an additional professor
will contribute a fraction of the average
product of each current professor
-this partial derivative gives us the MARGINAL
PRODUCT of labour
24
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
25
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/ ∂ O=Marginal Product of Other (MPO)
26
4.2.2 Cobb-Douglas Elasticities
Since the “Labor Elasticity” (LE) is defined as:
LE
= ∂ Q/ ∂ L(L/Q)
We can find that
LE =b(Q/L)(L/Q)
=b
The partial elasticity with respect to labor is b.
The partial elasticity with respect to capital is c
The partial elasticity with respect to other is f
27
4.2.2 Logs and Cobbs
We can highlight elasticities by using logs:
Q=aLbKcCf
Converts to
Ln(Q)=ln(a)+bln(L)+cln(k)+fln(C)
We now find that:
LE= ∂ ln(Q)/ ∂ ln(L)=b
Using logs, elasticities more apparent.
28
4.2.2 Logs and Demand
Consider 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
29
4.2.2 ilogs
Considering the demand for the ipad, assume:
Ln(Qdipad)=2.7 -1ln(Pipad)+4 ln(Ptablet)+0.1 ln(I)
We now find that:
Own Price Elasticity = -1, demand is unit elastic
Cross-Price Elasticity = 4, a 1% increase in the
price of tablets causes a 4% increase in
quantity demanded of ipads
Income Elasticity = 0.1, a 1% increase in income
causes a 0.1% increase in quantity demanded
30
for ipads
4.3 Total Derivatives
Often in econometrics, one variable is influenced by a
variety of other variables.
Ie: Happiness =f(sun, driving)
Ie: Productivity = f(labor, effectiveness)
Using TOTAL DERIVATIVES, we can examine how
growth of one variable is caused by growth in all
other variables
The following formulae will combine x’s impact on y
(dy/dx) with x’s impact on y, with other variables held
constant (δy/δx)
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4.3 Total Derivatives
Width
Assume you are increasing the square footage of a
house where
AREA = LENGTH X WIDTH
A=LW
dL
Length
If you increase the length,
the change in area is equal
Area
to the increase in length
times the current width:
Notice that:
δA/δL=W, (partial derivative, since width is constant)
Therefore the increase in area is equal to:
dA=(δA/δL)dL
32
4.3 Total Derivatives
Length
Width
A=LW
If you increase the width,
the change in area is equal
to the increase in width
times the current length:
Area
dW
Notice that:
δA/δW=L, (partial derivative, since length is constant)
Therefore the increase in area is equal to:
dA=(δA/δW)dW
Next we combine the two effects:
33
4.3 Total Derivatives
Width
A=LW
An increase in both length
and width has the following
impact on area:
dW
Length
Area
Now we have:
dA=(δA/δL)dL+(δA/δW)dW+(dW)dL
dL
But since derivatives always deal with instantaneous
slope and small changes, (dW)dL is small and ignored,
resulting in:
dA=(δA/δL)dL+(δA/δW)dW
34
4.3 Total Derivatives
dA=(δA/δL)dL+(δA/δW)dW
Width
Length
Area
dW
dL
Effectively, we see that change in the dependent
variable (A), comes from changes in the independent
variables (W and L). In general, given the function
z=f(x,y) we have:
f ( x, y )
f ( x, y )
z
z
dz 
dx 
dy  dx  dy
x
y
x
y
35
4.3 Total Derivative Example
In a joke factory,
QJokes=workers(funniness)
You employ 500 workers,
each of which can create
100 funny jokes an hour.
How many more jokes could
you create if you increase
workers by 2 and their
average funniness by 1
(perhaps by discovering any
joke with an elephant in it is
slightly more funny)?
q
q
dq 
df 
dw
f
w
dq  wdf  fdw
dq  500(1)  100( 2)
dq  500  200  700
36
4.3 Total Derivative Extension
The key advantage of the total derivative is it takes
variable interaction into account.
The partial derivative (δz/δx) examines the effect of x
on z if y doesn’t change. This is the DIRECT EFFECT.
However, if x affects y which then affects z, we might
want to measure this INDIRECT EFFECT.
We can modify the total derivative to do this:
f ( x, y )
f ( x, y )
z
z
dz 
dx 
dy  dx  dy
x
y
x
y
dz z dx z dy z z dy


 
dx x dx y dx x y dx
37
4.3 Total Derivative Extension
f ( x, y )
f ( x, y )
z
z
dz 
dx 
dy  dx  dy
x
y
x
y
dz z dx z dy z z dy


 
dx x dx y dx x y dx
Here we see that x’s total impact on z is broken up into
two parts:
1) x’s DIRECT impact on z (through the partial
derivative)
2) x’s INDIRECT impact on z (through y)
Obviously, if x and y are unrelated, (δy/δx)=0, then the
total derivative collapses to the partial derivative38
4.3 Total Derivative Example
Assume Happiness=Candy+3(Candy)Money+Money2
h=c+3cm+m2
Furthermore, Candy=3+Money/4 (c=3+m/4)
The total derivative of happiness with regards to money:
dh h h dc


dm m c dm
dh
 [3c  2m]  [(1  3m)(1 / 4)]
dm
dh
 0.25  3c  2.75m
dm
39
4.3 Total Derivative and Elasticity
Total derivatives can also give us the relationship
between elasticity and revenue that we found in
Chapter 2.2.3:
TR  PQ
TR
TR
dTR 
dP 
dQ
P
Q
dTR
dP
dQ
Q
P
dP
dP
dP
dTR
P dQ
 Q(1 
)
dP
Q dP
dTR
 Q(1   ) (where  is price elasticity of demand)
40
dP
4.4 Unconstrained Optimization
Unconstrained optimization falls into two
categories:
1) Optimization using one variable (ie:
changing wage to increase productivity,
working conditions are constant)
2) Optimization using two (or more)
variables (ie: changing wage and
working conditions to maximize
productivity)
41
4.4 Simple Unconstrained Optimization
For a multivariable case where only one variable is
controlled, optimization steps are easy:
Consider the function z=f(x)
1) FOC:
Determine where δz/δx=0 (necessary condition)
2) SOC:
δ2z/δx2<0 is necessary for a maximum
δ2z/δx2>0 is necessary for a minimum
3) Determine max/min point
Substitute the point in (2) back into the original
equation.
42
4.4 Simple Unconstrained Optimization
Let productivity = -wage2+10wage(working conditions)2
P(w,c)=-w2+10wc2
If working conditions=2, find the wage that maximizes
productivity
P(w,c)=-w2+40w
1) FOC:
δp/δw =-2w+40=0
w=20
2) SOC:
δ2p/δw2= -2 < 0, a maximum exists
43
4.4 Simple Unconstrained Optimization
P(w,c)=-w2+10wc2
w=20 (maximum confirmed)
3) Find Maximum
P(20,4)=-202+10(20)(2)2
P(20,4)=-400+800
P(20,4)=400
Productivity is maximized at 400 when wage is 20.
44
4.4 Complex Unconstrained Optimization
For a multivariable case where only two variable are
controlled, optimization steps are more in-depth:
Consider the function z=f(x,y)
1) FOC:
Determine where δz/δx=0 (necessary condition)
And
Determine where δz/δy=0 (necessary condition)
45
4.4 Complex Unconstrained Optimization
For a multivariable case where only two variable are
controlled, optimization steps harder:
Consider the function z=f(x,y)
2) SOC:
δ2z/δx2<0 and δ2z/δy2<0 are necessary for a maximum
δ2z/δx2>0 and δ2z/δy2>0 are necessary for a minimum
Plus, the cross derivatives can’t be too large compared
to the own second partial derivatives:
2
  z   z    z 
 2  2   
  0
 x  y   xy 
2
2
2
46
4.4 Complex Unconstrained Optimization
2
  z   z    z 
 2  2   
  0
 x  y   xy 
2
2
2
If this third SOC requirement is not fulfilled, a SADDLE
POINT occurs, where z is a maximum with regards
to one variable but a minimum with regards to the
other. (ie: wage maximizes productivity while
working conditions minimizes it)
Vaguely, even though both variables work to increase
z, their interaction with each other outweighs this
maximizing effect
47
4.4 Complex Unconstrained Optimization
Let P(w,c)=-w2+wc-c2 +9c , maximize productivity
1) FOC:
δp/δw =-2w+c=0
2w=c
δp/δc=w-2c+9=0
w=2c-9
w=2(2w)-9
-3w=-9
w=3
2w=c
6=c
48
4.4 Complex Unconstrained Optimization
P(w,c)=-w2+wc-c2 +9c
δp/δw =-2w+c=0
δp/δc=w-2c+9=0
w=3, c=6 (possible max/min)
2) SOC:
δ2p/δw2= -2 < 0
δ2p/δc2= -2 < 0, possible max
2
  p   p    p 
 2  2   
  (2)( 2)  12
 w  c   wc 
2
2
2
2
  p   p    p 
 2  2   
  3  0
 w  c   wc 
2
2
2
Maximum confirmed
49
4.4 Complex Unconstrained Optimization
P(w,c)=-w2+wc-c2 +9c
w=3, c=6 (confirmed max)
3) Find productivity:
p( w, c)   w2  wc  c 2  9c
p( w, c)  32  (3)6  6 2  9(6)
p( w, c)  9  18  36  54
p( w, c)  27
Productivity is maximized at 27 when wage=3 and
working conditions=6.
50
4.5 Constrained Optimization
Typically constrained optimization consists of
maximizing or minimizing an objective function with
regards to a constraint, or
Max/min z=f(x,y)
Subject to (s.t.): g(x,y)=k
Where k is a constant
51
4.5 Constrained Optimization
Often economic agents are not free to make any
decision they would like. They are
CONSTRAINED by factors such as income, time,
intelligence, etc.
When optimizing with constraints, we have two general
methods:
1) Internalizing the constraint
2) Creating a Lagrangeian function
52
4.5 Internalizing Constraints
If the constraint can be substituted into the equation to
be optimized, we are left with an unconstrained
optimization problem:
Example:
Bob works a full week, but every Saturday he has
seven hours left free, either to watch TV or read.
He faces the constrained optimization problem:
Max. Utility=7TV-TV2+Read (U=7TV-TV2+R)
s.t. 7=TV+Read (7=TV+R)
53
4.5 Internalizing Constraints
Max. U=7TV-TV2+R
s.t. 7=TV+R
We can solve the constraint:
R=7-TV
And substitute into the objective function:
U=-TV2+7TV+(7-TV)
U=-TV2+6TV+7
54
4.5 Internalizing Constraints
Max. U=7TV-TV2+R
s.t. 7=TV+R
U=-TV2+6TV+7
We can then perform unconstrained optimization:
FOC:
δU/ δTV=-2TV+6=0
TV=3
R=7-TV
R=7-3
R=4
55
4.5 Internalizing Constraints
Max. U=7TV-TV2+R
s.t. 7=TV+R
U=-TV2+6TV+7, TV=3, R= 4
δU/ δTV=-2TV+6
SOC:
δ2U/ δTV2=-2<0, concave max.
Evaluate:
U=7TV-TV2+R
U=7(3)-32+4
U=21-9+4=16
56
4.5 Internalizing Constraints
Max. U=7TV-TV2+R
s.t. 7=TV+R
U=-TV2+6TV+7, TV=3, R= 4
δU/ δTV=-2TV+6
δ2U/δTV2=-2<0, concave max.
U=21-9+4=16
Utility is maximized at 16 when Bob watches 3 hours of
TV and reads for 4 hours.
57
4.5 Internalizing Constraints
Substituting the constraint into the objective function
may not be applicable for a variety of reasons:
1) The substitution makes the objective function unduly
complicated, or substitution is impossible
2) You want to evaluate the impact of the constraint
3) The constraint is an inequality
4) Your exam paper asks you to do so
In this case, you must construct a Lagrangian function.
58
4.5 The Lagrangian
Given the optimization problem:
Max/min z=f(x,y)
s.t. g(x,y)=k (Where k is a constant)
The Lagranean (Lagrangian) function becomes:
L=z*=z(x,y)+λ(k-g(x,y))
Where λ is known as the Lagrange Multiplier.
We then continue with FOC’s and SOC’s.
59
4.5 The Lagrangian
L=z*=z(x,y)+λ(k-g(x,y))
FOC’s:
L
 0,
x
L
 0,
y
L
0

Note that the third FOC simply returns the constraint,
g(x,y)=k
Typically, one will solve for λ in the first two conditions
to find a relationship between x and y, then use this
relationship with the third condition to solve for x
and y.
60
4.5 The Lagrangian
L=z*=z(x,y)+λ(k-g(x,y))
After finding FOC’s, to confirm a maximum or
minimum, the SOC is employed.
This SOC must be negative for a maximum and
positive for a minimum
Note that for more terms, this function becomes
exponentially complicated.
SOC’s:
 δg 
SOC   
 δx 
2
 δ z   δg 
 2    
 δy   δy 
2
2
 δ 2 z   δg  δg  δ 2 z 
 2   2  

 δx   δx  δy  δxδy

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4.5 Lagrangian example
Max. U=7TV-TV2+R
s.t. 7=TV+R
L=z*=z(x,y)+λ(k-g(x,y))
L=7TV-TV2+R+λ(7-TV-R)
FOC:
L
L
L
 7 - 2TV -   0
 1-   0
 7 - TV - R  0
TV
R

  7 - 2TV
 1
7  TV  R
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4.5 Lagrangian example
(1)  7 - 2TV
( 2)   1
(3)7  TV  R
(1)  (2)
7  2TV  1
TV  3
(3) :
7  TV  R
7  3 R
4R
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4.5 Lagrangian Example
2
 δg   δ z   δg 
SOC  
  2   

 δTV   δR   δR 
2
2
 δ 2 z   δg  δg  δ 2 z 

  2



2 
 δTV   δTV  δR  δTVδR 
SOC  1 0  1 - 2  2110
2
2
SOC  2  0
Since the second order condition is negative, the points
found are a maximum.
Notice that we found the same answers as internalizing
the constraint.
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4.5 The Lagrange Multiplier
The Lagrange Multiplier, λ, provides a measure of how
much of an impact relaxing the constraint would
make, or how the objective function changes if k of
g(x,y)=k is marginally increased.
The Lagrange multiplier answers how much the
maximum or minimum changes when the constraint
g(x,y)=k increases slightly to g(x,y)=k+δ
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4.5 Lagrangian example
  7 - 2TV
TV  3
R4
  7 - 2TV  7 - 2(3)  1
This means that if Bob gets an extra hour, his maximum
utility will increase by approximately 1.
(Alternately, if Bob loses an hour of leisure, his
maximum utility will decrease by approximately 1.)
Check:
If 8=TV+R, TV=3.5, R=4.5, U=16.75
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(utility increases by approximately 1)