1 4. models with multiple explanatory variables chapter 2 assumed that the dependent variable (y) is...
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4. Models with MultipleExplanatory 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
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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.)
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4. The Partial DerivativeIt 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.
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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
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4.1 Partial DerivativesConsider 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 DerivativesIn 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/∂x2) ∂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
in economic shorthand)
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4.1 Partial DerivativesLet 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 DerivativesLet 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)
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4.1.1 Higher Partial DerivativesHigher 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
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4.1.1 Young’s TheoremFrom 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∂xThis 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 our world, 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 hungry.
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4.2.1 Corn ExampleConsider the following formula for corn
production:
Corn = 500+100Rain-Rain2+50Scare*Fertilizer
Corn = bushels of cornRain = centimeters of rainScare=number of scarecrowsFertilizer = tonnes of fertilizer
Explain this formula
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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
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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
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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
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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
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4.2.1 Corny Example
9) ∂ 2Corn/∂Fertilizer ∂Scare =50
-Additional scarecrows increase fertilizer’s impact on the corn harvest by 50
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4.2.1 Demand ExampleConsider 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
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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 UniversityConsider a production function for university
degrees:
Q=aLbKcAf
Where
L=Labour (ie: professors), K=Capital (ie: classrooms)A=Administration
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4.2.2 Average and Marginal ProductsFinding 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
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4.2.2 Cobb-Douglas ProfessorsFor 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 ProductConsider 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)
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4.2.2 Cobb-Douglas ElasticitiesSince the “Labor Elasticity” (LE) is defined as:
LE = ∂ Q/ ∂ L(L/Q)
We can find thatLE =b(Q/L)(L/Q)
=b
The partial elasticity with respect to labor is b.The partial elasticity with respect to capital is cThe partial elasticity with respect to other is f
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4.2.2 Logs and CobbsWe 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.
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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
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4.2.2 ilogsConsidering 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 for ipads
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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)
Assume you are increasing the square footage of a house where
AREA = LENGTH X WIDTH
A=LW
If you increase the length,
the change in area is equal
to the increase in length
times the current width:
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4.3 Total Derivatives
Length
Area
Width
dL
Notice that:δA/δL=W, (partial derivative, since width is constant)Therefore the increase in area is equal to:dA=(δA/δL)dL
A=LW
If you increase the width,
the change in area is equal
to the increase in width
times the current length:
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4.3 Total Derivatives
Length
Area
Width
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:
A=LW
An increase in both length
and width has the following
impact on area:
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4.3 Total Derivatives
Length
Area
Width
dW
Now we have:dA=(δA/δL)dL+(δA/δW)dW+(dW)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
dL
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4.3 Total DerivativesLength
Area
Width
dW
dA=(δA/δL)dL+(δA/δW)dW
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:
dL
dyy
zdxx
zdy
y
yxfdx
x
yxfdz
),(),(
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)? 36
4.3 Total Derivative Example
700200500
)2(100)1(500
dq
dq
fdwwdfdq
dww
qdff
qdq
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:
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4.3 Total Derivative Extension
dx
dy
y
z
x
z
dx
dy
y
z
dx
dx
x
z
dx
dz
dyy
zdxx
zdy
y
yxfdx
x
yxfdz
),(),(
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4.3 Total Derivative Extension
dx
dy
y
z
x
z
dx
dy
y
z
dx
dx
x
z
dx
dz
dyy
zdxx
zdy
y
yxfdx
x
yxfdz
),(),(
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 derivative
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4.3 Total Derivative ExampleAssume 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:
mcdm
dh
mmcdm
dhdm
dc
c
h
m
h
dm
dh
75.2325.0
)]4/1)(31[(]23[
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4.3 Total Derivative and ElasticityTotal derivatives can also give us the relationship between elasticity and revenue that we found in Chapter 2.2.3:
demand) of elasticity price is (where )1(
)1(
QdP
dTR
dP
dQ
Q
PQ
dP
dTRdP
dQP
dP
dPQ
dP
dTR
dQQ
TRdP
P
TRdTR
PQTR
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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)
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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.
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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
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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.
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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)
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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:
022
2
2
2
2
yx
z
y
z
x
z
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4.4 Complex Unconstrained Optimization
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
022
2
2
2
2
yx
z
y
z
x
z
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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
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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
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1)2)(2(
22
2
2
2
2
2
22
2
2
2
2
cw
p
c
p
w
p
cw
p
c
p
w
p Maximum confirmed
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4.4 Complex Unconstrained Optimization
P(w,c)=-w2+wc-c2 +9c
w=3, c=6 (confirmed max)
3) Find productivity:
27),(
5436189),(
)6(966)3(3),(
9),(22
22
cwp
cwp
cwp
ccwcwcwp
Productivity is maximized at 27 when wage=3 and working conditions=6.
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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
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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
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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)
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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
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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
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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
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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.
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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.
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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.
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4.5 The Lagrangian
L=z*=z(x,y)+λ(k-g(x,y))
FOC’s:0 ,0 ,0
LLL
yxNote 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.
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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:
δxδy
zδ
δy
δg
δx
δg2
δx
zδ
δy
δg
δy
zδ
δx
δg 2
2
22
2
22
SOC
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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:
2TV-7
0-2TV-7
TV
L
1
0-1
R
L
RV
T
T7
0R-V-7L
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4.5 Lagrangian example
RTV
7)3(
1)2(
2TV-7)1(
3
127
)2()1(
TV
TV
R
R
RV
4
37
T7
:)3(
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4.5 Lagrangian Example
02
01122-101
δTVδR
zδ
δR
δg
δTV
δg2
δTV
zδ
δR
δg
δR
zδ
δTV
δg
22
2
2
22
2
22
SOC
SOC
SOC
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
12(3)-7 2TV-7
4
3
2TV-7
R
TV
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 (utility increases by approximately 1)