4. Models with Multiple Explanatory Variables

1
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

2
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.)

3
4. The Partial Derivative

• It is often impossible analyze ONE variables
  impact if ALL variables are changing.
• Instead, we analyze one variables 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.

4
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

5
4.1 Partial Derivatives

• Consider the function zf(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).

6
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
• ?y/?x
• ?f(x,z)/?x
• fx(x,z)
• Note dydx is equivalent to ?y/?x if yf(x) ie
  if y only has x as an explanatory variable.
• (Therefore often these are used interchangeably
  in economic shorthand)

7
4.1 Partial Derivatives

• Let y 2x23xz8z2
• ?y/ ?x 4x3z0
• ?y/ ?z 03x16z
• (0s are dropped)
• Let y xln(zx)
• ? y/ ? x ln(zx) zx/zx
• ln(zx) 1
• ? y/ ? z x(1/zx)x
• x/z

8
4.1 Partial Derivatives

• Let y 3x2zxz3-3z/x2
• ? y/ ? z3x23xz2-3/x2
• ? y/ ? x6xzz36z/x3
• Try these
• zln(2yx3)
• Expensessin(a2-ab)cos(b2-ab)

9
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 3x2zxz3-3z/x2
• ? y/ ? z3x23xz2-3/x2
• ? 2y/ ? z26xz
• ? 2y/ ? z ? x6x3z26/x3

10
4.1.1 Youngs Theorem

• From before
• Let y 3x2zxz3-3z/x2
• ? y/ ? x6xzz36z/x3
• ? 2y/ ? x26z-18z/x4
• ? 2y/ ? x ? z6x3z26/x3
• Notice that d2y/dxdzd2y/dzdx
• This is reflected by YOUNGS THEOREM order of
  differentiation doesnt matter for higher order
  partial derivatives

11
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

12
4.2.1 Interpreting Coefficients

• Given a function af(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)

13
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
  foods effect on happiness (or how food affects
  tvs effect on happiness). For example, watching
  TV may not increase happiness if someone is
  hungry.

14
4.2.1 Corn Example

• Consider the following formula for corn
  production
• Corn 500100Rain-Rain250ScareFertilizer
• Corn bushels of corn
• Rain centimeters of rain
• Scarenumber of scarecrows
• Fertilizer tonnes of fertilizer
• Explain this formula

15
4.2.1 Corny Example

• 1) Intercept 500
• -if it doesnt rain, there are no scarecrows and
  no fertilizer, the farmer will harvest 500
  bushels
• 2) ?Corn/?Rain100-2Rain
• -positive until Rain50, 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

16
4.2.1 Corny Example

• 3) ?2Corn/?Rain2-2lt0, (concave)
• -More rain has a DECREASING impact on the corn
  harvest OR
• -More rain DECREASES rains impact on the corn
  harvest by 2
• 4) ?Corn/?Scare50Fertilizer
• -More scarecrows will increase the harvest 50
  for every tonne of fertilizer
• -if no fertilizer is used, scarecrows are
  useless

17
4.2.1 Corny Example

• 5) ? 2Corn/?Scare20 (straight line, no
  curvature)
• -Additional scarecrows have a CONSTANT impact on
  corns harvest
• 6) ? 2Corn/?Scare?Fertilizer50
• -Additional fertilizer increases scarecrows
  impact on the corn harvest by 50

18
4.2.1 Corny Example

• 7) ?Corn/?Fertilizer50Scare
• -More fertilizer will increase the harvest 50
  for every scarecrow
• -if no scarecrows are used, fertilizer is
  useless
• 8) ? 2Corn/?Fertilizer20, (straight line)
• -Additional fertilizer has a CONSTANT impact on
  corns harvest

19
4.2.1 Corny Example

• 9) ? 2Corn/?Fertilizer ?Scare 50
• -Additional scarecrows increase fertilizers
  impact on the corn harvest by 50

20
4.2.1 Demand Example

• Consider the demand formula
• Q ß1 ß2 Pown ß3 Psub ß4 INC
• (Quantity demanded depends on a products own
  price, price of substitutes, and income.)
• Here ? Q/ ? Pown ß2 the impact on quantity
  when the products price changes
• Here ? Q/ ? Psub ß3 the impact on quantity
  when the substitutes price changes
• Here ? Q/ ? INC ß4 the impact on quantity when
  income changes

21
4.2.3 Partial Elasticities

• Furthermore, partial elasticities can also be
  calculated using partial derivatives
• Own-Price Elasticity ? Q/ ? Pown(Pown/Q)
• ß2(Pown/Q)
• Cross-Price Elasticity ? Q/ ? Psub(Psub/Q)
• ß3(Psub/Q)
• Income Elasticity ? Q/ ? INC(INC/Q)
• ß4(INC/Q)

22
4.2.2 Cobb-Douglas Production Function

• A favorite function of economists is the
  Cobb-Douglas Production Function of the form
• QaLbKcOf
• Where Llabour, KCapital, and OOther
  (education, technology, government, etc.)
• This is an attractive function because if
  bcf1, the demand function is homogeneous of
  degree 1. (Doubling all inputs doubles outputsa
  happy concept)

23
4.2.2 Cobb-Douglas University

• Consider a production function for university
  degrees
• QaLbKcCf
• Where
• LLabour (ie professors), KCapital (ie
  classrooms)CComputers

24
4.2.2 Cobb-Douglas University

• Finding partial derivatives
• ? Q/ ? L abLb-1KcCf
• b(aLbKcCf)/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

25
4.2.2 Cobb-Douglas Professors

• For example, if 20 professors are employed by the
  department, and 500 students graduate yearly, and
  b0.5
• ? 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

26
4.2.2 Marginal Product

• Consider the function Qf(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/ ? LMarginal Product of Labour (MPL)
• ? Q/ ? KMarginal Product of Capital (MPK)
• ? Q/ ? OMarginal Product of Other (MPO)

27
4.2.2 Cobb-Douglas Elasticities

• Since the Professor Elasticity (or PE) is
  defined as
• PE ? Q/ ? L(L/Q)
• We can find that
• PE b(Q/L)(L/Q)
• b
• The partial elasticity with respect to labour is
  b.
• The partial elasticity with respect to capital is
  c
• The partial elasticity with respect to other is f

28
4.2.2 Logs and Cobbs

• We can highlight elasticities by using logs
• QaLbKcCf
• Converts to
• Ln(Q)ln(a)bln(L)cln(k)fln(C)
• We now find that
• PE ? ln(Q)/ ? ln(L)b
• Using logs, elasticities more apparent.

29
4.2.2 Logs and Demand

• Consider a log-log demand example
• Ln(Qdx)ln(ß1) ß2 ln(Px) ß3 ln(Py) ß4 ln(I)
• We now find that
• Own Price Elasticity ß2
• Cross-Price Elasticity ß3
• Income Elasticity ß4

30
4.2.2 ilogs

• Considering the demand for the ipad, assume
• Ln(Qdipad)2.7 -1ln(Pipad)4 ln(Pnetbook)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 netbooks 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

31
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 xs impact on
  y (dy/dx) with xs impact on y, with other
  variables held constant (dy/dx)

32
4.3 Total Derivatives

• Assume you are increasing the square footage of a
  house where
• AREA LENGTH X WIDTH
• ALW
• If you increase the length,
• the change in area is equal
• to the increase in length
• times the current width

dL
Length
Width
Area
Notice that dA/dLW, (partial derivative, since
width is constant) Therefore the increase in area
is equal to dA(dA/dL)dL
33
4.3 Total Derivatives

• ALW
• If you increase the width,
• the change in area is equal
• to the increase in width
• times the current length

Length
Width
Area
dW
Notice that dA/dWL, (partial derivative, since
length is constant) Therefore the increase in
area is equal to dA(dA/dW)dW Next we combine
the two effects
34
4.3 Total Derivatives

• ALW
• An increase in both length
• and width has the following
• impact on area

Length
Width
Area
dW
dL
Now we have dA(dA/dL)dL(dA/dW)dW(dW)dL But
since derivatives always deal with instantaneous
slope and small changes, (dW)dL is small and
ignored, resulting in dA(dA/dL)dL(dA/dW)dW
35
4.3 Total Derivatives
Length
dA(dA/dL)dL(dA/dW)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 zf(x,y) we
have
Width
Area
dW
dL
36
4.3 Total Derivative Example
In a joke factory, QJokesworkers(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)?
37
4.3 Total Derivative Extension
The key advantage of the total derivative is it
takes variable interaction into account. The
partial derivative (dz/dx) examines the effect of
x on z if y doesnt 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
38
4.3 Total Derivative Extension

• Here we see that xs total impact on z is broken
  up into two parts
• xs DIRECT impact on z (through the partial
  derivative)
• xs INDIRECT impact on z (through y)
• Obviously, if x and y are unrelated, (dy/dx)0,
  then the total derivative collapses to the
  partial derivative

39
4.3 Total Derivative Example
Assume HappinessCandy3(Candy)MoneyMoney2 hc3c
mm2 Furthermore, Candy3Money/4 (c3m/4) The
total derivative of happiness with regards to
money
40
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
41
4.4 Unconstrained Optimization

• Unconstrained optimization falls into two
  categories
• Optimization using one variable (ie changing
  wage to increase productivity, working conditions
  are constant)
• Optimization using two (or more) variables (ie
  changing wage and working conditions to maximize
  productivity)

42
4.4 Simple Unconstrained Optimization

• For a multivariable case where only one variable
  is controlled, optimization steps are easy
• Consider the function zf(x)
• 1) FOC
• Determine where dz/dx0 (necessary condition)
• 2) SOC
• d2z/dx2lt0 is necessary for a maximum
• d2z/dx2gt0 is necessary for a minimum
• 3) Determine max/min point
• Substitute the point in (2) back into the
  original equation.

43
4.4 Simple Unconstrained Optimization

• Let productivity -wage210wage(working
  conditions)2
• P(w,c)-w210wc2
• If working conditions2, find the wage that
  maximizes productivity
• P(w,c)-w240w
• 1) FOC
• dp/dw -2w400
• w20
• 2) SOC
• d2p/dw2 -2 lt 0, a maximum exists

44
4.4 Simple Unconstrained Optimization

• P(w,c)-w210wc2
• w20 (maximum confirmed)
• 3) Find Maximum
• P(20,4)-20210(20)(2)2
• P(20,4)-400800
• P(20,4)400
• Productivity is maximized at 400 when wage is 20.

45
4.4 Complex Unconstrained Optimization

• For a multivariable case where only two variable
  are controlled, optimization steps are more
  in-depth
• Consider the function zf(x,y)
• 1) FOC
• Determine where dz/dx0 (necessary condition)
• And
• Determine where dz/dy0 (necessary condition)

46
4.4 Complex Unconstrained Optimization

• For a multivariable case where only two variable
  are controlled, optimization steps harder
• Consider the function zf(x,y)
• 2) SOC
• d2z/dx2lt0 and d2z/dy2lt0 are necessary for a
  maximum
• d2z/dx2gt0 and d2z/dy2gt0 are necessary for a
  minimum
• Plus, the cross derivatives cant be too large
  compared to the own second partial derivatives

47
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

48
4.4 Complex Unconstrained Optimization

• Let P(w,c)-w2wc-c2 9c , maximize productivity
• 1) FOC
• dp/dw -2wc0
• 2wc
• dp/dcw-2c90
• w2c-9
• w2(2w)-9
• -3w-9
• w3
• 2wc
• 6c

49
4.4 Complex Unconstrained Optimization

• P(w,c)-w2wc-c2 9c
• dp/dw -2wc0
• dp/dcw-2c90
• w3, c6 (possible max/min)
• 2) SOC
• d2p/dw2 -2 lt 0
• d2p/dc2 -2 lt 0, possible max

Maximum confirmed
50
4.4 Complex Unconstrained Optimization

• P(w,c)-w2wc-c2 9c
• w3, c6 (confirmed max)
• 3) Find productivity

Productivity is maximized at 27 when wage3 and
working conditions6.
51
4.5 Constrained Optimization

• Typically constrained optimization consists of
  maximizing or minimizing an objective function
  with regards to a constraint, or
• Max/min zf(x,y)
• Subject to (s.t.) g(x,y)k
• Where k is a constant

52
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
• Internalizing the constraint
• Creating a Lagrangeian function

53
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. Utility7TV-TV2Read (U7TV-TV2R)
• s.t. 7TVRead (7TVR)

54
4.5 Internalizing Constraints

• Max. U7TV-TV2R
• s.t. 7TVR
• We can solve the constraint
• R7-TV
• And substitute into the objective function
• U-TV27TV(7-TV)
• U-TV26TV7

55
4.5 Internalizing Constraints

• Max. U7TV-TV2R
• s.t. 7TVR
• U-TV26TV7
• We can then perform unconstrained optimization
• FOC
• dU/ dTV-2TV60
• TV3
• R7-TV
• R7-3
• R4

56
4.5 Internalizing Constraints

• Max. U7TV-TV2R
• s.t. 7TVR
• U-TV26TV7, TV3, R 4
• dU/ dTV-2TV6
• SOC
• d2U/ dTV2-2lt0, concave max.
• Evaluate
• U7TV-TV2R
• U7(3)-324
• U21-9416

57
4.5 Internalizing Constraints

• Max. U7TV-TV2R
• s.t. 7TVR
• U-TV26TV7, TV3, R 4
• dU/ dTV-2TV6
• d2U/dTV2-2lt0, concave max.
• U21-9416
• Utility is maximized at 16 when Bob watches 3
  hours of TV and reads for 4 hours.

58
4.5 Internalizing Constraints

• Substituting the constraint into the objective
  function may not be applicable for a variety of
  reasons
• The substitution makes the objective function
  unduly complicated, or substitution is impossible
• You want to evaluate the impact of the constraint
• The constraint is an inequality
• Your exam paper asks you to do so
• In this case, you must construct a Lagrangian
  function.

59
4.5 The Lagrangian

• Given the optimization problem
• Max/min zf(x,y)
• s.t. g(x,y)k (Where k is a constant)
• The Lagranean (Lagrangian) function becomes
• Lzz(x,y)?(k-g(x,y))
• Where ? is known as the Lagrange Multiplier.
• We then continue with FOCs and SOCs.

60
4.5 The Lagrangian

• Lzz(x,y)?(k-g(x,y))
• FOCs

Note that the third FOC simply returns the
constraint, g(x,y)k Typically, one will solve
for ? 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.
61
4.5 The Lagrangian

• Lzz(x,y)?(k-g(x,y))
• After finding FOCs, 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.
• SOCs

62
4.5 Lagrangian example

• Max. U7TV-TV2R
• s.t. 7TVR
• Lzz(x,y)?(k-g(x,y))
• L7TV-TV2R?(7-TV-R)
• FOC

63
4.5 Lagrangian example
64
4.5 Lagrangian Example
Since the second order condition is negative, the
points found are a maximum. Notice that we found
the same answers as internalizing the
constraint.
65
4.5 The Lagrange Multiplier

• The Lagrange Multiplier, ?, 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)kd

66
4.5 Lagrangian example

• 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 8TVR, TV3.5, R4.5, U16.75 (utility
  increases by approximately 1)

67
4.5.1 Inequality Constraints

• Often constraints are INEQUALITIES
• ie You can spend UP TO 200 on Christmas
  presents.
• When the constraint is an inequality, one must
  ask, IS THE CONSTRAINT BINDING?
• Example One Constraint is binding
• Maximize utility subject to a budget 100.
• Assume that ?2.5 an extra 1 in the Christmas
  budget will increase utility by 2.5
• The constraint IS BINDING your answer is
  correct.

68
4.5.1 Inequality Constraints

• Example Two Constraint is not binding
• Maximize utility subject to time 10 hours.
• Assume that ?-5 an extra hour would DECREASE
  utility by 5. OR
• One fewer hour would INCREASE utility by 5. You
  dont WANT to use all the available hours
• The constraint IS NOT BINDING therefore
• Redo the problem WITHOUT the constraint.

69
4.5.1 Inequality Constraints

• Inequality procedures can be summarized as
  follows (Table 4.1 in text)

g(x,y)k g(x,y)k
Maximize f(x,y) ?lt0, NOT BINDING ?gt0, BINDING ?lt0, BINDING ?gt0, NOT BINDING
Minimize f(x,y) ?lt0, BINDING ?gt0, NOT BINDING ?lt0, NOT BINDING ?gt0, BINDING