1. Utility Maximization

2. A Utility Function Mathematically Representing Preferences
U=U(x, y)
U(A) Y
U(B)
Utility functions have A
Indifference curves
describe bundle-ordering
preferences B X

3. Consumption Opportunities: The Budget Constraint
Assume that an individual has I dollars to allocate between good x and good y
pxx + pyy  M y
The individual can afford
to choose only combinations
of x and y in the shaded
triangle x

4. The Budget Constraint
MC of consuming one more unit of x, the amount of y that must be foregone.
The slope of the budget line is this MC. y
The slope is the change
in y for a one unit increase
in the consumption of x.
If Px = 10 and Py = 5, then
consuming one more x means
consuming two less y. x

5. Not an indifference curve!
Maximizing Utility
Keep buying x until the MB(x) = MC(x)
Interaction of…
Preferences, diminishing MB because of diminishing MRS. MB = MRS
MB in terms of y willing to be given up
In dollars, MB = py*MRS
MC of x = px/py
MC in terms of Y given up
In dollars, MC = px.
MB, MC X X*
Not an indifference curve! MC MB

6. Optimization Principle
To maximize utility, given a fixed amount of income, an individual will buy the goods and services:
That exhaust total income
Savings or borrowing is allowed (if we modify the budget constraint to include a temporal component)
So long as MB(x) ≥ MC(x), MB(y) ≥ MC(y), etc.
Or, until MB(x) = MC(x), MB(y) = MC(y)

7. Intuition
MRS is the maximum amount of y the person is willing to give up to consume more x; the definition of MB.
px/py tells us the number of units of y that must be given up to consume one more x; the definition of MC. y x A B
At “A”, MRS>Px/Py (MB > MC),
You are willing to pay more than you have to, consumer surplus increases.
Utility and consumer surplus
can be increased by consuming more x.
At “B”, MRS<Px/Py, (MB < MC)
Utility and consumer surplus
can be increased by consuming
less x.
Px = 10
Py = 5
Slope of budget line = -2 U0

8. Intuition
At “C”, the MB = MC for the last unit of both goods consumed.
That is, at “C”, MRS = px/py, or y A C U1 B U0 x

9. Optimization
Unconstrained optimization is a lot easier to solve than constrained optimization.
Substitution: maximize the cross section of U along the budget line
Lagrange method

10. Substitution
This turns the constrained optimization into an unconstrained problem.
Find the equation for the cross section of the U=U(x,y) above the budget line and maximize it -- i.e. find the top of the parabola U y y* x* x

11. Substitution
Substitute and maximize
And substitute
again

12. Problem with this method
It can get very mathematically complicated very quickly.
Even U=xαyβ gets very tricky to solve.

13. LaGrange Method
LaGrange knew that unconstrained optimization (like profit max) is relatively simple compared to constrained optimization.
Taking what he knew unconstrained optimization he attempted to simplify the constrained maximization problem by making it mimic the unconstrained problem.

14. Unconstrained Optimization Example
Profit maximization is an example of this. We maximize the difference between two functions: π=R(q) – C(q).

15. Lagrange Method
LaGrange wanted to find a way that was simpler than constrained optimization and more workable than simple substitution.
He wanted to make constrained optimization take the form of the simpler unconstrained problem.
First, let’s look at a simpler problem.

16. Maximize Utility - Expenditure
Maximize utility minus the cost of buying bundles. Think about a one-good world. U
U=U(x)
slope = Ux
Expenditure = E = pxx
slope = Ex= px
Problems:
Ux not measured in $ like E.
E is not constrained, we can spend as much as we like. x x*

17. Maximizing Utility - Expenditure
First change the expenditure function by multiplying px by λ. Now call that function EU.
We want λ to measure the marginal utility of $1.
In that case, units of x consumed would cost us utility and both U(x) and EU(x) would be measured in the same units. U
U=U(x)
slope = Ux
EU=λpxx
Slope = EUx= λpx x x*

18. Maximizing Utility - Expenditure
Problem, an infinite number of λ choices that will each solve this with a different x*
EUx = λ1 px U
U=U(x)
EUx= λ2 px
EUx = λ3 px x x* x* x*
Now the slope of the expenditure function and expenditure are
measured in utils, not dollars. But we are not constraining x yet.

19. LaGrange Method
So first subtract λM from the expenditure function to get
EL = λpxx - λM U
U=U(x)
slope = Ux
EU = λpxx
EL = λpxx - λM x* x
slope = ELx= λpx
-λM

20. LaGrange Method
We know we want to find the x* such that that distance between U(x) and EL(x*) = U(x*). That is, where EL(x*) = 0
So we maximize v = U(x) - 0
Substitute λpxx – λM = 0 in for 0, to constrain x* to our budget. U
U=U(x)
slope = Ux
U=U(x*)-0
EL = λpxx - λM x x*
slope = ELx= λ px
-λI

21. LaGrange Method
Our optimization becomes an unconstrained problem by including the requirement that λpxx = λM.
λ is chosen along with x to maximize utility so that λ = the marginal utility of $1. That is, λpx = Ux. U
U=U(x)
slope = Ux
U=U(x*)-0
EL = λ(pxx – M) x x*
slope = ELx= λ px
-λM

22. LaGrange Method

23. Two Goods: Lagrange’s Manufactured Plane
To maximize utility, maximize the height of the utility function above the plane
EL = λpxx + λpyy – λM
Such that
λpxx + λpyy – λM = 0 U
U = U(x,y) y
When
x = 0 and
y = 0,
U = - λM
ELy= λ py
LaGrange Plane
EL=g(x,y)
EL= λpxx+ λpyy- λM
g’x=ELx= λ px
g’y=ELy= λ py
ELx= λ px x

24. Lagrange Method U
U = U(x,y) y
UL=g(x,y) = 0
λ(pxx+pyy – M) = 0 x

25. Basic Demand Analysis
Using Lagrangian to generate ordinary (Marshallian) demand curves.
FOCs necessary
SOCs sufficient (check that they hold)
Ordinary (Marshallian) demand curves
Inverse demand curves
Meaning of λ
Indirect Utility
Expenditure Function
Comparative Statics General Results

26. Demand Functions using Lagrange’s Method
Set up and maximize:
λ* chosen so that the constraint plane is
parallel to the utility function.
Any x* and y* that maximizes utility
will also have to exhaust income.

27. FOCs for an Optimum
For utility to be maximized, it is necessary that the indifference curve is tangent to the budget constraint (as above).
But it is not sufficient, we also need a diminishing MRS.
FOCs satisfied
Utility Maximized y y y
Utility Maximized
FOCs satisfied x x x

28. SOCs for an Optimum Sufficient condition for a maximum to exist
If the MRS is non-increasing (utility function quasi concave) for all x, that is sufficient for a maximum to exist – but it may not be unique.
If the MRS is diminishing (utility function strictly quasi concave) for all x, that is sufficient for a unique maximum to exist. Need this to satisfy second order conditions for maximization.
SOCs do not hold
Utility Maximized y y y
SOCs satisfied x x x

29. Expenditure Minimization: SOC
The FOC ensure that the optimal consumption bundle is at a tangency.
The SOC ensure that the tangency is a minimum, and not a maximum by ensuring that away from the tangency, along the budget line, utility falls. Y
U*>U’
U=U*
U=U’ X

30. Checking SOC: utility function strictly quasi-concave
The second order conditions will hold if the utility function is strictly quasi-concave
A function is strictly quasi-concave if its bordered Hessian is negative definite. That is:
A function is strictly quasi-concave if:
-UxUx < 0
2UxUxyUy - Uy2Uxx - Ux2Uyy > 0
Binger and Hoffman, page 115

31. Checking SOC: Constrained Maximization
The second order condition for constrained maximization will hold if the following bordered Hessian matrix is negative definite:
Note:
So this Hessian and
The last only differ
by 

32. Ordinary Demand Curves
And from the FOC:

33. Inverse Demand Curves
Starting with the ordinary demand curves:

34. Utility and Indirect Utility
Maximum Utility, a function of quantities
Indirect Utility a function of price and income

35. Optimization : Expenditure Function
Start with indirect utility
Solve for M:
This equation determines the expenditure needed to generate Ū, the expenditure function:

36. Digression: Envelope Theorem
Say we know that y = f(x; ω)
We find y is maximized at x* = x(ω)
So we know that y* = y(x*=x(ω),ω)).
Now say we want to find out
So when ω changes, the optimal x changes, which changes the y* function.
Two methods to solve this…

37. Digression: Envelope Theorem
Start with: y = f(x; ω) and calculate x* = x(ω)
First option:
y = f(x; ω), substitute in x* = x(ω) to get y* = y(x(ω); ω):
Second option, turn it around:
First, take then substitute x* = x(ω)
into yω(x ; ω) to get
And these two answers are equivalent:

38. Envelope Result, 1st option to get
Plug the optimal values into the LaGrangian
Silberburg, first edition, page has a similar treatment.

39. Optimization: Envelope Result
Plug the optimal values into the LaGrangian
Silberburg, first edition, page has a similar treatment.

40. Optimization: Comparative Statics
If we have a specified utility function and we derive the equations for the demand functions, the comparative statics are easy.
Take the derivatives to calculate the changes in x and y when prices or income change.
However, what if all we know is U = U(x, y) and we feel safe only assuming:
Ux > 0 Uy > 0 Uxx < 0 Uyy < 0
Can we get anything from that?

41. Optimization: Comparative Statics
Start with:

42. Comparative Statics: Utility Maximizing x*, y*, λ*

43. Tells us that if income increases by $1, so will total expenditure.
Comparative Statics: Effect of a change in M Differentiate (1), (2), (3) w.r.t. M
Side note:
Tells us that if income increases by $1, so will total expenditure.

44. Comparative Statics: Effect of a change in M Put in Matrix Notation
Solve for

45. Comparative Statics: Effect of a change in I Put in Matrix Notation
Solve for

46. Comparative Statics: Effect of a change in px Differentiate (1), (2), (3) w.r.t. px

47. Comparative Statics: Effect of a change in px Put in Matrix Notation
Solve for

48. Comparative Statics: Effect of a change in px Put in Matrix Notation… AGAIN
Solve for

49. Comparative Statics: Preview of income and substitution effects
Rearrange
these
Sub in
these
Income
effect
matters

50. Specific Utility Functions
Cobb-Douglas
CES
Perfect Compliments

51. Cobb-Douglas: Utility Max
Problem:
Set up the LaGrangian
FOC

52. Cobb-Douglas: Demand FOC Imply, to maximize utility, these must hold.
Plug into the budget constraint to get the ordinary (Marshallian) demand functions:
Note, demand only a function of own price changes (one Cobb-Douglas weakness)

53. Cobb-Douglas: Demand
Preferences are homothetic (only a function of the ratio of y:x). When income rises, optimal bundle along a ray from the origin.
Expenditure a constant proportion of income
Income elasticities are = 1

54. Cobb-Douglas: Indirect Utility
Plug x* and y* into the utility function

55. Cobb-Douglas: Expenditure Function
Start with indirect utility function
Solve for M, and then rename it E

56. CES: Utility Max
Problem:
Set up the Lagrangian
FOC

57. CES: Demand
FOC Imply
Plug into the budget constraint and solve:

58. CES: Indirect Utility
Plug x* and y* into the utility function

59. CES: Expenditure Function
Solve for M, then rename E.

60. CES: Expenditure Function (cont)
Solve for M, then rename E.

61. Perfect Compliments: Utility Max
Problem:
No Lagrangian, just exhaust income such that:
So plug this condition into the budget equation
Essentially, we substitute the expansion path into the budget line equation.

62. Perfect Compliments: Demand
Demand equations

63. Perfect Compliments: Indirect Utility
Since utility from x = utility from y at utility max:

64. Perfect Compliments: Expenditure Function
Since the utility from consumption of each must be equal,

65. Bonus Topics Money Metric Utility Function Homogeneity
Corner Solutions (Kuhn-Tucker)
Lump-Sum Principle
MRS and MRT (Marginal Rate of Transformation – slope of PPF)

66. Money Metric Utility Function
Start with an expenditure function and replace with Ū with U=U(x,y)
Now we know the minimum expenditure to get the same utility as the bundle x’, y’.
That is, for an x’, y’, this function tells us the cost of being at the tangency on the same indifference curve. As all bundles on the same curve get the same min. expenditure and prefernece ordering is preserved, this is a utility function.

67. Evaluating Housing Policy
Assume you find the poor generally expend 1/3 of income on housing.
The government wants to double the quality of housing the poor consume at the same 1/3 their income.
How to evaluate?

68. Pre-Public Housing Y
If expenditure on housing is generally 1/3 of income, assume
U=h1/3y2/3
ph=$1
px=$1
M=$1,000
667 H
333
667

69. Public Housing Y ph=$1 px=$1 M=$1,000 Qualified citizens get 667
housing units for 1/3 of income ($333)
667 H
333
667

70. Public Housing: Money Metric Utility
ph=$1
px=$1
M=$1,000
Qualified citizens get 667
housing units for 1/3 of income ($333)
667
UEPH=1,261
UE=1,000 H
333
667
The extra housing has a value to the poor of $261. Depending on the cost to the
government of providing the housing, the program can be evaluated.

71. xi* = xi(p1,p2,…,pn,M) = xi(tp1,tp2,…,tpn,tM)
Homogeneity
If all prices and income were doubled, the optimal quantities demanded will not change
the budget constraint is unchanged
xi* = xi(p1,p2,…,pn,M) = xi(tp1,tp2,…,tpn,tM)
Individual demand functions are homogeneous of degree zero in all prices and income
To test, replace all prices and income with t*p and t*M. The quantity demanded should be unaffected (all the “t” should cancel out).

72. Corner Solutions
Non-corner solutions: Optimal bundle will be where x > 0, y > 0 and MRS = px/py Y
Two corner solutions: Optimal bundle will be where x =0 X

73. Corner Solutions Y A X
At “A”, MRS = px/py, but the optimal quantity of X = 0

74. Corner Solutions Y B X
At “B”, the tangency condition holds where x* < 0. Given the budget line, the optimal feasible x is where x = 0 and MRS < px/py

75. Corner Solution
To develop these conditions as part of a Lagrangian equation, we add non-negativity constraints: 𝑥= 𝑠 2 (we could add 𝑦= 𝑡 2 if we wanted to be really thorough).
Lagrangian:
L=𝑈(𝑥,𝑦)+𝜆(M− 𝑝 𝑥 𝑥− 𝑝 𝑦 𝑦)+𝜇(𝑥− 𝑠 2 )
Requiring x = s2 is simply a way of ensuring that x ≥ 0.

76. Corner Solutions
FOC

77. Corner Solutions Kuhn-Tucker Condition This tell us that either:
μ =0 (the optimum is at a tangency)
s = 0 (the optimum is where x = 0)
μ =0 and s = 0 (the optimum is at a tangency where x = 0)

78. Corner Solutions
If s > 0 and μ = 0 Y X
The usual assumption is that the optimal bundle will be where x>0, y>0 and

79. Corner Solutions At “A”, , but the optimal quantity of x = 0
If s = 0 and μ = 0 Y A X
At “A”, , but the optimal quantity of x = 0

80. Corner Solutions At B’, the tangency condition holds where x* < 0.
If s = 0 and μ > 0 Y B B’ X
At B’, the tangency condition holds where x* < 0.
The optimum is where x=0 and

81. Kuhn-Tucker Example
Utility: U=xy+20y, M = 40, px = $4 and py = 1.

82. Looks Like
Tangency where x=-5, y=60 Y X

83. How about a feasible optimum?
Tangency where x=-5, y=60 Y X

84. Kuhn-Tucker Set-up
Utility: U=xy+20y, M = 40, px = $4 and py = 1

85. Kuhn-Tucker Result

86. Kuhn-Tucker Result
In this example, at x = 0, y = 40,   0 :

87. How about a feasible optimum?
Optimum where x=0, y=40 Y X

88. Solving Kuhn-Tucker
If you find that the optimal bundle is not on the budget constraint, check all corners for a maximum utility.

89. Lump Sum Tax Taxing a good vs taxing income Tax on x only
Lump sum tax (income tax)

90. Lump Sum Tax
Difference in the budget lines: sales tax

91. Lump Sum Principle A tax on x rotates the budget line to have: Y y*

92. Lump Sum Tax
Difference in the budget lines: income tax

93. Lump Sum Principle
Tax paid = x*τ. Alternatively, an income tax of that same amount would shift the budget line so that the consumer can just afford the same bundle they chose under the tax on x. Y y* X x*

94. Lump Sum Principle
When
Indirect Utility function is:

95. Lump Sum Principle
Set, I=60, py = 2, px=1
Utility is:

96. Lump Sum Principle With a $1 tax on x, Utility is:
And tax revenue is $20.

97. Lump Sum Principle
With a $20 income tax,
Utility is:
68.85 > 63.24

98. The Lump Sum Principle: Perfect Compliments
If the utility function is U = Min(x,4y) then plug x=4y into the budget equation to get:
The indirect utility function is then:

99. The Lump Sum Principle: Perfect Compliments
Set, I=60, py = 2, px=1
Utility is:

100. The Lump Sum Principle: Perfect Compliments
With a $1 tax on x,
Utility is:

101. The Lump Sum Principle: Perfect Compliments
With a $24 income tax,
Utility is:
Since preferences do not allow substitution, the consumer makes the same choice either way and the tax does not affect utility.

102. MRS=MRT Using Varian’s example about milk and butter
B* and M* may be different for all consumers.
However, the MRS at the tangency is the same for ALL consumers, no matter the income or preferences.
Milk
Pb=3; Pm=1
MRS=3
MRS=3
MRS=3
Butter

103. Marginal Rate of Transformation (a.k.a Rate of Product Transformation)
Varian is sort of implicitly assuming the PPF is linear. So there is a constant trade of in producing butter or milk as resources are reallocated.
Milk
MRS=3
MRT=3
Social Indifference curve
Butter

104. Marginal Rate of Transformation
With a more realistic PPF, the MRT rises as more butter and less milk is produced.
Milk
MRT=3
Butter

105. MRS=MRT
In the long run (π=0), the cost of producing butter must be 3 times the cost of producing milk. That is, the tradeoffs in consumption = the tradeoff in production
Milk
px/py=3
Butter

106. Back to Varian’s treatment
A new technology that allows you to produce butter with 4 gallons of milk is not going to be a winner as everyone would choose the cheaper butter previously offered.
Milk
MRT=4
Pb=4; Pm=1
Butter

107. However, a new technology that allows you to produce 1 pound of butter with 2 gallons of milk IS going to be a winner!
Milk
MRS=MRT=2
Pb=2; Pm=1
Butter

108. MRS=MRT
Here is what improved butter making technology does with a more standard PPF.
Milk
px/py=2
px/py=3
Butter