1. Utility Maximization and Choice
Chapter 4
Utility Maximization and Choice

2. Consumer Behavior
● Theory of consumer behavior
Description of how consumers allocate incomes among different goods and services to maximize their well-being.
Consumer behavior is best understood in these distinct steps:
Consumer Preferences
Utility
Budget Constraints
Consumer Choices

3. Key Definitions Budget Set: The set of baskets that are affordable
Budget Constraint:
The set of baskets that the consumer may purchase given the limits of the available income.
Budget Line:
The set of baskets that one can purchase when spending all available income.

4. The Budget Constraint PXX + PYY ≤ I
Assume only two goods available: X and Y
Price of good x: Px ; Price of good y: Py
Income: I
Total expenditure on basket (X,Y): PxX + PyY
The Basket is Affordable if total expenditure does not exceed total Income:
PXX + PYY ≤ I

5. The Budget Constraint pxx + pyy  I
Assume that an individual has I amount of income to allocate between good x and good y
pxx + pyy  I
The individual can afford
to choose only combinations
of x and y in the shaded
triangle
Quantity of y
If all income is spent
on y, this is the amount
of y that can be purchased
If all income is spent
on x, this is the amount
of x that can be purchased
Quantity of x

6. A Budget Constraint Example
Two goods available: X and Y
I = $10
Px = $1
Py = $2
All income spent on X → I/Px : units of X bought
All income spent on Y → I/Py : units of Y bought
Budget Line 1:
1X + 2Y = 10 Or
Y = 5 – X/2
Slope of Budget Line
= -Px/Py
= -1/2

7. • B • • A Budget Constraint Example Y Budget line = BL1 I/PY= 5 A
-PX/PY = -1/2 B • • C X
I/PX = 10

8. Budget Constraint Shifting of Budget line: 1) Change in income
2) Change in price of good X
3) Change in price of good Y
4) Change in the price of both good X and Y.

9. Budget Constraint
Location of budget line shows what the income level is.
Increase in Income will shift the budget line to the right.
More of each product becomes affordable
Decrease in Income will shift the budget line to the left.
less of each product becomes affordable

10. A Budget Constraint ExampleY
Shift of a budget line
I = $12
PX = $1
PY = $2
Y = 6 - X/2 …. BL2
If income rises, the budget line shifts parallel to the right (shifts out)
If income falls, the budget line shifts parallel to the left (shifts in) 6 5
BL2
BL1 10 12 X

11. A Budget Constraint ExampleY
Rotation of a budget line
If the price of X rises, the budget line gets steeper and the horizontal intercept shifts in
If the price of X falls, the budget line gets flatter and the horizontal intercept shifts out
I = $10
PX = $1
PY = $3
Y = X/3 …. BL2 6
BL1 5
3.33
BL2 10 X

12. Consumer Choice Assume:  Only non-negative quantities
"Rational” choice:
The consumer chooses the basket that maximizes his satisfaction given the constraint that his budget imposes.
Consumer’s Problem:
Max U(X,Y)
Subject to: PxX + PyY < I

13. Constrained Consumer Choice
Consumers maximize their well-being (utility) subject to their budget constraint.
The highest indifference curve attainable given the budget is the consumer’s optimal bundle.
When the optimal bundle occurs at a point of tangency between the indifference curve and budget line, this is called an interior solution / Interior optimum.

14. Interior Optimum MRSx,y = MUx/MUy = Px/Py
Interior Optimum: The optimal consumption basket is at a point where the indifference curve is just tangent to the budget line.
A tangent: to a function is a straight line that has the same slope as the function…therefore….
MRSx,y = MUx/MUy = Px/Py
“The rate at which the consumer would be willing to exchange X for Y is the same as the rate at which they are exchanged in the marketplace.”

15. • • • Interior Consumer Optimum Y B Preference Direction
Optimal Choice (interior solution) IC C • BL X

16. Interior Consumer Optimum
Utility is maximized where the indifference curve is tangent to the budget constraint
Quantity of y B U2
Quantity of x

17. Interior Consumer Optimum

18. Interior Consumer Optimum: Assumptions
U (X,Y) = XY and MUx = Y while MUy = X
I = $1,000
P = $50 and P = $200
Suppose Basket A contains (X=4, Y=4)
Suppose Basket B contains (X=10, Y=2.5)
Question:
Is either basket the optimal choice for the consumer?
Basket A:
MRSx,y = MUx/MUy = Y/X = 4/4 = 1
Slope of budget line = -Px/Py = -1/4
Basket B:
MRSx,y = MUx/MUy = Y/X = 1/4

19. • Interior Consumer Optimum Example Y 50X + 200Y = I U = 25 2.5 X 1010
Chapter Four

20. Contained Optimization
What are the equations that the optimal consumption basket must fulfill if we want to represent the consumer’s choice among three goods or more than three goods?

21. Optimization Principle
To maximize utility, given a fixed amount of income, an individual will buy the goods and services:
that exhaust total income
for which the MRS is equal to the rate at which goods can be traded for one another in the marketplace

22. A Numerical Illustration
Assume that the individual’s MRS = 1
willing to trade one unit of x for one unit of y
Suppose the price of x = $2 and the price of y = $1
The individual can be made better off
trade 1 unit of x for 2 units of y in the marketplace

23. Second Order Conditions for a Maximum
The tangency rule is necessary but not sufficient unless we assume that MRS is diminishing
if MRS is diminishing, then indifference curves are strictly convex
if MRS is not diminishing, we must check second-order conditions to ensure that we are at a maximum

24. SOCs for a Maximum The tangency rule is only a necessary condition
we need MRS to be diminishing
Quantity of y U1 B U2 A
There is a tangency at point A,
but the individual can reach a higher level of utility at point B
Quantity of x

25. Corner Solutions
Individuals may maximize utility by choosing to consume only one of the goods
At point A, the indifference curve
is not tangent to the budget constraint
Quantity of y U2 U1 U3 A
Utility is maximized at point A
Quantity of x

26. Consumer Maximization: Corner Solution
At point e, the indifference curve
is not tangent to the budget constraint
Utility is maximized at point e
, Quantity r e 25 Y I 3 I 2
Budget line I 1 50 X
, Quantity

27. Example for quasilinear corner solution
David is considering his purchases of food (x) and clothing (y).
He has a utility function U(x,y) = xy + 10x.
His income is I = 10.
He faces a price of Px = $1 and a price of clothing Py = $2.
What is David’s optimal basket.

28. Constrained Consumer Choice with Quasilinear
Preferences
If the relative price of one good is too high and preferences are quasilinear, the indifference curve will not be tangent to the budget line and the consumer’s optimal bundle occurs at a corner solution.

29. Constrained Consumer Choice with Perfect Substitutes
With perfect substitutes, if the marginal rate of substitution does not equal the marginal rate of transformations, then the consumer’s optimal bundle occurs at a corner solution, bundle b.

30. Example for perfect substitutes
Sara views chocolate and vanilla ice cream as perfect substitute.
She likes both and is willing to trade one scoop of chocolate for two scoops of vanilla ice cream.
Q. if the price of a scoop of chocolate ice cream is (Pc) is three times the price of vanilla (Pv), will Sara buy both types of ice cream?
If not which will she buy?

31. Rational Choice and Marginal Utility
1) If MUx/Px > MUz/Pz,
consume more of good x.
2) If MUy/Py > MUz/Pz,
consume more of good y.

32. Constrained Consumer Choice with Perfect Complements
The optimal bundle is on the budget line and at the right angle (i.e. vertex) of an indifference curve.

33. ℒ = U(x1,x2,…,xn) + (I - p1x1 - p2x2 -…- pnxn)
The n-Good Case
The individual’s objective is to maximize
utility = U(x1,x2,…,xn)
subject to the budget constraint
I = p1x1 + p2x2 +…+ pnxn
Set up the Lagrangian:
ℒ = U(x1,x2,…,xn) + (I - p1x1 - p2x2 -…- pnxn)

34. The n-Good Case FOCs for an interior maximum:
ℒ/x1 = U/x1 - p1 = 0
ℒ /x2 = U/x2 - p2 = 0 •
ℒ /xn = U/xn - pn = 0
ℒ / = I - p1x1 - p2x2 - … - pnxn = 0

35. Interpreting the Lagrangian Multiplier
 is the marginal utility of an extra dollar of consumption expenditure
the marginal utility of income

36. Interpreting the Lagrangian Multiplier
At the margin, the price of a good represents the consumer’s evaluation of the utility of the last unit consumed
how much the consumer is willing to pay for the last unit

37. ℒ/xi = U/xi - pi  0 (i = 1,…,n)
Corner Solutions
A corner solution means that the first-order conditions must be modified:
ℒ/xi = U/xi - pi  0 (i = 1,…,n)
And If
ℒ/xi = U/xi - pi  0,
then xi = 0
This means that
any good whose price exceeds its marginal value to the consumer will not be purchased

38. Cobb-Douglas Demand Functions With constant utility
Cobb-Douglas utility function:
U(x,y) = xy
Setting up the Lagrangian:
ℒ = xy + (I - pxx - pyy)
Let us assume that  +  = 1
FOCs:
ℒ/x = x-1y - px = 0
ℒ/y = xy-1 - py = 0
ℒ/ = I - pxx - pyy = 0

39. Cobb-Douglas Demand Functions
First-order conditions imply:
y/x = px/py
Since  +  = 1:
pyy = (/)pxx = [(1- )/]pxx
Substituting into the budget constraint:
I = pxx + [(1- )/]pxx = (1/)pxx

40. Cobb-Douglas Demand Functions
Solving for x yields
Solving for y yields
The individual will allocate  percent of his income to good x and  percent of his income to good y

41. Example of Cobb – Douglas Utility
Suppose a person has a utility function given as U = x0.8y0.2
Given that Px = 5 and Py = 3 and Income = 75.
Calculate the amount of utility the consumer derives.
Suppose there is 1% increase in income, what will be the changed utility.

42. CES Demand
To illustrate cases in which budget shares are responsive to economic circumstances,
3 specific examples of CES Function.
1) Assume that  = 0.5
2) If  = -1
3) If  = -

43. CES Demand (y/x)0.5 = px/py 1) Assume that  = 0.5
U(x,y) = x0.5 + y0.5
Setting up the Lagrangian:
ℒ = x0.5 + y0.5 + (I - pxx - pyy)
FOCs:
ℒ/x = 0.5x px = 0
ℒ/y = 0.5y py = 0
ℒ/ = I - pxx - pyy = 0
This means that
(y/x)0.5 = px/py

44. CES Demand
Substituting into the budget constraint, we can solve for the demand functions
In these demand functions, the share of income spent on either x or y is not a constant
depends on the ratio of the two prices
The higher is the relative price of x, the smaller will be the share of income spent on x

45. U(x,y) = -x -1 - y -1 = -(1/x) – (1/y)
CES Demand
2) If  = -1,
U(x,y) = -x -1 - y -1 = -(1/x) – (1/y)
Though Utility function sounds strange the marginal utilities are positive and diminishing.
First-order conditions imply that
y/x = (px/py)0.5
The demand functions are
These demand functions are less price responsive.
As Px ↑, the individual cuts back only modestly on good x, so the spending on x ↑.

46. I = pxx + pyy = pxx + py(x/4)
CES Demand
3) If  = -,
U(x,y) = Min(x,4y)
The person will choose only combinations for which x = 4y
This means that
I = pxx + pyy = pxx + py(x/4)
I = (px py)x

47. CES Demand Hence, the demand functions are
Share of the person’s budget to good x ↑as Px ↑.
Why?
X and y must be consumed in fixed proportion.

48. Indirect Utility Function
It is often possible to manipulate first-order conditions to solve for optimal values of x1,x2,…,xn
These optimal values will be •
x*n = xn(p1,p2,…,pn,I)
x*1 = x1(p1,p2,…,pn,I)
x*2 = x2(p1,p2,…,pn,I)

49. Indirect Utility Function
We can use the optimal values of the x’s to find the indirect utility function
maximum utility = U(x*1,x*2,…,x*n)
maximum utility = V(p1,p2,…,pn,I)
The optimal level of utility will depend indirectly on prices and income

50. The Lump Sum Principle
Taxes on an individual’s general purchasing power are superior to taxes on a specific good
an income tax allows the individual to decide freely how to allocate remaining income
a tax on a specific good will reduce an individual’s purchasing power and distort his choices

51. The Lump Sum Principle
A tax on good x would shift the utility-maximizing choice from point A to point B B U2
Quantity of y A U1
Quantity of x

52. The Lump Sum Principle
An income tax that collected the same amount would shift the budget constraint to I’ I’
Quantity of y
Utility is maximized now at point C on U3 U3 C A B U1 U2
Quantity of x

53. The Lump Sum Principle
1) If the utility function is Cobb-Douglas with  =  = 0.5, we know that
The indirect utility function is

54. The Lump Sum Principle If a tax of $1 was imposed on good x
the individual will purchase x* = 2
indirect utility will fall from 2 to 1.41
An equal-revenue tax will reduce income to $6
indirect utility will fall from 2 to 1.5

55. The Lump Sum Principle
2) If the utility function is fixed proportions with U = Min(x,4y), we know that
The indirect utility function is

56. The Lump Sum Principle If a tax of $1 was imposed on good x
indirect utility will fall from 4 to 8/3
An equal-revenue tax will reduce income to $16/3
Since preferences are rigid, the tax on x does not distort choices

57. Expenditure Minimization
Dual minimization problem for utility maximization
allocate income to achieve a given level of utility with the minimal expenditure
the goal and the constraint have been reversed

58. Expenditure Minimization
Point A is the solution to the dual problem
E2 provides just enough to reach U1
Quantity of y E3
E3 will allow the individual to reach U1 but is not the minimal expenditure required to do so E2 E1
E1 is too small to achieve U1 U1
Quantity of x
E1, E2, E3, are different expenditure level

59. total expenditures = E = p1x1 + p2x2 +…+ pnxn
Expenditure Minimization
The individual’s problem is to choose x1,x2,…,xn to minimize
total expenditures = E = p1x1 + p2x2 +…+ pnxn
subject to the constraint
utility = Ū = U(x1,x2,…,xn)
The optimal amounts of x1,x2,…,xn will depend on the prices of the goods and the required utility level

60. minimal expenditures = E(p1,p2,…,pn,U)
Expenditure Function
The expenditure function shows the minimal expenditures necessary to achieve a given utility level for a particular set of prices
minimal expenditures = E(p1,p2,…,pn,U)
The expenditure function and the indirect utility function are related
both depend on market prices but involve different constraints

61. Expenditure Minimization with Calculus
Minimize expenditure, E, subject to the constraint of holding utility constant:
The solution of this problem, the expenditure function, shows the minimum expenditure necessary to achieve a specified utility level for a given set of prices:

62. Duality
The mirror image of the original (primal) constrained optimization problem is called the dual problem.
Min PxX + PyY
(X,Y) subject to: U(X,Y) = U*
where: U* is a target level of utility.
If U* is the level of utility that solves the primal problem, then an interior optimum, if it exists, of the dual problem also solves the primal problem.

63. • Optimal Choice Example: Expenditure MinimizationY
Example: Expenditure Minimization •
Optimal Choice (interior solution)
U = U*
Decreases in
expenditure level
PXX + PYY = E* X

64. Expenditure Function Two ways to compute expenditure function.
1) Cobb- Douglas case
The indirect utility function in the two-good
If we interchange the role of utility and income (expenditure), we will have the expenditure function
E(px,py,U) = 2px0.5py0.5U

65. Expenditure Function 2) Fixed – Proportions case
The indirect utility function is
If we again switch the role of utility and expenditures, we will have the expenditure function
E(px,py,U) = (px py)U