1. Budgetary and Other Constraints on Choice

2. Consumption Choice Sets
A consumption choice set is the collection of all consumption choices available to the consumer.
What constrains consumption choice?
Budgetary, time and other resource limitations.

3. Budget Constraints
A consumption bundle containing x1 units of commodity 1, x2 units of commodity 2 and so on up to xn units of commodity n is denoted by the vector (x1, x2, … , xn).
Commodity prices are p1, p2, … , pn.

4. Budget Constraints
Q: When is a consumption bundle (x1, … , xn) affordable at given prices p1, … , pn?

5. Budget Constraints
Q: When is a bundle (x1, … , xn) affordable at prices p1, … , pn?
A: When p1x1 + … + pnxn £ m where m is the consumer’s (disposable) income.

6. Budget Set and Constraint for Two Commoditiesx2
Budget constraint is
p1x1 + p2x2 = m.
m /p2
m /p1 x1

7. Budget Set and Constraint for Two Commoditiesx2
Budget constraint is
p1x1 + p2x2 = m.
m /p2
m /p1 x1

8. Budget Set and Constraint for Two Commoditiesx2
Budget constraint is
p1x1 + p2x2 = m.
m /p2
Just affordable
m /p1 x1

9. Budget Set and Constraint for Two Commoditiesx2
Budget constraint is
p1x1 + p2x2 = m.
m /p2
Not affordable
Just affordable
m /p1 x1

10. Budget Set and Constraint for Two Commoditiesx2
Budget constraint is
p1x1 + p2x2 = m.
m /p2
Not affordable
Just affordable
Affordable
m /p1 x1

11. Budget Set and Constraint for Two Commoditiesx2
Budget constraint is
p1x1 + p2x2 = m.
m /p2
the collection of all affordable bundles.
Budget
Set
m /p1 x1

12. Budget Set and Constraint for Two Commoditiesx2
p1x1 + p2x2 = m is
x2 = -(p1/p2)x1 + m/p2
so slope is -p1/p2.
m /p2
Budget
Set
m /p1 x1

13. Budget Constraints
For n = 2 and x1 on the horizontal axis, the constraint’s slope is -p1/p2. What does it mean?

14. Budget Constraints
For n = 2 and x1 on the horizontal axis, the constraint’s slope is -p1/p2. What does it mean?
Increasing x1 by 1 must reduce x2 by p1/p2.
See Notes
Notas:
1)Demostrar que los precios relativos (la pendiente de la recta de presupuesto) es una tasa de cambio, por ende es un costo de oportunidad.

15. Budget Sets & Constraints; Income and Price Changes
The budget constraint and budget set depend upon prices and income. What happens as prices or income change?

16. How do the budget set and budget constraint change as income m increases?x2
Original
budget set x1

17. Higher income gives more choicex2
New affordable consumption choices
Original and
new budget
constraints are
parallel (same
slope).
Original
budget set x1

18. How do the budget set and budget constraint change as income m decreases?x2
Original
budget set x1

19. How do the budget set and budget constraint change as income m decreases?x2
Consumption bundles
that are no longer
affordable.
Old and new
constraints
are parallel.
New, smaller
budget set x1

20. Budget Constraints - Income Changes
Increases in income m shift the constraint outward in a parallel manner, thereby enlarging the budget set and improving choice.
Decreases in income m shift the constraint inward in a parallel manner, thereby shrinking the budget set and reducing choice.

21. Budget Constraints - Income Changes
No original choice is lost and new choices are added when income increases, so higher income cannot make a consumer worse off.
An income decrease may (typically will) make the consumer worse off.

22. Budget Constraints - Price Changes
What happens if just one price decreases?
Suppose p1 decreases.

23. How do the budget set and budget constraint change as p1 decreases from p1’ to p1”?x2
m/p2
-p1’/p2
Original
budget set
m/p1’
m/p1” x1

24. How do the budget set and budget constraint change as p1 decreases from p1’ to p1”?x2
m/p2
New affordable choices
-p1’/p2
Original
budget set
m/p1’
m/p1” x1

25. How do the budget set and budget constraint change as p1 decreases from p1’ to p1”?x2
m/p2
New affordable choices
Budget constraint
pivots; slope flattens
from -p1’/p2 to
-p1”/p2
-p1’/p2
Original
budget set
-p1”/p2
m/p1’
m/p1” x1

26. Budget Constraints - Price Changes
Reducing the price of one commodity pivots the constraint outward. No old choice is lost and new choices are added, so reducing one price cannot make the consumer worse off.

27. Budget Constraints - Price Changes
Similarly, increasing one price pivots the constraint inwards, reduces choice and may (typically will) make the consumer worse off.
The rate of exchange chance (see an example).

28. Uniform Ad Valorem Sales Taxes
A uniform sales tax levied at rate t changes the constraint from p1x1 + p2x2 = m to (1+t)p1x1 + (1+t)p2x2 = m

29. Uniform Ad Valorem Sales Taxes
A uniform sales tax levied at rate t changes the constraint from p1x1 + p2x2 = m to (1+t)p1x1 + (1+t)p2x2 = m i.e p1x1 + p2x2 = m/(1+t).
Case for tax quantity
Case for tax on both sales tax and income

30. Uniform Ad Valorem Sales Taxes
p1x1 + p2x2 = m x1

31. Uniform Ad Valorem Sales Taxes
p1x1 + p2x2 = m
p1x1 + p2x2 = m/(1+t) x1

32. Uniform Ad Valorem Sales Taxes
Equivalent income loss is x1

33. Uniform Ad Valorem Sales Taxes
Case for tax quantity
Case for sales tax and income tax
Quantity subsidy (see notes)
Value Subsidy (see notes)
Lump-sum tax or subsidy (see notes)
Budget set with rationing (see notes)

34. Preferences

35. Rationality in Economics
Behavioral Postulate: A decisionmaker always chooses its most preferred alternative from its set of available alternatives.
So to model choice we must model decisionmakers’ preferences.

36. Preference Relations
Comparing two different consumption bundles, x and y:
strict preference: x is more preferred than is y.
weak preference: x is as at least as preferred as is y.
indifference: x is exactly as preferred as is y.

37. Preference Relations
Strict preference, weak preference and indifference are all preference relations.
Particularly, they are ordinal relations; i.e. they state only the order in which bundles are preferred.

38. Preference Relations p p
denotes strict preference; x y means that bundle x is preferred strictly to bundle y. p p

39. Preference Relations p p f ~ f ~
denotes strict preference so x y means that bundle x is preferred strictly to bundle y.
~ denotes indifference; x ~ y means x and y are equally preferred.
denotes weak preference; x y means x is preferred at least as much as is y. p p ~ f ~ f

40. Assumptions about Preference Relations
Completeness: For any two bundles x and y it is always possible to make the statement that either x y or y x. ~ f ~ f

41. Assumptions about Preference Relations
Reflexivity: Any bundle x is always at least as preferred as itself; i.e x x. ~ f

42. Assumptions about Preference Relations
Transitivity: If x is at least as preferred as y, and y is at least as preferred as z, then x is at least as preferred as z; i.e. x y and y z x z. ~ f ~ f ~ f

43. Indifference Curves
Take a reference bundle x’. The set of all bundles equally preferred to x’ is the indifference curve containing x’; the set of all bundles y ~ x’.

44. Indifference Curves x2
x’ ~ x” ~ x”’ x’ x”
x”’ x1

45. Indifference Curves x2
z x y p p x z y x1

46. Indifference Curves I1 All bundles in I1 are
strictly preferred to all in I2. x2 x z I2
All bundles in I2 are strictly preferred to all in I3. y I3 x1

47. Indifference Curves x2 x I(x’) I(x) x1
WP(x), the set of bundles weakly
preferred to x. x
I(x’)
I(x) x1

48. Indifference Curves x2 x I(x) x1 WP(x), the set of bundles weakly
preferred to x. x
WP(x) includes I(x).
I(x) x1

49. Indifference Curves x2 x I(x) x1 SP(x), the set of bundles strictly
preferred to x, does not include
I(x). x
I(x) x1

50. Indifference Curves Cannot Intersect
From I1, x ~ y. From I2, x ~ z.
Therefore y ~ z. x2 I1 x y z x1

51. Indifference Curves Cannot Intersect
From I1, x ~ y. From I2, x ~ z.
Therefore y ~ z. But from I1 and I2 we see y z, a contradiction. x2 I2 I1 p x y z x1

52. Slopes of Indifference Curves
When more of a commodity is always preferred, the commodity is a good.
If every commodity is a good then indifference curves are negatively sloped.

53. Slopes of Indifference Curves
Good 2
Two goods a negatively sloped indifference curve.
Better
Worse
Good 1

54. Slopes of Indifference Curves
If less of a commodity is always preferred then the commodity is a bad.

55. Slopes of Indifference Curves
Good 2
One good and one bad a positively sloped indifference curve.
Better
Worse
Analizar el ejemplo
Bad 1

56. Utility Functions
A preference relation that is complete, reflexive, transitive and continuous can be represented by a continuous utility function.
Continuity means that small changes to a consumption bundle cause only small changes to the preference level.

57. Utility Functions f ~ p p
A utility function U(x) represents a preference relation if and only if: x’ x” U(x’) > U(x”) x’ x” U(x’) < U(x”) x’ ~ x” U(x’) = U(x”). ~ f p p

58. Utility Functions Utility is an ordinal (i.e. ordering) concept.
E.g. if U(x) = 6 and U(y) = 2 then bundle x is strictly preferred to bundle y.

59. Utility Functions & Indiff. Curves
Consider the bundles (4,1), (2,3) and (2,2).
Suppose (2,3) (4,1) ~ (2,2).
Assign to these bundles any numbers that preserve the preference ordering; e.g. U(2,3) = 6 > U(4,1) = U(2,2) = 4.
Call these numbers utility levels. p

60. Utility Functions & Indiff. Curves
An indifference curve contains equally preferred bundles.
Equal preference  same utility level.
Therefore, all bundles in an indifference curve have the same utility level.

61. Utility Functions & Indiff. Curves
So the bundles (4,1) and (2,2) are in the indiff. curve with utility level U º 4
But the bundle (2,3) is in the indiff. curve with utility level U º 6.
On an indifference curve diagram, this preference information looks as follows:

62. Utility Functions & Indiff. Curvesx2
(2,3) (2,2) ~ (4,1) p
U º 6
U º 4 x1

63. Utility Functions & Indiff. Curves
Another way to visualize this same information is to plot the utility level on a vertical axis.

64. Utility Functions & Indiff. Curves
3D plot of consumption & utility levels for 3 bundles
Utility
U(2,3) = 6
U(2,2) = 4 U(4,1) = 4 x2 x1

65. Utility Functions & Indiff. Curves
This 3D visualization of preferences can be made more informative by adding into it the two indifference curves.

66. Utility Functions & Indiff. Curvesx2
Higher indifference curves contain
more preferred bundles. x1

67. Utility Functions & Indiff. Curves
Comparing more bundles will create a larger collection of all indifference curves and a better description of the consumer’s preferences.

68. Utility Functions & Indiff. Curvesx2
U º 6
U º 4
U º 2 x1

69. Utility Functions & Indiff. Curves
As before, this can be visualized in 3D by plotting each indifference curve at the height of its utility index.

70. Utility Functions & Indiff. Curvesx2
U º 3
U º 2
U º 1 x1

71. Utility Functions & Indiff. Curves
The collection of all indifference curves for a given preference relation is an indifference map.
An indifference map is equivalent to a utility function; each is the other.

72. Utility Functions
There is no unique utility function representation of a preference relation.
Suppose U(x1,x2) = x1x2 represents a preference relation.
Again consider the bundles (4,1), (2,3) and (2,2).

73. Utility Functions
U(x1,x2) = x1x2, so U(2,3) = 6 > U(4,1) = U(2,2) = 4; that is, (2,3) (4,1) ~ (2,2).
Definir transformación monótona (ver notas) p

74. Utility Functions p U(x1,x2) = x1x2 (2,3) (4,1) ~ (2,2).
Define V = U2. p

75. Utility Functions p p U(x1,x2) = x1x2 (2,3) (4,1) ~ (2,2).
Define V = U2.
Then V(x1,x2) = x12x22 and V(2,3) = 36 > V(4,1) = V(2,2) = 16 so again (2,3) (4,1) ~ (2,2).
V preserves the same order as U and so represents the same preferences. p p

76. Goods, Bads and Neutrals
A good is a commodity unit which increases utility (gives a more preferred bundle).
A bad is a commodity unit which decreases utility (gives a less preferred bundle).
A neutral is a commodity unit which does not change utility (gives an equally preferred bundle).

77. Goods, Bads and Neutrals
Utility
Utility function
Units of water are goods
Units of water are bads
Water x’
Around x’ units, a little extra water is a neutral.

78. Some Other Utility Functions and Their Indifference Curves
Instead of U(x1,x2) = x1x2 consider
V(x1,x2) = x1 + x2.
What do the indifference curves for this “perfect substitution” utility function look like?

79. Perfect Substitution Indifference Curvesx2
x1 + x2 = 5 13
x1 + x2 = 9 9
x1 + x2 = 13 5
V(x1,x2) = x1 + x2. 5 9 13 x1

80. Perfect Substitution Indifference Curvesx2
x1 + x2 = 5 13
x1 + x2 = 9 9
x1 + x2 = 13 5
Ver propiedad Propiedades
V(x1,x2) = x1 + x2. 5 9 13 x1
All are linear and parallel.