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Indifference Curves. An indifference curve shows a set of consumption bundles among which the individual is indifferent. Quantity of Y. Combinations (X 1 , Y 1 ) and (X 2 , Y 2 ) provide the same level of utility. Y 1. Y 2. U 1. Quantity of X. X 1. X 2. Marginal Rate of Substitution.

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indifference curves
Indifference Curves
  • An indifference curve shows a set of consumption bundles among which the individual is indifferent

Quantity of Y

Combinations (X1, Y1) and (X2, Y2)

provide the same level of utility

Y1

Y2

U1

Quantity of X

X1

X2

marginal rate of substitution
Marginal Rate of Substitution
  • The negative of the slope of the indifference curve at any point is called the marginal rate of substitution (MRS)

Quantity of Y

Y1

Y2

U1

Quantity of X

X1

X2

marginal rate of substitution1

At (X1, Y1), the indifference curve is steeper.

The person would be willing to give up more

Y to gain additional units of X

At (X2, Y2), the indifference curve

is flatter. The person would be

willing to give up less Y to gain

additional units of X

Marginal Rate of Substitution
  • MRS changes as X and Y change
    • reflects the individual’s willingness to trade Y for X

Quantity of Y

Y1

Y2

U1

Quantity of X

X1

X2

indifference curve map

Increasing utility

Indifference Curve Map
  • Each point must have an indifference curve through it

Quantity of Y

U1 < U2 < U3

U3

U2

U1

Quantity of X

transitivity
Transitivity
  • Can two of an individual’s indifference curves intersect?

The individual is indifferent between A and C.

The individual is indifferent between B and C.

Transitivity suggests that the individual

should be indifferent between A and B

Quantity of Y

But B is preferred to A

because B contains more

X and Y than A

C

B

U2

A

U1

Quantity of X

convexity
Convexity
  • A set of points is convex if any two points can be joined by a straight line that is contained completely within the set

Quantity of Y

The assumption of a diminishing MRS is

equivalent to the assumption that all

combinations of X and Y which are

preferred to X* and Y* form a convex set

Y*

U1

Quantity of X

X*

convexity1
Convexity
  • If the indifference curve is convex, then the combination (X1 + X2)/2, (Y1 + Y2)/2 will be preferred to either (X1,Y1) or (X2,Y2)

Quantity of Y

This implies that “well-balanced” bundles are preferred

to bundles that are heavily weighted toward one

commodity

Y1

(Y1 + Y2)/2

Y2

U1

Quantity of X

X1

(X1 + X2)/2

X2

utility and the mrs
Utility and the MRS
  • Suppose an individual’s preferences for hamburgers (Y) and soft drinks (X) can be represented by
  • Solving for Y, we get

Y = 100/X

  • Solving for MRS = -dY/dX:
  • MRS = -dY/dX = 100/X2
utility and the mrs1
Utility and the MRS

MRS = -dY/dX = 100/X2

  • Note that as X rises, MRS falls
    • When X = 5, MRS = 4
    • When X = 20, MRS = 0.25
marginal utility
Marginal Utility
  • Suppose that an individual has a utility function of the form

utility = U(X1, X2,…, Xn)

  • We can define the marginal utility of good X1 by

marginal utility of X1 = MUX1 = U/X1

  • The marginal utility is the extra utility obtained from slightly more X1 (all else constant)
marginal utility1
Marginal Utility
  • The total differential of U is
  • The extra utility obtainable from slightly more X1, X2,…, Xn is the sum of the additional utility provided by each of these increments
deriving the mrs
Deriving the MRS
  • Suppose we change X and Y but keep utility constant (dU = 0)

dU = 0 = MUXdX + MUYdY

  • Rearranging, we get:
  • MRS is the ratio of the marginal utility of X to the marginal utility of Y
diminishing marginal utility and the mrs
Diminishing Marginal Utility and the MRS
  • Intuitively, it seems that the assumption of decreasing marginal utility is related to the concept of a diminishing MRS
    • Diminishing MRS requires that the utility function be quasi-concave
      • This is independent of how utility is measured
    • Diminishing marginal utility depends on how utility is measured
  • Thus, these two concepts are different
marginal utility and the mrs
Marginal Utility and the MRS
  • Again, we will use the utility function
  • The marginal utility of a soft drink is

marginal utility = MUX = U/X = 0.5X-0.5Y0.5

  • The marginal utility of a hamburger is

marginal utility = MUY = U/Y = 0.5X0.5Y-0.5

examples of utility functions
Examples of Utility Functions
  • Cobb-Douglas Utility

utility = U(X,Y) = XY

where  and  are positive constants

    • The relative sizes of  and  indicate the relative importance of the goods
examples of utility functions1

U3

U2

U1

Examples of Utility Functions
  • Perfect Substitutes

utility = U(X,Y) = X + Y

Quantity of Y

The indifference curves will be linear.

The MRS will be constant along the

indifference curve.

Quantity of X

examples of utility functions2

U3

U2

U1

Examples of Utility Functions
  • Perfect Complements

utility = U(X,Y) = min (X, Y)

Quantity of Y

The indifference curves will be

L-shaped. Only by choosing more

of the two goods together can utility

be increased.

Quantity of X

examples of utility functions3
Examples of Utility Functions
  • CES Utility (Constant elasticity of substitution)

utility = U(X,Y) = X/ + Y/

when   0 and

utility = U(X,Y) = ln X + ln Y

when  = 0

    • Perfect substitutes   = 1
    • Cobb-Douglas   = 0
    • Perfect complements   = -
examples of utility functions4
Examples of Utility Functions
  • CES Utility (Constant elasticity of substitution)
    • The elasticity of substitution () is equal to 1/(1 - )
      • Perfect substitutes   = 
      • Fixed proportions   = 0
homothetic preferences
Homothetic Preferences
  • If the MRS depends only on the ratio of the amounts of the two goods, not on the quantities of the goods, the utility function is homothetic
    • Perfect substitutes  MRS is the same at every point
    • Perfect complements  MRS=  if Y/X > /, undefined if Y/X = /, and MRS= 0 if Y/X < /
nonhomothetic preferences
Nonhomothetic Preferences
  • Some utility functions do not exhibit homothetic preferences

utility = U(X,Y) = X + ln Y

MUY = U/Y = 1/Y

MUX = U/X = 1

MRS = MUX / MUY = Y

  • Because the MRS depends on the amount of Y consumed, the utility function is not homothetic
important points to note
Important Points to Note:
  • If individuals obey certain behavioral postulates, they will be able to rank all commodity bundles
    • The ranking can be represented by a utility function
    • In making choices, individuals will act as if they were maximizing this function
  • Utility functions for two goods can be illustrated by an indifference curve map
important points to note1
Important Points to Note:
  • The negative of the slope of the indifference curve measures the marginal rate of substitution (MRS)
    • This shows the rate at which an individual would trade an amount of one good (Y) for one more unit of another good (X)
  • MRS decreases as X is substituted for Y
    • This is consistent with the notion that individuals prefer some balance in their consumption choices
important points to note2
Important Points to Note:
  • A few simple functional forms can capture important differences in individuals’ preferences for two (or more) goods
    • Cobb-Douglas function
    • linear function (perfect substitutes)
    • fixed proportions function (perfect complements)
    • CES function
      • includes the other three as special cases