Transcript Indifference Curves • An indifference curve shows a set of consumption bundles among which the individual is indifferent Quantity of Y Combinations (X1, Y1) and.

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
• The negative of the slope of the
indifference curve at any point is called
the marginal rate of substitution (MRS)
Quantity of Y
dY
MRS  
dX
Y1
Y2
U1
Quantity of X
X1
X2
U U1
Marginal Rate of Substitution
• MRS changes as X and Y change
– reflects the individual’s willingness to trade Y
for X
Quantity of Y
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
Y1
Y2
U1
Quantity of X
X1
X2
Indifference Curve Map
• Each point must have an indifference
curve through it
Quantity of Y
Increasing utility
U3
U2
U1
U1 < U2 < U3
Quantity of X
Transitivity
• Can two of an individual’s indifference
curves intersect?
Quantity of Y
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
C
B
A
U2
But B is preferred to A
because B contains more
X and Y than A
U1
Quantity of X
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
X*
Quantity of X
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
X1 (X1 + X2)/2 X2
Quantity of X
Utility and the MRS
• Suppose an individual’s preferences for
hamburgers (Y) and soft drinks (X) can
be represented by
utility  10  X  Y
• Solving for Y, we get
Y = 100/X
• Solving for MRS = -dY/dX:
MRS = -dY/dX = 100/X2
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
• 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 Utility
• The total differential of U is
U
U
U
dU 
dX 1 
dX 2  ... 
dX n
X1
X 2
X n
dU  MUX dX1  MUX dX 2  ...  MUX dXn
1
2
n
• 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
• Suppose we change X and Y but keep
utility constant (dU = 0)
dU = 0 = MUXdX + MUYdY
• Rearranging, we get:
dY

dX
MU X U / X


MU Y U / Y
U constant
• MRS is the ratio of the marginal utility of
X to the marginal utility of Y
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
• Again, we will use the utility function
utility  X  Y  X Y
0.5
0.5
• The marginal utility of a soft drink is
marginal utility = MUX = U/X = 0.5X0.5Y0.5
• The marginal utility of a hamburger is
0.5Ymarginal utility = MUY = U/Y =0.50.5X
0.5
0.5
dY
MU X .5 X Y
Y
MRS  



0.5 0.5
dX Uconstant MUY .5 X Y
X
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 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.
U3
U1
U2
Quantity of X
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.
U3
U2
U1
Quantity of X
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 Functions
• CES Utility (Constant elasticity of
substitution)
– The elasticity of substitution () is equal to
1/(1 - )
• Perfect substitutes   = 
• Fixed proportions   = 0
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
• 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:
• 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 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 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