Transcript Utility Functions and Indifference Curves: Getting Into the

Utility Functions and Indifference Curves:
Getting Into the Consumer’s Head
A utility function indicates how a consumer
orders different bundles of goods: bread,
eggs, wine.
Suppose U(b,x,y) = 3b + 2x + 1y
then
U(1,4,6) = 3 + 8 + 6 = 17
U(2,1,3) = 6 + 2 + 3 = 11
 The first bundle is preferred to the second
 The values of U(b,x,y) = 17 and 11 don’t
mean anything by themselves.
– They just indicate the order of preferences.
Getting into the Consumer’s Head

You can’t argue with a consumer’s valuations of
different bundles of goods, no matter how
bizarre it seems.
– If
U(b,x,y) = b3 x2 y1
• This consumer prefers any bundle that
contains a bit of each good to a bundle with
lots of two of the goods but none of the other
(Man cannot live by bread alone … gotta have
eggs and wine too)
• This consumer is indifferent between a bundle
with lots of bread and eggs but no wine and a
bundle with just a little bread and eggs but no
wine: U(B,X,0) = U(b,x,0) = 0
Getting Into the Consumer’s Head
In theory, we can tease out the
consumer’s utility function by
observing her choices when offered
different combinations of goods, her
revealed preferences.
 In practice, we only have to accept the
idea that utility functions and
indifference maps exist.
– We can then get some powerful
insights into consumer behavior.

Getting Into the Consumer’s Head
An Indifference Map
Wine(y)
u2
u1
Eggs(x)
All combinations on U2 are equally preferred
All combinations on U2 are preferred to
combinations on U1
MORE BEATS LESS!!!
Getting Into the Consumer’s Head
Slope of the indifference curve = dy/dx
= Rate at which wine must be
substituted for eggs to keep the
consumer at the same level of
indifference

The fewer eggs the consumer has, the
more wine she needs to compensate for
taking away yet another egg.
• Indifference curves are convex to the origin
• The marginal utility of eggs increases as the
number of eggs the consumer has decreases:
MUx  when x 
Getting Into the Consumer’s Head
The Consumer’s Budget Line
(Budget = $B)
Wine(y)
B/py
u2
u1
Eggs(x)
B/px
Slope of Budget Line = - (B/py)/(B/px) = - px/py
dy/dx = - px/py
Getting Into the Consumer’s Head

At point of highest attainable indifference
– the budget line just touches the highest
indifferent curve
– the budget line is tangent to the highest
indifference curve
– the slope of the budget line equals the
slope of the indifference curve
The consumer’s willingness to
substitute y for x along her highest
attainable indifference curve equals
her ability to substitute y for x along
her budget line.
Getting Into the Consumer’s Head
An important insight:

At the point of highest attainable
indifference, the consumer’s willingness to
substitute y for x along her indifference
curve equals her ability to substitute y for x
along her budget line
– her ability to substitute y for x is the price
ratio - px/py … the slope of the budget line

Since everyone faces the same price ratio,
dy/dx = - px/py, the slope of everyone’s
budget line is the same
– everyone’s relative valuation of eggs
and wine (x and y) is the same.
Getting Into the Consumer’s Head

Along an indifference curve,
- MUx dx = MUy dy
Reduced utility from less x=Increased utility from more y
So on indifference curve, dy/dx = -MUx/MUy
Consumer is in equilibrium when slope of her
indifference curve equals slope of budget line
- MUx/MUy = dy/dx = - px/py
or MUx/ px = MUy/ py = … MUz/ pz
Get equal bang per buck from all
goods!
Utility and Demand

Consumer gets equal bang per buck from all
goods: apples, bread, eggs, wine, zolo toys
MUa/pa = MUb/pb = … MUx/px = MUy/py = … MUz/pz
If px  , MUx must fall
Since MUx  when x 
, px   x 
LAW OF DEMAND!
More eggs are bought when the price
of eggs declines.
Demand and Consumer Surplus

More eggs are bought when the price of
eggs declines.

Demand curve reflects price, px, consumer is
willing to pay for each quantity of x
Consumer is willing to pay more for the first
egg than for subsequent eggs
… the first egg will be used most
productively, e.g., to bake a cake.
 But consumer need only pay the same market
price for each egg she buys.
– This is less than she is willing to pay for all but
the last egg she buys
• she’d never pay more than she’s willing to pay.
Consumer realizes consumer surplus in market
Consumer Surplus in Pictures
Price(px)
Price consumer is willing to pay for
each quantity of x
C.S.
egg
Market price consumer
has to pay for each
she buys
Eggs(x)
Advertising  Irrational Behavior?

Advertising gets you to buy more of a good
at each price than you did before the ad
– Successful advertising increases demand

Does this violate the idea that a consumer
has a utility function relating satisfaction to
specific quantities of goods, a,b,…x,y,… z?
…or does advertising change the utility
function?
…does advertising change the product itself?
• Is an advertised bobblehead different from a
bobblehead you encounter with no prior
information about its virtues?