Lesson-14
Consumer Behavior-- Indifference Curve
Indifference Curve Analysis
We can begin by examining the two good, single consumer case. Each consumer starts
with a budget constraint, representing how one's income is spent on a set of goods and
services. We'll assume that there are only two goods to consider in the typical consumer
budget and that all of this consumer's income is spent on these goods.
The Budget Constraint is:
I = P1Q1 + P2Q2
(where I = income, P = price, Q = quantity for goods 1 and 2)
We can take this equation and rearrange it to get:
Q1 = -(P2/P1)(Q2) + (I/P1)
What can we say about the rearranged budget constraint equation? First, we may notice
that this rearranged budget constraint is an equation for a line (with a negative slope P2/P1
and vertical intercept I/P1). Intuitively, we may recognize that the ratio of prices
represents a comparison of the cost to consumers of one unit of each good. Therefore, in
a sense, we can say that P2/P1 is the ratio of the marginal cost of goods 1 and 2
respectively. Recalling our macroeconomic discussion of price indexes, we see that I/P1
is a measure of our good 1 purchasing power (i.e. how much of good 1 our income can
buy). If P1 falls, I/P1 gets bigger - which means that we can buy more of good 1.
While the budget constraint represents how much a consumer is able to spend, we also
need to know how much a consumer wants to spend on each good. That is, we need some
information about this consumer's preferences regarding each good.
This information is found in an indifference curve. Indifference curves are drawn with
two basic ideas in mind:
(a) Within certain limits, consumers always prefer more of everything to less (e.g. I'd
prefer receiving 3 boxes of Cocoa Puffs and 2 boxes of Honeycomb to 2 and 1
box respectively); and
(b) It is possible to derive the same satisfaction out of a variety of potential purchase
combinations (e.g. when considering a potential cereal purchase, a consumer may
be indifferent between buying 3 boxes of Cocoa Puffs and 2 boxes of Honeycomb
versus 2 and 3 boxes respectively).
Therefore, by considering one's preferences, we see that consumers make purchasing
decisions which depend upon the satisfaction (more formally, the utility) derived from a
particular good. Each unit consumed (e.g. each box of cereal) in a given time period
yields some sort of satisfaction. When we examine the amount of satisfaction derived
from each unit consumed, we are considering something called marginal utility (MU).
The slope of the indifference curve may be expressed as a ratio of the marginal utilities
associated with each good (MU2/MU1). Rather than write this ratio, however, we can
simplify things by calling it the marginal rate of substitution between goods 1 and 2
(MRS).
Where does equilibrium occur?
Equilibrium occurs where the slopes of the indifference curve and budget constraint are
equal. Mathematically, this occurs where MRS = P2/P1. This is an equilibrium point
because at this point there is no reason to move away. The marginal rate of substitution
can also be thought of as a ratio of marginal benefit that each good provides our
consumer. Therefore, equilibrium in this setting involves equating the marginal benefit
for two goods with their marginal cost. In simpler terms, we're saying that our consumer
is getting out of each good exactly what they're worth.
We can demonstrate equilibrium graphically as well (see the graph below). Consider two
different indifference curves: IC (the red curve) and IC' (the blue curve). Every point on
IC (and IC') represents a different potential purchase of goods 1 and 2. As mentioned
above, on each indifference curve our consumer is indifferent about purchasing any of the
potential combinations along that curve. Consequently, along a particular curve, the only
difference between each point is the amount of goods 1 and 2 that are purchased. The
consumer is just as satisfied with any of the points on a given curve. Two things will
determine which point gets selected: the consumer's income and the price of each good.
To find out where the equilibrium is, if one exists, we want to see if there is one point
that is always preferred to every other point. We can begin by starting at a specific point
(the one we pick isn't important). To keep things simple, we'll continue to assume that our
consumer spends their entire income on these two goods. Start at point B, at the top of the
Budget Constraint. Based on our discussion above, we know that points A and B provide
this consumer with equal levels of satisfaction. That is, the consumer is indifferent
between points A and B.
Although this consumer is indifferent between points A and B, this is not the case with
points A and C. Point C is clearly better than point A for one important reason. At point
C, our consumer gets more of both goods. As mentioned above, the basic idea behind
these indifference curves (where each good's MU is greater than zero) is that "more is
better." When comparing two points, like A and C, this is always true. When you get
more of one good but less of the other, it may be true but not necessarily so (e.g. our
consumer is not better off when moving from A to B).
Thus far we know that our consumer is indifferent between A and B, but prefers C to A.
Therefore, logic dictates that our consumer must also prefer C to B. No matter which
point we start with, our result would be the same. In the end we realize that, if "all roads
lead to point C," point C must be the equilibrium.
Indifference Curves and the Consumer Equilibrium
Let’s assume that a representative consumer named Homer Simpson consumes beer and
pork rinds in varying amounts. Assume further that the overall utility he derives from
consuming these goods can be described by the utility function below. Note that this is
just one possible example of a utility function, that there are many other possible
functions we could have used instead.
(1)
We can use this utility function to derive Homer's indifference curve. By setting
(1) equal to a specific number, we are saying that there are various combinations
of B and R that yield a level of utility equal to that specific number. For example,
suppose we set Homer's utility function equal to 100. We derive the indifference
curve allowing 100 units of utility (i.e. utils) by rearranging the equation as
follows.
(1a)
Now, solve (1a) for B by squaring both sides to get:
(1b)
Second, we divide both sides of (1b) by the variable R.
(1c)
B = 10,000/R
This is the equation for one indifference curve. As stated above, (1c) tells us the
various combinations of beer and pork rinds that will provide Homer with 100
utils of satisfaction.
For example, if Homer consumes 10 units of beer, he needs to consume 1,000
units of pork rinds to get 100 utils of satisfaction. Of course, this equation also
tells us that Homer would be indifferent between consuming that bundle of goods
(10 units of beer and 1,000 units of pork rinds) and another one with 100 units of
beer and 100 units of pork rinds. This is because both bundles provide 100 utils of
satisfaction.
The graph that goes with (1c) is pictured below. The two different consumption
points we just discussed are pictured too (with their coordinates reported as (R,
B). Both are on the indifference curve, both yield 100 utils of satisfaction.
Not knowing whether Homer will actually consume at either of these points, or
whether he’ll even consume on this indifference curve, we turn now to figuring
out where Homer’s consumption will actually occur. To do this we need a couple
pieces of missing information: (a) the slope of the indifference curve, and (b) the
budget constraint equation.
In a model where we examine two goods simultaneously, the slope of the
indifference curve is going to be the marginal utility related to consuming more of
one good divided by the marginal utility related to consuming less of the other
good. While the utility along any indifference curve is constant, the marginal
utility is not.
The marginal utility (MU) for each good above is given as:
The slope of the indifference curve, called the marginal rate of substitution, will
be MUR/MUB. Note that the slope of this curve is negative (to see this
mathematically, consider (1c)), which means we write the marginal rate of
substitution for pork rinds and beer (MRSR, B) as:
(2)
MRSR, B = -B/R
We’ll assume that the price of beer is $4 and that the price of pork rinds is $2.
Assume further that Homer’s income is $200. The budget constraint is then given
as:
(3)
4B + 2R = 200
Rearranging (3), by solving for B, we get the following (rearranged budget
constraint):
(3a)
B = -0.5R + 50
Noting that (3a) is the equation of a line (slope of –0.5, vertical intercept of 50),
we can graph the indifference curve and budget constraint together. Equilibrium is
attained where the (blue) indifference curve is tangent to the (red) budget
constraint.
This
point
is
included
in
the
graph.
The graph enables us to visually determine equilibrium, but also note the two
conditions which must simultaneously occur when we are at this equilibrium
point. Those conditions are:
•
•
The slope of the budget constraint must equal the slope of the indifference
curve (i.e. MRSR, B = -PR/PB)
Our consumer must be on their budget constraint (i.e. 4B + 2R = 200)
With this in mind, we can now solve for equilibrium here. Substitute the values of
the slopes into the first condition.
(4)
-B/R = -0.5
Solve (4) for B.
(4a)
B = 0.5R
Substitute (4a) into the budget constraint (for B).
(5)
4(0.5R) + 2R = 200
Solve (5) for R. This is the equilibrium value for R (i.e. R*).
R* = 50
Plug R* into the original budget constraint (or (4)), and solve for B. This is the
equilibrium value for B (i.e. B*).
4B + 2(50) = 200
B* = 25
Given Homer’s budget constraint and utility function, Homer should consume 25
units of beer and 50 units of pork rinds. If he does this, then his overall utility will
be:
That is, Homer will experience about 35.4 utils of satisfaction from his 25 units of
beer and 50 units of pork rinds.
Utility Max Application of the Implicit Function Theorem
Assume that a consumer named Homer Simpson consumes varying amounts of Duff beer
and pork rinds
Let:
•
•
Units of beer consumed = B
Units of pork rinds consumed = R
Homer derives his utility from consuming these goods in accordance with the
following utility function (where U = utility):
(1)
U = f (B, R)
Homer's purchasing decision is limited by the following budget constraint (where
pi is the price of good i, and I is Homer's income):
(2)
pBB + pRR = I
Note that (2) can be rearranged to become:
(2a)
Utility maximization leads us to the following equilibrium condition (which says
that the slope of the indifference curve equals the slope of the budget constraint):
(3)
(Where MUi = marginal utility of good i; which equals the derivative of the utility
function with respect to good i)
Let us first take the total derivative of (1), the utility function. Upon doing so, we
have:
(4)
(Where k is a constant equal to some overall level of utility, such that k
0)
Dividing both sides of (4) by dB yields:
(5)
Because dB/dB = 1, and dk/dB = 0, we can simplify (5) to get:
(5a)
Solving (5a) for dR/dB yields:
(5a)
At this point, we need to stop and ask what we've got thus far. In doing so, let's
recall a couple of points made above. First, we note that the marginal utility of
good i can be expressed as the first derivative of the utility function taken with
respect to good i. Second, we note that an indifference curve's slope is equal (in
the two-good case) to the ratio of the marginal utilities.
Because the right-hand side of (6) involves the ratio of two derivatives of the
utility function (each taken with respect to one of the goods consumed by Homer),
the right-hand side of (6) must be the slope of Homer's indifference curve. If the
slope of Homer's indifference curve was set equal to the slope of his budget
constraint, then we would have the consumer equilibrium expression given in (3).
To take the actual derivatives just mentioned, however, we need to assume a
functional form for the utility function in (1). Let's assume a linear (additive)
utility function for this example, the function given below (where
is a
parameter that's greater than zero,
is a parameter that's between 0 and 1, and
ln(i) = natural log of good i):
(1a)
U =
+
)ln(R)
ln(B) + (1 -
If we take the derivatives described in (6) and substitute those derivatives into (3),
then we have (recall that if y = ln(x), then dy/dx = 1/x):
(7)
The two equations which describe the tangency point between Homer's
indifference curve and his budget constraint are (7) and (2a). Using these
equations together, we can solve for B* and R*. In their present form, those
solutions are:
If we wish to go further and assume numerical values for the parameters in this
model, then we could assume the following:
= 0.5
= 100
pB = $4
pR = $2
I = $200
Substituting into our solution above, the numerical values for B* and R* are:
B* = 25
R* = 50
These are the amounts of beer and pork rinds that will give Homer his maximum
utility.
Substitution and Income Effects in the Indifference Curve model
Homer Simpson, our representative consumer, consumes varying amounts of beer and
pork rinds. Assume that B = quantity of beer consumed, and that R = quantity of pork
rinds consumed. Homer’s utility function is given as: U ( B, R ) = B ⋅ R .
The marginal rate of substitution (which is the slope of Homer’s indifference curve)
MRS R ,B = − B
R . Recall that
between beer and pork rinds is given in absolute value as:
this can be derived from Homer’s utility function. If we use a different utility function,
then we get a different MRSR,B.
Assume further that the price of beer is $4, the price of pork rinds is $2, and that Homer’s
income is $200. We can obtain Homer’s budget constraint from this information, which
we can rearrange as: B = -0.5R + 50.
Consumer equilibrium occurs in the graph below at pt. X1, where the (blue) indifference
curve is tangent to the (red) budget constraint.
B
X1
R
It is possible to calculate the quantities of beer and pork rinds at this consumer
equilibrium. After doing so, we would find that B* = 25 units and R* = 50 units.
How is the graph above affected when the price of pork rinds increases from $2 to $4?
This change is shown on the graph below. The budget constraint becomes steeper and
Homer moves to a new (pink) indifference curve and a lower level of utility at pt. X2. If
we calculate the new consumer equilibrium at pt. X2, we would get B* = 25 and R* = 25.
B
X2
X1
R
Notice, however, that the price change included two actions. The movement from pt. X1
to pt. X2 involved a change in the marginal rate of substitution (i.e. a change in the slope
of the indifference curve), and a change in utility (i.e. a change from the blue indifference
curve to the pink indifference curve). This is different from a change in income, which
only involves one change – a change in utility. These two actions form the analytical
basis for what we call the substitution effect and the income effect.
The Substitution and Income Effects
When prices rise, consumers lose purchasing power. What if the price of pork rinds goes
up, but the government offers to compensate Homer for this loss of purchasing power.
That is, Mayor Quimby offers to mail Homer a check, in an effort to keep Homer from
feeling worse off. Homer still faces the higher pork rinds price, but doesn’t experience a
change in utility. That is, for Homer to be no worse off after the price increase, the
government check must be large enough to keep Homer on his original indifference
curve.
If the government check allows Homer to remain on his original indifference curve, will
he just return to pt. X1 and go back to buying 50 units of pork rinds? No. Even though
Homer would return to his original indifference curve, he would also still face a different
pair of prices. Therefore, we know that Homer must be located at a different point on
that original indifference curve.
By taking the second graph above, and drawing a “hypothetical budget constraint”, we
can find this new point. This new constraint must satisfy two criteria. First, the
constraint must be parallel to the new prices (where beer and pork rinds each cost $4).
Second, the constraint must be tangent to the original indifference curve.
The dotted line in the graph below satisfies these criteria, and so represents this new
constraint. This line is tangent to Homer’s original indifference curve at pt. W. This
point reveals the quantities of beer and pork rinds that Homer would buy after receiving
his government check (the check that keeps his utility constant). Of course, in real life,
Homer would never get a check from the Mayor, but we will use pt. W to distinguish
between the two actions (or effects) we noted as occurring with every price change.
B
W
X2
X1
R
How much would Homer consume at pt. W? The calculation is somewhat involved.
First, note that the slope of Homer’s new constraint is -1. Consequently, at pt. W, the
slope of his original indifference curve equals -1. If R/B = 1 at pt. W, then B = R at pt.
W also. That is, we can ascertain that Homer will buy an equal amount of beer and pork
rinds at pt. W.
Homer’s original level of utility is 25 2 (i.e. plug the original consumer equilibrium
values
of B = 25 and R = 50 into Homer’s utility function). To maintain Homer’s original level of
utility, then
B ⋅ R = 25 2 . That is, Homer will buy some combination of B and R that
makes his utility function equal to 25 2 . Recall that, at pt. W, Homer will buy an equal
amount of beer and pork rinds.
Therefore, we can rewrite
B ⋅ B = 25 2 , which simplifies to B* = 25 2 .
B ⋅ R = 25 2 as
If B* = 25 2 , and B* = R*, then
R* = 25 2 .
The new (hypothetical) budget constraint would be given as 4B + 4R = 200 + I, where
I is the change in income necessary to keep Homer’s utility constant. Plugging in B*
and R* from the paragraph above, we find that I = $82.84. That is, if Homer receives a
check for $82.84, then Homer can continue to receive his original level of utility (i.e.
25 2 utils) even though pork rinds are $2 more expensive now.
What are the substitution and income effects? The two effects are separated by pt. W.
As the quantity of pork rinds changes between pt. W and pt. X1 we observe the
substitution effect. At pt. X1, Homer consumes 50 units of pork rinds. At pt. W, Homer
consumes 25 2 units of pork rinds (i.e. about 35.36 units). The substitution effect
associated with this price increase is represented by a decrease in quantity. That is, the
substitution effect reveals a negative relationship between the price and quantity change.
In fact, with every price change, we find this negative relationship within the substitution
effect.
The income effect is measured as the quantity change attributed to moving from pt. W to
pt. X2. Between these two points, only utility changes, there is no change in the slope of
the budget constraint. At pt. X2, Homer consumes 25 units of pork rinds. The difference
between pts W and X2 becomes 25 2 − 25 , about 10.36 units.
Note that, like the substitution effect, there is a decrease in quantity within the income
effect. Unlike the substitution effect, however, a negative relationship between price and
quantity does not always arise within the income effect. For normal goods, the income
effect reveals a negative relationship between price and quantity changes. That is, price
increases lead to the income effect involving a decrease in quantity, and price decreases
lead to the income effect involving an increase in quantity. Obviously, Homer considers
pork rinds to be a normal good.
For inferior goods, we get the opposite result – the income effect involves a positive
relationship between price and quantity changes. Any increase in price (decrease) would
lead to the income effect yielding an increase in quantity (decrease).
Suppose the inferior good is highly inferior. For example, suppose we have a good
where any small increase in price leads to a large, positive income effect. This would
explain why a fairly large price change leads to an insignificant (overall) change in
quantity. The inferior good’s large income effect moves in the opposite direction of the
substitution effect, causing the overall change (i.e. the sum of the two effects) to be very
small.
In some cases, if a good is inferior enough, the positive income effect may be so large
that it leads to price increases (decreases) being accompanied by overall quantity
increases (decreases). When this occurs, we are dealing with a special (and rare) type of
good known as a Giffen good. Giffen goods are so inferior that the income effect
overwhelms the substitution effect, leading to the perverse result described above –
where there is an overall positive relationship between price and quantity changes.
Application of Indifference Curves: Lump Sum vs. Per Unit Taxes
Assume we have a new representative consumer, Marge Simpson, who derives different
levels of utility from buying varying quantities of beer and pork rinds (QB = quantity of
beer, QR = quantity of pork rinds).
Her utility can be calculated from the following utility function: U = Q B ⋅ Q R
This utility function implies that her indifference curves, IC, have a slope that is
nonlinear. That slope, called the marginal rate of substitution between beer and pork
rinds, can be calculated by plugging different quantities into the equation below:
MRS B, R = −
QR
QB
Let’s assume further that Marge faces the following prices and that she has the income
given below as well.
Price of beer = $4
Price of pork rinds = $2
Income = $200
We can start our analysis by asking this question:
How many units of beer and pork rinds should Marge buy?
We know, from previous work/handouts, that there are two equations which will help us
determine the answer to our question. Furthermore, we know that both must be true
simultaneously. Those equations are:
(1) 4QB + 2QR = 200
(Marge must be somewhere on her budget constraint,
BC)
(2) - 4/2 = - QR/QB
(the slope of Marge’s BC must equal MRSB,R at
equilibrium)
Solving (1) and (2) simultaneously (e.g. using algebraic substitution), we find that Marge
will buy 25 units of beer and 50 units of pork rinds. Her indifference curve graph appears
as:
QR
50
IC1
25
QB
Let’s assume that the government is considering two different types of tax. The first tax
is a per unit tax. That is, a tax that is levied on the number of units of a specific good that
are purchased by Marge. The second tax is a lump sum tax. That is, a tax of some fixed
amount that does not correspond with the number of units Marge decides to buy of either
good.
Suppose that a $1 per unit tax is levied on beer. This changes the actual price that Marge
pays for each unit of beer. For each unit of beer that she purchases, she pays the sum of
the equilibrium price and tax on that unit. For example, if the price is $4 per unit and the
tax is $1 per unit sold, then Marge will pay $5 for each unit she buys. That changes the
budget constraint equation (above) and, of course, the slope of the budget constraint as
follows:
(1a) 5QB + 2QR = 200
(2a) - 5/2 = - QR/QB
Again, solving (1a) and (2a) simultaneously, we find that Marge will buy 20 units of beer
and 50 units of pork rinds. The budget constraint shifts inward, and she moves to a new,
lower indifference curve as follows:
QR
50
IC1
IC2
20
25
QB
We note that, with this per unit tax on beer in place, the government will raise $20 in tax
revenue from Marge (i.e. Marge pays $1 tax for each of the 20 units she buys).
If the government decides to go with the lump sum tax, then Marge (and all other
consumers) must pay (instead) a specific lump sum amount. It seems safe to assume that
if the government raises the same amount of tax revenue from Marge with either tax, then
the government will not have a preference as to which tax is used.
An important second question, however, is whether Marge has a preference. To answer
this question, let’s assume that the government levies a lump sum tax of $20 on Marge
(equal to the amount she’d pay with the per unit tax).
The new lump sum tax would reduce her Marge’s income by $20, but, unlike the per unit
tax, will leave the price of beer unchanged. With her new, lower post-tax income ($180,
instead of $200), Marge faces a new budget constraint. The budget constraint doesn’t
change in slope, but does shift inward because of a change in each intercept. This is
reflected in the following equations:
(1b) 4QB + 2QR = 180
(2b) - 4/2 = - QR/QB
Again, solving (1b) and (2b) simultaneously, we find that Marge will buy 22.5 units of
beer and 45 units of pork rinds. On a graph, we see the budget constraint shift in parallel
as Marge moves to a new, lower indifference curve (IC3):
QR
45
IC1
IC3
22.5
QB
The most straightforward method for determining whether Marge prefers one tax over the
other is to calculate her utility in both situations. To do so, we utilize the utility function
given at the beginning of this handout and then compare the utility associated with the
per unit tax case and the lump sum tax case by plugging in the appropriate equilibrium
values for QB and QR as follows:
Per unit tax:
U = 20 ⋅ 50 = 31.62
Lump Sum tax:
U = 22.5 ⋅ 45 = 31.82
Marge’s utility is higher when the government raises $20 in tax revenue from a lump sum
tax than when the government raises $20 from a per unit tax.
Why do we get this result? In other material, we have noted that price changes have a
dual effect on a consumer’s purchasing decision. There is both a change in relative
prices, called the substitution effect, and a purchasing power change, called the income
effect. When income changes, there is only one effect – the income effect. The per unit
tax is comparable to a price change and so consumers react more with this tax than with
the lump sum tax - when there is only a change in income. The difference in utility
between these two cases represents a type of utility loss for Marge. Because deadweight
loss arises when consumers substitute their consumption away from a particular good, the
lump sum tax is thought to be more efficient, because the lump sum tax does not induce
this kind of behavior (i.e. there is no substitution effect with a lump sum tax).