Download constructing an indifference curve

APPENDIX TO CHAPTER 6
Indifference Curve Analysis
This appendix explains another version of consumer choice theory based on indifference curves
and budget lines.
CONSTRUCTING AN INDIFFERENCE CURVE
Let’s begin with an experiment to find out a consumer’s consumption preferences for quantities
of two goods. The consumer samples a number of pairs of servings with various ounces of steak
and lobster tail (surf and turf). Each time the same question is asked, “Would you prefer serving
A or serving B?” After numerous trials, suppose the consumer states indifference between eating
choices A -D shown in Exhibit A-1. This means the consumer is just as satisfied having either
seven ounces of steak and four ounces of lobster (A), or three ounces of steak and eight ounces
of lobster (D), or either of the two other combinations of B or C. Interpretation of the curve
connecting these points is that each of these choices yields the same total utility because no
choice is preferred to any other choice. Since, as explained in the chapter, there is no such thing
as a utility meter, this approach is actually a method for determining equal levels of satisfaction
or total utility for different bundles of goods without an exact measure of utils. The curve derived
from this experimental data is called an indifference curve. Note that not only points A, B, C,
and D, but all other points on the smooth curve connecting them are equally satisfactory
combinations to the consumer.
[Exhibit A-1]
Why Indifference Curves Are Downward-sloping and Convex
If total utility is the same at all points along the indifference curve, then consuming more of one
good must mean less of the other is consumed. Given this condition, movement along the
indifference curve generates a curve with a negative slope. Suppose a consumer moves in
marginal increments between any two combination points in Exhibit A-1. For instance, say the
consumer decides to move from point A to B and consume an extra ounce of lobster. To do so,
the consumer increases total utility (+MU) by consuming an extra quantity of lobster. However,
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An Indifference Schedule for a Consumer
Choice
A
B
C
D
Steak
(ounces)
7
5
4
3
Lobster
(ounces)
4
5
6
8
Exhibit A-1 A Consumer’s Indifference Curve
Points A, B, C, D and each point along the curve represents a combination of steak and lobster
that yields equal total utility for a given consumer. Stated differently, the consumer is indifferent
between consuming servings having quantities represented by all points composing the
indifference curve.
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since by definition total utility is constant everywhere along the curve, the consumer must give
up a quantity of steak (2 ounces) in order to reduce total utility (-MU) by precisely enough to
offset the gain in total utility from the extra lobster.
The inverse relationship between goods along the downward-sloping indifference curve
means that the absolute value of the slope of an indifference curve equals what is called the
marginal rate of substitution (MRS). The MRS is the rate at which a consumer is willing to
substitute one good for another with no change in total utility. Begin at A and move to B along
the curve. The slope and MRS of the curve is -2/1 or simply 2 when the minus sign is removed to
give the absolute value. This is the consumer’s subjective willingness to substitute A for B. At
point A, the consumer has a substantial amount of steak and relatively little of lobster. Therefore,
the consumer is willing to forgo or “substitute” 2 ounces of steak to get 1 more ounce of lobster.
In other words, the marginal utility of losing each ounce of steak between A and B is low
compared to the marginal utility of gaining each ounce of lobster.
Now suppose the consumer moves from B to C, and the slope changes to 1/1 (MRS = 1).
Between these two points, the consumer is willing to substitute 1 ounce of steak for an equal
quantity of lobster. This means that between B and C the marginal utility lost per ounce of steak
equals the marginal utility gained from each ounce of lobster, while total utility remains constant.
Finally, assume the consumer moves from C to D. Here the slope equals ½ (MRS = ½) because
the consumer at point C has a substantial amount of lobster and relatively little steak.
Consequently, the marginal utility lost from giving up 1 ounce of steak equals twice the marginal
utility gained from an additional ounce of lobster.
As we see in this example, as the quantity of lobster increases along the horizontal axis,
the marginal utility of additional ounces of lobster decreases. Correspondingly, as the quantity of
steak decreases along the vertical axis, its marginal utility increases. So moving down the curve
means the consumer is willing to give up smaller and smaller quantities of steak on the vertical
axis to obtain each additional ounce of lobster on the horizontal axis.
CONCLUSION
The slope of the indifference curve is negative and equal to the
marginal rate of substitution (MRS), which declines as one moves downward along the
curve. The result is a curve with a diminishing slope that is convex (bowed inward) to the
origin.
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The Indifference Map
As explained above, any point selected along an indifference curve gives the same level of
satisfaction or total utility. Actually, there are other indifference curves that can be drawn for a
consumer. As shown in Exhibit A-2, indifference curves I1 to I6 also exist to form an
indifference map, which is a selection of a consumer’s indifference curves. In fact, if all
possible curves are drawn, they would completely fill the space between the axes because they
would be so many. And it is important to note that each curve reflects a different level of total
utility. Also, each curve moving from the origin in the northeasterly direction from I1 to I6, and
beyond yields higher total utility. This is reasonable because at each higher indifference curve
the consumer is able to select more of both goods and therefore be better off. To verify this
concept, select a combination on one indifference curve in Exhibit A-2 and then select a
combination of more of both goods on a higher indifference curve.
CONCLUSION Each consumer has a set of indifference curves that form a map. Since
consumers wish to achieve the highest possible total utility, they always prefer curves
farther from the origin where they can select more quantities of two goods.
[Exhibit A-2]
The Budget Line
Having considered the consumer’s preferences for steak and lobster in the indifference map, the
next step is to see what the consumer can afford. The consumer’s ability to purchase steak and
lobster is limited or constrained by the amount of money (income) that the consumer has to
spend and the prices of the two goods. Suppose the consumer likes to go to a fine restaurant and
budgets $10 per week to dine on surf and turf. The price of steak is $1 per ounce and the price of
lobster is $2 an ounce. If the consumer spends the entire budget on steak, 10 ounces of steak can
be purchased. At the other extreme, the whole budget could be spent for lobster and 5 ounces
would be purchased. As shown in Exhibit A-3, and given the consumption possibilities of buying
fractions of ounces of steak and lobster, there are a range of choices that can be purchased along
the budget line ranging from 10 ounces on the steak axis to 5 ounces on the lobster axis. The
table in this exhibit computes selected whole-unit combinations that each equal $10 total
expenditure.
[Exhibit A-3]
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Exhibit A-2 A Consumer’s Indifference Map
An indifference map is a selected set of indifference curves. Along curves farther from the
origin, it is possible to select more of both goods compared to indifference curves closer to the
origin. Therefore, curves further from the origin are preferred because they yield higher levels of
total utility.
6A-5
Selected Whole-unit Consumption Possibilities of Steak and Lobster
Affordable with $10 Budget
Choice
A
B
C
D
E
Ounces of steak
(price = $1/ounce)
10
8
4
2
0
Ounces of lobster
(price = $2/ounce)
0
1
3
4
5
Total
Expenditure
$10 ($10 + $0)
$10 ($8 + $2)
$10 ($4 + $6)
$10 ($2 + $8)
$10 ($0 + $10)
(TGW)
Exhibit A-3 A Consumer’s Budget Line
A budget line shows all the possible combinations of goods that can be purchased with a given
budget and given prices for these goods.
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CONCLUSION The budget line represents various combinations of goods that a
consumer can purchase at given prices with a given budget.
To compute the slope of the budget line requires some shorthand mathematical notation.
Let B represent the amount of money the consumer has to spend on steak and lobster. Also, let Pl
and Ps represent the prices of lobster and steak, respectively, and L and S represent the ounces of
lobster and steak, respectively. Given this notation, the budget line may be expressed as
Pl L + PsS = B
In order to express the equation in slope-intercept form, divide both sides by Ps and get
Pl L + S = B
Ps
Ps
Solving for S yields
S = B - Pl L
Ps Ps
Note that S is a linear function of L with a vertical intercept of B/ P s and a slope of -Pl/Ps.
Since the price of lobster is $2 an ounce and price of steak is $1 per ounce, the slope of the
budget line is -2 or 2 as an absolute value.
CONCLUSION The slope of the budget line equals the ratio of the price of good X on the
horizontal axis divided by the price of the good Y on the vertical axis. Expressed a
formula:
slope of budget line = Px/Py
A Consumer Equilibrium Graph
Exhibit A-4 shows the budget line from Exhibit A-3 superimposed on an indifference map. This
allows us to easily compare consumer preferences to affordability. The utility-maximizing
combination is the equilibrium point X where the budget line is just tangent to the highest
attainable indifference curve I2, and the quantity of steak purchased is 4 ounces and quantity of
lobster is 3 ounces.
[Exhibit A-4]
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Exhibit A-4 Consumer Equilibrium
Consumer equilibrium is at point X where the budget line is tangent to the highest attainable
indifference curve I2. Only at this point does the marginal rate of substitution (MRS) along I2
equal the slope of the budget line, which equals the price ratio Pl/Ps. Although point Y is on a
higher curve I3 and would yield a greater total utility than X, point Y is beyond the budget line
and therefore is unattainable by the consumer. Point Z is on a lower indifference curve, but it
will not be chosen. The consumer can move upward along the budget line by shifting dollars
from purchases of lobster to purchases of steak and reach point X on the higher indifference
curve I2.
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As explained in Exhibit A-5 of the appendix to Chapter 1, the slope of a straight line
tangent to a curve is equal to the slope of the curve at that point. This mathematical relationship
translates into the economic interpretation of the tangency condition in this appendix. At the
point of tangency, the MRS equals the slope of the budget line. At point X, the slope from
Exhibit A-3 is the price ratio Pl/Ps = 2 and therefore it follows that the MRS = 2 at point X on the
curve I2. At any other combination, the MRS will not be equal to the price ratio and the consumer
will reallocate the budget until equilibrium is achieved at point X.
CONCLUSION Consumer equilibrium occurs where the budget line is tangent to the
highest attainable indifference curve. At this unique point, MRS = slope (price ratio of
Px/Py).
Derivation of the Demand Curve
This appendix concludes with Exhibit A-5, which shows how the demand curve for lobster can
be derived from a map of two indifference curves. The table in this exhibit reproduces the table
from Exhibit A-3 and adds column (4) with the price of lobster equal to $1 per ounce rather than
$2 per ounce. Now compare columns (3) and (4) in the table. Holding the price of steak constant
at $1 per ounce and the budget equal to $10, the consumer can afford to purchase more lobster at
each choice except A where the entire budget is spent on steak. The top graph is drawn from
Exhibit A-4 where at point X the price of lobster is $2 per ounce and the quantity demanded of
lobster is 3 ounces. In the bottom graph of Exhibit A-5, this is one point on the demand curve for
lobster at X. To find another point on the demand curve, we decrease the price of lobster to $1
per ounce and trace the new budget line points from column (4) of the table onto the top graph.
The budget line swings outward along the horizontal axis with the original vertical intercept
remaining unchanged. The reason the vertical axis remains at 10 ounces of steak is because the
price of steak remains at $1 per ounce, and if the consumer spends the entire $10 on steak, then
the price of lobster is irrelevant to the vertical intercept. However, at a lower price for lobster, the
consumer can afford more lobster moving downward along the new budget line with the same
$10 budget because lobster is cheaper.
[Exhibit A-5]
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Selected Whole-unit Combinations of Steak and Lobster
Affordable with $10 Budget
(1)
Choice
A
B
C
D
E
(2)
Ounces of steak
(price = $1/ounce)
10
8
4
2
0
(3)
Ounces of lobster
(price = $2/ounce)
0
1
3
4
5
(4)
Ounces of lobster
(price = $1/ounce)
0
2
6
8
10
(5)
Total
Expenditure
$10 ($10 + $0)
$10 ($8 + $2)
$10 ($4 + $6)
$10 ($2 + $8)
$10 ($0 + $10)
Exhibit A-5 Deriving the Demand Curve
In the top part of this exhibit, the initial budget line intersects the highest attainable indifference curve I 2 at point X
with the price of steak equal to $1 per ounce and the price of lobster equal to $2 per ounce. Holding the budget and
the price of steak constant at $10 and $1, respectively, the price of lobsters drops to $1 per ounce. As a result, the
budget line shifts to intersect the higher indifference curve I 3 at point Y.
The bottom part of the exhibit corresponds points X and Y to points X and Y. At $2 per ounce for lobster, the
consumer buys 3 ounces. At $1 per ounce for lobster, the consumer buys 5 ounces. Connecting these two points
results in a downward-sloping demand curve.
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Given the new budget line shown in the top graph in Exhibit A-5, the consumer finds that higher
indifference curve I3 is now attainable. As a result, consumer equilibrium moves from point X to
point Y where 5 ounces of lobster are purchased. This gives the corresponding second point Y
shown in the lower graph. Connecting these two points allows us to draw the consumer’s
demand curve for lobster. And consistent with the law of demand-Voila! The demand curve is
indeed downward-sloping.
SUMMARY

An indifference curve shows the different combinations of two goods that give the same
satisfaction or total utility to a consumer.
Indifference curve
(repeat Exhibit A-1)

The marginal rate of substitution (MRS) is the rate at which a consumer is willing to
substitute one good for another with no change in total utility.

An indifference map is a selection of a consumer’s indifference curves.
Indifference map
(Repeat Exhibit A-2)
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
The budget line gives the different combination of two goods that a consumer can
purchase with a given amount of money and the relative prices for the goods. The slope
of the budget line equals the price of the good on the horizontal axis divided by the price
on the horizontal axis.
Budget line
(Repeat graph from Exhibit A-3)

Consumer equilibrium occurs where the budget line is tangent to the highest possible
indifference curve as shown originally at point X. If the price of one good falls, then the
consumer equilibrium changes to point Y on a higher indifference curve. Connecting the
two corresponding price-quantity points X and Y generates a downward-sloping demand
curve.
Consumer equilibrium (Repeat Exhibit A-5)
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KEY CONCEPTS
Indifference curve
Marginal rate of substitution (MRS)
Indifference map
budget line
SUMMARY OF CONCLUSION STATEMENTS

The slope of the indifference curve is negative and equal to the marginal rate of
substitution (MRS), which declines as one moves downward along the curve. The result
is a curve with a diminishing slope that is convex (bowed inward) to the origin.

Each consumer has a set of indifference curves that form a map. Since consumers wish to
achieve the higher possible total utility, they always prefer curves farther from the origin
where they can select more quantities of two goods.

The budget line represents various combinations of goods that a consumer can purchase
at given prices with a given budget.

The slope of the budget line equals the ratio of the price of good on the horizontal axis
divided by the price of good Y on the vertical axis. Expressed as a formula: slope = Px/Py.

Consumer equilibrium occurs where the budget line is tangent to the highest attainable
indifference curve. At this unique point, MRS = slope (price ratio of Px/Py).
STUDY QUESTIONS AND PROBLEMS
1.
Suppose a consumer’s marginal rate of substitution is three slices of pizza for one coke. If
the price of a coke is $1.00 and the price of three slides of pizza is $2.00, would the
consumer change his or her consumption combination?
2.
Let M represent the quantity of medical services, such as the number of doctor visits, and
let O represent the quantity of other goods purchased by a consumer in a given year. Let
the budget (B) be in thousand of dollars, and the price of medical services and other
goods be in terms of dollars per unit, with B = 60, Pm = 6, and Po = 10.
a.
Graph the budget line and determine the slope.
b.
Show the result if the price of medical services (Po) decrease t0 $5.
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c.
Add two indifference curves to the graph that are tangent to the budget line and
explain the result.
3.
a.
Assume that the data in the following table is an indifference curve for a consumer:
Choice
Units of Y
Units of X
A
2
10
B
4
6
C
6
4
D
12
2
Graph this indifference curve and label quantity of Y on the vertical axis and quantity of
X on the horizontal axis. Label each point A-D.
b.
Assume the consumer’s budget is $12 and the prices of X and Y are $1 and $1.50,
respectively. Draw the budget line in the above graph.
c.
What combination of X and Y will the consumer purchase?
d.
What is the value of the MRS and the slope (Px/Py) at consumer equilibrium?
e.
Beginning with the graph drawn in a. above, explain and draw graphs to derive a demand
curve for X.
6A-14
Practice Quiz
For a visual explanation of the correct answers, visit the tutorial at http://tucker.swlearning.com.
Exhibit A-6 A Consumer’s Budget Line and Indifference Curves
1.
At point A in Exhibit A-6, consumers would be:
a.
b.
c.
d.
spending all of their income but not maximizing total utility.
spending all of their income and maximizing total utility.
maximizing total utility without spending all of their income.
none of the above.
2.
The consumer equilibrium shown in Exhibit A-6 is located at point:
a.
b.
c.
d.
A.
B.
C.
D.
3.
In Exhibit A-6 point D is:
a.
b.
c.
d.
a consumer equilibrium.
unattainable given the consumers’ current budget constraint.
a point that does not exhaust all of the consumers’ income.
none of the above.
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4.
Assume that a consumer’s preference is for two goods X and Y in Exhibit A-6. By
holding the price of Y and money income constant while varying the price of X, it is
possible to derive:
a.
b.
c.
d.
the demand curve for X.
the demand curve for Y.
the demand curve for both X and Y.
none of the above.
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