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,
An Indifference Schedule for a Consumer
Choice
/ Steak(ounces) / Lobster
(ounces)
A / 7 / 4
B / 5 / 5
C / 4 / 6
D / 3 / 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.
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.
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]
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.
Selected Whole-unit Consumption Possibilities of Steak and Lobster
Affordable with $10 Budget
Choice / Ounces of steak(price = $1/ounce) / Ounces of lobster
(price = $2/ounce) / Total
Expenditure
A / 10 / 0 / $10 ($10 + $0)
B / 8 / 1 / $10 ($8 + $2)
C / 4 / 3 / $10 ($4 + $6)
D / 2 / 4 / $10 ($2 + $8)
E / 0 / 5 / $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.
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/ Ps 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]
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.
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]
Selected Whole-unit Combinations of Steak and Lobster
Affordable with $10 Budget
(1)Choice / (2)
Ounces of steak
(price = $1/ounce) / (3)
Ounces of lobster
(price = $2/ounce) / (4)
Ounces of lobster
(price = $1/ounce) / (5)
Total
Expenditure
A / 10 / 0 / 0 / $10 ($10 + $0)
B / 8 / 1 / 2 / $10 ($8 + $2)
C / 4 / 3 / 6 / $10 ($4 + $6)
D / 2 / 4 / 8 / $10 ($2 + $8)
E / 0 / 5 / 10 / $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 I2 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 I3 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.
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)
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 Xand Y generates a downward-sloping demand curve.
Consumer equilibrium (Repeat Exhibit A-5)
KEY CONCEPTS
Indifference curveMarginal rate of substitution (MRS)
Indifference mapbudget 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.