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Properties Of Indifference Curve
The properties of indifference curves in economics describe the characteristics of the relationship between two goods that provide the same level of satisfaction to a consumer. These properties include the downward slope, convex shape, non-intersecting nature, and the higher level of satisfaction associated with curves further from the origin. Indifference curves are essential for understanding consumer preferences and decision-making.
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7 Key excerpts on "Properties Of Indifference Curve"
eBook - ePub- Andrew Barkley, Paul W. Barkley(Authors)
- 2020(Publication Date)
- Routledge(Publisher)
Figure 7.4 ) violates the definition of “indifference.” Point B shows more of both goods than point A, but since it lies on the same indifference curve as point A, it seemingly produces the same level of utility. This cannot be true. This reasoning applies to all combinations of two goods, and it follows that all real-world indifference curves are downward sloping. Put another way, the property of nonsatiation (more is preferred to less) ensure that indifference curves must be downward sloping. A consumer must give up some of one good in order to get the other good. The slope of the indifference curve represents the consumer’s willingness to trade, or sacrifice, one good for another.- 2 Everywhere Dense . This property means that there is an indifference curve through every single point in the positive quadrant. Every combination of the two goods produces some level of satisfaction. The term “everywhere dense” means that there are an infinite number of isoquants in the plane.
QUICK QUIZ 7.7 Why do we only draw some of the indifference curves in the graphs?- 3 Cannot Intersect . Indifference curves cannot intersect, since that would mean that two different levels of utility were equal to each other at the point of intersection. To untangle this problem, assume that two indifference curves intersect, as in Figure 7.5 .
Figure 7.5 Proof of why indifference curves cannot intersect
First, notice that points A and B are on the same indifference curve (I1 ). Each point provides the same level of utility. Next, notice that points B and C are on the same indifference curve (I2 ), so they each represent the same level of utility. If A and B have equal levels of utility, and B and C have equal levels of utility, then it follows that A and C must have equal levels of utility (A ∼ B and B ∼ C, so A ∼ C). However, Figure 7.5 shows that combination A produces a higher level of utility than combination C, since A has more of each good than C (A ⊱ C).Therefore, indifference curves cannot intersect. A contradiction follows if they do. The equations A ∼ C and A ⊱ C cannot both be true at the same time. Therefore, indifference curves must not touch, since each curve represents a different level of utility.- 4 Convex to Origin. This property states that the indifference curves must bend inward toward the origin (be convex to the origin). This is due to the law of diminishing marginal utility: the first unit of a good is the most satisfying! The graph in Figure 7.6 shows this.
Figure 7.6
eBook - ePubMicroeconomic Foundations I
Choice and Competitive Markets
- David M. Kreps(Author)
- 2012(Publication Date)
- Princeton University Press(Publisher)
x. At the start of this chapter, several characteristics of this diagram are listed, which can now be justified.1. The indifference classes are “thin” curves; they have no depth; they contain no ball of positive radius. This is a consequence of local insatiability: Suppose there was a ball of radius ∈ wholely contained within some indifference curve. Let x denote the center of the ball. If preferences are locally insatiable, there is some y within ∈/2—that is, within the ball—that is strictly preferred to x, a contradiction.2. The indifference curves are strictly decreasing as we move from left to right. In symbols, if y = (y1 , y2 ) ~ x = (x1 , x2 ) and y1 > x1 , then y2 < x2 . This is true if preferences are strictly monotone: If y1 > x1 and y2 ≥ x2 (and preferences are strictly monotone), then (y1 , y2 ) (x1 , x2 ).3. The indifference curves are continuous and don’t “run out” or end abruptly except on the boundaries of X. That is, if we start at any point x ≥ x′ and follow a continuous path to a point x″ ≤ x′, we cross the indifference curve of x′. This is true if preferences are monotone and continuous; if they are monotone, x ≥ x′ ≥ x″ implies x x′ x″; hence Lemma 2.10 applies. Continuity of preferences alone is insufficient; see Problem 2.7.4. The indifference curves are strictly convex. That is, if distinct points x and y lie along the same indifference curve and a ∈ (0, 1), then ax + (1 – a)y lies on a higher indifference curve. This, of course, is strict convexity.The indifference curves in Figure 2.1 have one further property: They are smooth, without kinks or sudden changes in derivatives. Nothing that we have said or done in this chapter gets us to this sort of property, although we see in later chapters that this property has some nice consequences.2.5. Weak and Additive Separability
eBook - ePub- Andrew Barkley, Paul W. Barkley(Authors)
- 2016(Publication Date)
- Routledge(Publisher)
Therefore, indifference curves cannot intersect. A contradiction follows if they do. The equations A ~ C and A > C cannot both be true at the same time. Therefore, indifference curves must not touch, since each curve represents a different level of utility.- 4. Convex to Origin. This property states that the indifference curves must bend inward toward the origin (be convex to the origin). This is due to the law of diminishing marginal utility: the first unit of a good is the most satisfying! The graph in Figure 7.6 shows this.
Figure 7.6 The law of diminishing marginal utilityThe law of diminishing marginal utility is used to show that if a consumer has many pairs of pants (point A: 6 pairs of pants, 1 shirt), she is willing to trade 3 pairs of pants for one additional shirt (point B: 3 pairs of pants, 2 shirts). On the other hand, if the consumer had 5 shirts and only 1 pair of pants (point C), she would be willing to give up 2 shirts for the second pair of pants (point D: 2 pairs of pants and 3 shirts). A consumer’s willingness to trade one good for another depends on how much of each good he or she has. The first unit provides the higher level of satisfaction, and consumption of subsequent units provides less additional utility, as shown in Figure 7.6 .7.4.5 Indifference curves for substitutes and complements
- Perfect Substitutes (consumption) = goods that are completely substitutable, so that the consumer is indifferent between the two goods.
The indifference curve for perfect substitutes is a straight line with a constant slope. In Figure 7.7 , the consumer is indifferent between any combination of blue and green shirts that adds up to three shirts. This indifference curve is a special case, since it is not convex to the origin. The consumer is willing to trade one good for the other at a constant rate, so the goods are, in a way, the same good, “shirts.” The opposite case of perfect substitutes is Perfect Complements
- Zahid A. Khan, Arshad N. Siddiquee, Brajesh Kumar, Mustufa H. Abidi(Authors)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
Fig. 2.11 shows the indifference map. 0 Q y Q x A B Figure 2.11 The indifference map 2.8.4 Rate of Commodity Substitution Both commodities on x- and y-axis can be substituted one for another. This becomes the basis of indifference curve being rectangular hyperbolic and downward sloping. The slope 70 Engineering Economics with Applications of the indifference curve at any point on it is negative, which is known as the marginal rate of substitution or rate of commodity substitution. For any line passing tangent to the indifference curve, at the point of tangency, the slope of the indifference curve is equal to y xy xy x dQ MRS RCS dQ - = = . The concept of marginal utility is implicit in the MRS xy or RCS xy which is equal to the ratio of the marginal utilities derived from the commodity x and y. Symbolically, x xy xy y MU MRS RCS MU = = or y yx yx x MU MRS RCS MU = = This implies that the diminishing marginal rate of substitution and indifference curve is convex to the origin. 2.8.5 Properties of ICs Properties Of Indifference Curves The properties of usual indifference curve are: higher indifference curves are preferred to lower ones; indifference curves are downward sloping; indifference curves do not cross; and indifference curves are bowed inward (that is, convex to the origin)). Property 1: Higher indifference curves are preferred to lower ones. Remark: Consumers usually prefer more of something rather than less of it. The higher indifference curves represent larger quantities of goods than the lower indifference curves. Property 2: Indifference curves are downward sloping. Remark: A consumer is willing to give up one good only if he gets more of the other good in order to remain equally happy. If the quantity of one good is reduced, the quantity of the other good must increase. For this reason, most indifference curves slope downward. Property 3: Indifference curves do not cross. 0 Q y Q x A B C Figure 2.12 Intersecting indifference curve
- Roberto Serrano, Allan M. Feldman(Authors)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
We show this in Figure 2.5 below. We call preferences well behaved when indifference curves are downward sloping and convex. In reality, of course, indifference curves are sometimes concave. There are many examples we can think of in which a consumer might like two goods, but not in combination. You may like sushi and chocolate ice cream, but not together in the same dish; you may like classical music and hip-hop, but not in the same evening; you may like pink clothing and orange clothing, but not in the same outfit. Again, if the goods are defined generally enough, like classical music consumption per year, hip-hop consumption per year, pink and orange clothing worn this year, the assumption of 2.2 The Consumer’s Preference Relation 13 Y Good 2 Good 1 X X/ 2 + Y/ 2 Figure 2.5 Convexity of preferences means that indifference curves are convex, as in the figure, rather than concave. This means that the consumer prefers averaged bundles over extreme bundles. For example, the bundle made up of 1 / 2 times X plus 1 / 2 times Y ; that is, X / 2 + Y / 2 is preferred to either X or Y . This is what we normally assume to be the case. Y Good 2 Good 1 X X/ 2 + Y/ 2 Figure 2.6 A concave indifference curve. This consumer prefers the extreme points X and Y to the average X / 2 + Y / 2. 14 2 Preferences and Utility convexity of indifference becomes very reasonable. We show a concave indifference curve in Figure 2.6 above. 2.3 The Marginal Rate of Substitution The marginal rate of substitution is an important and useful concept because it describes the consumer’s willingness to trade consumption of one good for consumption of the other. Consider this thought experiment. The consumer gives up a unit of good 1 in exchange for getting some amount of good 2. How much good 2 does she need to get in order to end up on the same indifference curve? This is the quantity of good 2 that she needs to replace one unit of good 1. Or, consider a slightly different thought experiment.
eBook - PDF- Martha L. Olney(Author)
- 2015(Publication Date)
- Wiley(Publisher)
It slopes down because to keep total utility constant, more beef (B) will be offset by fewer apples (A). The indifference curve is not a straight line because of the law of diminishing marginal utility. If you are not consuming much beef, then you’re willing to give up many apples to gain a little bit of beef. But if you are already consuming a lot of beef, you won’t give up many apples at all to gain more beef because yet more beef gives you very little additional satisfaction. Economists describe the shape of the curves by saying: Indifference curves are convex to the origin. The slope of the indifference curve depends on the marginal utilities of apples and beef. The slope changes as you move along the indifference curve. Figure 5.3a shows the slope. Remember: Total utility is constant along any (a) (b) Quantity of A rise run y x Quantity of B Quantity of B Quantity of A I 1500 (total utility = 1500 utils) I 1000 (total utility = 1000 utils) Slope = Indifference Curve −MU B MU A Figure 5.3 Indifference curves. An indifference curve such as the one shown in Figure 5.3a shows the many combinations of goods A and B that each provide the same total utility. The indifference curve is down- ward sloping because more B is offset by less of A. It is convex to the origin because of the law of diminishing marginal utility. The slope of the indifference curve is −MU B /MU A , which economists call the marginal rate of substitution. The indifference curve labeled I 1000 in Figure 5.3b shows the combination of goods A and B that provide total utility of 1,000 utils. Combinations of goods A and B that provide more utility are further from the origin. The indifference curve I 1500 shows combinations of A and B that provide 1,500 utils of satisfaction. 70 Chapter 5 Consumer Theory indifference curve. When you move between points x and y, total utility does not change.
eBook - PDF- Steven Landsburg(Author)
- 2013(Publication Date)
- Cengage Learning EMEA(Publisher)
© Cengage Learning 58 CHAPTER 3 Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Because this consumer always selects a corner solution, he consumes either zero units of X or zero units of Y. But goods that consumers choose to purchase none of are not very interesting from the viewpoint of economics. So now we have our addi-tional reason for assuming that indifference curves are convex. They might not be — but in this case one of the goods in question would not be consumed at all, and we would prefer to turn our attention to goods that are consumed. Therefore, we usu-ally confine our attention to convex indifference curves. 3.3 Applications of Indifference Curves Now let ’ s put our new tools to use. In this section, we ’ ll see several applications of indifference curve analysis. Standards of Living Economic conditions change all the time. Incomes go up and down, and so do prices. How do we tell which changes are good for the consumer and which are bad? Sometimes it ’ s easy. If your friend Harold ’ s income goes up while prices remain unchanged, his life has certainly improved. If his income stays fixed while all prices rise, he ’ s worse off than before. But what if some prices rise while others fall? Is that good or bad for Harold? Sometimes there ’ s not enough information to answer that question. Other times there is. Let ’ s take an example: Harold consumes goods X and Y. Their prices are P X ¼ $3 and P Y ¼ $4. He chooses to buy 4 units of X and 2 of Y, exhausting his income of $20.
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