PAPER NO.1 : MICROECONOMICS ANALYSIS MODULE NO.6 : INDIFFERENCE CURVES

Transcription

1 Subject Paper No and Title Module No and Title Module Tag 1: Microeconomics Analysis 6: Indifference Curves BSE_P1_M6 PAPER NO.1 : MICRO ANALYSIS

2 TABLE OF CONTENTS 1. Learning Outcomes 2. Introduction 3. Consumer Preferences 4. Indifference Curves and Indifference Map 5. Marginal Rate of Substitution 5.1 Principle of Diminishing MRS 5.2 Marginal Rate of Substitution and Marginal Utility 6. Properties of Indifference Curves 7. Exceptional Shapes of Indifference Curves 7.1 Perfect Substitutes and Perfect Complements 7.2 Good, Bad and Neutral commodities 8. Summary PAPER NO.1 : MICRO ANALYSIS

3 1. Learning Outcomes After studying this module, you shall be able to learn about Meaning of consumer preferences Concept of indifference curve and indifference map The Law of Diminishing Marginal Rate of Substitution Relationship between Marginal Rate of Substitution and Marginal Utility Properties of Indifference curves Perfect Substitutes and Perfect Complements Good, Bad and Neutral commodities 2. Introduction Indifference curve analysis is a very popular method used to explain consumer behavior. The technique of indifference curve was first invented by Edgeworth (1881) and then by Fischer (1892). Later on the Italian economist Pareto (1906) put it to extensive use and the results were subsequently extended by Soviet economist Slutsky (1915). We have already explained the concepts of cardinal and ordinal utility theories. Modern indifference curve approach uses the concept of ordinal utility. The indifference curve analysis assumes that the consumer have complete information about all the aspects of economic environment. Further, the consumer is assumed to act rationally, so given the money income and prices of goods, he will choose the combination from among various alternatives that gives him maximum satisfaction. 3. Consumer Preferences Preference means choosing one alternative over others. To analyze the preferences under two dimensional set up, the commodities, which a consumer can buy, can be divided into two groups as good X and good Y. The consumer s choice among various alternatives depends upon his preferences, that is, the ranking he gives to various alternatives. Other factors, for example income and prices also influence his choice of an alternative. In indifference curve approach of consumer behavior, certain important assumptions about the nature of consumer s preference are the following. 1. Completeness: Under this assumption, the consumer is capable of comparing all the alternative combinations and ranks them according to utility. Given two bundles P and Q, he can decide whether he prefers P to Q, Q to P or he is indifferent between the two. 2. Transitivity: Transitivity of preference means that if a person prefers a combination P to Q and also prefers Q to R, then he will prefer P to R. Thus transitivity implies that consumer s taste and preferences are consistent. PAPER NO.1 : MICRO ANALYSIS

4 3. Non Satiation: More of a good is always preferable to less of that good. Put differently, more is better, other things remaining constant. The assumption implies that the individual is not already over supplied or over satiated with any good and that the good is desirable. 4. Indifference Curves 4. Indifference Curves An indifference curve represents the various alternative combinations of two commodities, which give the same level of satisfaction to the consumer so the consumer is indifferent between these combinations. The indifference curve is a graphical representation of indifference schedule, where all the alternative combination of commodities gives exactly the same satisfaction level or utility to the consumer. To explain it further, consider the example below: Table 1 Combination Unit of good X Unit of good Y P Q 2 45 R 3 25 S 4 15 T 5 10 The given table shows that the consumer is indifferent between the given five alternative combinations (P, Q, R, S and T) of two goods (X and Y). When all the combinations are represented in the form of a graph, we obtain an indifference curve as shown in the figure 1. [Figure 1: An indifference curve] PAPER NO.1 : MICRO ANALYSIS

5 Figure 1 shows the quantities of commodity X on the horizontal axis and of commodity Y on the vertical axis. All of the five combinations (shown by points P, Q, R, S and T) fall on same indifference curve indicating that these alternative combinations give equal utility to the consumer. The indifference curve is also called iso-utility curve ( iso means same) as it represents equal level of satisfaction at every point on the curve. Note, as we move down the indifference curve, an increase in quantity of good X is offset by a decrease in quantity of good Y, so as to maintain the same level of satisfaction along the curve. 4.1 Indifference Map One indifference curve shows consumption bundles providing single level of satisfaction. We can now think of different indifference curves for each given satisfaction level. In fact, we can draw any number of indifference curves representing different level of utility. An indifference map is the whole set of indifference curves representing the consumer s taste and preferences (figure 2). Higher indifference curve showing greater quantities of both commodities represent higher satisfaction levels as more is better. On the other hand lower indifference curves showing lesser quantities represents a lower satisfaction level. [Figure 2: Indifference map] In the above diagram indifference curve shown as IC 1 represents the lowest satisfaction level while the curve IC 4 represents the highest satisfaction level. Along any given indifference curve the level of satisfaction remains same at all points. PAPER NO.1 : MICRO ANALYSIS

6 5. Marginal Rate of Substitution Marginal rate of substitution (MRS) is one of the most basic concepts of the indifference curve approach. It shows the rate at which a consumer is willing to exchange commodities X and Y along a given indifference curve. More specifically, marginal rate of substitution of X for Y (MRS X, Y) is the amount of good Y, which the consumer is willing to give up as he gets one additional unit of good X, so that his satisfaction level remains the same. In our example (table-1), when a consumer moves from P to Q, the consumer sacrifices 55 unit of Y for one extra unit of X and still remains on the same indifference curve. Therefore, at this stage, MRS X,Y is equal to 55. The movement is shown below in Figure 3. [Figure 3: Marginal Rate of Substitution] When a consumer moves from point P to Q on this indifference curve he sacrifices ΔY for ΔX and remains at the same satisfaction level. This means the marginal rate of substitution of X for Y (MRS X,Y) is ΔY/ ΔX (=PA/AQ). MRS X,Y = (change in Y) / (Change in X) Now, if the points P and Q are very close, the values of ΔY and ΔX will be very small. Then the MRS X,Y will be given by the slope of the tangent to the indifference curve at that point (Figure 5). 5.1 Principle of Diminishing Marginal Rate of Substitution In our example (table 1) we observe that, when the consumer moves from point P to T, he does not have to sacrifice same amount of good Y for each extra unit of good X. Rather, he sacrifices lesser and lesser amount of commodity Y for each extra unit of X as he moves towards point T. This behavior of the consumer is said to be satisfying the principle of diminishing marginal rate of substitution which says that the MRS X,Y diminishes as more and more good X is substituted for PAPER NO.1 : MICRO ANALYSIS

7 good Y. The principle of diminishing marginal rate of substitution is illustrated diagrammatically in figure 4a and figure 4b. [Figure 4a and figure 4b: Diminishing Marginal Rate of Substitution] It is evident from the figure 4a that, as the consumer move downwards along the IC, the length of ΔY becomes shorter and shorter while ΔX is kept the same. Put differently, the MRS X,Y falls as the consumer gets more of good X and less of good Y. The diminishing MRS X,Y can also be demonstrated by drawing tangents at different points on the same indifference curve. The MRS X,Y at any point is given by the slope of the tangent at that point on the indifference curve. It is clear from the Figure 4b that the slope of the tangent declines as we go down the indifference curve. From the law of diminishing marginal utility we know that, as consumption of X increases, its marginal utility decreases. Therefore, the consumer sacrifices lesser amount of good Y for each additional unit of good X. Thus MRS X,Y falls when the amount of X is increased. 5.2 Marginal Rate of Substitution and Marginal Utility It can be shown mathematically that MRS X,Y between two commodities is equal to the ratio of marginal utilities of commodities X and Y. Since utility on each point of the indifference curve remains the same we can represent an indifference curve by U(x, y) = a (i) Where a represent the constant utility along indifference curve and x and y are the amount of commodities X and Y respectively. Taking total differentiation of (i), we get U U dx + x y dy = 0 PAPER NO.1 : MICRO ANALYSIS

8 dy U dx = x U U/ x and U/ y are marginal utilities of goods X and Y respectively. Thus y dy dx = MUx = MRSx, y MUy Thus the MRS between two goods is equal to the ratio between the marginal utility of two goods. 6. Properties of Indifference Curves The main properties of indifference curves can be deduced from the assumptions about the nature of preferences upon which indifference curve analysis is based. (1) Indifference curves slope downward to the right: This means that indifference curves are negatively sloped. This follows from the assumption of non-satiation of preferences, that is, more of a good is always preferable to less of that good. Negatively sloped indifference curve shows that an increase in the amount of a good must be accompanied by the decrease in amount of the other, so that satisfaction level remains same. Let us consider a change in combination from point A to point B in a way that the quantity of good X increases, while Y remains same. This will make B more preferable in comparison to the original combination A and therefore the combination A and B will not be on the same indifference curve. Instead, of being negatively sloped if indifference curves were horizontal, the consumer would be indifferent between the above two combinations, point A and point B. But this would violate our assumption of non-satiated preferences, ruling out horizontal indifference curves when the two goods are normal. The situation is illustrated in figure 5a. PAPER NO.1 : MICRO ANALYSIS

9 [Figure 5a: Horizontal straight line indifference curve] [Figure 5b: Vertical straight line indifference curve] Likewise, with two normal goods an indifference curve cannot be represented by a vertical straight line (figure 5b) as in this case, combinations A and B yield same satisfaction level having same amount of X, but B contains more Y than A. Further, an upward or positively sloped indifference curve (figure 5c) means that the combinations A and B yield equal satisfaction, though B contains more of goods X and Y. But this cannot be so, if both the goods have positive marginal utilities. [Figure 5c: Upward sloping indifference curve] [Figure 5d: Normal indifference curve (downward sloping)] In all the three situations discussed above, the level satisfaction of the consumer rises, as he moves from point A to point B since he starts to consume more of at least one commodity. Therefore normal indifference curves cannot be horizontal, vertical or upward sloping. Thus, as a last possibility, an indifference curve must be downward sloping as shown in figure 5d. Negatively sloped indifference curve indicates that, for a consumer to get same satisfaction level from various combinations, as the quantity of good X is increased, the quantity of Y must be reduced. PAPER NO.1 : MICRO ANALYSIS

10 (2) Indifference curves are convex to the origin: indifference curves are usually convex to the origin which means that the left portion of an indifference curve is relatively steeper while the right portion is relatively flatter. Thus, with movement from left to right, down the indifference curve the value of its slope (absolute) declines. This property of the indifference curve follows from the principle of diminishing marginal rate of substitution discussed in section 5.1. A convex shaped indifference satisfies the condition of diminishing MRS X,Y as is clear from the figures 6a where, as the consumer moves down along the IC from A to F, he gives up lesser and lesser amount of Y for each additional unit of X. This is because, when the consumer reduces his consumption of good Y while increasing his consumption of good X, the urge for more units of X declines continuously, making the consumer sacrifice fewer Y for each extra unit of X. Instead, if the indifference curve was concave this would imply that MRS X,Y increases as more and more units of X are substituted for Y. [Figure 6: Diminishing MRS XY on a convex indifference curve] PAPER NO.1 : MICRO ANALYSIS

11 [Figure 7: Increasing MRS XY on a concave indifference curve] It is clear from figure 7 that, for each extra unit of good X, the consumer is ready to sacrifice more and more quantity of Y, that is, MRS X,Y increases as more quantity of good X is substituted for good Y. Thus, an indifference curve that is concave to the origin violates the principle of diminishing marginal rate of substitution. Therefore, in the presence of diminishing marginal rate of substitution, we cannot have concave indifference curves. (3) Two indifference curves cannot intersect or touch each other: Intersection of two indifference curves, representing two different levels of satisfaction, is not possible. This property follows from the assumptions of transitivity and non-satiation of preferences. This property can be proved through the method of contradiction. Suppose two indifference curves intersect (or touch) each other at point P (figure 8 and 9). Now consider point A on indifference curve IC 1 and point B on IC 2 which is vertically above point A. Consumer will be indifferent between combinations A and P since both are lying on the indifference curve IC 1. Further, the combinations B and P will provide equal utility to the consumer as they both lie on same indifference curve IC 2. Applying transitivity assumption, it follows that the combination A will be equivalent to B in terms of satisfaction. But combination B lies on a higher indifference curve and contains more of good Y than combination A while the amount of X is same in both the combinations. Thus, the consumer will definitely prefer B to A if the assumption of non-satiation of preferences holds. But the intersection of two indifference curves leads to a conclusion of A being equal to B in terms of utility or satisfaction which is not possible. Therefore it can be concluded that the indifference curves cannot intersect or touch each other. PAPER NO.1 : MICRO ANALYSIS

12 [Figure 8: Indifference curves cannot intersect] [Figure 9: Indifference curves cannot touch] (4) Higher indifference curves represent higher level of satisfaction: A higher indifference curve will represent a higher satisfaction level than a lower indifference curve. Put differently, the combinations lying on higher indifference curve will provide greater satisfaction than the combinations lying on lower indifference curve. In the figure 10, IC 2 is a higher indifference curve than IC 1. Therefore, the combination B which lies on indifference curve IC 2 will provide the consumer more satisfaction than combination A which lies on IC 1. This is because the combination B contains more of both commodities X and Y than the combinations A. Applying the assumption of non-satiety, the consumer must prefer B to A. Now, using transitivity assumption it can be said that the consumer will prefer any combinations on IC 2 to any combination on IC 1. We therefore conclude that a higher indifference curve represents a higher level of satisfaction. PAPER NO.1 : MICRO ANALYSIS

13 [Figure 10: Higher indifference curve shows higher level of satisfaction] 7. Exceptional shapes of indifference curves As discussed in the previous sections, standard indifference curves are convex as implied by the principle of diminishing marginal rate of substitution. However, under certain circumstances, the shape of indifference curves may not necessarily be strictly convex as discussed below. 7.1 Perfect Substitutes and Perfect Complements Two commodities are said to be perfect substitute, if one good can be substituted for the other good at a constant rate. Put differently, the consumer is ready to sacrifice same quantity of one commodity for each additional unit of other commodity. This means that the marginal rate of substitution between the two commodities is constant and thus the indifference curve is linear. Thus a straight line indifference curve implies that MRS XY remains constant as more and more unit of X substituted for Y. As explained in figure 11, on a straight line indifference curve consumer is willing to give up same amount of Y for each additional unit of X (ΔY 1 = ΔY 2 = ΔY 3 = ΔY 4) as more and more unit of X substituted for Y. Since MRS XY is equal to the slope of indifference curve at a point on it, and because a straight line has same slope throughout, therefore the straight line indifference curve will mean the same MRS XY throughout. However, in real life, goods tend to be imperfect substitutes and therefore the indifference curves are not normally a straight line. PAPER NO.1 : MICRO ANALYSIS

14 [Figure 11: Perfect substitutes] [Figure 12: Perfect complements] Now, we consider the case of perfect complements, which are consumed together in a fixed proportion. Perfect complements cannot at all be substituted for each other and are therefore marginal rate of substitution between such goods is zero. A consumer needs fixed quantities of both the goods (for example left and right shoe) to give him satisfaction. He cannot maintain satisfaction by substituting one commodity for the other. In figure 12, the indifference curve for perfectly complementary goods will be L-shaped or right angled, with the vertex of L showing the proportion in which the two complementary commodities are used. Here the satisfaction level can only be PAPER NO.1 : MICRO ANALYSIS

15 increased by simultaneous increase in the quantity of the two commodities in the fixed proportion which leads the consumer to a higher indifference curve. 7.2 Good, Bad and Neutral commodities If a commodity is good, more of the commodity is always preferable to less of that commodity. We have explained the indifference curves of such commodities which slope downward and are convex to the origin. On the contrary, when a commodity is a bad, more of it will lower the consumer s satisfaction. Such bads (for example, pollution) are undesirable and provide disutility to the consumer. Therefore the indifference curves in this case assume a different shape. Suppose a bad is represented on X-axis and a good is represented on Y-axis, then the indifference curve will be positively sloped as shown in figure 13a. This is because, a movement towards the right along the indifference curve reduces the consumer s satisfaction level due to increased consumption of bad and therefore, to keep his satisfaction level constant, the amount of good has to be increased. Thus the direction of preference in this case is upward to the left (Figure 13a). If commodity X is a good and commodity Y is a bad the indifference curves are upward sloping to the right as shown in figure 13b. [Figure 13a: Indifference curves between bad and good ] [Figure 13b: Indifference curves between good and bad ] When commodity X and Y both are bads indifference curves will be downward sloping and concave to the origin as shown in figure 14. If the consumption of one bad is increased, the consumption of other bad must fall so that the fall in satisfaction level due to increase in consumption of one bad can be compensated by increase in the satisfaction level due to decrease in consumption of other bad. As the consumer goes down the indifference curve absolute slope increases and hence the MRS X,Y increases. This means that equal successive increase in the amount of X will have to be compensated by more and more reduction in quantity of Y. PAPER NO.1 : MICRO ANALYSIS

16 [Figure 14: Indifference curves when both commodities are bad ] In the case of neuters or neutral goods the consumer does not care whether he has more or less of that commodity. In other words, marginal utility of such commodity is zero and they do not provide any satisfaction to the consumer. If a commodity X is a neuter and Y a normal commodity, the indifference curves will be horizontal straight lines as shown in the figure 15a indicating that a higher satisfaction level can only be achieved from increasing the consumption of the normal good. On the other hand, if commodity Y is neuter and commodity X is a normal good, then the indifference curves will be vertical straight lines as depicted in figure 15b. In this case, to increase the satisfaction and therefore to reach a higher indifference curve, the consumer must acquire more of normal good X as commodity Y does not provide any utility to the consumer. PAPER NO.1 : MICRO ANALYSIS

17 [Figure 15a: Indifference curves between a neuter and good ] [Figure 15b: Indifference curves between good and neuter ] A commodity may be a good up to a point, called the point of satiation and then becomes bad if the consumer is forced to increase his consumption beyond that point. In short, far too much of any good may be bad. Let us suppose that the commodity X becomes bad as its consumption increases beyond X 1 unit and commodity Y becomes bad beyond the quantity Y 1. This case is represented in figure 16 where circular indifference curves are obtained. Point B represents the highest satisfaction level that can be achieved under this condition therefore is the point of satiation or bliss. In the figure, different shapes of indifference curves are shown in four possible situations. In zone I both commodities are goods and thus the indifference curves have normal shape. In zone II, commodity X is a bad and Y is a good and therefore the indifference curves are upward sloping. In zone III, X is a good while Y is a bad and shown by upward sloping indifference curves. In zone IV both the commodities are bad and thus indifference curves slope downward. PAPER NO.1 : MICRO ANALYSIS

18 [Figure 16: Different zones of goodness and badness and point of satiation] When the consumer has excess of a commodity (so that it becomes a bad ), his satisfaction can be increased by reducing that bad. For example, the consumer can increase his satisfaction by moving from point P to Q. In general, the preference direction of the consumer will be towards point B i.e. the closer the consumer is to point B, the better off he is. PAPER NO.1 : MICRO ANALYSIS

19 8. Summary Indifference curve analysis is based on the concept of ordinal utility theory. In indifference curve analysis, consumer s preferences are assumed to satisfy the properties of completeness, transitivity and non-satiation. Indifference curves represent the same level of satisfaction or utility to the consumer and are also called iso-utility curves. Marginal rate of substitution of X for Y (MRS X,Y) represents the amount of Y which the consumer has to give up for the gain of one additional unit of X so that his level of satisfaction remains the same. MRS X,Y diminishes as more and more of good X is substituted for good Y. The MRS between two goods is equal to the ratio between the marginal utility of two goods. Indifference curves slope downward to the right. Indifference curves are convex to the origin. Two indifference curves cannot touch or intersect each other. Higher indifference curves represent higher level of satisfaction. Under certain circumstances, the shape of indifference curves may not be strictly convex and downward sloping. Exceptional shapes of indifference curves can be obtained in case of perfect substitutes, perfect complements and in the presence of bads and neutral goods. PAPER NO.1 : MICRO ANALYSIS