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Here is an elaborated discussion on ‘Indifference Curves’, highlighting:- 1. Introduction to Indifference Curves 2. Risk Aversion and Indifference Curves 3. Testing the Properties of Indifference Curves.
Introduction to Indifference Curves:
The modern theory of consumer choice is formulated in terms of preferences that satisfy the first three axioms — completeness, reflexive-ness and transitivity. A consumer’s tastes and preferences are represented graphically by using a technique called indifference curves, which are essentially boundary lines separating preferred consumption bundles from inferior ones.
Fig. 4.1 shows a typical indifference curve (IC). Here the two axes represent the consumption of two goods x1 and x2. The curve IC is a locus of points containing bundles such as A, B,… among which the consumer is indifferent, i.e., he is indifferent among (x̅1, x̅2) as indicated by point D and all other points on the boundary line IC.
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In Fig. 4.1 we show a weakly preferred set- bundles weakly preferred to x̅1, x̅2, i.e., bundles which he prefers to x̅1, x̅2 or among which he is indifferent. The shaded area in this figure contains all the bundles, i.e., (alternative combinations of (x1, x2) which are at least as good as bundle (x̅1x̅2).
In Fig 4.1 we choose a particular bundle at random such as (x̅1, x̅2). Then we shade in all of the consumption bundles that are weakly preferred to (x̅1, x̅2). These bundles form the weakly preferred set. The locus of the bundles on the boundary of the set is the indifference curve. It is because the curve represents the bundles for which the consumer is just indifferent to (x̅1, x̅2).
It is possible to draw an indifference curve through any consumption bundle chosen at random. The indifference curve through any consumption bundle that we pick up consists of all bundles of goods that leave the consumer indifferent to the given bundle.
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If X = (x1, x2) and x1 and x2 are goods (not bads). X0 is preferred to X’ if;
either x01 > x’1 and x02 > x’2
or x02 > x’2 and x01 > x’1.
(In either case X0 dominates x’). In utility terms u (x01, x02) > u(x’1 x’2) if these conditions are satisfied. In Fig. 4.2 all points in the area a are better than X0 and those in area b are worse than it. So what is the slope of the indifference curve through X0? It cannot, go through a or b and must therefore slope from north-west to south-east. Thus an indifference curve slopes downward.
What is not Shown by Indifference Curves?
The main problem with indifference curves to describe preferences is that they do not show the consumer which bundles are better and which are worst. They just show the bundles among which the consumer is indifferent. The bundles among which the consumer is indifferent is a matter of perception. An indifference curve shows the consumer the bundles that he perceives as being indifferent to one another.
Non-Intersecting Indifference Curves:
An important property of indifference curves is that they, representing distinct levels of preferences cannot meet or intersect. This means that an indifference curve has to lie consistently above or below another. This important principle about indifference curves is proved in Fig. 4.3. There are bundles of goods X, Y and Z.
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Here X lies on IC2, Y on IC1 and Z at the intersection of both. By assumption the indifference curves represent distinct levels of preference. This means that one if, of the bundles such as X is strictly preferred to the other bundle Y. Here we see that X ~ Z because both are on IC2 and Z ~ Y because both are on IC1.
The axiom of transitivity, therefore, implies that X ~ Y. But this contradicts the assumption that X > Y and establishes the result that indifference curves cannot cross, as at point Z in Fig 4.3. An indifference curve showing a particular level of preference must either lie above or below others showing other levels of preferences.
Indifference curves are a nice way of describing preferences. Preferences do vary. So we may now examine what kind of preferences give rise what shapes of indifference curves.
Risk Aversion and Indifference Curves:
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The extent of an individual’s risk aversion can be shown by using indifference curves that relate expected income (measured by the mean along the vertical axis) to the variability of expected income (measured by the standard deviation along the horizontal axis). Each indifference curve shows all the combinations of standard deviation and expected income that give the individual the same level of utility or satisfaction.
Since a higher variability of income (risk) must be compensated by a higher expected income, these indifference curves are positively sloped. Fig. 4.15 shows two sets of indifference curves. The indifference curves in part (a) are steep and refer to an individual who has a strong aversion to risk, while those in part (b) are flat for a less risk-averse individual.
Specifically, indifference curve IC2 in part (a) shows that the individual is indifferent among the standard deviation of 0.5 and the expected income of Rs. 80 (point A) and the standard deviation of 1.0 and the expected income of Rs. 120 (point B) and the standard deviation 1.5 and the expected income of Rs. 180 (point C).
Thus, indifference curve IC2 shows that, starting from point A, the individual requires an additional Rs. 60 in expected income to just compensate him for an increase in standard deviation from 0.5 to 1.0 (and reach point B) and the individual requires an additional Rs. 60 in expected income to compensate him for an increase in the standard deviation from 1.0 to 1.5 (and reach point C).
On the other hand, starting at point B on IC2 and the standard deviation of 1.0, the higher expected income of Rs. 160 puts the individual at point D on higher indifference curve IC3, while the lower expected income of Rs. 80 puts him at point E on a lower indifference curve IC1.
Finally, an increase in the standard deviation from 0.5 to 1.0 at the expected income of Rs. 80 shifts the individual from point A on IC2 to point E on a lower indifference curve IC1.
Part (b) of Fig. 4.15 shows the indifference curves for an individual who is less risk averse than the individual in the part (a). For example, indifference curve IC2 shows that the individual is indifferent among the expected income of Rs. 40 and standard deviation of 0.5 (point A), the expected income of Rs. 60 and standard deviation of 1.0 (point B), and the expected income of Rs. 100 and standard deviation of 1.5 (point C).
On the other hand, for the standard deviation of 1.0, higher expected income of Rs. 80 puts the individual at point D on higher indifference curve IC3 (from point B on IC2), while the lower expected income of Rs. 40 puts him at point E on a lower indifference curve ICV.
Finally, an increase in the standard deviation from 0.5 to 1.0 at the expected income of Rs. 40 shifts the individual from point A on IC2 to point E on the lower indifference curve IC1.
Testing the Properties of Indifference Curves:
1. Density:
Indifference curves are always dense. In a particular commodity space we can draw any number of indifference curves since; by assumption the commodity space is continuous (due to the non-existence of indivisible commodities). The curves will lie very close to one another and may even become indistinguishable from one another. All these curves together constitute the consumer’s indifference map.
2. Non-Intersection:
Two indifference curves cannot meet or intersect. If they do then the axiom of dominance will be violated. See Fig. 4.29. By the axiom of dominance we have T P R but R and S lie on the same indifference curve so that R I S. By dominance S P Q.
Hence, by the axiom of transitivity, T P Q. But T and Q are on the same indifference curve. The results are contradictory. Thus intersecting indifference curves lead to the violation of the two axioms used to ensure their very existence.
If the convexity axiom is relaxed altogether, each indifference curve will be concave to the origin, as shown in Fig. 4.30. If the indifference curves are L-shaped, the axiom of dominance will again be violated. This happens if two commodities are perfect substitutes of each other rather than just substitutes, in which case indifference curves are straight lines with constant MRS.
For analytical purposes, it is convenient if u(x1, x2) can be assumed to be differentiable. It is possible, however, for continuous preferences not to be represented by a differentiable utility function. The simplest example, shown in Fig. 4.31, is the case of Leontief preferences where A” > A’ iff min. {x”1, x”2} ≥ {x’1, x’2}. The non-differentiability arises because of the kink in indifference curves when x1 = x2.
As Fig. 4.31 shows, a point like B is indifferent to a point such as A, even though B contains more of x1 and the same amount of x2. Permitting A and B to be indifferent amounts to relaxing the axiom of dominance. Fig. 4.31 permits ‘weak’ dominance (a point such as C cannot be indifferent to A) but not ‘strong’ dominance. Such a situation is encountered when two commodities are perfect complements, so that they are consumed in fixed proportions.
Here we assume utility functions to be twice continuously differentiable. A condition purely in terms of preferences that implies this property is that indifference sets are smooth surfaces that fit together nicely so that the rates at which commodities are substituted for each other depend differentiable on the consumption levels.
Satiation in One Commodity:
If the consumer reaches satiation in respect of one commodity, the indifference curve will have upward sloping segments like the one shown in Fig. 4.32. In this case a point such as C is as good as point A. But we also have B P C instead of BP C which is what the axiom of dominance suggests. In this case the consumer has reached a point of satiation in x2 at point C.
From C upwards through B, the consumer requires more x1 to tolerate more of x2. Similarly, D is a point of satiation in respect of x1 so that after D the indifference curve begins to bend upwards through E. This type of situation is encountered if the axiom of dominance (which rules out satiation in all commodities) is violated.
Concluding Comments:
As a result of these axioms, we can represent the preference ordering of the consumer by a set of continuous convex-to-the-origin indifference curves, such that each consumption bundle lies on one and only one of them. Moreover, as a result of the axiom of non-satiation we can say that bundles on a higher indifference curve are preferred to those on a lower one.
Thus, the best consumption bundle open to a consumer is the one lying on the highest possible (attainable) indifference curve.
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