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We may now consider a few examples of preferences in order to throw some light on different types of preferences and optimal choice in each case.
1. Perfect Substitutes:
In Fig 6.5 we show three possible situations. If p2 > p1, then the slope of the budget line is less than that of the indifferences curves. In this case the consumer makes optimal choice by spending all his money on x1. If p1 > p2 then he buys only x2 at point A. Finally if p1 = p2, any combination of x1 and x2 that satisfies the budget constraint indicates optimal choice.
Thus the demand function corresponding to the above three situations may be expressed as follows:
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(i) When p1 < p2, x1 = m/p1.
(ii) When p1 = p2, x1 = any number between 0 and m/p1.
(iii) When p1 > p2, x1 = 0.
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The basic point to note here is that if x1 and x2 are perfect substitutes, then a rational consumer will purchase the cheaper one. If the prices of the two goods are the same, whether the consumer will buy x1 or x2 is not worth his bother. He does not care which one he ends up buying.
If the budget line coincides with IC3 we get an infinite number of equilibrium points because there is a whole range of optimal choices. Any combination of x1 and x2 will be an optimal combination.
2. Perfect Complements:
In case of perfect complements, illustrated in Fig. 6.6 the optimal choice will always lie in the diagonal. This means that the consumer is buying equal amounts of x1 and x2.
For example, people buy shoes and socks in pairs, not in single numbers.
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Algebraic Treatment of Optimal Choice:
Since the consumer, in this case, is purchasing the same quantity of x1 and x2, we can denote this amount as x. So the budget constraint is p1x + p2x = m.
Solving for x gives us the optimal choices of x1 and x2 we get:
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x1 = x2 = x = m/p1 + p2
It is as if the consumer is buying a composite good x whose price is p1 + p2. In this case the demand function will be x = f(p), where p = p1 + p2. Since x1 and x2 are consumed together (and in equal amounts) it is as if the consumer is spending all his money on a composite commodity whose price is p1 + p2 = p.
3. Neutrals and Bads:
The consumer spends all his income on non-neutral goods. The same thing is true if one commodity is a bad like polluted air. Thus, if x1 is a good and x2 is a bad then the demand function will be
x1 = m/p1.
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x2 = 0.
4. Discrete Goods:
Suppose x, is a discrete good like a motor car which is not available in fractional amounts while x2 is money implying expenditure on all other goods. If the consumer buys 1, 2, 3 … cars, he will choose the consumption bundle (1, m – p1), (2, m – 2p1), 3(m – 3p1) and so on. The utility of each of the bundles can be compared to see which gives him maximum utility or satisfaction.
In terms of Fig. 6.7 the optimal bundle is on the highest attainable indifference curve. At a very high price of the consumer will not buy it at all in Fig. (a). As its price falls the consumer will find it optimal to consume 1 unit of x1 in Fig. (b). As p, continues to fall the consumer will choose to consume more of x1.
5. Concave Preferences:
In Fig. 6.8, showing concave to the origin indifference curves, optimal choice occurs at a point like Z, which is a boundary (corner) point. Non-convex preferences mean that if the consumer is having sufficient money to purchase tea and biscuits and if he does not like to consume both at a time, he will spend all his money on one or the other.
In Fig. 6.8 the optimal occurs at the boundary point, Z, not the interior tangency point, X because Z is on a higher indifference curve (IC3) than X(IC2). Such a situation is one of “monomania”, which implies corner solution.
6. Cobb-Douglas Preference:
The optimal choices for the utility function of the Cobb-Douglas form u(x1, x2) = x1αx1β are:
x1 = (α/α + β) (m/p1)
x2 = (β/α + β) (m/p2)
Property:
Cobb-Douglas preferences have an important property: the Cobb-Douglas consumer always spends a fixed proportion of his income on both x1 and x2, the size of which is determined by the exponents α and β. The expenditure on x1 is p1x1. So the proportion of his total income spent on x is p1x1/m. If we substitute the demand function for x1 we have,
p1x1/m = (- p1/m) (α/α + β) (m/p1) = α/α + β
Similarly the proportion of his income spent on x2 is β/α + β.
Since a consumer’s expenditure is equal to his income we choose a Cobb-Douglas utility function for a particular specification, i.e., one in which the exponents sum to 1, i.e., if
u(x1, x2) = x1αx1β, then s1 = α/α + β and s2 = β/α + β and the two sum to 1.
7. Satiation in Commodities:
If satiation exists with respect to both goods. The result will be as shown in Fig. 6.9: the indifference map will be ‘closed’. The optimum will exist at point X. Here we not only have the ordinary segment CD and the two segments CB and DE, but an added segment BE which closes the indifference curve making it almost a circle.
The section BE is odd because it implies that the consumer has enough of both commodities. As he moves from B towards E he gains an undesired quantity of x1 and is indifferent to the extra amount of x1 as long as he can get rid of some of x2.
Here the indifference curve through B and E is not superior to one through K. Since higher indifference curve lie inside, the best point is X. If the budget line is one through A, then it is an optimum. But if it is through K, then although there is tangency at K, it is not an optimum. The consumer is better off if he does not spend all his income and stays at point X, which is an interior point in the attainable set.
The budget line could be vertical (e.g., through C) or horizontal (through D).
The former would imply a zero price for x2 and the latter a zero price x1: they would be ‘free goods’. Between C and B, ready prices are negative. If we exclude the lunch possibility, we find situations where prices are either positive or zero, but not negative.
8. Non-Linear Budget Lines:
The budget line is normally taken as a straight line because prices are assume to remain fixed. In practice consumers are at times able to influence the prices of the goods they purchase by exercising some ‘monopsonistic’ power. Thus if the consumer can force the price of x1 down by purchasing more of x1, the ratio p2/p1 would increase as more of x1 is purchased.
The budget line will then be concave as in Fig. 6.10. If, however, the budget line was strictly convex — that is bent inwards as x1 increases then the consumer has to pay higher and higher prices for x1 as he buys more. Fig. 6.11 shows a case where we find multiple optimum because the budget line and the indifference curve are, in part, coincident.
Finally Fig. 6.12 shows a corner solution because the budget line has a different slope to the indifference curve (strict convexity has been relaxed since IC1 cuts the x2 axis.
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