Partial and general equilibrium, law of demand and demand analysis
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- The Weak Axiom of Revealed Preference
TheoryIf a person chooses a certain bundle of goods (ex. 2 apples, 3 bananas) while another bundle of goods is affordable (ex. 3 apples, 2 bananas), then we say that the first bundle is revealed preferred to the second. It is then assumed that the first bundle of goods is always preferred to the second. This means that if the consumer ever purchases the second bundle of goods then it is assumed that the first bundle is unaffordable. This implies that preferences are transitive. In other words if we have bundles A, B, C, ...., Z, and A is revealed preferred to B which is revealed preferred to C and so on then it is concluded that A is revealed preferred to C through Z. With this theory economists can chart indifference curves which adhere to already developed models of consumer theory. The Weak Axiom of Revealed PreferenceThe Weak Axiom of Revealed Preference (WARP) is a characteristic on the choice behavior of an economic agent. For example, if an individual chooses A and never B when faced with a choice of both alternatives, they should never choose B when faced with a choice of A,B and some additional options. More formally, if A is ever chosen when B is available, Revealed preference theory, pioneered by American economist Paul Samuelson, is a method by which it is possible to discern the best possible option on the basis of consumer behavior. Essentially, this means that the preferences of consumers can be revealed by their purchasing habits. Revealed preference theory came about because the theories of consumer demand were based on a diminishing marginal rate of substitution (MRS). This diminishing MRS is based on the assumption that consumers make consumption decisions based on their intent to maximize their utility. While utility maximization was not a controversial assumption, the underlying utility functions could not be measured with great certainty. Revealed preference theory was a means to reconcile demand theory by creating a means to define utility functions by observing behavior. If a person chooses a certain bundle of goods (ex. 2 apples, 3 bananas) while another bundle of goods is affordable (ex. 3 apples, 2 bananas), then we say that the first bundle is revealed preferred to the second. It is then assumed that the first bundle of goods is always preferred to the second. This means that if the consumer ever purchases the second bundle of goods then it is assumed that the first bundle is unaffordable. This implies that preferences are transitive. In other words if we have bundles A, B, C, ...., Z, and A is revealed preferred to B which is revealed preferred to C and so on then it is concluded that A is revealed preferred to C through Z. With this theory economists can chart indifference curves which adhere to already developed models of consumer theory. then there can be no optimal set containing both alternatives for which B is chosen and A is not. This characteristic can be stated as a characteristic of Walrasian demand functions as seen in the following example. Let pa be the price of apples and pb be the price of bananas, and let the amount of money available be m=5. If pa =1 and pb=1, and if the bundle (2,3) is chosen, it is said that that the bundle (2,3) is revealed preferred to (3,2), as the latter bundle could have been chosen as well at the given prices. More formally, assume a consumer has a demand function x such that they choose bundles x(p,w) and x(p',w') when faced with price-wealth situations (p,w) and (p',w') respectively. If p·x(p',w') ≤ w then the consumer chooses x(p,w) even when x(p',w') was available under prices p at wealth w, so x(p,w) must be preferred to x(p',w'). 3.6 Slutsky theorem The Slutsky's Equation breaks down a change in demand due to price change into the substitution effect and the income effect. The equation takes the form: dx/dp = dh/dp - x dx/dm The term on the left is the change in demand when price changes, where x is the (Marshallian) demand for a good and p is the price. The term h is the Hicksian or the compensated demand. The term dh/dp measures the substitution effect. The term m is the income, and x*dx/dm measures the income effect. See below for more explanations and the derivation of the equation. We can make sense of the substitution and the income effects by this intuitive story. Suppose a consumer is consuming the optimal amount of two goods x and y, given his income and suddenly the price of x drops. The consumer will respond to this price change in two ways. First, as x becomes relatively cheaper the consumer will shift some of his consumption of y to x (assume x and y are not perfect complement). Second, as the price of x drops, even if the consumer does not make any consumption shift from y to x, he has more purchasing power because of the savings that results from the price drop in x. This savings allows the consumer to buy more goods (x or y). The shift in consumption from y to x is the substitution effect, and the increase in purchasing power due to the savings is the income effect. When we read the Slutsky's equation, the term dh/dp is the substitution effect. This is because the compensated depend h(p1, p2, u) fixes the consumer's utility level, and when the consumer's purchasing power remains constant, the term dh/dp only measures the shift in consumption when the price changes. On the other hand, the income effect depends on the amount of good the consumer is consuming (x), and the consumer's reaction to an income change (dx/dm) that comes from the "savings". Thus the term x*dx/dm measures the income effect. Derivation Notice that in equilibrium, the (Marshallian) demand and the compensated demand are the same. That is, x(p1, p2, m) = h(p1, p2, v(p1, p2, m)), where v is the value function of the utility maximization problem. To simplify notation, we write u = v(p1, p2, m), a fixed level of utility, and we write the budget constraint as p1x1 + p2x2 - m = 0. Now equate the two demands as above, x(p1, p2, m) = h(p1, p2, u) Without loss of generality, differentiate with respect to p1, dx/dp1 + dx/dm*dm/dp1 = dh/dp1 Note that the budget constraint in the Marshallian demand depends on p, so we have to use total derivative when differentiating the left side of the equation. The second term is merely an application of the chain rule. The term dm/dp1 is the derivative of the budget constraint p1x1 + p2x2 - m with respect to p1, ie, dm/dp1 = x1. Substitute this in and the equation becomes the Slutsky's equation dx/dp1 + dx/dm*x1 = dh/dp1 Endowment income effect
When the consumer is endowed with the goods instead of a fixed income, the budget constraint is p1x1 + p2x2 - p1w1 - p2w2 = 0. Writing the budget constraint this way, and by differentiating the budget constraint with respect to p1, it is easy to see that the Slutsky's equation becomes dx/dp1 = dh/dp1 - dx/dm*(x1 - w1) In other words, the substitution effect remains the same, but the income effect applies to the excess demand rather than the demand itself. Other Slutsky equations
Given the two examples and the derivation above, we can see that the Slutsky's equation always has the same format, and each format is different only because the budget constraint is different. Students can try deriving Slutsky's equations for other situations, such as one with labour supply, or intertemporal choices. 3.7 The Hick’s Theory Hicks’s work in this area was first published in 1934 in an article he wrote jointly with R. G. D. Allen (1934a), but it is more comprehensively set out in Chapters 1-3 of Value and Capital (1939a). Their work in this area was a development of earlier work by Edgeworth, Pareto and Slutsky, and they established that ordinal utility (elaborated in terms of indifference curves and budget lines) could derive the same propositions as cardinal utility (elaborated in terms of measurable marginal utilities) but that the former achieved the same results more clearly and more precisely. However, they accepted that the two theories were saying the same thing, e.g. ‘Tangency between the price line and an indifference curve is the expression … of the proportionality between marginal utilities and prices’ (Hicks, 1939a, p. 17). This can be demonstrated conveniently as in Figure 2. The slope of indifference curve, I, can be written as dY/dX and measures the marginal rate of substitution (MRS) of X for Y. However, along an indifference curve the total utility of the consumer is constant. Therefore it is possible to demonstrate that the slope of an indifference curve can also be defined as the ratio of the marginal utilities (MU) of X for Y.2 Thus the slope of indifference curve, I = dY/dX = MUx/MUy = MRS of X for Y. However, the slope of the budget line, AB = Px/Py. Therefore at the point of tangency MUx/MUy = Px/Py = MRS of X for Y, i.e. the ordinal and cardinal conditions for a maximum are equivalent. It is also of interest to note the similarities between this analysis and our previous discussion of isoquants. The elasticity of substitution is a property of an isoquant but the same general principle holds for an indifference curve. Specifically the marginal rate of substitution diminishes along a convex indifference curve and the marginal rate of technical substitution (MRTS) diminishes along a convex isoquant. Figure 2 can then be relabelled to demonstrate the cost minimising output level, i.e. measuring capital along the Y axis and labour along the X axis, cost minimisation occurs where PL/Pk = MRTS of L for K. In fact, it was the realisation of this symmetry that led Hicks in the direction of the 1934 Hicks-Allen article. 
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