It is important to update the way we teach indifference curves. The standard version is still being taught to millions of students annually, despite the discovery more than three decades ago of a crucial inconsistency in its conceptualisation, namely that it fails to indicate the reference point or the current level of consumption (Knetsch and Sinden 1984, Kahneman et al. 1990, Knetsch et al. 2012).
According to conventional indifference curve diagrams, when deciding between two goods – say, food and clothing – it is as though we’ve never consumed them before. Thus, we are assumed to come to the problem in a pristine state, without indicating the amount of the goods in question we consumed in the prior period or are adapted to. However, this is contradictory, because if we have not consumed these items before, how are we supposed to know how much utility we should expect from them?
Hence, the customary indifference curve depends on the implicit assumption that choices along indifference curves are reversible. That is, if an individual owns x and is indifferent between keeping it and trading it for y, then when owning y the individual should be indifferent about trading it for x. If loss aversion is present, however, this reversibility will no longer hold (Knetsch 1989, Kahneman et al. 1991). Knetsch and Sinden (1984) were the first to point out that the standard assumption pertaining to the equivalence of losses and gains is contradicted by the experimental evidence: “the compensation measure of value seems to exceed significantly the willingness to pay measure, which would appear to call into some question … interpretations of indifference curves”.
Indifference curves with reference-dependent utility
Thus, the mainstream representation of indifference curves is outdated, inconsistent, and misleading, because it overlooks the ample empirical evidence that current consumption (or the current endowment) matters for subsequent consumption decisions, as it becomes a reference point to which other states of the world are compared (Rabin 2008). The endowment effect implies that there is an extra discomfort associated with giving something up, in addition to the loss of the pleasure associated with consuming it. Let us suppose that the current level of consumption is (Qx1, Qy1) (Figure 1). Then point a becomes the origin of the coordinate system and the relevant reference point for the current period 1. While recently there has been some discussion about how to define the reference point in various circumstances (Heffetz and List 2013), in this example it is straightforward – it is simply the current level of consumption at point a.
Figure 1. Behavioural indifference curves in period 1 showing initial endowment
We divide the plane into four quadrants (numbered anticlockwise) with the axis going through the origin at point a. In quadrant 1 the reference point is irrelevant as both x and y are increasing. In this quadrant the standard convex-to-the-origin indifference curve is unchanged. However, x decreases in quadrant 2 while y increases; in quadrant 3 both x and y decrease; and in quadrant 4 x increases while y decreases. (All changes are relative to the axes that go through the initial reference point a in Figure 1).
Thus, lowering consumption of x below the initial level, Qx1, requires a larger amount of a compensating good y in order to maintain the same level of utility than the amount of y required to be given up if there were an identical increase in x beyond Qx1. In other words, at point a the loss in marginal utility of giving up a unit of x is larger (in absolute value) than the marginal utility of obtaining a unit of x – decreasing one’s consumption from the current level is more painful than increasing consumption from the current level is beneficial. This is critical, because it implies that the indifference curves are kinked at the axis that go through point a, with slopes steeper in quadrant 2 than in quadrant 4 – a factor overlooked in conventional treatments of indifference curves. David Just (2014: 81) works out the properties of such behavioural indifference maps with straight lines, i.e. with constant marginal rates of substitution, while Knetsch et al. (2012) demonstrate with indifference curves the discrepancy of evaluating welfare in the domains of gains and losses.
To demonstrate the impact of the endowment effect on the indifference map with declining marginal rates of substitution, let us suppose that the standard marginal rate of substitution along an indifference map is:
and the endowment effect of x at a point i is given by εxi and that of y is given by εyi, where ε>0 is the extra price (in terms of the other good) required to give up an object above the price for which it would be acquired. Then the marginal rate of substitution of the behavioural indifference curve in quadrant 2 relative to the reference point a is:
in quadrant 3 is:
and in quadrant 4 is:
Hence, in quadrant 2 the slope of the indifference curve is steeper than the standard indifference curve, because in order to give up 1 unit of x one would need a greater amount of y as compensation on account of the pain of giving up x relative to the level to which one is accustomed. Similarly in quadrant 4, except in this case the indifference curve is flatter than the standard indifference curve because in this case it is more difficult to give up y. In quadrant 3 the slope of the behavioural indifference curve relative to the standard one is ambiguous, depending on the sizes of εxi and εyi; the curve is drawn in this quadrant in such a way so that the endowment effects cancel each other and the standard indifference curve obtains.
The implication is that there is a kink in the behavioural indifference curves as they cross the axis from one quadrant to another. This implies that the utility function is not differentiable everywhere and that preferences are not homothetic. Moreover, budget lines cannot be tangent to the indifference curve along the axis that divides the plane into four quadrants. For instance, budget lines 1 and 2 in Figure 2 show that changes in price will not bring about any change in the consumption bundle at point a, contrary to conventional analysis. This may well explain the oft-found stickiness in adjustment to changes in wages, prices, and interest rates (Anderson 1998, Carlton 1986, Ausubel 1991).
Figure 2. Behavioural indifference curves in period 1 with several budget constraints
Furthermore, let us suppose that in period 1 the actual budget line 3 is tangent to the indifference curve at b in quadrant 4 (b is not on an axis in period 1). Thus, Figure 2 shows that with budget constraint 3 the new consumption bundle becomes (Qx2, Qy2) at point b. Once choosing to consume at point b in period 1, however, the origin of the new axis of the behavioural indifference map shifts to b and, in turn, that becomes the new reference point in period 2. This implies that the two sets of indifference maps cross over time, even if the consumer’s tastes do not change over time.1
Moreover, the new indifference map of period 2 is superimposed on the previous one of period 1 (Figure 3). However, the budget constraint that was tangent to the old indifference curve at b is no longer tangent to the new indifference curve at b (Figure 4). Therefore, the tangency with the new set of indifference curves will be elsewhere, implying that consumption will change in period 2 even if prices, income, and tastes remain unchanged. Thus, the consumption bundle can change even if there is no fundamental change in either the economy or in the consumer’s preferences. In other words, the adjustment to the new budget constraint occurs in two steps – the first step uses the initial reference point in order to choose the optimal bundle, and having made that choice the reference point also shifts, implying that the whole indifference curve shifts. This, in turn, displaces the optimal consumption bundle once again to c, even if there are no other changes in the relevant parameters.
Figure 3. In period 2, behavioural indifference curves shift the origin from a to the new reference point at b
Figure 4. In period 2, consumption changes to point c even if there is no change in tastes or the budget constraint
In sum, behavioural indifference curves are relative to a reference point. The endowment effect implies that people are willing to give up an object only at a higher price than the price at which they are willing to buy it, i.e. it is psychologically more difficult to give up an object than to acquire it. This changes the shape and properties of the indifference map, which has far-reaching implications not only in classrooms, but also in applied areas such as the evaluation of welfare states and the stickiness of economic variables such as wages, prices, and interest rates (Knetsch et al. 2012). This salient issue ought no longer to be ignored, and needs a much wider research agenda than hitherto allotted to it at the margins of the discipline.
Even at this stage it is important to incorporate behavioural indifference curves into the curriculum and stop teaching outdated concepts. If you think that behavioural indifference curves would be too complicated for beginners then I would urge you not to teach the conventional ones until the students are ready for the current version, because one should not mislead students by teaching inappropriate concepts. If the straight-talking Nobel-prize-winning physicist Richard Feynman (1918–1988) were still with us, he would concur with this view. In his famous 1974 commencement address at the California Institute of Technology, he beseeched the graduating class to practice scientific integrity, utter honesty, and to lean over backwards so as not to fool themselves (and of course others) (Feynman 1985). I believe that the same is true for us – teachers of economics, it is time to start leaning over backwards and to stop teaching the standard indifference curves.
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1 That indifference curves can intersect has been experimentally verified in a different setting (Kahneman et al. 1991: 197).