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In my professors lecture notes (it's more like a book), he states four properties of preferences relations:

  • monotonicity
  • transitivity
  • continuity
  • completeness

He then goes on to discuss utility levels and indifference curves. He describes indifference curves as level sets and that consumption bundles that the consumer is indifferent between lie on the same indifference curve.

My Question:

For a given set of consumption bundles, how do we know there are any bundles that the consumer is indifferent between? I don't see how we can define indifference curves without knowing that.

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We can obtain the result that Indiference Sets are not singletons, i.e. that consumption bundles that are equivalent under the preference relation must exist, by using a Consumption Set $X$ defined over $\mathbb R_+^n$, i.e. over real vectors, together with the property of Continuity of preferences (alternative-weaker prerequisites may exist though).

An equivalent way to state the property of Continuity of preferences (see for example Mas-Collel et al, ch 3.C), is that $\forall x \in X$ the upper contour set $UCS$ and the lower contour set $LCS$ are both closed, i.e they include their boundaries. The contour sets are sets that include consumptions bundles "at least as preferred as $x$" (upper) and "at most as preferred as $x$" (lower). In order for them to "include" their boundary", a boundary must exist in the first place. But their boundary is the Indifference Set, the set containing all bundles as preferred as $x$.

So your question essentially boils down to Is it possible that the boundary of the upper or lower contour sets of $x$, sets that are defined over real vectors, is a single point?

I believe that envisioning this in two-dimensions will show that it is impossible, but let's prove it.

Ad absurdum, assume that this holds for some $x \in X$. Then for any other consumption bundle in $X$, say $x'$, we will have either $x'>_{pr} x$ or $x'<_{pr} x$. In other words, any other point in $X$ will belong either in $UCS_x$ or in $LCS_x$, but not in both.

So consider two such points , $x''>_{pr}x>_{pr}x'$. Assume that $x'=\{\mathbf y_{(n-1)}, y_n\}$, $x''=\{\mathbf y_{(n-1)}, y_n+\epsilon(k)\}$, i.e. that they differ in only one of the goods in Goods Space, $\epsilon(k)>0$.

Transform now these two points into two sequences of points, $\{x''_k\}$ and $\{x'_k\}$, indexed by $k= 1,...$ and by constructing the sequence $\{\epsilon_k\}$ such that $\{\epsilon_k\} >0\; \forall k$ but also $\lim_{k\rightarrow \infty} \epsilon_k = 0$. The consumption space as defined permits these constructions. We note of course that $\{x'_k\}$ is a constant sequence since its value does not change as $k$ changes, but that's perfectly legal.

Consider now the sequence of pairs, for each $k$, $\{(x''_k, x'_k)\}_{k=1}^{\infty}$, and it holds that

$$ x''_k >_{pr} x'_k\;\;\;\forall k \tag{1}$$ $$\lim_{k\rightarrow \infty} x''_k = x'$$ $$\lim_{k\rightarrow \infty} x'_k = x'$$

Under Continuity of preferences we should obtain $$ (1) \implies \lim_{k\rightarrow \infty} x'' >_{pr} \lim_{k\rightarrow \infty} x'$$ but this is obviously not possible since these limits are identical. So by assuming that there exists an $x$ in the Consumption Set to which no other bundle is equivalent in terms of the preference relation, we have violated the Continuity property.

Alternatively, since the bundles are defined over real vectors, and both $UCS$ and $LCS$ are closed due to the Continuity property, then every Cauchy sequence in each of them that converges has its limit in them. So the sequence $\{x''_k\}$ has its limit in $UCS_x$, while the sequence $\{x'_k\}$ must have its limit in $LCS_x$. But this limit is the same as shown above, and equal to $x'$. But the only common point of these two sets is $x$. So it must be the case that $x'=x$. But this cannot hold since we have started by assuming $x>_{pr} x'$.

An example where indifference sets are singletons is the case of Lexicographic Preferences. And Lexicographic Preferences are not continuous.

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  • $\begingroup$ So are you saying the Continuity axiom implies, for a given single point in the Consumption set, there must exist other points (ie bundles) for which the consumer is indifferent when compared with our single point? In other words, assuming no other equivalent bundle exists means leads to a contradiction. And hence there must exist other bundles equivalent to the given bundle we have chosen. $\endgroup$ – Stan Shunpike Apr 7 '15 at 4:19
  • $\begingroup$ Yes that's the essence of the thing. Continuity implies that Indifference Sets are not singletons. $\endgroup$ – Alecos Papadopoulos Apr 7 '15 at 4:23
  • $\begingroup$ But if the in the Lexicographic case case continuity doesn't apply and the indifference sets are singletons, does that mean the graph of an indifference curve is a single point and a set of indifference curves a lattice? $\endgroup$ – Stan Shunpike Apr 8 '15 at 23:25
  • $\begingroup$ Like I'm getting confused because I'm not used to thinking about manifolds where the space we are working with isn't continuous. So I am having trouble imagining how a curve would be represented in say 2 dimensions for lexicographic preferences. Furthermore, I asked my professor about how curves work and he didn't really give a satisfactory explanation as to whether lexicographic preference can still be defined over $\Bbb{R}^n$. $\endgroup$ – Stan Shunpike Apr 8 '15 at 23:53
  • $\begingroup$ @StanShunpike They are indeed singletons Stan, when preferences are lexicographic. Try a simple diagram with two goods and you will see that no point in the plane can represent the exact same utility as any other. In other words, there are no curves here. $\endgroup$ – Alecos Papadopoulos Apr 9 '15 at 0:15

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