Preferences (after Rubinstein, question A1)
Let \(X = \mathbb{R}_+ \times \{0,1,2,\dots\}\), where \((x,t)\) is interpreted as receiving \(x\) at time \(t\). A rational preference relation \(\succsim\) on \(X\) has the following properties:
- There is indifference between receiving \(0\) at time \(0\) and receiving \(0\) at any other time.
- It is better to receive any positive amount of money as soon as possible.
- Money is desirable.
- The preference between \((x,t)\) and \((y,t+1)\) is independent of \(t\).
- Continuity: for any \((x,t)\) and \((y,s)\) with \((x,t) \succ (y,s)\), there is \(\epsilon > 0\) such that for any \(x'\) with \(|x-x'| < \epsilon\) and any \(y'\) with \(|y-y'| < \epsilon\), \((x',t) \succ (y',s)\).
(a) Show by construction that the preference relation \(\succsim\) has a utility representation.
Hint: First show that for any pair \((x,t)\), there is a unique number \(v(x,t) \in \mathbb{R}_+\) such that \((x,t) \sim (v(x,t),0)\).
Solution to (a)
Conventions
Rational means complete and transitive. Hence \(\succ\) is asymmetric and \(\sim\) is an equivalence relation, and we may freely substitute indifferent bundles for one another (for example, \(a \succ b\) and \(b \sim c\) imply \(a \succ c\)). I read the properties as follows:
- (ii) if \(x > 0\) and \(s < t\), then \((x,s) \succ (x,t)\);
- (iii) if \(x > y\), then \((x,t) \succ (y,t)\) for every date \(t\).
Plan. Define \(v(x,t)\) as the amount received today (at \(t=0\)) that is equally good as \((x,t)\). We show that it exists (Step 1), that it is unique (Step 2), and that it represents \(\succsim\) (Step 3).
Step 1 — Existence
Fix \((x,t)\). We must find \(z \in \mathbb{R}_+\) with \((z,0) \sim (x,t)\).
Trivial cases.
- If \(t = 0\), take \(z = x\).
- If \(x = 0\), property (i) gives \((0,0) \sim (0,t)\), so take \(z = 0\).
Main case: \(x > 0\) and \(t > 0\).
(a) Bracketing. Three facts:
- By (ii), \((x,0) \succ (x,t)\).
- By (iii), \((x,t) \succ (0,t)\), and by (i), \((0,t) \sim (0,0)\). Hence \((x,t) \succ (0,0)\).
So the target bundle sits strictly between two bundles that are received today: \[(x,0) \;\succ\; (x,t) \;\succ\; (0,0).\]
(b) Two sets. Define, for amounts received today,
\[A = \{\, z \in \mathbb{R}_+ : (z,0) \succ (x,t) \,\}, \qquad B = \{\, z \in \mathbb{R}_+ : (x,t) \succ (z,0) \,\}.\]By (a), \(x \in A\) and \(0 \in B\), so both sets are non-empty. They are disjoint because \(\succ\) is asymmetric.
(c) Both sets are open (in \(\mathbb{R}_+\)), by continuity (v).
- Let \(z \in A\), i.e. \((z,0) \succ (x,t)\). Apply (v) to this pair and keep the second bundle fixed (\(y' = x\)). There is \(\epsilon > 0\) with \((z',0) \succ (x,t)\) whenever \(|z - z'| < \epsilon\). So a neighbourhood of \(z\) lies in \(A\).
- Let \(z \in B\), i.e. \((x,t) \succ (z,0)\). Apply (v) keeping the first bundle fixed (\(x' = x\)). There is \(\epsilon > 0\) with \((x,t) \succ (z',0)\) whenever \(|z - z'| < \epsilon\). So a neighbourhood of \(z\) lies in \(B\).
(d) Connectedness. \(\mathbb{R}_+\) is an interval, hence connected, so it cannot be written as the union of two non-empty, disjoint, open sets. Therefore \(A \cup B \neq \mathbb{R}_+\): there is some \(z^* \notin A \cup B\).
(e) Indifference. Since \(z^* \notin A\), we do not have \((z^*,0) \succ (x,t)\); by completeness, \((x,t) \succsim (z^*,0)\). Since \(z^* \notin B\), we do not have \((x,t) \succ (z^*,0)\); by completeness, \((z^*,0) \succsim (x,t)\). Both weak preferences together mean \[(z^*,0) \sim (x,t).\]
Step 2 — Uniqueness
Suppose \(z_1 > z_2\) both satisfy \((z_i,0) \sim (x,t)\). By transitivity of \(\sim\), \((z_1,0) \sim (z_2,0)\). But (iii) gives \((z_1,0) \succ (z_2,0)\), a contradiction. Hence the amount is unique, and we may define
\[v(x,t) \;=\; \text{the unique } z \in \mathbb{R}_+ \text{ with } (z,0) \sim (x,t).\]By construction, \(v(x,0) = x\) and \(v(0,t) = 0\).
Step 3 — \(v\) represents \(\succsim\)
Lemma. For \(z, w \in \mathbb{R}_+\): \((z,0) \succsim (w,0) \iff z \ge w\).
Proof. If \(z = w\), the bundles are identical, so \((z,0) \succsim (w,0)\) by reflexivity. If \(z > w\), then \((z,0) \succ (w,0)\) by (iii). Conversely, if \(z < w\), then \((w,0) \succ (z,0)\) by (iii), so \((z,0) \succsim (w,0)\) fails by asymmetry. \(\square\)
Now take any two bundles. Because \((x,t) \sim (v(x,t),0)\) and \((y,s) \sim (v(y,s),0)\), we can substitute these indifferent bundles on each side:
\[(x,t) \succsim (y,s) \;\iff\; (v(x,t),0) \succsim (v(y,s),0) \;\iff\; v(x,t) \ge v(y,s),\]where the first equivalence uses transitivity of \(\succsim\) (in both directions) and the second is the Lemma. This is exactly the statement that \(U(x,t) = v(x,t)\) is a utility representation of \(\succsim\). \(\blacksquare\)
Remarks
- Sanity check. \(v(x,t) \le x\): if \(v(x,t) > x\), then \((v(x,t),0) \succ (x,0) \succ (x,t)\) by (iii) and (ii), contradicting \((v(x,t),0) \sim (x,t)\). Waiting never makes an amount more valuable, which is the discounting intuition.
- Unused property. Property (iv) was not needed for existence of a representation. It restricts how \(v\) behaves across dates (stationarity), so it matters for further structure on \(v\), not for the construction above.
- Where each property was used. (i): the cases \(x=0\) and the lower bracket; (ii): the upper bracket and the sanity check; (iii): bracketing, uniqueness and the Lemma; (v): openness of \(A\) and \(B\), which, together with connectedness of \(\mathbb{R}_+\), delivers existence.