Taste uncertainty and status quo effects in consumer choice

Abstract

We use reference-dependent expected utility theory to develop a model of status quo effects in consumer choice. We hypothesise that, when making their decisions, individuals are uncertain about the utility that will be yielded by their consumption experiences in different ‘taste states’ of the world. If individuals have asymmetric attitudes to gains and losses of utility, the model entails acyclic reference-dependent preferences over consumption bundles. The model explains why status quo effects may vary substantially from one decision context to another and why some such effects may decay as individuals gain market experience.

Appendix: Proofs

1.1 Proof of Result 1

Suppose that, for all states s _h , u _h (.) is continuous, strictly increasing and strictly concave. It is straightforward to show that these assumptions imply that v(x, z) is continuous in x and z, strictly increasing in x, strictly decreasing in z, and strictly concave in x, thus satisfying Well-Behavedness. We now prove that Acyclicity is satisfied.

We define a function w(.) in the following way: for all bundles $x,w(x) = \sum\nolimits_h {\pi \left( {s_h} \right){u_h}(x)}$. Thus w(x) can be interpreted as the expected value of the subjective experience of consuming x. Notice that this value is independent of the reference point. Thus, w(.) determines a reference-independent ordering of bundles.

We fix a parameter r such that r > 0 and, for all real numbers q, rq ≥ φ(q). Since φ(0) = 0 and φ(.) is weakly concave, such a parameter must exist. (If φ(.) is differentiable, r must equal φ′(0).) Hence, for all bundles x and z, $v\left( {x,z} \right) = \sum\nolimits_h {\pi \left( {s_h} \right)\varphi \left[ {{u_h}(x) - {u_h}(z)} \right]} \leqslant r\sum\nolimits_h {\pi \left( {s_h} \right)\left[ {{u_h}(x) - {u_h}(z)} \right] = r\left[ {w(x) - w(z)} \right]}$. Thus, v(x, z) ≥ 0 implies w(x) ≥ w(z), and v(x, z) > 0 implies w(x) > w(z). However, $v\left( {x,z} \right) \succ 0 \Leftrightarrow x \geqslant z\left| z \right.$, and $v\left( {x,z} \right) > 0 \Leftrightarrow x > z\left| z \right.$. Thus, x ≽ z | z implies w(x) ≥ w(z), and x ≻ z | z implies w(x) > w(z). So, if the move from z to x is weakly choosable, x is ranked at least as highly as z in the reference-independent ordering defined by w(.), while if the move is strictly choosable, x is ranked above z. It follows immediately from this result that Acyclicity is satisfied.

1.2 Proof of Result 3

Let $U = \left( {{U_{1}},\,...,{U_m}} \right)$ and V = (V ₁ , ..., V _m ) be vectors of state-conditional marginal utilities for goods 1 and 2 respectively, such that ${U_1}/{V_1} \geqslant {U_2}/{V_2} \geqslant ... \geqslant {U_m}/{V_m}$. Units of the two goods are normalised so that $\sum\nolimits_h {\pi \left( {s_h} \right){U_h}} = \sum\nolimits_h {\pi \left( {s_h} \right){V_h} = 1}$. Let $U\prime = \left( {U{\prime_1},\,...,U{\prime_m}} \right)$ be an alternative vector for good 1, such that the switch from U to U′ is a mean-preserving disalignment of u(.) with respect to v(.). Let z be any bundle, and let x be the bundle which differs from z by containing one less marginal unit of good 1 and q additional units of good 2. Set the value of q so that, when marginal utilities are given by U and V, x ~ z | z (i.e. q = r ^WTA₂₁ ). For each state h = 1, ..., m, we define ${A_h} = { \max }\left[ {\pi \left( {s_h} \right)\left( {{U_h} - {V_h}q} \right),\,0} \right]$, ${B_h} = { \max }\left[ {\pi \left( {s_h} \right)\left( {{V_h}q - {U_h}} \right),\,0} \right]$, $A{\prime_h} = { \max }\left[ {\pi \left( {s_h} \right)(U{\prime_h} - {V_h}q),\,0} \right]$, and $B{\prime_h} = { \max }\left[ {\pi \left( {s_h} \right)\left( {{V_h}q - U\prime h} \right),\,0} \right]$. Notice that because of our normalisations, and because of the definition of ‘mean-preserving disalignment’, $\sum\nolimits_h {\left( {{B_h} - {A_{h}}} \right)} = \sum\nolimits_h {\left( {B{\prime_h} - A{\prime_{h}}} \right) = q - 1}$. (In terms of the notation in Fig. 1, the areas A and B are equal to $\sum\nolimits_h {A_h} \,{\text{and}}\,\sum\nolimits_h {B_h}$ respectively.)

Because states are indexed in descending order of U _h /V _h , we can define an integer K such that, for each $h = 1,\,...K,{U_h}/{V_h} \geqslant q$, while for each $h = K + 1,\,...,m,{U_h}/{V_h} < q$. Since x ~ z | z, there must be at least one state in which U _h /V _h ≥ q; thus 1 < K ≤ m. It follows from the definitions of ${A_{h}}$ and B _h that for each h = 1, ..., K, A _h ≥ 0 and B _h = 0, while for each $h = K + 1,\,...,m,{A_h} = 0\,{\text{and}}\,{B_h} > 0$. (In terms of Fig. 1, in which K = 3, the whole of area A lies to the left of the cumulative probability of states 1 to 3, while the whole of area B lies to the right.) It follows from the definition of ‘mean-preserving disalignment’ that $\sum\nolimits_{h = 1} {^K\pi \left( {s_h} \right)(U{\prime_{h}} - {U_h}) > 0}$. But, for all h, $\pi \left( {s_h} \right)\left( {U{\prime_{h}} - {V_h}q} \right) \equiv \left( {A{\prime_h} - B{\prime_h}} \right) - \left( {{A_h} - {B_h}} \right)$. For all h = 1, ..., K, B _h = 0 (see above) and B′ _h ≥ 0 (by definition). Thus, we have $\sum\nolimits_{h = 1} {^K} \left( {A{\prime_h} - {A_h}} \right) > 0$. For all $h = K + 1,\,...,m,{A_h} = 0$ (see above) and A′ _h ≥ 0 (by definition). Thus, $\sum\nolimits_{h} {\left( {A{\prime_h} - {A_h}} \right) > 0}$. Recall that $\sum\nolimits_{h} {\left( {A{\prime_h} - {A_h}} \right)} = \sum\nolimits_{h} {\left( {B{\prime_h} - {B_h}} \right)}$. So the expected value of utility loss in states in which z gives weakly more utility than x (and, equivalently, the expected value of utility gain in states in which z gives strictly less than x) is greater in absolute value when marginal utilities are given by U′ than when they are given by U. Thus if β > 1, the change from U to U′ implies a change in preference from x ~ z | z to z ≻ x | z. To restore indifference, there must be an increase in q. Thus, the change induces an increase in r ^WTA₂₁ .

A symmetrical argument shows that (with β > 1) the same change in marginal utilities induces an increase in r ^WTA₁₂ . Thus, it induces an increase in Q ₂₁.