Would you prefer your retirement income to depend on your life expectancy?
A pension scheme is constructed where the consumption (pension) depends on additional information on the survival probability. There are two periods. The agent learns about her state at the beginning of period one. For each of the states there is a survival probability \(\pi_i\). At the end of period one, the agent learns whether she survives or not. Depending on the state, there is a (maximal) consumption rate \(y_1^i\). In period two, the maximal consumption rate is \(y_2^{i j}\) with \(y_2^{i 0} = 0\) (the agent is dead) and \(y_2^{i 1} > 0\). There is a budget constraint that limits the expected outgo for pensions. The value of the chosen consumption is measured by a utility function \(U(\{(c_1^i,c_2^{i j})\})\). It is assumed that there are \(N\) states \(i \in \{1,2,\ldots, N\}\) chosen at the beginning of period one with equal probabilities. The problem becomes \[ \max_{(y_1^i, y_2^i)} \max_{(c_1^i,c_2^{i j})} U\left(\left\{(c_1^i,c_2^{i j})\right\}_{i ,j} \right)\;, \] subject to \[ c_1^i \le y_1^i\;,\qquad c_2^{i1} \le y_2^i + (1+r)(y_1^i - c_1^i)\; \] \[ \frac1N \sum_{i=1}^N \Bigl(y_1^i + \frac{\pi_i y_2^{i 1}}{1+r}\Bigr) \le B\;. \] The function \(U\) should be increasing in each of the variables. Under the natural assumptions on \(U\), one gets \(c_1^i = y_1^i\), such that the optimisation problem becomes \[ \max_{(c_1^i,c_2^i)} U\left(\left\{(c_1^i,c_2^i)\right\}_{i ,j} \right)\;, \] subject to \[ \frac1N \sum_{i=1}^N \Bigl(c_1^i + \frac{\pi_i c_2^{i 1}}{1+r}\Bigr) = B\;. \] Note that it is possible that the agent may prefer death after period one to a low consumption \(c_1^{i 1}\). The preferences are now defined as \(V_2^i = u_2\) (in the final period) and \(V_1 = u_1 + \beta \mathcal{I}(\{V_2^i\})\) in the first period. The certainty equivalent \(\mathcal{I}\) is assumed to be continuous, increasing, symmetric, translation equivalent and super-modular. \(\beta\) is a time preference parameter and \(u_k\) are utility functions. Examples for the function \(\mathcal{I}\) are \(\mathcal{I}(\{V_2^i\}) = N^{-1} \sum V_2^i\) or \(\mathcal{I}(\{V_2^i\}) = -k^{-1} \ln (N^{-1} \sum e^{-k V_2^i})\) for some \(k \ne 0\). Some basic properties are proved and an explicit example is discussed.
scientific article; zbMATH DE number 7314454