The Economic Theory of Annuities by Eytan Sheshinski_1

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The Economic Theory of Annuities by Eytan Sheshinski_1 August 3, 2007 Time: 04:49pm chapter15.tex C H A P T E R 15 Bundling of Annuities and Other Insurance Products 15.1 Introduction It is well-known that monopolists who sell a number of products may find it profitable to “bundle” the sale of some of these products, that is, to sell “packages” of products with fixed quantity weights (see, for example, Pindyck and Rubinfeld (2007) pp. 404–414). In contrast, in perfectly competitive equilibria (with no increasing returns to scale or scope), such bundling is not sustainable. The reason is that if some products are bundled by one or more firms at prices that deviate from marginal costs, other firms will find it profitable to offer the bundled products separately, at prices equal to marginal costs, and consumers will choose to purchase the unbundled products in proportions that suit their preferences. This conclusion has to be modified under asymmetric information. We shall demonstrate below that competitive pooling equilibria may include bundled products. This is particularly relevant for the annuities market. The reason for this outcome is that bundling may reduce the extent of adverse selection and, consequently, tends to reduce prices. In the terminology of the previous chapter, consider two products, X1 and X2 , whose unit costs when sold to a type α individual are c1 (α ) and c2 (α ), respectively. Suppose that c1 (α ) increases while c2 (α ) decreases in α . Examples of particular interest are annuities, life insurance, and health insurance. The cost of an annuity rises with longevity. The cost of life insurance, on the other hand, typically depends negatively (under positive discounting) on longevity. Similarly, the costs of medical care are negatively correlated with health and longevity. Therefore, selling a package composed of annuities with life insurance or with health insu- rance policies tends to mitigate the effects of adverse selection because, when bundled, the negative correlation between the costs of these products reduces the overall variation of the costs of the bundle with individual attributes (health and longevity) compared to the variation of each product separately. This in turn is reflected in lower equilibrium prices. Based on the histories of a sample of people who died in 1986, Murtaugh, Spillman, and Warshawsky (2001), simulated the costs of August 3, 2007 Time: 04:49pm chapter15.tex 132 • Chapter 15 bundles of annuities and long-term care insurance (at ages 65 and 75) and found that the cost of the hypothetical bundle was lower by 3 to 5 percent compared to the cost of these products when purchased sepa- rately. They also found that bundling increases significantly the number of people who purchase the insurance, thereby reducing adverse selection. Bodie (2003) also suggested that bundling of annuities and long-term care would reduce costs for the elderly. Currently, annuities and life insurance policies are jointly sold by many insurance companies though health insurance, at least in the United States, is sold by specialized firms (HMOs ). Consistent with the above studies, there is a discernible tendency in the insurance industry to offer plans that bundle these insurance products (e.g., by offering discounts to those who purchase jointly a number of insurance policies). We have been told that in the United Kingdom there are insurance companies who bundle annuities and long-term medical care but could not find written references to this practice. 15.2 Example Let the utility of an type α individual be u(x1 , x2 , y; α ) = α ln x1 + (1 − α ) ln x2 + y, (15.1) where x1 , x2 , and y are the quantities consumed of goods X1 and X2 and the numeraire, Y. It is assumed that α has a uniform distribution in the population over [0, 1]. Assume further that the unit costs of X1 and X2 when purchased by a type α individual are c1 (α ) = α and c2 (α ) = 1 − α , respectively. The unit costs of Y are unity (= 1). Suppose that X1 and X2 are offered separately at prices p1 and p2 , respectively. The individual’s budget constraint is p1 x1 + p2 x2 + y = R, (15.2) where R(>1) is given income. Maximization of (15.1) subject to (15.2) yields demands x1 ( p1 ; α ) = ˆ α/ p1 , x2 ( p2 ; α ) = (1 − α )/ p2 and y = R − 1. The indirect utility, u, is ˆ ˆ ˆ therefore α 1−α α 1−α u( p1 , p2 ; α ) = ln + R − 1. ˆ ...