Academic journal article Journal of Risk and Insurance

Wealth, Income, and Optimal Insurance

Academic journal article Journal of Risk and Insurance

Wealth, Income, and Optimal Insurance

Article excerpt

ABSTRACT

This article considers the decision to purchase insurance against possible losses of a property or wealth. The decision involves a standard economic trade-off between the benefit of protection against loss and the cost of insurance premium. The premium is paid out of the income and decreases the consumption of other goods and services, rather than out of wealth and decreases the property or wealth. The demand for insurance depends mainly on the income and preferences. As a result, unlike in the standard model, a fair premium is neither necessary nor sufficient for the optimality of full coverage insurance. Rather, the individuals with higher incomes purchase full coverage insurance even at unfair prices of insurance while the individuals with lower income purchase partial coverage insurance at a fair price.

INTRODUCTION

A large body of literature has studied optimal insurance (e.g., Mossin, 1968; Smith, 1968; Schlesinger, 2000).(1) The standard model considers a risk-averse individual who wishes to insure her property against a possible loss. The analyses concern the level of insurance coverage that maximizes her expected utility of wealth, a random variable. Her wealth equals the initial value of the property minus the insurance premium when no loss occurs, and includes additionally the loss and the indemnity when a loss occurs. The standard model predicts that full coverage insurance is optimal with a fair price, and partial coverage insurance is optimal otherwise. This conclusion holds regardless of the preferences or income or wealth. An implication of the standard prediction is that everyone switches from full coverage insurance to partial coverage insurance when the loading factor increases slightly from zero to a small positive number, a dramatic but unrealistic event.

The key assumption in the literature is that utility depends only on wealth. This assumption simplifies the analyses and enables one to obtain clear-cut results. However, this assumption appears unrealistic in one important respect. The insurance premium is in practice paid out of one's income, rather than from wealth or property. Stated differently, the insurance premium reduces the consumption of other goods and services, not the property. To reflect this realism, the article assumes that utility depends on income (or composite good consumption) and wealth, more precisely the consumption of service flows generated by wealth. For instance, one cares about composite good consumption and housing consumption, as in the housing economics literature (Papageorgiou and Pines, 1999, Ch. 6; Zabel, 2004).

In this model, the purchase of an additional insurance coverage involves a standard economic trade-off. The coverage provides an additional protection against a possible loss when it occurs, but it increases the premium and decreases the income available for the consumption of other goods and services. The decision to purchase an additional coverage essentially depends on the valuation of the consumption of service flows generated by the property relative to the valuation of composite good consumption, or depends on the valuation of wealth relative to that of income. As a result, higher income individuals tend to purchase more insurance if both goods are normal. The reason is that as income increases, the composite good becomes less important, given the value of the property, and it increases the willingness to pay a larger premium and hence to buy more insurance. Thus, an individual may purchase full coverage insurance at unfair prices while she may purchase partial coverage insurance at a fair price, depending on her income and preferences. This conclusion appears to conform with real world observations (Beenstock, Dickinson, and Khajuria, 1988; Grace, Klein, and Kleindorfer, 2000).(2)

Making use of the model with income and wealth, the article reexamines a number of important propositions in the insurance economics literature: (i) full coverage insurance is optimal at a fair price while partial coverage is optimal at unfair prices (Mossin, 1968); (ii) no deductible is optimal at a fair price while a positive deductible is optimal at unfair prices (Mossin, 1968; Smith, 1968); (iii) with noninsurable loss, more than full coverage is optimal when the insurable loss and the noninsurable loss are positively correlated while less than full coverage is optimal when they are negatively correlated (Doherty and Schlesinger, 1983); and (iv) with risk-averse insurers, no deductible is optimal if the provision of insurance involves no deadweight loss and a deductible is optimal otherwise (Raviv, 1979). …

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