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We consider a continuous-time model of optimal consumption and pension insurance for a consumer with an uncertain lifetime. In the model, the consumer earns a stochastic wage income during her working life and optimally allocates her income between personal consumption, pension insurance, and securities with a deterministic dynamic return. Due to the weak development of the stock market in developing countries, employees' income comes mainly from wages and interest on savings from banks, that are discussed in this paper. The consumer's utility and bequest functions are constant absolute risk aversion (CARA). By characterizing the optimality condition of the consumer's problem using the Hamilton-Jacobi-Bellman equation, we find the optimal consumption and pension insurance as a function of wealth in closed form. We consider an application of the model while estimating its key elements using real-life data on age-specific population size, labor income, and interest rates. We show that as the absolute risk aversion for consumption increases, consumption and wealth move in the opposite direction. We also present a novel finding that wealth and consumption can be negatively related across consumers with different levels of consumption risk aversion.
Terbish et al. (Mon,) studied this question.