two motives exist for a consumer unit to spend more or less on consumption than its income: The first is to smooth its consumption expenditures through appropriate timing of borrowing and lending; and the second is either to realize interest earnings on deposits if the relevant rate of interest is positive, or to benefit from borrowing if the interest rate is negative. According to the PIH, the distribution of consumption across consecutive periods is the result of an optimizing method by which each consumer tries to maximize his utility. At the same time, whatever ratio of income one devotes to consumption in each period, all these consumption expenditures are allocated in the course of an optimization process—that is, consumer units try to optimize not only across periods but within each period. Calculation of income and consumption Friedman's 1957 book also made an argument for an entirely new way of calculating income (income is represented by the variable ) by differentiating between transitory and permanent income (which was also taken to include ordinal elements like human capital and talents). In A Theory of Consumption Function, Friedman develops: as a formula. In an earlier study, Friedman, Kuznets (1945), he proposes the idea of transitory and permanent income. Friedman also developed a consumption formula, , with meaning the permanent component of consumption, with being the transitory component. Friedman also drew a distinction between and . Transitory consumption can be interpreted as surprising or unexpected bills, such as a high water bill, or unexpected doctor's visit, which, in Friedman's mind, cannot be spurred by , because unexpected or 'surprise' consumption is not often financed through windfall gains. Simple model Consider a (potentially infinitely lived) consumer who maximizes his expected lifetime utility from the consumption of a stream of goods between periods and , as determined by one period utility function . In each period , he receives an income , which he can either spend on a consumption good or save in the form of an asset that pays a constant real interest rate in the next period. The utility of consumption in future periods is discounted at the rate . Finally, let denote the expected value conditional on the information available in period . Formally, the consumer's problem is then subject to Assuming the utility function is quadratic, and that , the optimal consumption choice of the consumer is governed by the Euler