# Interest Income Tax and Saving

## The Life Cycle Saving Hypothesis

Next, we investigate the effect of interest income tax on saving. First, let us consider how interest income is produced from economic activities. Since interest income is a return on saving, we need to explain saving in the context of the optimizing behavior of households.

Saving is conducted in order to attain the optimal intertemporal allocation of consumption. A main reason why households save is that labor income in the future is not enough to prepare for future living needs. Saving means that a part of current consumption is transferred to future consumption. Optimal saving is determined so as to equate the marginal benefit of saving to the marginal cost of saving.

The standard saving hypothesis is the life cycle saving theory. We explain this theory with a simple two-period model, as in Chap. 3. Imagine that income occurs only in the first period. For example, when young the agent works and earns labor income, while she or he retires when old. The household optimally allocates consumption in two periods. If it consumes all the current income, it cannot have any income in the future period to consume. If it saves part of the current income, it earns interest income used for consumption in the future.

The budget constraint in each period is given respectively as

where c_{1} is present consumption, c_{2} is future consumption, Y_{1} is present (labor) income, s is saving, and r is the rate of interest. From Eqs. (8.7) and (8.8), by eliminating s, we have the lifetime budget constraint,

The household maximizes its lifetime utility U = U(c_{1}, c_{2}) subject to the budget constraint (8.9) by choosing present and future consumption.

Figure 8.4 presents future consumption in its vertical axis and current consumption in its horizontal axis. The line AB shows the budget line (8.9) with the slope corresponding to the rate of interest. Since labor income occurs only in the present period, in the horizontal axis OA corresponds to the size of labor income. The indifference curve I refers to a combination of present and future consumption in order to maintain fixed lifetime utility and is concave toward point O.

**Fig. 8.4 Optimal saving**

The household chooses the highest utility point on the budget line. This point is at E, where the budget line and indifference curve are tangent. Alternatively, mathematically we have as the optimality condition

where U_{c1} means the marginal utility of c_{1} and U_{c2} the marginal utility of c_{2}.

If the utility function is specified to an additively separable type,

Equation (8.10) reduces to

where p denotes the time preference and V_{c1} means the marginal utility of c_{1}. V_{c2} is the marginal utility of c_{2}. The left-hand side of Eq. (8.10) or (8.10') is the slope of the budget line, while the right-hand side of Eq. (8.10) or (8.10') is the slope of the indifference curve. If r = p, the optimal condition means c_{1} = c_{2}. Point E is on the 45-degree line and consumption smoothing is desirable. The size of saving, AF, denotes the optimal saving for the household.

The slope of the indifference curve, 1 + p, means the marginal cost of saving, while the slope of the budget line, 1 + r, means the marginal benefit of saving. The marginal benefit of saving refers to the extent by which future consumption increases when current consumption is relinquished for saving. The marginal cost refers to how much a decline in current consumption for saving costs in monetary terms. The former depends on the rate of interest and the latter depends on the rate of time preference. The rate of interest is the rate of return on saving. If this rate is high, a decline of current consumption produces a large amount of future consumption.

The time preference rate refers to how much the household evaluates future consumption in terms of current consumption. A high time preference rate means that the household needs a large amount of future consumption at a given decline of current consumption. If this rate is high, it depresses saving.