# The Permanent Income Hypothesis

If the rate of interest r is equal to the rate of time preference p, as shown in Fig. 3.3, the optimal point E, which is a tangent to the indifference curve on AB, is also on the 45-degree line. The slope of an indifference curve is given as This value reduces to 1 + p on the 45-degree line where C1 = C2. If r = p, this value is equal to 1 + r. Hence, the slope of an indifference curve is equal to the slope of the budget line at point E. Namely, the optimality condition is satisfied at point E. Thus, we have as the consumption function,  Fig. 3.3 Consumption and saving

In other words, households consume permanent disposable income in each period. Further, the optimal level of consumption is the same over time. This is consumption-smoothing behavior.

Note that the rate of time preference corresponds to the slope of an indifference curve, while the rate of interest corresponds to the slope of the budget line. The time preference rate p refers to the way in which the consumer evaluates present utility compared with future utility. The higher the time preference rate, the longer the period of the future that is evaluated.

Generally speaking, the time preference and the interest rate are not always equalized. If the interest rate becomes higher, future consumption is evaluated more than present consumption. However, if the time preference rate is higher, present consumption is evaluated more than future consumption. If both rates are almost the same, consumption does not vary much over time. In reality, we observe that consumption fluctuations are not significant compared with income fluctuations in the short run. The permanent income hypothesis can explain effectively such stable consumption movements over time by using a simple model.

In the long run, consumption may increase as economic growth occurs. As explained in Chap. 5, if the rate of interest is greater than the rate of time preference, consumption rises.

In the permanent income hypothesis, the marginal propensity to consume from increases in permanent income is close to 1. This is not inconsistent with the Keynesian model, which states that the marginal propensity to consume from current income Y1 is less than 1. In the Keynesian model, an increase in Y1 raises both C1 and S; thus, an increase in C1 is less than an increase in Y1. This reaction is qualitatively true, as in the permanent income hypothesis. Note that an increase in Yp is normally less than an increase in Y1. Hence, a change in C1 is almost the same as a change in Yp. The significant difference to the Keynesian consumption function is that consumption depends only upon permanent disposable income.

From this point on, we use Eq. (3.9) as the consumption function of the neoclassical model.