2.3  The basic model: jointly determining replacement and saving rates[6]

In this section we develop a basic model built on the life cycle approach to consumption and saving.[7] This underpins our use of consumption smoothing as a basis for assessing accuracy. In the absence of uncertainty, the life cycle saving and consumption patterns can be simply illustrated as in Figure 1. Income rises through working life reaching a peak (typically at around 55 years) and declining somewhat in later life. In this simple model the household chooses a level of consumption that can be financed from income over the working life, and then from savings during retirement. This implies (ignoring interest for the moment) that savings (the area ABC) is equal to consumption needs in retirement (depicted as the rectangle CDEF).

As shown, consumption typically exceeds income during the early years (eg during tertiary education) implying the need to finance consumption by borrowing against future income. This simple life cycle pattern of income consumption and savings is modified when we allow for uncertainty. As shown by Moore and Mitchell (1997) when life expectancy is uncertain consumption will tend to rise until retirement and fall subsequently, rather than remaining uniform throughout (see Figure 1, part (b)). However, the basic pattern of earnings and savings reaching a peak prior to retirement and wealth decumulation throughout retirement to finance consumption is left unaltered.[8]

In the case of complete certainty a person may or may not plan to leave a bequest. However, in the face of uncertainty, some precautionary savings may be accumulated, which if not needed (because of lower than expected costs or premature death) may, by default, lead to bequests. Conversely, if accumulated savings prove inadequate due to unforeseen events, some other source of income in retirement would be required (typically either from family, the state or charitable agencies).

Figure 1 – A simple life-cycle model of income, savings and consumption

(a) No Uncertainty

(b) With Uncertainty

Source: Adapted from Moore and Mitchell (1997)

In the model we apply here we assume there are no sources of uncertainty.[9] Specifically this means that an individual of a given age plans to retire at a certain age (and does so); does not engage in the work force after retirement; knows exactly what their income until retirement will be; can accurately project the rate of return on investments; has a known life expectancy at the age of retirement (and lives for exactly that number of years); knows with certainty the amount of NZ Superannuation (NZS) that they will receive; plans and executes whatever bequests they wish to make; has no unexpected changes in health status that would affect income or expenditures and assumes tax rates and other policies remain unchanged. We further assume that the retirement phase for couples begins when the older partner reaches the NZS qualifying age (the younger partner is assumed to continue earning an income, which may affect the value of NZS received by the qualifying spouse).

Abstracting from uncertainty has the advantage of significantly simplifying the analysis. Clearly the results cannot be interpreted as applying to a particular individual whose incomes, expenditures, returns on assets and life expectancy are all subject to shocks. However in the case that these shocks are both unanticipated and distributed equally among both positive and negative changes, then the outcomes illustrated here can be interpreted as expected values for any given population group. For example, in our empirical analysis (Section 2.5) we use life expectancies at retirement age by ethnic group and gender. Other things equal, our results will show the income, saving, wealth and consumption levels that could be expected for, say, Māori and Pacific Island women aged 45-55 as a group, rather than for a specific individual in that group.

A graphical illustration of the model we apply is given in Figure 2.[10] At the current time a household has a net worth (depicted as W_a) as measured in the HSS. This is projected to grow to an amount denoted W_p by the time they reach a predetermined retirement age (here we assume 65). In order to have a given level of income in retirement they would need to have accumulated retirement wealth depicted in Figure 2 as the stock, W_r. Part of their retirement income is provided by NZS. The stock of wealth at retirement equivalent to the flow of income from NZS is incorporated in W_r and W_p.

The difference between the required wealth (W_r) and the projected wealth W_p is labelled as the shortfall and is the amount which would need to be accumulated between now and retirement in order to add to the projected stock and hence support an income in retirement of level (denoted Y_r). This additional amount, in the absence of inheritances or unanticipated windfall gains or losses in asset values, would need to be accumulated through savings. These flows are depicted in Figure 2(b).

The approach assumes that some fixed share of pre-retirement income will be saved (s=S/Y_p) and the replacement rate (R) is given by the ratio of gross income in retirement to gross income pre-retirement (ie, R= Y_r/Y_p). Under the New Zealand taxation system of TTE, post retirement taxes (denoted as T_r) are assumed to be zero, so real after tax consumption is equal to total pre-retirement income.[11]

Clearly some values of retirement income could imply a substantial shortfall in retirement wealth, which might in turn require unrealistic or unfeasible levels of saving pre-retirement. It is for this reason that the saving and replacement rates are jointly determined.

A number of additional factors arise which are not depicted in Figure 2. Instead of a constant pre-retirement income we assume that income grows from its actual level (as observed in the survey) by a fixed annual growth rate of 1% chosen to approximate the average annual rate of labour productivity and real wage growth in the economy. The gross income at retirement (Y_p) is then based on the observed actual earnings plus a compound growth of 1% annually.[12] Pre-retirement tax rates are based on pre-retirement real income (Y_p). NZS payments are assumed to grow at 1% annually in real terms, matching the growth in average real wages. Bequests involve only the current equity in the principal residence and uncertainty is removed by assuming individuals predict their life expectancies.

Figure 2 – A stylised view of stocks and flows of income, savings and retirement wealth in a model of the joint determination of saving and replacement rates