B  Derivation of the model of joint determination of saving and replacement rates

The framework outlined in this appendix is drawn from Moore and Mitchell (1997). They argue that it is necessary to develop a model which allows the replacement rate and the pre-retirement saving rate to be jointly determined. The reasons for this are twofold. Firstly, in view of a household's actual and projected income and assets, the saving rate needed to achieve some pre-specified replacement rate may be infeasible. Secondly, the replacement rate depends in part on the rate of taxation in retirement, which in turn depends on the level of retirement income, itself a determinant of the replacement rate. Only when the tax rates in retirement were pre-determined would this second issue be avoided.

The starting point is the condition that real consumption (i.e., income net of taxes and saving) be equal before and after retirement, as given by:

(1)     Y_p– T_p– S = Y_r– T_r

where:

Y_p = pre-retirement gross income;

T_p = pre-retirement taxes;

s = savings;

Y_r = retirement gross income;

T_r = retirement taxes.

Next define

s = pre-retirement saving rate = (S/Y_p )

and

R = replacement rate = (Y_r/Y_p)

so that substituting these definitions in (1) and dividing by Y_p gives:`

(2)    1 - (T_p/Y_p )- s = R - (T_r/Y_p )

Now let T_p = t_pY_p and T_r = t_rY_r where t_p and t_r are the pre- and post-retirement proportional tax rates, so that:

(3)     s = (1 - t_p) - (1 - t_r)R

Equation (3) defines a set of combinations of s and R which satisfy the condition specified in (1). By first finding a value for R, we can then solve for the corresponding value of s that satisfies (3).

The retirement income flow (Y_r) can be converted to a lump sum at retirement by applying an annuity factor (α).[22] This expresses the stream of retirement income in terms of a stock in wealth at the time of retirement. In other words, were a person to have accumulated this amount they would be able to receive a lifetime annuity of Y_r. Denoting the `required' wealth needed to generate Y_r as W_r, then:

(4)   W_r= αY_r= α[(1- s) Y_p-T_p + T_r]

The amount of saving needed to reach this required level of retirement income W_r will depend on:

• the existing stock of net wealth W_p
• the expected returns on investment
• future income
• tax rates.

We define W_p as the projected level of wealth, so that the shortfall is:

(5)   W_r - W_p= α[(1- s) Y_p- T_p + T_r]- W_p

We are now in a position to derive the rate of saving needed to reach the required level of wealth. This rate is the share of pre-tax income the household would need to save in order to have the level of income in retirement.

The amount accumulated by retirement would then be:

(6)    

where:

Y_a = actual income in year t = 1,...T;;

T = number of years from the person's current age until the pre-determined age of retirement;

g= annual growth rate of income;

r = after-tax real rate of return on savings;

.

Using (5) and (6) we can solve for the saving rate:

(7)    

where Y_p= Y_a(1 + g)^T. Now dividing by Y_p gives:

(8)    

It is argued that in the context of the New Zealand system of income tax, private retirement saving is made from after-tax pre-retirement income Y_p- T_p, and the earnings on the investments are taxed. However, once those accumulated funds are withdrawn (in this case to purchase an annuity), there is no further taxation on the income received in retirement. Furthermore, NZS payments are received net of tax. Hence under this system, T_r= 0. With this simplification the saving rate is no longer dependent on the replacement rate:

(9)    

and from (3), the replacement rate can be derived as:

(10)    R = 1 - t_p - s