RDP 2004-01: The Impact of Superannuation on Household Saving Appendix B: A Small Theoretical Model

This appendix presents a simple model of saving based on the overlapping generations framework introduced by Samuelson (1958). Within this highly stylised framework we can analyse the effects on saving from financial deregulation, the introduction of superannuation, and (unexpected) increases in wealth. These effects are illustrated with numerical simulations.

A simple overlapping-generations model with three generations

Our model is based on a standard overlapping-generations model of a small open economy in which consumers live for three periods. Consumers are young in the first period, middle-aged in the second period and old in the third period. They optimise the (log-) utility they get from life-time consumption, discounting future consumption at the rate β with 0 ≤ β ≤ 1.

Consumers receive an endowment in each period, which could be thought of as labour income. When they are young they are assumed to have a low income , for instance, because they are in education; at middle age, during their working life, they have a high income ; and they have a lower (labour) income in old age. Although the endowment is not storable, consumers can buy (or borrow) financial investments at the (exogenous) world interest rate r_t. Thus they can go into debt or accumulate wealth in the first and second period. For simplicity, they are assumed to leave no bequests at the end of the third period, that is, their wealth at the end is zero and they have no initial wealth beyond their endowment.

In the first basic case, consumers are allowed to borrow at any point in time as long as they have no debt at the end of their life. The budget constraints for each period (that is, wealth measured at the end of each period) are then defined as:

The change in the asset position is equal to the consumers' saving, that is, an increase means that she has saved and a decrease means that she has borrowed. If the asset position is unchanged, consumption in each period equals the endowment (labour income) plus interest income. The constraints in Equation (B2) can be collapsed to an intertemporal budget constraint by recursive substitution of and :

The first-order conditions, together with the intertemporal budget constraint, yield the optimal consumption path for a consumer:

The savings profile and the consumption profile are of course a function of the endowment path, the time preference and the interest rate. Typically, we assume an endowment profile that is low when young and old, and high in middle age. This means, if there are no restrictions on the amount that can be borrowed, consumers will typically go into debt while they are young, pay off that debt and accumulate wealth while they are in working age and finally consume that wealth when they are retired (together with the endowment during that period).

At any point in time, there is a young, a middle-aged and an old generation, and aggregate consumption, net wealth and saving are the sum of the individual consumption decisions. For simplicity, we assume that each generation has an equal number of consumers, normalised to one (if the population grows or falls over time, aggregate consumption, saving and wealth will also grow or fall).

Table B1 illustrates the results of the basic model when we set the time discount rate β to one, the interest rate to 0.05, and the initial endowment for each generation to (1, 4, 1). The first panel shows the results of the basic model if consumers are allowed to borrow. Unrestricted borrowing in the first period allows the young generation to bring consumption forward, and – with a zero interest rate – would yield the optimal consumption path of (2, 2, 2). Since we have assumed a positive interest rate, it pays off to postpone some of the consumption and increase life-time consumption somewhat.

Now let us consider the case where households are not allowed to borrow. If we do not allow households to borrow we introduce the additional assumption that household wealth cannot become negative:^[26]

For our endowment path this means that the young generation will be able to consume just their endowment in the first period (see the second panel in Table B1). The middle-age generation will, however, accumulate wealth and save some of their income for retirement. Undiscounted life-time consumption is higher in this case, but the consumers consume less when they are young and more when they are older.^[27] Of course, consumers may prefer to consume more when they are young, even if it means sacrificing some of the life-time consumption. The stock of aggregate wealth is also higher compared with the first scenario, since the young generation is not allowed to incur debt.

We will now modify this model in order to illustrate the effects of financial deregulation, the introduction of a compulsory superannuation scheme, and an (unexpected) increase in the value of assets held by households.

Financial deregulation and saving

We first analyse the impact of financial deregulation on saving in our model. This analysis borrows from Bayoumi (1993), who modelled the effect of financial deregulation on saving and the current account for the UK.

In a world of financial regulation, consumers face borrowing constraints. The extreme assumption of a ‘no borrowing’ constraint underlies the model described in the second scenario in Table B1. In this scenario, young consumers need to postpone consumption to middle and old age.

After financial deregulation, consumers face no borrowing constraints (at least in our stylised model world), described in first scenario in Table B1. Saving in both scenarios is the same, but the aggregate stock of wealth in the world with financial constraints is higher, mainly since the young generation does not contribute negative wealth, or debt. It should be noted that for the parameters chosen in our model simulations, in aggregate consumers would prefer to hold less net wealth, or more debt, which would allow them to smooth consumption more evenly through time.^[28]

Table B2 illustrates how our model changes when financial constraints get abolished in period 5, thus illustrating the effects of deregulation. While saving is unchanged in the long run, for a transition period (which takes two generations) saving is lower. This is because the middle and old generations, which were financially constrained in their youth, have postponed consumption and therefore remain on their ‘original’ consumption path. On the other hand, the young generation is already consuming on the ‘new’ path, which allows them to bring consumption forward. While the population still comprises consumers which were financially constrained in their youth, consumption will be higher (and saving will be lower) than in steady state. The amount of net wealth (which includes the young generation's debt) will gradually fall to the new level while consumption adjusts back to the new level.

This exercise illustrates that after financial deregulation – for a transition period – saving can be expected to be lower while debt levels rise. Ultimately though, saving is expected to return to the pre-deregulation levels, but the transition period in our stylised model comprises two generations.

A model with superannuation

We now turn to an analysis of the effects of a compulsory superannuation scheme in our model. We consider two channels through which a superannuation scheme can affect our model outcome. First, compulsory superannuation can force households to save. Second, consumers may be uncertain about the value of some future variables, such as retirement income. Superannuation schemes may then provide a signal about the value of this future variable, leading possibly to revisions of the saving and consumption path. We model each channel in turn.

Superannuation and forced saving

We introduce compulsory superannuation in our model by assuming that a fixed percentage s of labour income (that is, the endowment) is not available for consumption in the first two periods but will be saved, and – together with the interest on the saving – is available for consumption in retirement. We also assume for simplicity that income of the ‘old’ generation is not subject to superannuation contributions (since this generation has to consume all its wealth in the same period). Note that the corresponding saving is the sum of superannuation contributions plus the part of disposable income that is not consumed.

Reducing Equation (B7) to the intertemporal budget constraint in Equation (B8), we can see that this is the same as Equation (B3). The desired consumption path with and without superannuation is therefore identical.

We can now distinguish three cases depending on whether forced saving is more (or less) than desired saving, and depending on whether the consumer can borrow.

If the superannuation contributions are less than what the consumer wants to save anyway, she will simply offset the compulsory superannuation by a reduction in other savings – assuming the rate of return on both types of saving are identical. Total saving will then remain unchanged.

If, on the other hand, the consumer wishes to save less than the superannuation contributions, she can offset the superannuation contributions through borrowing. This is the case in our model simulations, where consumers would like to borrow when they are young and bring consumption forward. Of course, borrowing will entirely offset the superannuation saving only if the interest rate on borrowing and on saving is the same, as assumed in our model. Table B3 shows the model simulations when a superannuation contribution rate of 10 per cent is introduced in period 5. Not surprisingly, in the case of unconstrained borrowing, the results are identical to those in Table B1.

The situation changes, however, when a consumer wishes to save less than the superannuation contributions but she cannot borrow (enough) to offset the saving in superannuation contributions. Our ‘constrained’ scenario in Table B3 assumes that borrowing is zero, which implies that wealth is now at least as much as the sum of superannuation contributions for each generation. The wealth constraints are now:

In our ‘constrained’ example, consumption in the first two periods will be lower, and consumption in retirement will be higher, leading to a higher aggregate stock of wealth in every period. Ultimately, aggregate saving is zero, since the old generation dissaves every period the amount of superannuation paid in by the young and middle-age generation. However, in the ‘changeover’ period (which, in our model, is two generations) saving is higher since the old generation, who did not pay superannuation contributions when they were young and/or middle-age, have a lower (individual) wealth and therefore do not withdraw as much superannuation as if they had accumulated superannuation over their entire lifetime.

Superannuation and uncertainty about model parameters

We will now consider a different scenario where superannuation might change the consumption path chosen by consumers. In this case rather than ‘constraining’, superannuation resolves some uncertainty around the adequate level of saving for retirement. This might happen if some households are myopic and underestimate the need to finance consumption in old age, or they overestimate available income in retirement. Superannuation could then serve to indicate the ‘appropriate’ level of saving necessary for adequate retirement provision. In this model, we do not need borrowing constraints to affect saving, as the desired consumption path changes.

In our model, the assumption of an overestimation of retirement incomes would imply an expected which is higher than the actual , and myopia would imply a time preference parameter that is too low (thus discounting future consumption needs by more). In both cases, as we can see from Equation (B4) consumption is being brought forward through time.^[29]

Table B4 shows our model simulation if the introduction of superannuation signals that the time preference parameter should be increased from 0.9 to 1 (we have only included the results for the model without borrowing). Similar to the introduction of superannuation which cannot be offset, consumption is postponed when the time preference increases. This leads to an increase in aggregate wealth (as the young generation needs to borrow less to finance the lower consumption). While saving returns to its starting level in the long run, it increases during the transitional period, while the younger generation postpones consumption, and the older generation (which has consumed more in their youth) also consumes less.

Of course, expectation adjustment could in principle also happen in the other direction, that is, consumers save more since they are uncertain how much saving is required for adequate retirement provision. A superannuation contribution rate of s might indicate the ‘right’ level, and consumers would reduce their retirement provisions (and consequently saving and wealth implications are reverse to those illustrated in Table B4).

To summarise, our simple model has highlighted that the introduction of superannuation is likely to have most effect on aggregate wealth and saving if consumers cannot offset the additional saving by either reducing other saving or increased borrowing, or if they do not wish to offset it, since superannuation provides a signal by which model parameters (such as expected retirement income, or time preference parameters) get affected. Of course, more realistic assumptions, such as different rates of return on superannuation saving and other saving (e.g., because of different tax treatment) or costs of borrowing that are higher than the return on savings, will affect our conclusions in that fully offsetting the superannuation – even if feasible – will be costly and thus might be undesirable. If superannuation saving is not offset, during an adjustment period (which in our model is two generations) aggregate saving will also be higher, but ultimately, when outflows from superannuation funds match the inflows, saving will return to the initial level.

Unexpected capital gains

Finally, we will use our model to illustrate the effect of an unexpected, temporary increase in capital gains from investment. This extension aims at providing some insights into the effect on saving, consumption and wealth of unexpected wealth effects, such as the rapid increases in the prices of some assets over the 1990s. Again, of course, our model is highly stylised, and thus can only provide insights into the basic mechanism at work with respect to saving rather than give a detailed account of all the effects such a boom has on the macroeconomy.

In our stylised scenario, we can model an increase in asset prices as an increase in the stock of wealth for those consumers who hold positive wealth by w_s per cent. We assume that the increase is unexpected, that is, ex ante consumption decisions are made on the basis of the original model parameters for endowment and interest. Note that the change in wealth due to the asset price increase is not counted as saving (at least in the definition used here), which is the excess of income over consumption and therefore does not include capital gains.

Formally, the consumption path (assuming the wealth shock happens in time t = T) is given by the following solution:

where are the consumption choices of the basic model in Equation (B4).

None of the consumption decisions before t = T are affected, since the wealth shock is not anticipated. The consumption path of the generation that is young in t = T remains unchanged, since the young generation does not profit from the wealth shock (initial wealth is assumed to be zero). The consumption of the generation that is working age in t = T is not affected if they are in debt after the first period (our ‘typical’ scenario). However, if the working-age consumer has positive net assets in time T, she will increase consumption in the current period and in the next period, when he is old. Similarly, the consumer that is old in t = T will increase consumption if she has positive net assets (as is the case in our ‘typical’ scenario) and has an unchanged consumption otherwise. The wealth shock therefore can increase aggregate consumption in T and T + 1.

Table B5 shows the model simulations of such a temporary increase when wealth (excluding debt) is increased by 20 per cent between period 4 and period 5, but only for those who have positive wealth at this point (for our model parameters this is the generation which moves from middle-age to old). Not surprisingly consumption of this generation increases, leading to a rise in aggregate consumption and a temporary fall in the saving rate. We show here only the results for the model with borrowing, since the general conclusions are not affected by this assumption (remember that in our simulation the young generation either has negative wealth or zero wealth at the end of the first period).

Our stylised model shows, not surprisingly, what is known in the literature as the ‘wealth effect’ on consumption: an increase in wealth allows higher consumption by those who own the asset. This will lead – at least temporarily – to a lower saving rate.

Footnotes

Since households can still lend (that is, they hold positive wealth) they are assumed to lend to foreigners (for instance, they could buy government bonds). [26]

In our model, the lower debt levels increase disposable income since less interest needs to be paid to service the debt. However, life-time consumption of some households could also fall permanently if geared investments earn higher returns on their asset portfolio, thus increasing lifetime income when households are allowed to borrow (see Deaton (1992) and Attanasio (1998)). [27]

This is partly because wealth serves mainly the purpose of allowing to choose the timing of consumption. Of course, the introduction of other ‘utility’ of wealth, such as allowing bequests, would alter our model results. [28]

Formally, this can be shown by examining the derivatives of the optimal consumption choices with respect to e⁰ and β:
Since and , where A is an expression that is positive, the consumer will increase consumption in the first two periods if the expected income in the last period is higher. However, in the third period, he will realise that the expectation was wrong and he will be forced to reduce consumption. With the benefit of hindsight, he would prefer to have consumed slightly less in the first two periods in order to be able to consume more when old.
Since and , where B is a positive term, a lower β (which discounts future consumption by more) implies that consumption when young increases, while consumption in the next two periods decreases. [29]