5 The social rate of time preference
The SOC approach tries to identify the rate of return on the next best alternative to a public project. The social rate of time preference (SRTP) approach takes the alternative route of attempting to directly specify preferences for trading-off the future when considering public projects.[14]
As mentioned above, there is no market for public projects, so there is no way of observing individual preferences for how to value the future in such cases. Nor is there any reliable way to survey and aggregate the preferences of individuals with regard to how time should be valued for public projects. As a result, the SRTP should not, despite the conventional language used, be interpreted as the preferences of 'society' in any aggregate sense. Rather, it is more appropriate to interpret the SRTP as reflecting the preferences of a decision-maker acting on behalf of society. The framework set out below can then be thought of as an attempt to formalise what considerations a rational decision-maker should take into account when setting the discount rate.
5.1 Determination of the SRTP
The most common way of approaching the SRTP is to consider how a rational decision-maker might optimise the consumption and investment path of the economy, supposing it could do so. Carrying out this thought experiment sheds light on what considerations such a decision-maker would take into account in setting a 'socially optimal' discount rate, and how those considerations depend on the preferences, or values, of the decision-maker.
This approach can be formalised by supposing that the decision-maker can choose a consumption and investment path for the economy so as to maximise some function that evaluates different consumption paths; that is, it attaches a numerical value or 'score' to each path. Specifically, suppose that the decision-maker takes the following steps.
- Assign a 'welfare score' to the level of consumption in each time period. This score reflects the decision-maker's preferences for consumption within that period. In particular, it is generally assumed that the decision-maker has some preference for smoothing consumption over time.[15]
- Discount these welfare scores (as described in section 2) according to the decision-maker's own specific rate of time preference. This reflects the decision-maker's own values about how future welfare should be valued relative to the present
- Sum these discounted scores together to formulate an aggregate measure of 'social welfare' (that is, an abstract measure of the overall score that the decision-maker assigns to a given consumption path). This is often referred to as the decision-maker's 'social welfare function'.
This evaluation or social welfare function is therefore considered to be additive (the marginal benefit from increasing consumption in one period does not depend on consumption in other periods). It can be written formally as follows:
Where:
- W(C) represents the aggregate 'social welfare' that the decision-maker seeks to maximise by choosing the consumption profile C = (c_{0}, c_{1}, ..., c_{T}).
- U(c_{t}) is the function that the decision-maker uses to assign a 'welfare score' to the level of consumption (c_{t}) in any given year t.[16]
- p represents the decision-maker's pure rate of time preference. It is important to stress that this parameter is different from the discount rate that we are seeking to specify. Namely this parameter p is used to discount welfare scores, whereas the discount rate that we are seeking to specify is used to discount cash flows. Nonetheless, p is a key determinant of the overall discount rate.
- T is the length of the decision-maker's evaluation horizon.
Importantly, the above expression differs from the present value formula given above. The term W(C) is the present value of the stream of U(c) values, discounted at the pure time preference rate.
The term, p, in particular attracts a great deal of controversy, as it implies that the decision-maker values future generations less than the present. As this is a crucial value judgement, it is not surprising that views differ regarding the value of p that 'should' be imposed for public projects, and that these views are often expressed in strong terms.
5.2 The social rate of time preference
Given the form of objective - the welfare function given above - the decision-maker is then assumed to choose a consumption and investment path for the economy so as to maximise the value of W(C) subject to the constraints of available resources, productivity, and so on. A further important assumption is made about the weighting function U(c_{t}): this is considered to be 'iso-elastic'. That is, the elasticity of U with respect to c is assumed to take the constant value, say θ. The maximisation problem can be solved to show that money values, that is the consumption values, c, rather than the 'welfare' values, U, are discounted at the rate, r, where:
r = p + θ_{g}
and:
- p is the decision-maker's pure rate of time preference (as defined above)
- g is the annual growth rate of per capita consumption, generally assumed to be constant.
- θ is a term that reflects the nature of the decision maker's preferences with respect to consumption smoothing. As mentioned above it represents the (assumed to be) constant elasticity of U with respect to variations in c. It is a parameter of the welfare scoring function, U. Loosely speaking, it captures how averse the decision maker is to unequal consumption streams across time, or its 'aversion to intertemporal inequality'.
Then r is the social rate of time preference. In cost-benefit analyses, the present value (in money terms) of a stream of annual net benefits (interpreted here as the stream, c_{t}) is evaluated using this social time preference rate, r.[17]
The equation above is known as the 'Ramsey equation', following Ramsey (1928).[18] In the last two decades, a number of countries have moved to using this equation as the basis for setting public sector discount rates according to a social rate of time preference approach: see section 6 for a summary of international approaches.
The equation states that the socially efficient consumption discount rate is equal to:
- the decision-maker's rate of pure time preference, p,
- plus a 'wealth effect'. This reflects the reasoning that if the decision-maker cares about equalising consumption over time, and per capita consumption is expected to grow over time, future outcomes should be discounted due to the fact that people in the future enjoy higher living standards. The size of this wealth discount effect depends on the decision-maker's aversion to intertemporal inequality, as captured by θ.
This implies that even if the decision-maker chooses to value the welfare of all time-periods and generations equally (that is, chooses a rate of pure time preference equal to zero) there are nevertheless other reasons to discount cash flows of consumption in the future because of the wealth effect.[19]
5.3 Specifying the parameters
The SRTP approach involves a number of assumptions and value judgements, particularly in relation to the choice of variables in the Ramsey equation. These value judgements need to be made as explicit as possible. Many commentators attempt to set the values of p and θ by reference to observed individual behaviour. However, these approaches involve a number of strong assumptions, and also make the implicit assumption that what is, is an appropriate signal of what the relevant values should be.
Therefore it is more appropriate to understand the implications of different choices, and treat these decisions as value judgements that have to be made openly. How this might be possible is considered in section 6. Table 2 illustrates some of the parameter choices that have been used internationally.
Country | p | θ | g | SRTP |
---|---|---|---|---|
UK baseline public sector discount rate | 1.5 | 1 | 2 | 3.5 |
France baseline public sector discount rate[20] | 1 | 2 | 1.5 | 4 |
Stern climate change review | 0.1 | 1 | 1.3 | 1.4 |
Nordhaus critique of the Stern review | 1.5 | 2 | 2 | 5.5 |
Harmonised European Approaches for Transport Costing | 1.5 | 1 | 1.5 | 3 |
Sources: HM Treasury (2003), Cropper et al (2014), Stern (2007), Nordhaus (2007), HEATCO (2006)
Notes
- [14]For further discussion and references, see Creedy and Guest (2008).
- [15]More accurately, it is assumed that the decision-maker has a decreasing marginal valuation for consumption in any given period. That is, the greater is consumption, the less that additional increments to consumption add to the decision-maker's score. This implies consumption smoothing, as the more a decision-maker reallocates from one year to another, the less willing they are to do so further. Thus the decision-maker is somewhat averse to unequal consumption streams over time. The degree of this aversion depends on the specification of preferences.
- [16]In the context of individual multi-period optimisation, U represents a utility function. In the present context it is instead a cardinal weighting function. In the individual context, the term , , is referred to as a 'utility discount rate'.
- [17]It is tempting to think that discounting U using is equivalent to discounting c using r. However, the two are not necessarily equivalent.
- [18]It plays an important part in optimal growth models involving a ‘representative agent’: see, for example, Blanchard and Fischer (1989) and Barro and Sala-i-Martin (1995).
- [19]Some economists argue for the inclusion of an additional term in the Ramsey equation to account for the chance that there will be some catastrophe event so devastating that the outcomes of the policy become irrelevant. In reality, this term is difficult to quantify, and is likely to be small enough to ignore.
- [20]The French guidance does not explicitly state the values used to derive its public sector discount rate: the 4 per cent baseline discount rate was chosen as a central value.