3  Calculating a social discount rate

In terms of the rates that can be estimated, the social opportunity cost is the easier of the social rate of time preference and social opportunity cost to estimate. Section 3.1 discusses estimation of the social rate of time preference. Section 3.2 discusses the social opportunity cost and its estimation.

3.1  Social rate of time preference

There are several approaches to estimating the social rate of time preference. Factors to take into account are morbidity, uncertainty, and prospects for economic growth with diminishing marginal utility of consumption.

One approach is to use the after-tax market interest rate as all individuals equate their discount rate to the market. This is reasonable if there are perfect current and future markets. If everyone has access to the market and takes market interest rates into account then in simple capital markets with a tax distortion, the after-tax rate of interest is a potential estimate of the social rate of time preference. Unfortunately, this assumption does not hold. Warner and Pleeter (2201) have observed individual discount rates that were higher than the after tax market interest rate.

A key consideration for other methods of estimating the social rate of time preference is the fact that growth in consumption over time is expected and needs to be taken into account in the discount rate. Boscolo et al (1998 p. 1) outline the social discount rate as being the sum of two components. One is the “pure” rate of time preference based on consumption now or later. The other component indicates, “how changes in consumption affect the marginal utility of consumption” (Boscolo et al p. 1). This may be represented as:

Social rate of time preference = r + μg (2)

Where r is the “pure” rate of time preference, g is the expected growth rate in per capital consumption and μ is the negative elasticity of marginal utility with respect to consumption. In addition, Henderson (1968) took into account consumption growth and the shape of the utility function to derive a social rate of time preference.

Another similar approach is to determine the social rate of time preference based on the optimal rate of growth. Marglin (1963) suggests working backwards from the rate of optimal rate of growth to the discount rate that ensures a level of investment that achieves this optimal level of growth. Unfortunately, this approach requires this rate of growth to be known, as well as a number of other assumptions, making it rather difficult to estimate.

None of these methods is without problems. This means that when estimating the social rate of time preference, the results should be used with caution. Diagram 2 indicates that the social rate of time preference and the social opportunity cost lead to the same result in a number of circumstances. When using the after-tax market interest rate approach to estimate the social rate of time preference, the social opportunity cost is a good proxy.

When thinking about the relationship between the social opportunity cost and social rate of time preference, it is useful to determine the relative values. Using diagram 1 the social opportunity cost is greater than the social rate of time preference when on the left hand side of the market-clearing equilibrium because of the distortion. This is likely when there is a tax or other distortion creating a wedge between what savers receive and what investments must return.

Given that there are estimation difficulties with the social rate of time preference, it may be preferable to use the social opportunity cost. This means that when using the social opportunity cost as a proxy and a positive net present value results, there should be a positive net present value from using the social rate of time preference. This is satisfactory for clear-cut results, but could lead to under-investment if net present value results close to zero are not investigated further. Any omission or underestimation of costs would offset this potential issue.

It is important to undertake sensitivity analysis, given that there are only estimates of the social rate of time preference available and not an actual value. Any dramatic changes in the conclusions from small changes in the discount rate should be investigated further.

3.2  Social opportunity cost

The social opportunity cost discount rate can be estimated using a number of different models. The models aim to work out what the market would expect to receive for a particular project. This is the rate of return to balance the social opportunity cost of undertaking the project in the public sector versus the next best alternative in the private sector where rates are observable.

The calculation needs to take into account whether the project it is replacing would have received a subsidy, which would lower the rate, or would have been taxed, thereby raising the rate. The calculation also needs to take into account risk as well as any social costs or benefits (externalities). If the project were to replace private sector projects with negative externalities then the rate would be lower. Alternatively, if the private sector project delivers positive externalities then the discount rate would need to be higher.

The discussion does not assume that the exact private sector project displaced is known. In the New Zealand system, there are few subsidies and few tax concessions to take into account, so looking at the general case is a good approximation. In general, the externalities generated by the private sector project will be the same as for the public sector project. Risk is dealt with explicitly in the model used.

The choice of underlying model can significantly alter the result obtained. The main models to choose from include the Capital Asset Pricing Model (CAPM), the Arbitrage Pricing Theory (APT), and Fama and French’s multi-factor model (1993). The various models are briefly described in section 4 below.

The results from these models are then used in the standard weighted average cost of capital (WACC) formula to get a discount rate. The discount rate would be the weighted average cost of capital. The formula is:[2]

(3)    WACC = (1-T_c) k_b D/(D+E) + k_e E/(D+E)

where T_c is the corporate tax rate, k_b is the return on debt calculated using CAPM, k_e is the return on equity calculated using CAPM, D is bonds or debt and E is equity (also called stock).

This formula needs to be adjusted to reflect that the government does not pay tax or get a tax break on paying interest. This ensures the rate reflects the tax situation for a public sector project. This requires the formula to be divided by (1-T_c), such that:

(4)    WACC = k_b D/(D+E) + K_e E/(D+E)

where K_e is the return on equity calculated using the CAPM adjusted for the fact that the government does not pay corporate tax or algebraically as:

(5)     K_e = k_e / (1 - T_c).

As suggested above, it is important to undertake sensitivity analysis, as any method of calculation of the discount rate will only provide an estimate and not the actual value.