# Social Cost Benefit Analysis

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Created: 2008-11-14

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Contents

## Introduction

Cost-benefit analysis is a process for evaluating the merits of a particular project or course of action in a systematic and rigorous way. **Social** cost-benefit analysis refers to cases where the project has a broad impact across society -- and, as such, is usually carried out by the government.

In simple situations costs and benefits may relate to goods and services that have a simple and transparent monetary measure (their price). However, more usually, and especially in the social case, this is not the case (for example what is the value of national defense or a national road network). It should therefore be emphasized that the costs and benefits considered by CBA should be construed in their widest sense, as encompassed in the standard meaning of utility or social welfare as used by economists (though these measures may be expressed in money-metric terms).

In its essence cost-benefit analysis is extremely, indeed trivially, simple: evaluate costs $$C$$ and benefits $$B$$ for the project under consideration and proceed with it if, and only if, benefits match or exceed the costs. That is:

$$ \[ \underbrace{B}_{\textrm{Benefits}} \geq \underbrace{C}_{\textrm{Costs}} \] $$

So what makes things more complex? There are a variety of factors:

- Benefits and costs may accrue to different sets of people. If this is so we need some way to aggregate and compare different benefits and costs across people.
- Benefits and costs may occur at different points in time. In this case we need to compare the value of outcomes at different points in time.
- Benefits and costs may relate to different types of goods and it may be difficult to compare their relative values. This usually occurs when one of the goods does not have an obvious and agreed upon price. For example, we may be spending standard capital goods today in order to obtain environmental benefits tomorrow.
- Benefits and costs may be uncertain.
- Benefits and costs may be difficult to calculate and, as a result, there may be widely differing views about their sizes. One might think this could be subsumed under uncertainty, however the two points are rather different: two people agreeing that an outcome follows some probability distribution is different from them arguing about its mean and variance.

Usually, in real-world cases the dominant issue is usually the last one: the basic job of calculating estimates for the projects costs and benefits. This especially true in the the 'social' case where the projects under consideration may involve costs and benefits that very difficult to quantify -- what is the benefit of the national security derived from military spending, how large are the benefits from education, etc etc. Necessarily this quantification only makes sense on a case-by-case basis. However, here we are concerned with general principles and we therefore focus only on the preceding four items and look at how they can be incorporated into the analysis in a general way.

## The Model

### Basic Setup

We are considering a project with known (though perhaps uncertain) benefit $$B$$ and cost $$C$$. Our task is to decide whether it is worthwhile. As already discussed, if $$B$$ and $$C$$ were denominated in exactly the same terms (i.e. the same good, at the same time) for a single person and with no uncertainty things would be straightforward: check whether $$B \geq C$$. However, this is unlikely to be the case so we will need to do more work. All of this work, in its essence involves converting benefits and costs into values that can be easily compared. Equivalently we need to have benefits and costs denominated in terms of some standard good or measure of value. We shall term this good or measure of value the *numeraire*.

In theory, this numeraire could be anything: apples, years of life, acres of rainforest etc. However, given that many (though by no means all) goods are already denominated in terms of money, it is often natural to use a numeraire that is money-metric. We do need to specify whose money and for our purposes the natural choice is money in the hands of government: *government funds*.

To make our lives easy, let's assume that the costs are very simple being (with certainty) one unit of government funds today. This means we can focus entirely on the benefits and no real generality is lost this way. We begin by ignoring temporal and uncertainty issues. What then remains is to specify how the benefits are distributed.

### Distributional Concerns

Let us suppose there are $$N$$ people or groups who with benefit to group $$i$$ being $$b_{i}$$ in terms of their *own income*. Individuals existing income/consumption is denoted by $$x_{i}$$ and those operating the project (the government) have a utilitarian welfare function: $$ \[ W = \sum_{i} u(x_{i}) \] $$

The change in welfare (assuming away any changes in behaviour) arising from the individual benefits is: $$ \[ \Delta W = \sum_{i} u(x_{i} + b_{i}) - u(x_{i}) \] $$

If the gains are small relative to existing income we may approximate the change in individual welfare using the derivative: $$u(x_{i} + b_{i}) - u(x_{i}) = u'(x_{i})b_{i}$$. Defining, $$w_{i} = u'(x_{i})$$ we then have a set of 'distributional weights', that is weightings for individuals such that the total (welfare) benefit is just the sum of the weights times the individual benefits: $$ \[ \Delta W = \sum_{i} w_{i} b_{i} \] $$

One last step remains: we need to convert utility back into our numeraire (government funds) via multiplication by some constant $$\theta$$ -- the overall $$B$$ will then be $$\theta \Delta W$$. To specify this constant we choose a benchmark project and then define its benefit $$B$$ to exactly one unit of the numeraire -- since costs are also 1 this implies this project is just worthwhile. The standard approach is for the benchmark to be a project which generates benefist equivalent to one unit equally divided equally among all groups, i.e. $$b_{i} = 1/N$$. This implies that: $$ \[ \theta \sum_{i} w_{i} \frac{1}{N} \equiv 1 \Rightarrow \theta = \frac{N}{\sum_{i} w_{i}} \] $$

Note that if income were already equally distributed so $$x_{i} = x$$ and utility (which is only defined up to a constant) were normalized so that that the marginal utility of income at the reference income $$x$$ were exactly one we would have $$w_{i} = w = 1 \Rightarrow \theta = 1$$. To summarize:

$$ \begin{eqnarray} N & = & \textrm{Number of beneficiaries} \\ b_{i} & = & \textrm{Benefits to group i} \\ x_{i} & = & \textrm{Income of group i} \\ w_{i} & = & \textrm{Weights = Marginal Utility of group i} \\ \Delta W & = & \textrm{Welfare benefit} = \sum_{i} w_{i} b_{i} \\ \theta & = & \textrm{Conversion factor from SW to Numeraire} \\ B & = & \textrm{Benefit} = \theta \sum_{i} w_{i} b_{i} \end{eqnarray} $$

### Calculating the Conversion Factor $$\theta$$

With a few assumptions on the form of the utility function and knowledge of the distribution of income we can calculate an actual figure for $$\theta$$. Assume that utility takes CES form: $$ \[ u(x) = \frac{x^{1-\alpha}-1}{1-\alpha} \] $$

Thus, the weights (equal to marginal utility) are $$w_{i} = x_{i}^{-\alpha}$$ and hence $$\theta^{-1} = E(x^{-\alpha})$$. At this point, it is useful to move to continuous rather than discrete variables so: $$ \[ \frac{1}{\theta} = \int w_{i}(x) dF(x) = \int x^{-\alpha} dF(x) = E(x^{-\alpha}) \] $$

Now, the distribution of income $$x$$ is (approximately) log-normal $$LN(\nu, \sigma)$$ in which case using the formula for the moment generating function of the normal we have: $$ \[ E(x^{-\alpha}) = e^{-\alpha \mu + \frac{\alpha^{2}\sigma^{2}}{2}} \] $$

### Example 1: Benefits in Proportion to Income

Suppose the project generates value $$V$$ which is then distributed in proportion to income. Let $$\lambda$$ be the ratio of individual benefit to income so $$b_{i} = \lambda x_{i}$$. Now $$\sum_{i} b_{i} = V \Rightarrow \lambda = V / \sum_{i} x_{i}$$. Thus using our formula from above the total benefit in terms of the numeraire is:

$$ \begin{eqnarray} B & = & \theta \sum_{i} w_{i} b_{i} \\ & = & \theta \lambda \sum_{i} w_{i} x_{i} \\ & = & \theta \frac{NV}{\sum_{i} x_{i}} \frac{1}{N} \sum_{i} w_{i} x_{i} \end{eqnarray} $$

Using a CES utility function so $$w_{i} x_{i} = x_{i}^{1-\alpha}$$ and using expectations we have: $$ \[ B = \frac{V E(x^{1-\alpha})}{E(x^{-\alpha}) E(x)} \] $$

Using a log-normal distribution for income and the expression for the MGF as before this further reduces to $$B = e^{-\alpha \sigma^{2}}$$. For log-utility, $$\alpha = 1$$, and a reasonable estimate of $$\sigma$$ is 0.47 (Newbery 2008) which implies $$B = 0.8 V$$. Thus each pound/euro/dollar distributed generates a benefit in terms of the numeraire of 0.8 and, if the project is to be worthwhile, it must have a direct yield of at least 25% ($$= 1/0.8$$).

## Discounting

We now come to the time factor: benefits of effort or expenditure today may not be realised until tomorrow. In the spirit of keeping things simple let us fix everything about the problem except the temporal aspect. In particular, ignore distributional issues, uncertainty and variations in the types of goods involved.

Assume that by giving up one unit of expenditure today we gain $$V$$ units at time $$T$$. Let expenditure today be $$x$$ and at time $$T$$ (in the absence of the project) $$x_{T}$$. There is a utility function $$u$$ which converts expenditure into *contemporaraneous* utility (i.e. utility in that period). Let the numeraire be present period utility normalized so that the marginal utility today equals 1, i.e. $$u'(x) = 1$$.

Thus there are two major factors to take into account. First, how to convert utility from period $$T$$ to now. Humans tend to pefer things sooner rather than later. Hence, even with all else equal, utility today is prefered to utility tomorrow. The measure of this is termed the level of 'pure time-preference' and we will denote it by $$\rho(t)$$. Second, the reference situation tomorrow (in terms of resources, consumption etc) may not be the same as today and marginal utility from gaining or losing a unit will differ across time, quite apart from time-preference.

Assuming changes in income are relatively small we can approximate utility changes using derivatives we have (normalizing marginal utility of consumption today):

$$ \begin{eqnarray} C & = & \textrm{Cost} = -u'(x) = -1 \\ b & = & \textrm{Benefit at } T = V u'(x_{T}) \\ B & = & \textrm{Benefit in today's utility} = \rho(T) b \end{eqnarray} $$

So the project is worthwhile if:

$$ \[ V \rho(T) u'(x_{T}) \geq 1 \] $$

This implicitly defines a discount factor $$\delta(T) = \rho(T) u'(x_{T})$$ with the payoff of $$V$$ at time $$T$$ valued at $$\delta(T) V$$ today.

### Example 2: Climate Change

Suppose we can spend money today to mitigate the effects of climate change that yields benefits at some point $T$ in the future. Suppose, in the absence of this project, the economy would grow at rate $g$ per year so consumption at time $T$ is $e^{gT}$ times consumption today. Suppose time preference takes an exponential form so $$\rho(T) = e^{-\rho T}$$ and we have CES utility as before. Then:

$$ \[ B = V e^{-\rho T} e^{-\alpha g T} = V e^{- (\rho + \alpha g) T} \] $$

Comparing this with a standard exponential discount rate $$e^{-\delta T}$$ gives an implied discount rate $$\delta = \rho + \alpha g$$. Note that this is identical to the real interest rate found in the Ramsey_Cass_Koopmans_Model.

## Uncertainty

The models discussed above involve no uncertainty: all relevant values, e.g. the project's payoff, future consumption levels etc, are known with complete certainty. This is clearly unrealistic and it is useful to be able to consider situations which do involve (known) uncertainty.