Cost–benefit analysis

French engineer and economist Jules Dupuit, credited with the creation of cost–benefit analysis

The concept of CBA dates back to an 1848 article by Jules Dupuit, and was formalized in subsequent works by Alfred Marshall.[9] Jules Dupuit pioneered this approach by first calculating "the social profitability of a project like the construction of a road or bridge"[10] In an attempt to answer this, Dupuit began to look at the utility users would gain from the project. He determined that the best method of measuring utility is by learning one's willingness to pay for something. By taking the sum of each user's willingness to pay, Dupuit illustrated that the social benefit of the thing (bridge or road or canal) could be measured. Some users may be willing to pay nearly nothing, others much more, but the sum of these would shed light on the benefit of it. It should be reiterated that Dupuit was not suggesting that the government perfectly price-discriminate and charge each user exactly what they would pay. Rather, their willingness to pay provided a theoretical foundation on the societal worth or benefit of a project. The cost of the project proved much simpler to calculate. Simply taking the sum of the materials and labor, in addition to the maintenance afterward, would give one the cost. Now, the costs and benefits of the project could be accurately analyzed, and an informed decision could be made.

The Corps of Engineers initiated the use of CBA in the US, after the Federal Navigation Act of 1936 mandated cost–benefit analysis for proposed federal-waterway infrastructure.[11] The Flood Control Act of 1939 was instrumental in establishing CBA as federal policy, requiring that "the benefits to whomever they accrue [be] in excess of the estimated costs."[12]

In health economics, CBA may be an inadequate measure because willingness-to-pay methods of determining the value of human life can be influenced by income level. Variants, such as cost–utility analysis, QALY and DALY to analyze the effects of health policies, may be more suitable.[36][37]

For some environmental effects, cost–benefit analysis can be replaced by cost-effectiveness analysis. This is especially true when one type of physical outcome is sought, such as a reduction in energy use by an increase in energy efficiency. Using cost-effectiveness analysis is less laborious and time-consuming, since it does not involve the monetization of outcomes (which can be difficult in some cases).[38]

It has been argued that if modern cost–benefit analyses had been applied to decisions such as whether to mandate the removal of lead from gasoline, block the construction of two proposed dams just above and below the Grand Canyon on the Colorado River, and regulate workers' exposure to vinyl chloride, the measures would not have been implemented (although all are considered highly successful).[39] The US Clean Air Act has been cited in retrospective studies as a case in which benefits exceeded costs, but knowledge of the benefits (attributable largely to the benefits of reducing particulate pollution) was not available until many years later.[39]

A generic cost–benefit analysis has the following steps:[40]

1. Define the goals and objectives of the action.
2. List alternative actions.
3. List stakeholders.
4. Select measurement(s) and measure all cost and benefit elements.
5. Predict outcome of costs and benefits over the relevant time period.
6. Convert all costs and benefits into a common currency.
7. Apply discount rate.
8. Calculate the net present value of actions under consideration.
9. Perform sensitivity analysis.
10. Adopt the recommended course of action.

CBA attempts to measure the positive or negative consequences of a project. A similar approach is used in the environmental analysis of total economic value. Both costs and benefits can be diverse. Costs tend to be most thoroughly represented in cost–benefit analyses due to relatively-abundant market data. The net benefits of a project may incorporate cost savings, public willingness to pay (implying that the public has no legal right to the benefits of the policy), or willingness to accept compensation (implying that the public has a right to the benefits of the policy) for the policy's welfare change. The guiding principle of evaluating benefits is to list all parties affected by an intervention and add the positive or negative value (usually monetary) that they ascribe to its effect on their welfare.

The actual compensation an individual would require to have their welfare unchanged by a policy is inexact at best. Surveys (stated preferences) or market behavior (revealed preferences) are often used to estimate compensation associated with a policy. Stated preferences are a direct way of assessing willingness to pay for an environmental feature, for example.[41] Survey respondents often misreport their true preferences, however, and market behavior does not provide information about important non-market welfare impacts. Revealed preference is an indirect approach to individual willingness to pay. People make market choices of items with different environmental characteristics, for example, revealing the value placed on environmental factors.[42]

The value of human life is controversial when assessing road-safety measures or life-saving medicines. Controversy can sometimes be avoided by using the related technique of cost–utility analysis, in which benefits are expressed in non-monetary units such as quality-adjusted life years. Road safety can be measured in cost per life saved, without assigning a financial value to the life. However, non-monetary metrics have limited usefulness for evaluating policies with substantially different outcomes. Other benefits may also accrue from a policy, and metrics such as cost per life saved may lead to a substantially different ranking of alternatives than CBA.

Another metric is valuing the environment, which in the 21st century is typically assessed by valuing ecosystem services to humans (such as air and water quality and pollution).[43] Monetary values may also be assigned to other intangible effects such as business reputation, market penetration, or long-term enterprise strategy alignment.

CBA generally attempts to put all relevant costs and benefits on a common temporal footing, using time value of money calculations. This is often done by converting the future expected streams of costs (${\displaystyle C}$) and benefits (${\displaystyle B}$) into a present value amount with a discount rate (${\displaystyle r}$) and the net present value defined as:

$${\displaystyle {\text{NPV}}=\sum _{t=0}^{\infty }{B_{t}-C_{t} \over {(1+r)^{t}}}}$$

The selection of a discount rate for this calculation is subjective. A smaller rate values the current generation and future generations equally. Larger rates (a market rate of return, for example) reflects human present bias or hyperbolic discounting: valuing money which they will receive in the near future more than money they will receive in the distant future. Empirical studies suggest that people discount future benefits in a way similar to these calculations.[44] The choice makes a large difference in assessing interventions with long-term effects. An example is the equity premium puzzle, which suggests that long-term returns on equities may be higher than they should be after controlling for risk and uncertainty. If so, market rates of return should not be used to determine the discount rate because they would undervalue the distant future.[45]

Methods for choosing a discount rate

For publicly traded companies, it is possible to find a project's discount rate by using an equilibrium asset pricing model to find the required return on equity for the company and then assuming that the risk profile of a given project is similar to that the company faces. Commonly used models include the capital asset pricing model (CAPM):

$${\displaystyle r=r_{f}+\beta \left[\mathbb {E} (r_{M})-r_{f}\right]}$$

and the Fama-French model:

$${\displaystyle r=\underbrace {r_{f}} _{\text{Risk-Free Rate}}+\beta _{M}\underbrace {\left[\mathbb {E} (r_{M})-r_{f}\right]} _{\text{Market Risk}}+\beta _{SMB}\underbrace {\left[\mathbb {E} (r_{S})-\mathbb {E} (r_{B})\right]} _{\text{Size Factor}}+\beta _{HML}\underbrace {\left[\mathbb {E} (r_{H})-\mathbb {E} (r_{L})\right]} _{\text{Value Factor}}}$$

where the ${\displaystyle \beta _{i}}$ terms correspond to the factor loadings. A generalization of these methods can be found in arbitrage pricing theory, which allows for an arbitrary number of risk premiums in the calculation of the required return.

Risk associated with project outcomes is usually handled with probability theory. Although it can be factored into the discount rate (to have uncertainty increasing over time), it is usually considered separately. Particular consideration is often given to agent risk aversion: preferring a situation with less uncertainty to one with greater uncertainty, even if the latter has a higher expected return.

Uncertainty in CBA parameters can be evaluated with a sensitivity analysis, which indicates how results respond to parameter changes. A more formal risk analysis may also be undertaken with the Monte Carlo method.[46] However, even a low parameter of uncertainty does not guarantee the success of a project.

Principle of maximum entropy

Suppose that we have sources of uncertainty in a CBA that are best treated with the Monte Carlo method, and the distributions describing uncertainty are all continuous. How do we go about choosing the appropriate distribution to represent the sources of uncertainty? One popular method is to make use of the principle of maximum entropy, which states that the distribution with the best representation of current knowledge is the one with the largest entropy - defined for continuous distributions as:

$${\displaystyle H(X)=\mathbb {E} \left[-\log f(X)\right]=-\int _{\mathcal {S}}f(x)\log f(x)dx}$$

where ${\displaystyle {\mathcal {S}}}$ is the support set of a probability density function ${\displaystyle f(x)}$. Suppose that we impose a series of constraints that must be satisfied:

1. ${\displaystyle f(x)\geq 0}$, with equality outside of ${\displaystyle {\mathcal {S}}}$
2. ${\displaystyle \int _{\mathcal {S}}f(x)dx=1}$
3. ${\displaystyle \int _{\mathcal {S}}r_{i}(x)f(x)dx=\alpha _{i},\quad i=1,...,m}$

where the last equality is a series of moment conditions. Maximizing the entropy with these constraints leads to the functional:[47]

$${\displaystyle J=\max _{f}\;\int _{\mathcal {S}}\left(-f\log f+\lambda _{0}f+\sum _{i=1}^{m}\lambda _{i}r_{i}f\right)dx}$$

where the ${\displaystyle \lambda _{i}}$ are Lagrange multipliers. Maximizing this functional leads to the form of a maximum entropy distribution:

$${\displaystyle f(x)=\exp \left[\lambda _{0}-1+\sum _{i=1}^{m}\lambda _{i}r_{i}(x)\right]}$$

There is a direct correspondence between the form of a maximum entropy distribution and the exponential family. Examples of commonly used continuous maximum entropy distributions in simulations include:

• Uniform distribution
  • No constraints are imposed over the support set ${\displaystyle {\mathcal {S}}\in [a,b]}$
  • It is assumed that we have maximum ignorance about the uncertainty
• Exponential distribution
  • Specified mean ${\displaystyle \mathbb {E} (X)}$ over the support set ${\displaystyle {\mathcal {S}}\in [0,\infty )}$
• Gamma distribution
  • Specified mean ${\displaystyle \mathbb {E} (X)}$ and log mean ${\displaystyle \mathbb {E} (\log X)}$ over the support set ${\displaystyle {\mathcal {S}}\in [0,\infty )}$
  • The exponential distribution is a special case
• Normal distribution
  • Specified mean ${\displaystyle \mathbb {E} (X)}$ and variance ${\displaystyle {\text{Var}}(X)}$ over the support set ${\displaystyle {\mathcal {S}}\in (-\infty ,\infty )}$
  • If we have a specified mean and variance on the log scale, then the lognormal distribution is the maximum entropy distribution

CBA has been criticized in some disciplines as it relies on the Kaldor-Hicks criterion which does not take into account distributional issues. This means, that positive net-benefits are decisive, independent of who benefits and who loses when a certain policy or project is put into place. Phaneuf and Requate (2016: p. 649) phrased it as follows "CBA today relies on the Kaldor-Hicks criteria to make statements about efficiency without addressing issues of income distribution. This has allowed economists to stay silent on issues of equity, while focusing on the more familiar task of measuring costs and benefits".[51]

The main criticism stems from the diminishing marginal utility of income.[52][53] Without using weights in the CBA, it is not the case that everyone “matters” the same but rather that people who value money less (who are by assumption people with more money) receive a higher weight. One reason for this is that for high income people, one monetary unit is worth less relative to low income people, so they are more willing to give up one unit in order to make a change that is favourable for them.[54] This means that there is no symmetry in agents. A second reason is that any welfare change, no matter positive or negative, affects people with a lower income stronger than people with a higher income.

Taken together, this means that not using weights is a decision in itself – richer people receive de facto a bigger weight. As to compensate for this difference in valuation and in order to take into account distributional issues, it is possible to use different methods. The two most common ones are taxation, e.g. through a progressive tax, and the addition of weights into the CBA itself. There are a number of different approaches for calculating these weights. Often, a Bergson-Samuelson social welfare function is used and weights are calculated according to the willingness-to-pay of people.[55][56]

• Benefit–Cost Analysis Center at the University of Washington's Daniel J. Evans School of Public Affairs
• Benefit–Cost Analysis site maintained by the Transportation Economics Committee of the Transportation Research Board(TRB).
• Intro to Cost–Benefit Analysis
• Engineering Risk-Benefit Analysis (MIT OpenCourseWare)