Category Archives: Economics

328 – Weitzman discounting

Martin Weitzman (1942-2019) was an environmental economist who thought laterally. He made important contributions to the field in at least three areas. Here I’ll explain one of his clever insights: that uncertainty about the discount rate has an impact on the effect of discounting.

At a function in his honour in 2018, Weitzman said “I’m drawn to things that are conceptually unclear, where it’s not clear how you want to make your way through this maze,” and described how he “took a decisive step in that direction a few decades ago…getting into the forefront rather than…following everything that went on.”

Martin Weitzman’s Contributions to Environmental Economics

He certainly did get to the forefront! Like a number of other environmental economists I’ve spoken to, I was disappointed that he didn’t win the Nobel Prize in 2018 when his work on climate change and discounting would have made him a perfect co-winner with William Nordhaus.

This PD is about discounting. To follow it, you’ll need to know what discounting is, and how it works. For some simple background, see PD33, and for some insights as to why discounting values from the distant future raises curly questions, see PD34.

You are probably aware that discounting at any rate likely to be recommended by an economist has the (perhaps uncomfortable) result that large benefits in the distant future count for little in the present. While there are arguments for accepting that this is in fact a reasonable and realistic result, it hasn’t stopped people looking for rationales to reduce the discount rate. Some really dodgy reasons have been proposed, including by economists (e.g. the Stern Report), but Weitzman came up with a simple idea that is obviously correct and has an effect equivalent to lowering the discount rate in the long run.

The insight was that, as we think about years further into the future, there is increasing uncertainty about what the discount rate should be in each year. This insight requires two breaks from the way that economists usually think about discount rates. The first is recognising that the appropriate discount rate to use is not necessarily constant over time. I remember thinking that it surely wasn’t constant when I first learnt about discounting, but then I just slipped into assuming that it is constant, like everybody else. Weitzman had the wit to remember that it didn’t have to be constant. [Technical note: I’m not talking about hyperbolic discounting here. In Weitzman’s conceptual model, the discount rate could go up or down from period to period.]

The second break from normal practice was to think about the discount rate for a given year as something that could be uncertain. It obviously is uncertain, but it had hardly ever been treated as such.

When economists want to represent uncertainty quantitatively, we usually do so by defining the value as a subjective probability distribution. To represent a discount rate about which we are increasingly uncertain in the more distant future, we would represent a probability distribution that has a wider variance as time passes.

Having done that, Weitzman showed that an uncertain discount rate is mathematically equivalent to a certain discount rate that declines over time. In the video below I show how this works.

The spreadsheet I use in the above video is available here.

The consequence, as described by Weitzman, is that ‘the ‘‘lowest possible’’ interest rate should be used for discounting the far-distant future part of any investment project’ (Weitzman 1998).

To get the declining-discount-rate result, you don’t even have to assume that uncertainty about the discount rate is increasing over time. As long as the rate is uncertain, even constant uncertainty will give that result.

The idea has been picked up in various ways, including in the guidelines for BCA published by the UK government. They don’t recommend doing all the uncertainty calculations explicitly, but they recommend using a discount rate that declines over time.

Note that to get the “lowest possible” discount rate, he really does mean “far-distant”. He’s talking about dates centuries into the future. The insight doesn’t have big implications for dates within about 50 years, which is about as far as many government Benefit: Cost Analyses go. For what I consider to be realistic representations of discount rate uncertainty, it would mainly affect the results for benefits and costs beyond 50 or 100 years in the future. (See the video for more on this.)

Note that uncertainty about discount rates in the distant future affects the impact of discounting in those distant future years. It doesn’t affect discounting in earlier years. As a result, even if the certainty-equivalent discount rate for year 100 falls to zero (i.e. the value discounted to year 99 is the same as in year 100), the values will still be discounted to express them as present values in year zero. So future benefits still get discounted quite a bit, just a bit less than they would have if you didn’t account for uncertainty. (See the video for more on this as well.)

Of course, the discount rate isn’t the only thing that gets more uncertain as we look further into the future – pretty much everything does. But Weitzman’s insight is still useful and relevant for some investments, even if you explicitly look at other types of uncertainty as well.

When would I suggest using Weitzman discounting? For a BCA that is capturing benefits and costs for 100 years of more. I would recommend combining it with strategies to represent uncertainty about other key variables in the analysis.

In other work, Weitzman focused on the possibility that the end result of climate change could be truly catastrophic. He called it a “fat tailed” problem, for reasons you can read about in Weitzman (2011) and Weitzman (2014). He concluded that this should “make economists less confident about climate change BCA and to make them adopt a more modest tone that befits less robust policy advice” (Weitzman 2011, p.291).

Further reading

Weitzman, M.L. (1998). Why the Far-Distant Future Should Be Discounted at Its Lowest Possible Rate, Journal of Environmental Economics and Management 36, 201-208. Paper * IDEAS page

Weitzman, M.L. (2011). Fat-tailed uncertainty in the economics of catastrophic climate change, Review of Environmental Economics and Policy 5(2), 275-292. Paper

Weitzman, M.L. (2014). Fat Tails and the Social Cost of Carbon, American Economic Review 104(5), 544-546. Paper

327 – Heterogeneity of farmers

Farmers are highly heterogeneous. Even farmers growing the same crops in the same region are highly variable. This is often not well recognised by policy makers, researchers or extension agents.

The variation between farmers occurs on many dimensions. A random sample of farmers will have quite different soils, rainfall, machinery, access to water for irrigation, wealth, access to credit, farm area, social networks, intelligence, education, skills, family size, non-family labour, history of farm management choices, preferences for various outcomes, and so on, and so on. There is variation amongst the farmers themselves (after all, they are human), their farms, and the farming context.

This variation has consequences. For example, it means that different farmers given the same information, the same technology choices, or facing the same government policy, can easily respond quite differently, and they often do.

Discussions about farmers often seem to be based on an assumption that farmers are a fairly uniform group, with similar attitudes, similar costs and similar profits from the same practices. For example, it is common to read discussions of costs and benefits of adopting a new farming practice, as if the costs and the benefits are the same across all farmers. In my view, understanding the heterogeneity of farm economics is just as important as understanding the average.

Understanding the heterogeneity helps you have realistic expectations about how many farmers are likely to respond in particular ways to information, technologies or policies. Or about how the cost of a policy program would vary depending on the target outcomes of the program.

We explore some of these issues in a paper recently published in Agricultural Systems (Van Grieken et al. 2019). It looks at the heterogeneity of 400 sugarcane farmers in an area of the wet tropics in Queensland (the Tully–Murray catchment). These farms are a focus of policy because nutrients and sediment sourced from them are likely to be affecting the Great Barrier Reef. “Within the vicinity of the Tully-Murray flood plume there are 37 coral reefs and 13 seagrass meadows”.

Our findings include the following.

  • Different farmers are likely to respond differently to incentive payments provided by government to encourage uptake of practices that would reduce losses of nutrients and sediment.
  • Specific information about this can help governments target their policy to particular farmers, and result in the program being more cost-effective.
  • As the target level of pollution abatement increases, the cost of achieving that target would not increase linearly. Rather, the cost would increase exponentially, reflecting that a minority of farmers have particularly high costs of abatement. This is actually the result that economists would generally expect (see PD182).

Further reading

Van Grieken, M., Webster, A., Whitten, S., Poggio, M., Roebeling, P., Bohnet, I. and Pannell, D. (2019). Adoption of agricultural management for Great Barrier Reef water quality improvement in heterogeneous farming communities, Agricultural Systems 170, 1-8. Journal web page * IDEAS page

325 – Ranking projects based on cost-effectiveness

Where organisations are unable or unwilling to quantify project benefits in monetary or monetary-equivalent terms, a common approach is to rank potential projects on the basis of cost-effectiveness. Just like ranking projects based on Benefit: Cost Ratio (BCR), this approach works in some cases but not others.

To rank projects based on cost-effectiveness, you choose the metric you will use to measure project benefits, estimate that metric for each project, estimate the cost of each project, and divide the benefit metric by the cost. You end up with a cost-effectiveness number for each potential project, and you use these numbers to rank the projects.

An advantage of this approach is that it sidesteps the challenges of having to measure all the benefits in monetary or monetary-equivalent terms, which is what you have to do calculate a BCR. A disadvantage is that it only works to compare projects that generate similar types of benefits, which can all be measured with the same metric.

Assuming that we are satisfied with your benefits metric and that the projects to be ranked are similar enough, the question is, in what circumstances is it appropriate to rank projects based on cost-effectiveness? (Assuming that the objective is to maximise the overall benefits across all the projects that get funded.) It is logical to ask this given that cost-effectiveness is closely related to the BCR (it has the same structure – it’s just that benefits are measured differently), and we’ve seen in PD322, PD323 and PD324 that ranking projects by BCR works in some situations but not others.

It turns out that the circumstances where it is logical to use cost-effectiveness to rank projects are equivalent to the circumstances where it is logical to rank projects using BCR.

(i) If you are ranking separate, unrelated projects, doing so on the basis of cost-effectiveness is appropriate. Ranking projects by cost-effectiveness implies that there is a limited budget available and you are aiming to allocate it to the best projects.

(ii) If you are ranking mutually exclusive projects (e.g. different versions of the same project), ranking on the basis of cost-effectiveness can be highly misleading. If there are increasing marginal costs and/or decreasing marginal benefits (which are normal), ranking by cost-effectiveness will bias you towards smaller project versions. In PD323, I said to rank such projects by NPV and choose the highest NPV you can afford with the available budget. If we are not monetising the benefits, there is no equivalent to the NPV — you cannot subtract the costs from a non-monetary version of the benefits. This means that, strictly speaking, you cannot rank projects in this situation (mutually exclusive projects) without monetising the benefits. If you absolutely will not or cannot monetise the benefits, what I suggest you do instead is identify the set of project versions that can be afforded with the available budget, and choose the project version from that set that has the highest value for the benefit metric. You don’t divide by the costs, but you do use the costs to determine which project versions you can afford. This is a fudge that only makes sense if you adopt the unrealistic assumption that any unspent money will not be available to spend on anything else, but it seems to me to be the best way to go, if monetising the benefits is not an option.

(iii) If you are ranking separate, unrelated projects, and there are multiple versions available for at least one of those projects, then cost-effectiveness does not work and the rule about choosing the highest-value benefit metric does not work either. Instead, you should build an integer programming model to simultaneously weigh up both problems: which project(s) and which project version(s). There is a brief video showing you how to do this in Excel in PD324. In the video, the benefits are measured in monetary terms, but the approach will work if you use non-monetary measures of the benefits.

There are a number of tools available for ranking projects based on cost-effectiveness (e.g. Joseph et al. 2009) but it is important to be clear that the approach only works in certain cases.

Even if you are using cost-effectiveness in the right circumstances (case (i) above), it has a couple of limitations relative to using BCR. One is that you cannot use it to rank projects with distinctly different types of benefits that cannot all be measured with the same metric. Another limitation is that cost-effectiveness provides no evidence about whether any of the projects would generate sufficient benefits to outweigh its costs.

Further reading

Joseph, L.N., Maloney, R.F. and Possingham, H.P. (2009). Optimal allocation of resources among threatened species: a project prioritization protocol. Conservation Biology, 23, 328-338.  Journal web site

Pannell, D.J. (2015). Ranking environmental projects revisited. Pannell Discussions 281. Here * IDEAS page

324 – NPV versus BCR part 3

In PD322 and PD323 we have been exploring whether to use Net Present Value (NPV) or Benefit: Cost Ratio (BCR) in Benefit: Cost Analysis (BCA) when assessing and comparing projects. I’ve presented some simple rules to follow when the projects are separate and unrelated, or when the projects are mutually exclusive (i.e., when you can only choose to do one of them). But what if the decision maker is faced with choosing from multiple separate projects, and there are multiple versions of at least one of the projects?

Before getting into the details, I want to clarify that, in all the examples I’m presenting in this series, I’m assuming that the objective is to maximise the total NPV across all funded projects. I should have emphasised that earlier. A key message from the three related blog posts is that to get the highest total NPV, you should not necessarily choose the projects with the highest individual NPVs. Sometimes that’s the case but in other cases it’s not.

Now, suppose that a decision maker is faced with selecting from three different versions of a project to protect Lake Antelope (call them projects A1, A2 and A3) and four versions of a project to protect Lake Giraffe (projects G1, G2, G3 and G4). In this situation, there is no simple rule to follow, like just choose the projects with the highest BCRs or the highest NPVs. Instead, this situation requires the use of a constrained optimisation algorithm. Assuming that you are choosing whole projects (i.e. you can’t choose 0.7 of a project), the required algorithm is called integer programming.

Fortunately, integer programming is available in the Excel spreadsheet software, and it is not difficult to use it to select the optimal portfolio of projects in this complex situation. I’ll present the numbers for a relatively simple example of this complex decision problem, and then I’ll show you how to solve it in a brief YouTube video.

First the example. The present values of benefits (B) and costs (C) for the project versions for Lake Antelope and Lake Giraffe are as follows: project A1: B=$180, C=$40; project A2: B=$360, C=$100; project A3: B=$400, C=$200; project G1: B=$200, C=$10; project G2: B=$400, C=$60; project G3: B=$600, C=$160; project G4: B=$800, C=$310. The available budget is $300. You can choose at most one of the Lake Antelope project versions and at most one of the Lake Giraffe project versions. Which project versions (if any) should you choose to maximise the overall net benefits?

Watch the video to see how to use integer programming to solve this in Excel. Within the capacity of the software, this approach will work for any number of projects and any number of project versions.

Note that the information you get from a model like this is not a simple ranking of the projects. Instead, it tells you how the optimal combination of projects and project versions changes depending on the available budget. To get this information, you change the program budget in the model and re-solve it. For this example, the results look like this.

Range of budget levelsOptimal version of A projectOptimal version of G project
$0 to $9nilnil
$10 to $49nilG1
$50 to $59A1G1
$60 to $99nilG2
$100 to $159A1G2
$160 to $259A2G2
$260 to $409A2G3
$410 and higherA2G4

 

The two project versions in each row of the table come as a package. There is no point in ranking them. If the budget was to shrink, you would not just drop one of these, you would change which project version you selected, as shown in the table.

NPV/BCR Rule 4: If selecting and ranking multiple projects, and at least one of the projects is available in multiple versions, don’t use NPV or BCR. Instead, use integer programming to optimise the selection of projects and project versions simultaneously.

Having to create a constrained optimisation model like this might seem like more bother than is worthwhile, but it isn’t difficult, it doesn’t take long, and it may result in substantially higher benefits being generated by the program, compared with the use of a simpler rule-of-thumb.

If you’ve read all three of these Pannell Discussions on NPV and BCR, your head may be spinning at the complexity of all this. Apologies for that, but the complexity is real and needs to be understood if analysts applying BCA are to be sure of giving sound advice to decision makers.

Appendix

Another realistic complexity that might be relevant in some cases is that there might be two separate constraints on the funding of projects. For example, there could be a limited budget available for initial project implementation and a separate limited budget available for ongoing maintenance. As with the example above, neither BCR nor NPV is sure to rank the projects correctly in this situation, and you need to use a method like integer programming to be sure of getting it right. You could build an Excel model like the one in the video, and include separate constraints for implementation costs and maintenance costs, rather than a single constraint for costs overall, which is what I did in the video.

On the other hand, in the numerical examples I’ve looked at, it is usually not too terrible to assume that implementation costs and maintenance costs are drawn from one limited budget. In other words, using BCR to rank separate, unrelated projects can still be OK even if there are two constraints on funding. But it is an approximation and it might not give you the best possible solution.

323 – NPV versus BCR part 2

In PD322 we looked at whether to use Net Present Value (NPV) or Benefit: Cost Ratio (BCR) in Benefit: Cost Analysis (BCA) when assessing and comparing separate, unrelated projects. What if they are not separate, unrelated projects?

The most common scenario where you have to go beyond the simple rules presented in PD322 is where you are comparing different versions of the same project (e.g. different scales or different actions, but addressing the same broad goal in the same region). They are not separate, unrelated projects – they are mutually exclusive. If you do one of them, it rules out doing any of the others.

It is good practice to assess multiple versions of the same project before settling on a particular version. Versions of a project with more ambitious targets can deliver greater benefits, but also incur greater costs, so it is usually not readily apparent how ambitious the project should be. (Project versions also vary on dimensions other than ambitiousness, such as their spatial targeting, or the specific actions they will include.) The first project version to be specified may or may not end up being the best version when several versions are compared.

However, if you are ranking projects, and the projects you are ranking consist of multiple versions of the same project, using BCR for the ranking process will probably not give you the correct result.

The following example should give you a feel for why BCR does not give you the correct project ranking in this situation.

Suppose that projects X1 and X2 are two versions of project X. If you did project version X2, you would have to bear the cost of doing X2, and in addition, you would bear the opportunity cost of not doing project X1 (i.e., you would miss out on the net benefits of doing X1). Similarly, the full cost of doing X1 should include the opportunity cost of not doing X2. That’s why the traditional BCRs of projects X1 and X2 do not provide a reliable ranking — there are additional costs that a BCR doesn’t capture.

The obvious solution is to include the opportunity cost of not doing the best alternative project when calculating the BCR. However, this is an inconvenient approach because the identity of the best alternative project depends on the available budget. Each time you generated results for a different budget level, you would have to recalculate all the BCRs based on different opportunity costs.

A much simpler approach (that gives the same answer) is to choose the project with the largest NPV that you can afford within the funds allocated for this project. For example, if you were faced with choosing from amongst the five project versions in the table below, and the available budget was $600, you could afford any of the projects and you would choose project 4, which provides the largest NPV of $490. If the budget was only $200, you would choose project 3, which provides an NPV of $440. (You could no longer afford project 4 because it costs $310.)

Note that projects 3 and 4 don’t offer the highest traditional BCRs, which is provided by project 1 (BCR = 20), but we don’t want project 1 because it has the lowest NPV (only $190), and the lowest adjusted BCR.

ProjectPV(Benefits)PV(Costs)NPVBCR
1$200$10$19020.0
2$400$60$3406.7
3$600$160$4403.8
4$800$310$4902.6
5$1000$560$4401.8

 

NPV/BCR Rule 3: If selecting from different versions of the same project, choose the project with the largest NPV that you can afford within the funds allocated for this project.

The above NPV rule is based on the assumption that you must choose one project version from a set of mutually exclusive project versions, and that the funding is committed to be used for one of these project versions and won’t be used for other projects.

The next Pannell Discussion looks at a more complex scenario where the decision-maker is faced with choosing from multiple separate projects, and there are multiple mutually-exclusive versions of at least one of the projects.