1.5 – should i care about my grandchildren’s future?

discounting basics

The choice of the social discount rate is one of the most influential components for understanding the costs and benefits of climate change. Because discounting compounds values over time, and GHG mitigation generates benefits over a long time horizon, small changes in the discount rate can have large impacts on the present value of future benefits.

As a reminder, the present value (PV) of a future value is calculated as: \[PV(b) = \frac{b}{(1 + r)^t}\] where \(b\) is the nominal value, \(r\) is the discount rate, and \(t\) is the number of years until the payment is received.

Imagine receiving a benefit of $1 million in 100 years.

• At a 2% discount rate, the PV of that benefit is ~$138,000.
• At a 3% discount rate, the PV of that benefit is ~$52,000.
• At a 7% discount rate, the PV of that benefit is ~$1150.

If the cost of achieving that benefit were $100,000 today, it is easy to see how the project makes sense at a 2% discount rate, but not at a 3% or 7% discount rate.

In the interactive calculator below, choose a value and a time horizon in years to see how the discount rate affects the present value of a benefit received in the future.

Code

viewof discountRate = Inputs.range([0, 0.1], {value: 0.025, step: 0.005, label: "discount rate"});
viewof timeHorizon = Inputs.range([1, 300], {value: 300, step: 1, label: "time horizon (years)"});
viewof initialValue = Inputs.range([1000, 10000], {value: 1000, step: 1000, label: "initial value"});

discountedValue = {
  const data = [];
  for (let t = 0; t <= timeHorizon; t++) {
    const discounted = initialValue / Math.pow(1 + discountRate, t);
    data.push({year: t, value: discounted});
  }
  return data;
}

chart = {
  const width = 600, height = 400;
  const svg = html`<svg width=${width} height=${height}></svg>`;
  
  const margin = {top: 20, right: 30, bottom: 30, left: 40};
  const x = d3.scaleLinear()
    .domain([0, timeHorizon])
    .range([margin.left, width - margin.right]);

  const y = d3.scaleLinear()
    .domain([0, initialValue])
    .nice()
    .range([height - margin.bottom, margin.top]);

  const xAxis = (g) => g
    .attr("transform", `translate(0,${height - margin.bottom})`)
    .call(d3.axisBottom(x).ticks(width / 80).tickSizeOuter(0));

  const yAxis = (g) => g
    .attr("transform", `translate(${margin.left},0)`)
    .call(d3.axisLeft(y));

  const line = d3.line()
    .x(d => x(d.year))
    .y(d => y(d.value));

  d3.select(svg).selectAll("*").remove(); // clear previous render

  const g = d3.select(svg)
    .append("g");

  g.append("g").call(xAxis);
  g.append("g").call(yAxis);

  g.append("path")
    .datum(discountedValue)
    .attr("fill", "none")
    .attr("stroke", "steelblue")
    .attr("stroke-width", 1.5)
    .attr("d", line);

  return svg;
}

approaches to discounting

One of the fundamental challenges of valuing climate impacts is that the benefits of GHG mitigation accrue over centuries. Some of those benefits will be enjoyed by the current generation, while others will be enjoyed by future generations. How should we value benefits to future generations?

There are two primary approaches to valuing benefits to future generations:

• The descriptive (or opportunity-cost) approach states that We should invest resources according real market rates of return. Taking $50M and investing in climate mitigation has an opportunity cost because we could take that money and invest it in other projects. Investing that same $50M in financial markets might earn, say, 5% returns per year. Therefore, discount rates should be determined by the opportunity cost of capital investments.

• The prescriptive (or normative) approach states that it is unethical to “discount” the welfare of future generations, so we should use a very low discount rate (e.g., 1% or even 0%). It is important to distinguish discount rates on goods vs. discount rates on welfare of future generations. Future generations will (very likely) be wealthier than us, so we might want to value consumption in our relatively poor generation more heavily than consumption in the future (even if we value future generations’ welfare equally)

the social discount rate

The social discount rate is the rate at which society discounts future benefits and costs. One useful tool for thinking about the social discount rate under a prescriptive approach to discounting is the Ramsey equation. The Ramsey equation is defined as follows:

\[ r = \delta + \eta g\]

where:

• \(r\) is the social discount rate
• \(\delta\) is the pure rate of time preference (e.g., our degree of impatience)
  • This term governs how consumers discount utility over time. Higher \(\delta\) raises marginal benefit of consuming now. Think of this as impatience.
  • If you weight future generations’ welfare the same as today, you’d set \(\delta = 0\).
  • Reasonable values of \(\delta\) typically fall between 0% and 2%.
• \(\eta\) is the elasticity of marginal utility of consumption
  • AKA: the coefficient of relative risk aversion or intergenerational inequality aversion
  • Governs how much we value an additional dollar as society grows wealthier. As \(\eta\) increases, the more we dislike inequality, which means we value today’s consumption more strongly.
  • Reasonable values of \(\eta\) typically fall between 1 and 4.
• \(g\) is the expected growth rate of consumption per capita
  • Long-run economic growth rates typically fall between 1% and 3% per year.

what social discount rate should we use?

Choosing a value for the social discount rate is a tricky business that can be informed just as much by ethics and philosophy as it can by economics.

• Historically, the federal government typically used two discount rates for benefit-cost analysis: 3% and 7%. The 3% rate is used to reflect the opportunity cost of consumption, while the 7% rate is used to reflect the opportunity cost of capital investment.

• Alternative rationales depend on moral or ethical considerations in choosing the parameters for the Ramsey equation. The 2007 Stern Review on the Economics of Climate Change used a social discount rate of 1.4%. These choices were made to reflect the following:

  • \(\delta=0.1\%\) reflects the idea that we should value future generations’ welfare almost equally to our own.
  • \(\eta=1\) reflects the idea that we should value an additional dollar of consumption equally, regardless of how wealthy we are. The choice of \(\eta=1\) was particularly conversial. Many economists believe that \(\eta\) should be larger than 1, which would imply that we should value an additional dollar of consumption more highly as we become wealthier. This would imply that we should use a higher discount rate.
  • \(g=1.3\%\) reflects the idea that we expect consumption per capita to grow at 1.3% per year.

In 2023, the US Government adopted new guidelines for discounting (and conducting benefit-cost analysis, more generally) in a very wonky document titled “Circular A-4”. Several arguments for discounting that pertain to climate change are presented in Circular A-4:

Circular A-4 arguments for a prescriptive approach to long-term discounting

Some believe that it is ethically impermissible to discount the utility of future generations. That is, government should treat all generations equally. Even under an approach that does not discount the utility of future generations, it is often appropriate to discount longterm consumption benefits and costs—although at a lower rate than the near-term effects more likely to fall on a single generation—if there is an expectation that future generations will be wealthier and thus will value a marginal dollar of benefits or costs by less than those alive today, or if there is a non-zero probability of sufficiently catastrophic risks. To account for these special ethical considerations, an extensive literature uses a “prescriptive” approach to long-term discounting, determining the appropriate degree of weight that society should place on the welfare of future generations. (Circular A-4, pp. 80)

Circular A-4 arguments for using a lower/declining rate for long-term discounting

A distinct reason for discounting the benefits and costs accruing to future generations at a lower rate is uncertainty about the appropriate value of the discount rate. Private market rates provide a reasonably reliable reference for determining the rate at which society is willing to trade consumption over time within a few decades, but for extremely long time periods no comparable private rates exist. Because future changes in the social rate of time preference are uncertain but correlated over time, the certainty-equivalent discount rate will have a declining schedule. The appropriate discount rate declines because it is the average of the cumulative discount factors, not an average of the discount rates, that matters. (Circular A-4, pp. 80)

uncertainty and the discount rate

Projecting future costs and benefits is inherently uncertain. When using the Ramsey equation, we need to make assumptions about the growth rate of consumption per capita over centuries. When focused on a descriptive approach, the discount rate might reflect changes in interests rates that may be driven by uncertain future monetary policy.

The key point to remember is that uncertainty in the discount rate provides justification for using a lower discount rate.

Imagine today, we use a 4% discount rate, but that discount rate has an equally likely chance of becoming 1% or 7% in 100 years. What discount rate should we use today to reflect that uncertainty?

We can work through an example to define the certainty-equivalent discount rate. The certainty-equivalent discount rate is the discount rate that would make us indifferent between that rate and taking the gamble on the uncertain rate.

The PV of receiving $1000 in 100 years for each rate is:

Table 1: Discount rates to calculate certainty equivalents

discount rate	nominal value	discounted value
1%	$1000	$367.88
4%	$1000	$18.32
7%	$1000	$0.91

Can we just use the average of the discounted values for the 1% and 7% rates (i.e., \(\bar{PV}_{100}\) ($367.88 + $0.91) / 2 = $184.40) as the certainty-equivalent discount rate? The short answer is no, and the reason depends on the shape of the present-value formula.¹

But! We can calculate the certainty-equivalent discount rate by comparing the average of the two discounted values across adjacent years (i.e., comparing year 100 vs. 101).

• PV of $1000 in 101 years at 1% discount rate: $364.22
• PV of $1000 in 101 years at 7% discount rate: $0.85
• Average of the two above values, \(\bar{PV}_{101}\): ($364.22 + $0.85)/2 = $182.53

Then, we can calculate the certainty-equivalent discount rate by dividing the average of the two discounted values across two adjacent years:

\[ \frac{\bar{PV}_{100}}{\bar{PV}_{101}} = \frac{\$184.40}{\$182.53} = 1.01024... = (1+r) \] \[ \implies r \approx 1.02\% \]

This value is the certainty-equivalent discount rate for the 50-50 chance of seeing a 1% or 7% discount rate in 100 years. This value is clearly much lower than 4%, which is the average of the two rates. If there is uncertainty in the discount rate, that leads to a much lower certainty-equivalent discount rate that we should use in our benefit-cost analysis.

resources and further reading

• Arrow et al., 2003. “Determining Benefits and Costs for Future Generations”, Science.

• Prest, Brian C., William Pizer, and Richard G. Newell, 2021, Improving Discounting in the Social Cost of Carbon, Resources Magazine, 2021.

Footnotes

1. This is an application of Jensen’s inequality, which states that \(f(E[x]) \leq E[f(x)]\) for any convex \(f\).