Deriving values of the social rate of time preference

Parker, Chris Working Paper Deriving values of the social rate of time preference New Zealand Treasury Working Paper, No. 25/01 Provided in Cooperation with: The Treasury, New Zealand Government Suggested Citation: Parker, Chris (2025) : Deriving values of the social rate of time preference, New Zealand Treasury Working Paper, No. 25/01, New Zealand Government, The Treasury, Wellington This Version is available at: https://hdl.handle.net/10419/311811 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ WORKING PAPER Deriving values of the social rate of time preference Chris Parker New Zealand Treasury Working Paper 25/01 February 2025 DISCLAIMER : The views, opinions, findings, and conclusions or recommendations expressed in this working paper are strictly those of the author(s). They do not necessarily reflect the views of the New Zealand Treasury or the New Zealand Government. The New Zealand Treasury and the New Zealand Government take no responsibility for any errors or omissions in, or for the correctness of, the information contained in this working paper. The paper is presented not as policy, but with a view to inform and stimulate wider debate. NZ TREASURY WORKING PAPER 25/01 Deriving values of the social rate of time preference MONTH/YEAR February 2025 AUTHOR Chris Parker The Treasury 1 The Terrace Wellington 6011 New Zealand Email [email protected] URL Treasury website at February 2025: https://www.treasury.govt.nz/publications/wp/wp-25-01 ACKNOWLEDGEMENTS Cory Davis, for assisting with Python; to Luke Symes, Tim Ng, Dr Zachary Turk, Dr Graeme Guthrie, and Professor Ben Groom for quality assurance/peer review; Kirsten Jensen, Dr Chris Thompson, Shane Domican, and Dominick Stephens for comment. All errors remain those of the author. NZ TREASURY New Zealand Treasury PO Box 3724 Wellington 6008 NEW ZEALAND Email Telephone Website [email protected] 64-4-472 2733 www.treasury.govt.nz WP 25/01 | Deriving values of the social rate of time preference i Abstract This report estimates the ‘social rate of time preference’ (SRTP) for New Zealand public policy appraisal using a Ramsey equation. It uses the Weitzman (1998) method to derive a declining discount rate schedule of 2% (real) for years 1–30, 1.5% for years 31–100, and 1% for years 101 on. The report does not compare and contrast social discount rate methods and how to use them in practice. JEL CLASSIFICATION H43, H50 KEYWORDS Discounting; social discount rate; time preference, Government WP 25/01 | Deriving values of the social rate of time preference ii Executive summary Constant social rate of time preference We derived a Ramsey equation based on a social welfare function of the consumption of goods and services over all future time. We required that the social rate of time preference (SRTP) is not less than the growth rate of consumption to ensure finiteness of net present values. To characterise our uncertainty about SRTP parameters we use triangular probability distributions of plausible values for the pure rate of time preference, annihilation risk, the elasticity of marginal social welfare, and the growth rate of real per capita consumption. These are set out below: 𝒓𝒓= 𝝆𝝆+𝜶𝜶+𝝁𝝁𝝁𝝁 >𝝁𝝁 Parameters Triangular distribution inputs Comment Minimum Mode Max Pure rate time preference 𝝆𝝆 0.00% 0.25% 2.0% Values used in major welfare economics studies Annihilation risk 𝜶𝜶 0.00% 0.05% 0.20% Extinction risk per century Elasticity marginal social welfare with respect to consumption 𝝁𝝁 0.25 0.75 1.5 Reasonably tolerable consumption loss from transfers Growth rate of real consumption per capita 𝝁𝝁 0.75% 1.25% 2.0% Historical per capita real consumption and projections Resulting output SRTP 𝒓𝒓 0.8% (Mean) 2.1% * 4.8% Range of restricted values for SRTP. Rounded to one d.p. * Between 1.2%–3.3%, with 95 percent confidence. We use Monte Carlo simulation to derive a range of SRTP values centring around 2%, and between 1.2%–3.3% with 95 percent confidence. We discarded about 15% of the draws because they resulted in a SRTP less than the growth rate of consumption. Declining SRTP Using the Weitzman (1998) method we derived the following declining certainty equivalent SRTP schedule (using half percentage point increments): Schedule of SRTPs Years 2% 1-30 1.5% 31-100 1% 101+ WP 25/01 | Deriving values of the social rate of time preference iii Contents Executive summary ............................................................................................ i 1. Introduction .................................................................................................... 1 2. Determining the ranges for parameters ....................................................... 2 2.1 Pure rate of time preference ................................................................. 2 2.2 Annihilation risk ..................................................................................... 3 2.3 The elasticity of marginal social value of consumption .......................... 3 2.4 Growth rate of consumption .................................................................. 6 3. Monte Carlo simulation of parameters ......................................................... 8 4. Declining ‘certainty equivalent’ SRTP ........................................................ 10 References ....................................................................................................... 15 Appendix 1 – Mathematical basis for SRTP ................................................... 18 The concept of the ‘social rate of time preference’ .................................... 19 Appendix 2 – Python code for Monte Carlo analysis .................................... 22 List of figures Figure 1: Annual growth of per capita real private consumption (GDP) ................ 6 Figure 2: Annual growth in real GDP per capita (percentage) .............................. 7 Figure 3: Monte Carlo simulation output (restricted set) ....................................... 9 Figure 4: Discount factors .................................................................................. 11 Figure 5: Average discount factor ...................................................................... 11 Figure 6: Marginal and average certainty equivalent SRTPs .............................. 13 Figure 7: Comparing discount factors (DFs) ....................................................... 13 Figure 8: Discount factors with and without declining SRTP............................... 14 WP 25/01 | Deriving values of the social rate of time preference 1 Deriving values of the social rate of time preference 1. Introduction As part of the Treasury review of the Public Sector Discount Rate (Treasury Circular 2024/15, Treasury 2024a), this note provides an estimate of the range of plausible values for the social rate of time preference (SRTP) based on the following form of Ramsey equation: 𝑟𝑟= 𝜌𝜌+𝛼𝛼+𝜇𝜇𝜇𝜇 (1) The SRTP (𝑟𝑟) estimate based on a Ramsey equation uses the parameters: • pure rate time preference 𝜌𝜌 • annihilation risk 𝛼𝛼 • elasticity of marginal social welfare with respect to consumption 𝜇𝜇 • growth rate of real consumption per capita 𝜇𝜇. Appendix 1 provides a detailed mathematical derivation of the equation and parameters above. We review the literature and review alternative formulations for the Ramsey equation. This note provides a range for an estimated constant SRTP, as well as a declining SRTP schedule based on a certainty-equivalent approach. This document does not canvass other approaches to setting the public sector discount rate.1 Creedy and Passi (2017) and Creedy (2007) explain that a SRTP can be derived based on a mix of empirical estimates and value judgements of decision makers acting on behalf of society using a ‘social welfare function’. They would assign ‘welfare scores’ (analogous to utility that an individual might assign to their own preferences) to different outcomes to determine which policy options are preferable to society, based on a set of ethical considerations. 1 More can be found in Grimes (2023), Abelson and Dalton (2023), NZIER (2024a), Turk (2024), Groom et al (2022), and Creedy and Passi (2017). WP 25/01 | Deriving values of the social rate of time preference 2 2. Determining the ranges for parameters For the general social welfare function of all future welfare we used for our analysis to be finite, the discount rate needs to be greater than the growth rate in perpetuity, ie, 𝑟𝑟>𝑔𝑔, or that:2 𝜌𝜌+𝛼𝛼+𝜇𝜇𝑔𝑔 >𝑔𝑔 (2) We do not place restrictions directly on each parameter as a function of the other parameters, but rather on the total resulting SRTP value. We implicitly assume each parameter value is not independent by discarding certain combinations of parameter values. For example, if the pure rate of time preference 𝜌𝜌 and the annihilation risk is thought to be very low, then we would require a somewhat higher value for marginal elasticity of marginal welfare 𝜇𝜇. Applying this restriction is not common in the literature, and nor is using Monte Carlo simulation to identify a range of possible outcomes whereby the possibility of 𝑟𝑟<𝑔𝑔 occuring is more apparent. While we are confident we should exclude combinations of values where 𝑟𝑟<𝑔𝑔, we consider results in the analysis both with and without excluding draws where 𝑟𝑟<𝑔𝑔. 2.1 Pure rate of time preference The pure rate of time preference is highly contentious and normative; eg, some argue it should be zero (eg, Ramsey 1928) on the basis of “intergenerational equity”. Creedy and Passi (2017) document ranges between 0.1% to 1.5% used by policy making authorities globally or by well-known welfare economics studies. Some other literature considers values of 0% (eg, HM Treasury 2024, Groom et al 2022). Sense Partners (2022) provides a Māori economic worldview from the iwi (tribe) Ngāi Tūhoe, suggesting the future should not be discounted. Drupp et al (2018) surveyed 262 global experts and 38 percent thought it should be zero, but the responses were substantially right-skewed with a mode of zero, median of 0.5%, and a mean of 1.10%. Turk replicated the survey for New Zealand economists and within a group of 33 self-identified experts found a mode of 0.1%, median of 1.0%, and a mean of 1.7%. Based on these studies, we assume it lies in the range 0% to 2%, and assume a continuous triangular distribution for it with a mode of 0.25% (implying a substantial right-skew); this results in a mean estimate of 0.75%. 2 That the discount rate needs to exceed the growth rate in perpetuity is described as an ‘economically natural assumption’ in Dixit (1993 p13) and Dixit and Pindyck (1994 p138). WP 25/01 | Deriving values of the social rate of time preference 9 Figure 3: Monte Carlo simulation output (restricted set) About 15% of the draws were discarded from the restricted set because the SRTP was less than the growth rate drawn, sometimes by as much as 1 percentage point. Discarding these led to the range of SRTP being higher than what one might suppose by considering the ranges of values for the individual variables in isolation. This is seen by comparing the two rows in Table 4 above. WP 25/01 | Deriving values of the social rate of time preference 10 4. Declining ‘certainty equivalent’ SRTP The two Treasury-commissioned reports of Grimes (2023) and Abelson and Dalton (2023) reviewed different approaches to declining discount rates. These experts advised us (in these reports and in workshops) that we use the Weitzman (1998) approach to derive a certainty equivalent declining discount rate over a certain future cashflow, which is the approach taken here.11 Weitzman’s insight was if one is uncertain which discount rate to use, rather than deduce some average of the discount rates one should instead average the present values of a (certain) dollar of impact in the future (ie, the discount factors) from each potential discount rate. The equivalent discount rate that gives the same expected net present value is not the simple average of the potential rates but is weighted towards the lower of the potential rates. The further in time the discounted cash flow, the closer the certainty equivalent discount rate is to the lowest potential one considered. One intuition is the larger rates annhilate themselves quicker, leaving the lowest to have an increasingly larger influence. Thus the certainty equivalent discount rate declines over time. This approach is suitable when there is permanent disagreement or uncertainty about ethical judgements that lead to heterogeneity in the appropriate SRTP to use and future benefit streams are certain. This, all else equal, serves to lower the hurdle rate for long-lived now or never projects. Guthrie (2021) considers how the Weitzman result applies to investments that are irreversible, have timing flexibility, and whose benefit streams are uncertain. He finds variation in discount rates also increases the option value of delaying investment that offsets — somewhat or possibly more than — the decrease in the hurdle rate for long-lived assets. Furthermore, when the decision maker is calculating the optimal time to exercise an option they need to consider what they are likely to do in that future point in time to ensure decisions are ‘dynamically consistent’. After considering the net effect of these matters, he suggests little would be lost in just ignoring discount rate heterogeneity for such investments. But if such heterogeneity were to be accounted for, he shows how to do it in an options valuation model that ensures investment policies are dynamically consistent. The range of possible outcomes for the SRTP illustrated in Figure 3 means that uncertainty about the present value of constant expected benefits and costs increases with distance in time. Figure 4 illustrates the discount factor 1/(1 + 𝑟𝑟)𝑡𝑡 for each discount rate 𝑟𝑟 from Table 4 as a function of time horizon over a 200 year horizon. Lower discount rates result in flatter and higher discount factor curves. 11 This approach was further clarified in Gollier and Weitzman (2010). WP 25/01 | Deriving values of the social rate of time preference 11 Figure 4: Discount factors We take the simple average of the two sets of deciles (from the restricted and unrestricted sets) to find an average discount factor — the orange curve in Figure 5. Figure 5: Average discount factor 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 0.76% 1.40% 1.58% 1.73% 1.87% 2.01% 2.16% 2.33% 2.54% 2.82%4.86% 0.30% 1.19% 1.40% 1.58% 1.74% 1.90% 2.06% 2.25% 2.46% 2.76% 4.86% 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 Average discount factor WP 25/01 | Deriving values of the social rate of time preference 12 The ‘marginal discount rate’ is the one that applies to the last year of cash flows (also known as the instantaneous discount rate, eg, Cropper et al 2014). Any declining discount rate (DDR) schedule should apply at the margin, rather than at the average, so that the discount factor curve is smooth and monotonically decreasing. For example, the HM Treasury’s (2008) schedule of discount rates is 3.5% for years 1-30 and 3% for years 31-75. If a project had, say, 31 year life, only the 31st year is discounted at 3%. That is, the HM Treasury’s marginal discount rate between years 31-75 is 3%. Figure 6 illustrates both, with the (black) average ‘certainty equivalent’ SRTP being the single constant value that would give the average discount factor as at that year. The (blue) marginal certainty equivalent SRTP is calculated as the rate of change of the average discount factor with respect to time — it lies under the average SRTP since the average is reducing. Rather than issue guidance on a continuously declining public sector discount rate we estimate a simple schedule of changes, much like the HM Treasury in the United Kingdom and increasingly other countries (Drupp et al 2018). This schedule aims to roughly approximate the blue marginal certainty equivalent SRTP, and is plotted as the grey curve, using fairly round numbers for both the SRTP and the period lengths for simplicity. If we start with the average of about 2% and keep that constant for 30 years and then reduce the marginal SRTP to 1.5% for years 31 to 100, then 1% thereafter, we find the resulting discount factor curve (the grey curve in Figure 7) is close enough to that required (the orange). The declining discount rate schedule leads to slight under-discounting after year 30. However, two minor considerations we excluded from our analysis might have reduced the discount rates in Figure 6 by about 0.15%–0.2% (increasing inequality in future following Emmerling et al (2017) discussed in footnote 13 of maybe 0.16%, and the adjustment by Gollier (2013) discussed in Appendix 1 of maybe up to 0.05%). If these two adjustments were added then the declining discount rate schedule would be a very close approximation. WP 25/01 | Deriving values of the social rate of time preference 13 Figure 6: Marginal and average certainty equivalent SRTPs Figure 7: Comparing discount factors 0.80% 0.85% 0.90% 0.95% 1.00% 1.05% 1.10% 1.15% 1.20% 1.25% 1.30% 1.35% 1.40% 1.45% 1.50% 1.55% 1.60% 1.65% 1.70% 1.75% 1.80% 1.85% 1.90% 1.95% 2.00% 2.05% 2.10% 2.15% 2.20% 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 Discount rate Yea r s CE (average) discount rate CE (marginal) discount rate DDR schedule 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 Average discount factor Discount factor with DDR WP 25/01 | Deriving values of the social rate of time preference 14 This declining SRTP schedule is summarised in Table 5. Note, most routine policy initiatives in practice may not have impacts after 30 years; only highly significant and transformative initiatives that do not have too much flexibility in waiting likely will be materially impacted by a declining SRTP. Table 5: Declining SRTP schedule Schedule for declining SRTP Years 2% 1-30 1.5% 31-100 1% 101+ This declining SRTP leads to somewhat significant differences in the discount factors in the very long-term, to help better reflect intergenerational issues, so it is perhaps worth using for intergenerational issues. Figure 8 augments Figure 7 by including the discount factor with a constant SRTP of 2%. There is an increasing gap between the grey discount factor curve with a declining SRTP and blue discount factor curve with a constant SRTP; a discount factor of 0.1 is pushed out 50 years, from about year 120 to year 170. The sum of a uniform cashflow of $1 over 200 years is about $50 at a constant 2% discount rate, and about $59 with a declining discount rate, which is a 19% increase. The impact of a declining SRTP would be proportionally much greater if an initiative had growing benefits over the long-term. Figure 8: Discount factors with and without declining SRTP 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 Average discount factor DF with DDR No DDR WP 25/01 | Deriving values of the social rate of time preference 15 References Abelson, Peter and Tim Dalton (2023) Declining Discount Rates in Cost Benefit Analysis, Applied Economics Pty Ltd report for the NZ Treasury, www.treasury.govt.nz/publications/commissioned-report/declining-discount-ratescost-benefit-analysis Atkinson, Anthony B. (1970) ‘On the Measurement of Inequality’, Journal of Economic Theory, 2, 244-263 Creedy, John (2007) ‘Policy Evaluation, Welfare Weights and Value Judgements: A Reminder’, Australian Journal of Labour Economics, Vol. 10, No. 1, March 2007, pp 1-15 Creedy, John and Passi, Hemant (2017) Public sector discount rates: A comparison of alternative approaches, New Zealand Treasury Working Paper 17/02 Cropper, Maureen, Mark Freeman, Ben Groom and William Pizer (2014) "Declining Discount Rates", American Economic Review, 104 (5): 538–43 Dixit, Avinash (1993) The Art of Smooth Pasting, Harwood Academic Publishers Dixit, Avinash, and Pindyck, Robert (1994) Investment Under Uncertainty, Princeton University Press Drupp, Moritz A., Mark C. Freeman, Ben Groom, and Frikk Nesje (2018) ‘Discounting Disentangled’, American Economic Journal: Economic Policy, 10(4): 109–134 https://doi.org/10.1257/pol.20160240 Edgeworth, F.Y. (1897) ‘The Pure Theory of Taxation’, Economic Journal. Reprinted in Papers Relating to Political Economy, 2, 63-125. London: Macmillan Emmerling J, Groom B, Wettingfeld T. (2017) ‘Discounting and the representative median agent’, Economics Letters, 161:78–81 Evans, Chris (2018) The Compliance Costs Of Taxing Capital Gains, report prepared for the Tax Working Group, Professor Chris Evans: The Compliance Costs of Taxing Capital Gains - 13 November 2018 - Information Release - Tax Working Group - New Zealand Evans, David (2005) ‘The Elasticity of Marginal Utility of Consumption: Estimates for 20 OECD Countries’, Fiscal Studies, 26, 197-224 Gollier, Christian, and Martin Weitzman (2010) ‘How should the distant future be discounted when discount rates are uncertain?’ Economics Letters 107:350–353 Gollier, Christian (2013) Pricing the Planet’s Future: The Economics of Discounting in an Uncertain World. Princeton, NJ: Princeton University Press Grimes, Arthur (2023) How Should the New Zealand Government Discount Future Payoffs? Motu Economic and Public Policy Research report for the NZ Treasury, www.treasury.govt.nz/publications/commissioned-report/how-should-new-zealandgovernment-discount-future-payoffs WP 25/01 | Deriving values of the social rate of time preference 16 Groom, Ben, and David Maddison (2013) Non-Identical Quadruplets: Four New Estimates of the Elasticity of Marginal Utility for the UK. Grantham Research Institute on Climate Change and the Environment Working Paper 121 Groom, Ben, and David Maddison (2019) ‘New Estimates of the Elasticity of Marginal Utility for the UK’, Environmental Resource Economics 72: 1155–1182 Groom, Ben, Moritz Drupp, Mark Freeman, Frikk Nesje (2022) The future, now: A review of social discounting. Annual Review of Resource Economics, 467-491 Guthrie, Graeme (2021) ‘Discounting, Disagreement, and the Option to Delay’, Environmental and Resource Economics, volume 80, 95–133 HM Treasury (2024) The Green Book - Central Government Guidance on Appraisal and Evaluation www.gov.uk/government/publications/the-green-book-appraisaland-evaluation-in-central-government/the-green-book-2020 Inland Revenue (2023) Annual Report 2023, Inland Revenue Annual Report, Te Tari Taake Pūrongo ā-Tau, 2022–23 (ird.govt.nz) NZIER (2024a) A Price On Time: Rethinking the Public Sector Discount Rate, NZIER Insight 114, www.nzier.org.nz/publications/a-price-on-time-rethinking-the-publicsector-discount-rate-nzier-insight-114 NZIER (2024b) Perceptions of fairness in New Zealand, Phase 1 report, NZIER report to the New Zealand Treasury, August 2024, www.nzier.org.nz/hubfs/Public%20Publications/Client%20reports/Fairness%20Pha se%201%20Report.pdf Pirttilä, Jukka; Uusitalo, Roope (2007) Leaky bucket in the real world: estimating inequality aversion using survey data, CESifo Working Paper, No. 2026, Center for Economic Studies and ifo Institute (CESifo), Munich Ramsey, Frank (1928) ‘A Mathematical Theory of Saving’, The Economic Journal, Volume 38, Issue 152, 1 December 1928, 543–559 Rees, Martin (2003) Our Final Hour: A Scientist's Warning: How Terror, Error, and Environmental Disaster Threaten Humankind's Future In This Century—On Earth and Beyond, Basic Books, New York City Sense Partners (2022) Tūhoe economic worldview – Mapping to an orthodox framework, Final Report to the Ministry of Business, Innovation and Employment, 23 August 2022, www.mbie.govt.nz/business-and-employment/economicdevelopment/tuhoe-economic-worldview-mapping-to-an-orthodox-framework Stern, Nicholas, S Peters, V Bakhshi, A Bowen, C Cameron, S Catovsky, D Crane, S Cruickshank, S Dietz, N Edmonson, S-L Garbett, L Hamid, G Hoffman, D Ingram, B Jones, N Patmore, H Radcliffe, R Sathiyarajah, M Stock, C Taylor, T Vernon, H Wanjie and D Zenghelis (2006) Stern review: the economics of climate change. London: HM Treasury WP 25/01 | Deriving values of the social rate of time preference 17 Thompson, Chris (2022) Equality, equity, and distributive justice. Background Paper to Te Tai Waiora: Wellbeing in Aotearoa New Zealand 2022. Treasury Analytical Paper 22/03, www.treasury.govt.nz/sites/default/files/2022-12/ap22-03.pdf Turk, Zachary (2024) Discounting Disentangled: Aotearoa, https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4830535 Treasury (2024a) Treasury Circular 2024/15: Updated Public Sector Discount Rates for Cost Benefit Analysis, www.treasury.govt.nz/publications/circulars/treasurycircular-2024-15-updated-public-sector-discount-rates-cost-benefit-analysis Treasury (2024b) Fiscal Strategy Model - BEFU 2024, www.treasury.govt.nz/publications/fsm/fiscal-strategy-model-befu-2024 Weitzman, Martin (1998) ‘Why the Far-Distant Future Should Be Discounted at Its Lowest Possible Rate’, Journal of Environmental Economics and Management, November, 36(3), 201-08 Weitzman, Martin (2007) ‘Subjective Expectations and Asset-Return Puzzles’, American Economic Review, 97(4), 1102–1130 WP 25/01 | Deriving values of the social rate of time preference 18 Appendix 1 – Mathematical basis for SRTP There are two key dimensions we will abstract away from in the analysis that follows. The first is that goods and services could vary between market and non-market items, with a particular distinction between natural environmental attributes and general market consumables. The second is individual citizens can have vastly different bundles of goods and services that they consume also, with inequality between the rich and poor. Here we suppose we can simply add together the value of aggregate per capita consumption 𝑐𝑐𝑡𝑡 in each period 𝑡𝑡 for all kinds of goods and services for all people. This means aversion of the decision maker to within-period inequality is ignored, and we do not here consider different intertemporal treatments to natural environment versus other commodities. These two simplications may not greatly affect what follows. If a particular natural species were to be increasingly threatened with extinction this could be represented through an escalating real price assigned to each item of that species, and so be implicitly captured in 𝑐𝑐𝑡𝑡. However, there could be a theoretical basis to consider having different discount rates across categories of consumption (Grimes 2023). If crosssectional matters of inequality, inequity, and distributional justice were important (as outlined in Thompson 2022) then the decision maker may be able to weight the impacts of consumption across people differently when deriving the 𝑐𝑐𝑡𝑡 term.12 However, if matters of inequality, inequity, and distributional justice were both cross-sectional and occured over time (such as intergenerational poverty traps / privilege) then more general approaches may be required.13 A policy intervention may change the amount of total consumption over time, and the purpose of the judgement is whether that is preferrable to not intervening. The decision maker may compare the social welfare in each state to determine which scores higher. 12 This is feasible for ‘welfarist’ or consequentionist theories of distributive justice, which assume that outcomes (and their distribution) are morally relevant. These theories include utilitarianism (maximising summed utilities), maxi-min (maximise the welfare of the worst-off person), prioritarianism (weighting the worse-off higher), and sufficientarianism (ensuring all have enough utility). Weighting would not address equity according to other theories of justice that take factors other than welfare and its distribution to be morally relevant. These other non-consequentionist theories include luck egalitarianism (equal opportunity rather than equal outcomes is important), relational egalitarianism (ensuring the moral equality of people), and libertarianism (just and fair processes and actions). These latter three seem to fit most with the ways New Zealanders think about fairness (NZIER 2024b). 13 For example, Emmerling et al (2017) adjusted the SRTP for growing intra-period inequality, which is akin to reducing the growth rate in consumption. New Zealand was not modelled, but for Australia they estimate the SRTP would be 0.16% points lower if 𝜇𝜇= 1 given the faster rate of growth of mean household income than median household income.