Prices and Taxes in a Ramsey Climate Policy Model under Heterogeneous Beliefs and Ambiguity
Abstract
:1. Introduction
“The trouble with the world is that the stupid are cocksure and the intelligent are full of doubt.” Bertrand Russell
- Unknown private beliefs
- (a)
- Political planner: The government strategically assumes private beliefs to be true. Its ignorance of private beliefs produces ambiguity.
- (b)
- Paternalistic planner: The government trusts and adheres to the true model. However, the presence of unknown private beliefs creates ambiguity for policy.
- (c)
- Pessimistic planner: Having doubts about the model and facing unknown private beliefs, the planner confronts two sources of ambiguity.
- Known pessimistic private beliefs
- (a)
- Pessimistic planner: The planner’s own doubts doubts about the model and the private sector’s pessimistic doubts that constrain the Ramsey planner’s policy produce two sources of ambiguity.
2. Multiplier Preferences
2.1. Skepticism (Random Belief Distortion)
2.2. Pessimism (Ambiguity Aversion)
3. Households
3.1. CRRA Preferences
3.2. The Household’s Budget Constraint
3.3. The Consumer’s Maximization Problem
3.4. Pessimistic Consumer
3.4.1. Inner Minimization
3.4.2. Outer Maximization with Implied Risk-Sensitive Recursion
3.5. Skeptical Consumer
3.6. Arrow–Debreu Prices under Belief Distortions
4. Firms
5. Two Versions of Hotelling’s Rule
6. The Ramsey Planner’s Constraints
6.1. National Income Identity and the Government Budget Constraint
6.2. The Ramsey Planner’s Implementability Constraints
7. Ramsey Planning in Four Belief Regimes
7.1. Political Planner
7.1.1. Inner Game with Nature
7.1.2. Outer Maximization with Implied Risk-Sensitive Recursion
7.1.3. Paternalistic Planner
7.1.4. Inner Game with Nature
7.1.5. Outer Maximization with Implied Risk-Sensitive Recursion
7.2. Pessimistic Planner, Skeptical Consumer
7.2.1. Inner Minimization
7.2.2. Outer Maximization with Implied Risk-Sensitive Recursion
- The consumer has rational expectations (). As noted earlier, this case has been widely treated in papers on robust climate policy.
7.3. Pessimistic Planner, Pessimistic Consumer
Outer Maximization with Implied Risk-Sensitive Recursion
8. The Equilibrium Price of Capital
9. A Comment about Methodology
10. The Social Cost of Carbon
11. The Carbon Tax
12. An Ex Ante Tax on Capital
13. Policy Implications of Belief Heterogeneity and Ambiguity
13.1. Preliminary Results
13.2. Homogeneous Beliefs
- 1.
- The social cost of carbon exceeds the standard formulation derived in (88) for a social planner, unless ,
- 2.
- The carbon tax premium is positive, unless ,
- 3.
- The ex ante rate on capital is negative, unless ,
13.3. Heterogeneous Beliefs: Skeptical Consumers
- 1.
- Social cost of carbon is higher than under RE, unless :
- 2.
- The carbon tax is higher than under RE, unless :
- 3.
- Ex ante capital tax rate is less than under RE, unless :
- 1.
- The social cost of carbon contains an ambiguity premium for both Ramsey and social plans:
- 2.
- The carbon tax is higher than under RE, unless :
- 3.
- Ex ante capital tax is lower than under RE, unless :
- 1.
- The social cost of carbon contains an ambiguity premium, unless :
- 2.
- The carbon tax contains an ambiguity premium, unless :
- 3.
- The ex ante capital tax contains an ambiguity subsidy, unless :
- 1.
- The social cost of carbon contains an ambiguity premium in both Ramsey and social plans:
- 2.
- The carbon tax contains a positive or negative ambiguity premium, unless :
- 3.
- The ex ante capital tax is a subsidy in both Ramsey and social plans:
13.4. Heterogeneous Beliefs: Pessimistic Consumers
- 1.
- The social cost of carbon contains an ambiguity premium in both Ramsey and social plans:
- 2.
- If , the premium on the carbon tax may be positive or negative, and is negative otherwise:
- 3.
- The ex ante capital tax rate may be positive or negative:
14. A Feedback from Taxes to Consumers’ Pessimistic Beliefs
15. Conclusions
Funding
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. The Household’s Intertemporal Budget Constraint
Appendix B. Derivation of Implementability Constraint (37)
Appendix C. The Social Cost of Carbon
- Skeptical private sector
- (a)
- Planner has no ambiguity about distorted private beliefs (, , )
- (b)
- Political planner (, )
- (c)
- Paternalistic planner (, , )
- (d)
- Pessimistic planner (, ,
- Pessimistic consumers
- (a)
- Pessimistic planner (, , )41Adding and subtracting
Appendix D. Sign of Carbon Tax Premium χ
- Skeptical private sector
- Pessimistic consumers
- (a)
- Pessimistic planner (, ; , )47,Adding and subtracting
Appendix E. Sign of Ex Ante Capital Tax
- Skeptical private sector
- (a)
- No ambiguity (, , , ).
- (b)
- Political planner (, , , ).
- (c)
- Paternalistic planner (, , , ).
- (d)
- Pessimistic planner (, , , )
- Pessimistic consumers
- (a)
- Pessimistic planner (, , , )Adding and subtracting
1 | The Yale Climate Opinion Maps 2020 (Marlon et al. 2020) is encouraging in that 72 percent of respondents said global warming is real and a threat to humanity and the planet, with 57 percent believing it to be human caused. |
2 | Uncertainty in climate modeling has been intensively treated in the literature; see Pindyck (2007, 2013b), Roe and Baker (2007), Weitzman (2007, 2009, 2013), Heal and Millner (2013), and lately Barnett et al. (2020). |
3 | Tipping points include shutdown of the Atlantic Meridional Overturning Circulation, West Antarctic ice sheet disintegration, Amazon rainforest dieback, West African monsoon shift, permafrost and methane hydrates, coral reef die-off, Indian monsoon shift, Greenland ice sheet disintegration, boreal forest shift, and permafrost and methane hydrates. Dietz et al. (2021) note that such “climate tipping points are subject to considerable scientific uncertainty in relation to their size, probability, and how they interact with each other. Their economic impacts are even more uncertain, and consequently, these are often ignored or given a highly stylized treatment that fails to accurately represent geophysical dynamics and is nearly impossible to calibrate. As a result, tipping points are only weakly reflected in the policy advice economists give on climate change, typically by way of caveats and contextualization, rather than an integral part of the modeling that gives rise to estimates of the social cost of carbon (SCC) and other economic metrics of interest. |
4 | |
5 | This is not the first paper to dwell on a symmetry between carbon and capital taxation arising from a difference between social and private discount factors. Barrage’s (2020) shows that the very logic leading to an optimal zero ex ante tax on capital derived in (Atkeson et al. 1999; Chari et al. 1994; Zhu 1992) implies a positive Pigou tax on carbon. |
6 | In a recent paper, Dietz and Niehörster (2019), estimate ambiguity loads—i.e., the extra insurance premiums due to ambiguity—and show how these depend on the insurer’s attitude to ambiguity. |
7 | |
8 | In Belfiori’s (2015) inter-generational model of altruism, the difference in discount factors is endogenous because each generation of households living in an infinite sequence of generations assigns a positive weight to the welfare of future generations, causing the government’s discount factor to rise above that of households and leading to a carbon tax that exceeds the standard Pigou tax. |
9 | An axiomatization of multiplier preferences can be found in Strzalecki (2011). |
10 | Throughout, the expectations operator denotes the mathematical expectations of some function , , with respect to the probability density : . |
11 | This means one cannot assign positive probability to events as functions of that have probability measure zero under the distribution of the approximating model or alternatively, the distorted and the approximating distributions are, at the very least, in agreement about which events have zero probability and which events are certain. |
12 | Previous systematic treatments of decision making under ambiguity include Klibanoff et al. (2005, 2009), and Traeger (2014). |
13 | Following examples by Anderson et al. (2013), Golosov et al. (2014), Nordhaus (2008), Li et al. (2016), and others, I omit leisure (or hours worked) to keep things simple. I also leave out any utility effects of environmental quality caused by climate change, since no conclusions relevant for this paper would be affected by their inclusion. See however, Barrage (2020). |
14 | See Hennlock (2009). While gets at early resolution of uncertainty, as noted by Bansal and Yaron (2004), the assumption of CRRA misses certain aspects of risk aversion that may create pricing issues. |
15 | For analytical convenience that will not affect conclusions, I will assume that energy production is costless, as is approximately true for oil (see Golosov et al. 2014). Accordingly, it is reasonable to fold the fossil energy extraction/production sector into the household, by making the consumer be the owner of the resource Q and the seller of energy . This assumption differs from the literature (see Golosov et al. 2014; Nordhaus 1993), where, instead, the consumer owns the energy producing firm and receives its profits. However, ownership of the firm is merely a financial veil for direct ownership of the resource itself, hence a distinction without a difference. |
16 | The implied relative risk aversion is that rises as . and . See Epstein and Zin (1991). |
17 | The formula for can be derived as a special case of Epstein and Zin (1989) and Weil (1990) preferences
|
18 | Readers will recognize Hansen and Sargent’s (1995) discounted risk-sensitive recursion related to recursive preferences introduced earlier by Epstein and Zin (1989) and Weil (1990). Risk-sensitivity as a concept was introduced by Jacobson (1973) and later familiarized by Whittle (1981). |
19 | Excellent expositions of the physics of climate change for economists are Hsiang and Kopp (2018) and Traeger’s (2018) description of the complex sets of channels involved in the transmission from carbon to temperature change. |
20 | Barnett et al. (2020) also adopt this measure of climate response. |
21 | Modeling damages resulting from climate change as negative total productivity shocks to the economy follows common practice, as for example in Golosov et al. (2014). The damaging effects of increased temperature on productivity in the world economy have been documented by Burke et al. (2015). Donadelli et al. (2017) used vector autoregressions to show that with a 50-year horizon, a one-standard deviation temperature shock lowers both cumulative output and labor productivity growth by 1.4 percentage points. Based on their model, they further show that temperature risk is associated with non-negligible welfare costs of 18.4% of the agent’s lifetime utility that grow exponentially with the size of the impact of temperature on TFP. |
22 | |
23 | For the derivation see Appendix A. |
24 | Henceforth, the n multiplier refers to the government’s belief, and m to the consumer’s belief distortion. |
25 | In this paper, I do not consider learning by either planner or private agents. An approach to doing so is available in Tetlow and von zur Muehlen (2009) who study robust monetary policy using structured singular value analysis when agents have misspecified models but are engaged in learning under the handicap that their learning of the reduced form of the economy is subject to potentially destabilizing parameter perturbations. |
26 | |
27 | The first application of risk-sensitive decision making to economic policy is van der Ploeg (1984). An early treatment of Knightian uncertainty in economics is von zur Muehlen (1982). In his analytical climate economy model, Traeger (2018) posits risk-sensitive preferences attributable to the planner representing the consumer. |
28 | |
29 | This belief regime is related to the robust fiscal policy model in Karantounias (2020). A fifth possible belief regime, wherein the planner has doubts but the private sector trusts the extant model, is not treated here having been widely discussed in the referenced literature. |
30 | |
31 | Current estimates of the social cost of carbon emissions are around USD75 per ton of carbon. Some consider this a gross underestimate and suggest the number is closer to USD220 per ton. (See Moore and Diaz (2015) and Than (2015)). |
32 | Initial wealth is a function of and and the initial tax rates and . It is well known that since initial capital is supplied inelastically, the government has an incentive to raise the initial capital tax as high as possible. Likewise, with given, there is nothing (i.e., no welfare criterion) to prevent the planner from expropriating the energy sector by setting . As is conventional, I fix both and at 0. |
33 | By definition, next period’s stock is given, i.e., determined by previous , hence not influenced by . |
34 | A positive warming shock reduces both consumption and productivity and therefore the value of future welfare . |
35 | Wilbanks et al. (2008) present empirical evidence that the net effect of declining consumption on wealth is negative. |
36 | Proofs are provided in Appendix C, Appendix D and Appendix E, and utilize the preceding assumptions and lemmas, particularly the results that , , , , , , and , being positively correlated with x, are positively correlated with with each other. |
37 | Pascal argued that a rational person should live as though God exists and seek to believe in God. If God does not exist, such a person will have only a finite loss (some pleasures, luxury, etc.), whereas if God does exist, he stands to receive infinite gains (as represented by eternity in Heaven) and avoid infinite losses (an eternity in Hell). |
38 | This section is inspired by Karantounias (2013) who studied a fiscal authority’s ability to manage pessimistic expectations. |
39 | See for example, Traeger (2018). Other features not addressed here include (i) Arrhenius’ Greenhouse Law describing radiative forcing that connects carbon with temperature change, as described in Hassler et al. (2016), (ii) tipping points analyzed by Lemoine and Traeger (2016) and (Cai et al. 2013), describing abrupt nonlinear climate changes that pose a potentially existential threat to humanity in ways that may override concerns with belief and skepticism, and (iii) polar amplification analyzed by Brock and Xepapadeas (2017). |
40 | The derivations in this and the next two Appendices utilize these facts: Let represent four random, possibly related variables, and a a non-random variable or constant. Then
|
41 | The derivation uses (77)
|
42 | By the law of iterated expectations . |
43 | Where called for, the proofs use the result that for any martingale process m, , if
|
44 | Constancy of follows from (71). |
45 | In the following, is purely random and independent of any other variable in the economy. Likewise, is independent, where from footnote 42, , and . |
46 | In the following, I use the facts that for each m, and also that . In addition, note that
|
47 | In the following, I use
|
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m | ||||||
---|---|---|---|---|---|---|
Skeptical consumers | ||||||
Political planner | 1 | |||||
Paternalistic planner | 1 | |||||
Pessimistic planner | ||||||
Pessimistic consumers | ||||||
Pessimistic planner |
I. Homogeneous Beliefs (RE) | ||||
Ramsey | >0 | ≥ | ||
Social | 0 | 0 | 0 | |
II. Heterogeneous Beliefs | ||||
1. Skeptical consumers | ||||
Ambiguity absent | ||||
Ramsey | >0 | ≥ | ≥ | ≤ |
Social | 0 | ≥0 | ≤0 | |
Ambiguity present | ||||
Political planner | ||||
Ramsey | >0 | ≥ | ≥ | ≤ |
Social | 0 | ≥ | ≥0 | ≤0 |
Paternalistic planner | ||||
Ramsey | >0 | ≥ | ≥ | ≤ |
Social | 0 | ≥ | ≥0 | ≤0 |
Pessimistic planner | ||||
Ramsey | >0 | ≥ | ⋛ | ⪋ |
Social | 0 | ≥ | ≥ | ≤ |
Effect of planner’s ambiguity | ||||
Ramsey | >0 | ↑ | ↑ | ↓ |
Social | 0 | ↑ | ↑ | ↓ |
Effect of consumer’s ambiguity | ||||
Ramsey | >0 | ↑ | ↓ | ↓ |
Social | 0 | ↑ | ↓ | ↓ |
2. Pessimistic consumers | ||||
Pessimistic planner | ||||
Ramsey | >0 | ≥ | ⋛ | ⪋ |
Social | 0 | ≥ | ⋛ | ⪋ |
Effect of planner’s ambiguity | ||||
Ramsey | >0 | ↑ | ||
Social | 0 | ↑ | ||
Effect of consumer’s ambiguity | ||||
Ramsey | >0 | ↑ | ||
Social | 0 | ↓ |
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von zur Muehlen, P. Prices and Taxes in a Ramsey Climate Policy Model under Heterogeneous Beliefs and Ambiguity. Economies 2022, 10, 257. https://doi.org/10.3390/economies10100257
von zur Muehlen P. Prices and Taxes in a Ramsey Climate Policy Model under Heterogeneous Beliefs and Ambiguity. Economies. 2022; 10(10):257. https://doi.org/10.3390/economies10100257
Chicago/Turabian Stylevon zur Muehlen, Peter. 2022. "Prices and Taxes in a Ramsey Climate Policy Model under Heterogeneous Beliefs and Ambiguity" Economies 10, no. 10: 257. https://doi.org/10.3390/economies10100257
APA Stylevon zur Muehlen, P. (2022). Prices and Taxes in a Ramsey Climate Policy Model under Heterogeneous Beliefs and Ambiguity. Economies, 10(10), 257. https://doi.org/10.3390/economies10100257