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“A Dynamic Analysis of Human Welfare in a Warming Planet”

Humberto Llavador, Universitat Pompeu Fabra

John E. Roemer, Yale University

Joaquim Silvestre, University of California, Davis

10th International Meeting of the Society for Social Choice and Welfare

State University – Higher School of Economics, Moscow

July 2010

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Need to “stabilize” GHG concentrations, understood as change in temperature slow enough so that the effects of climate change on human welfare can be attenuated via

* Mitigation (e. g., levees)

* Adaptation (e. g., moving North)

• Recent climate science supports the view that stabilization will require very low GHG emissions in the future

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Our Approach

• Postulate a path of low global GHG emissions which, according to current climate science, is broadly consistent with the stabilization of GHG concentrations

(IPCC 2007 stark targets)

Very low emissions after 2050

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Postulated Path for CO2 Emissions

Implied Path for CO2 Concentrations

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Discounted Utilitarianism• Many economists take intergenerational welfare to

be the discounted sum of utilities of the present and all future generations (Nordhaus,Weitzman, Stern). The dispute among them concerns what discount rate to use.

• We find discounted utilitarianism ethically unacceptable, at least for the low pure-time discount factors typically used, which put a weight on the utility of future generations much lower than that of the present generation.

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• The only ethical justification for putting a lower weight on the welfare of future generations in the utilitarian calculus should be based on a positive probability of extinction of mankind.

• As argued in the Stern Review, this rationale would perhaps support a discount rate of = 0.001 = 0.1% per annum (discount factor of 0.999001 per annum) associated with a 0.905 probability of mankind’s surviving 100 years.

• A rigorous development of this idea requires a model of uncertainty (Llavador et al., JME, in press).

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Sustainability• Most non-economists (other social scientists,

physical scientists) advocate some notion of sustainability, rather than discounted utilitarianism.

• Discounted utilitarianism has a long history in economic theory, but little popular resonance.

• Our work does take sustainability as the basis for evaluating social welfare.

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Our Normative Criteria

* Pure Sustainability Optimization

= Intergenerational Maximin (Leximin)

Zero growth of welfare (in our model)

* Sustainable Growth Optimization

= Max welfare of first generation

subject to maintaining welfare growth

at a pre-specified rate

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• We do not propose a rule for adjudicating among various values of : but our calculations suggest that values of g of 1% to 2% per year (28% to 64% per generation) are ethically more attractive than the Maximin path at = 0.

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We find that the low-emission path that we postulate is compatible with

* Sustainable levels of human welfare

* Steady growth in human welfare

NOT SURPRISINGLY, THERE IS A TRADEOFF:

HIGHER GROWTH RATES

IMPOSE LOWER UTILITY ON THE FIRST GENERATION

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It turns out that:

• The proposed paths require doubling the resources devoted to the creation of knowledge

• Faster growth rates demand substantially heavier investments in education

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Method• Define human welfare measure

• Construct & calibrate intergenerational model

• Develop algorithm to compute paths for the economic variables satisfying our normative criteria

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Disclaimers

* The present paper focuses upon intergenerational equity Our current research extends the analysis to the

intragenerational conflict between North and South

* We ignore population issues, and uncertainty

Accordingly, we postulate one representative agent for each generation, to be thought of as the average person

[One date t = 1, 2, … per generation = 25 years]

* We calibrate our technology parameters and reference values by US data representative agent interpreted as average US person

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Human welfare

• We define human welfare, or utility, at each date (or generation) as involving much more than commodity consumption -- which is one of its elements

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* Commodity consumption ct (per capita)

* Leisure (per capita), which is more valuable if the person has more education

* The stock of knowledge (per capita) a public good

* The quality of the biosphere (global),

a public good, inversely related to GHG concentrations or global temperature

The utility of Generation t is a functionˆ( ) ( ) ( ) ( )c n mn m m

t t t tc x S S S

tx

ˆ( )m mtS S

ntS

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Stock-Flow Dynamics

(1) At each date t, the current generation (t ) inherits from the past:

* A biospheric quality (GHG concentrations)

* A stock of knowledge

* A level of education (per capita) which was provided by its parental generation (t – 1)

* A stock of physical capital (per capita)

1e

tx

ktS

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(2) Generation t allocates its labor into four uses (in efficiency units):

* The education of children

* The production of output, to be divided between consumption and the investment in physical capital

* Investment in knowledge (R&D)

* Leisure time

, 1e c nt t t t tx x x x x t

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(3) The activities of Generation t result in:

* GHG emissions, et which impact on the biosphere, inducing a new level of GHG concentration

* A new level of knowledge

* A new level of capital stock

* A level of education for Generation t + 1

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(4) The inherited stocks & the activities of Generation t determine its welfare

(5) The new stocks (of educated adults, capital, knowledge, and biospheric quality) are passed on to the next generation, which again engages in these activities, and so on

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Welfare of Generation t

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1

( , , , , )

( ) ( ) ( ) ( ) ( )

,

( , , , ) 0, 0

c k n e m

c k n mt t t t t

c k n mt t t t t

t t

c k n e m

f x S S e S

k x S S e S

c i

=

³ +

<?

* Aggregate production function

>

>>

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Laws of Motion

Law of motion of physical capital

1 2(1 ) , 1k k kt t tS k i S t-- + ³ ³

Law of motion of stock of knowledge

1 3(1 ) , 1n n n nt t tS k x S t-- + ³ ³

Education production function

4 1 , 1et tk x x t

> >

> >

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Objective 1: Pure Sustainability Optimization

* Defined as the sustainability of welfare at the highest possible level

* We seek the path of activities over time which maximizes the level of welfare that can be enjoyed by every generation forever

* Motivated by the moral irrelevance of the date of birth

* max subject to and s. to postulated GHG emission path, production

relations, laws of motion, resource constraints, and initial conditions of stocks in the base year (2000).

ˆ( ) ( ) ( ) ( ) , 1 c n mn m mt t t tc x S S S t

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Objective 2: Sustainable Growth Optimization * We seek the path of activities over time which maximizes

the welfare of Generation 1 constrained to subsequently maintaining a given rate of growth in the welfare of generations 2, 3, …

* Given > 0, max subject to:

and s. to postulated GHG emission path, feasibility & initial conditions

* Motivated by the public-good character of human development (we all desire mankind’s progress)

1ˆ( ) ( ) ( ) ( ) (1 ) , 1 c l n ml n m m tt t t tc x S S S t

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• Consider a pair such that the constant sequence is an environmentally feasible flow-stock path

• Consider a rate of utility growth per generation.

Computational strategy based on the Ray Optimization Theorem below

* *( , )me S

* *1(( , ))

m

te S

Optimization Program

Ray Optimization TheoremGiven a triple (, e*, Sm*), (e*, Sm*) is a CO2 emission-stock pair,

there exists a ray (, e*, Sm*) 3+ such that, if the

initial conditions belong to this ray, then at the solution to Program E(, e*, Sm*) utility grows at rate , emissions and concentrations remain constant at and all economic variables grow at rate g (per generation), where 1 + = (1 + g)m

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Algorithm: Fix [rate of growth & long run emissions and concentrations]

Ray

t = 1

Initial Condition

t = 2

( , *, *) me S2000 2000 2000( , , )e k nx S S

1 1 1( , , )e k nx S S

2 2 2( , , )e k nx S S

etx

,k nt tS S

( , *, *) me S

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Calibration

• We calibrate our parameters by US data

• GHG stocks and emissions are global; all other variables are in per capita terms

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Two scenarios on future US emissions

Scenario 1. US continues to emit 24% of globalOptimistic path as far as the welfare of US residents(upper bound) It is unreasonable to suppose that the US will

emit more than its present share of GHGs

Scenario 2. US emits its per capita share (approximately 4% of global)

Pessimistic path for US (lower welfare bound) It is likely that trading in permits will result in the US’s

emitting more than its per capita share

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Results. Table 4 (Top)

Sustainable Welfare (Zero Growth)

Note: Increased physical capital & knowledge

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Results. Table 4 (Bottom)Sustainable Welfare (Zero Growth)The allocation of labor time

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Results. Table 5 (Top)

Sustainable Growth1% annual growth or 28% generational growth

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Results. Table 5 (Bottom)

Sustainable Growth1% annual growth or 28% generational growth

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Results. Table 6 (Top)

Sustainable Growth2% annual growth or 64% generational growth

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Results. Table 6 (Bottom)

Sustainable Growth2% annual growth or 64% generational growth

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Table 7.

The sacrifice of the first generation for the sake of growth

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Sustainable Growth

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The Allocation of Labor TimeYear 2000 Reference

• Education: 0.0333

• Production of consumption: 0.2150

• Investment in physical capital: 0.0683

• Creation of knowledge: 0.0167

• Leisure: 0.6667

1.0000

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Table 8 indicates how these fractions should be

modified in the proposed solutions

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Conclusion 1

Human quality of life can be sustained forever at a level substantially higher than the year-2000 reference level (26% higher in Scen. 1, or 17% higher in Scen. 2)

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Conclusion 2

For the low US emissions of Scenario 2, the sustainable consumption is lower than the year-2000 reference consumption level

The increase in utility is then due to the increases in the stock of knowledge and in the quality of leisure

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Conclusion 3

Moderate growth rates can be achieved at the cost of a small reduction in the utility of the first generation, which stays well above the year 2000 reference level.

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Conclusion 4

The most important change required by the implementation of the desirable paths is the (approximate) doubling the reference fraction of labor devoted to the creation of knowledge, whereas the fractions of labor allocated to consumption and leisure are similar to those of the reference year 2000.

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Conclusion 5Higher growth rates require * Substantial increases in the fraction of labor

devoted to education (of the order of a 30% increase for each additional 1% of annual growth)

* Moderate increases in the fractions of labor devoted to knowledge and the investment in physical capital (of the order of a 5% increase for each additional 1% of annual growth).

* Minor decreases in the amount of labor-time devoted to the production of consumption goods and to leisure (of the order of a 2% decrease for each additional 1% of annual growth).

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Comparison with the Literature

Nordhaus (2008a)

Maximizes an intergenerational social welfare function. He does not fix a path of emissions. Instead, he proposes a law of motion of biospheric degradation, and optimizes over not only the paths of consumption, investment, and capital, but also of emissions.

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• Nordhaus’ social welfare function in is discounted utilitarian. He maximizes the discounted sum of generational utility levels,

where Lt is the number of people in generation t.• He calls the and of “central” and “unobserved

normative parameters,” reflecting “the relative importance of the different generations.”

• (Note that the our Pure Sustainability Optimization objective function could be viewed as a limit case for Lt = 1, = 0 and ).

1

1

1 1( )

1 (1 )

T

t t ttL c

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• Nordhaus chooses = 2 and discount rate corresponding to a per year rate of

calibrated from the rate of time impatience of existing consumers, and calculated via the Ramsey equation.

ˆ 0.015

,

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• The utility function depends only on the consumption of a produced commodity.

• Accordingly, emissions and biospheric quality affect human welfare only indirectly, through their impact on production.

• Nordhaus proposes an exogenous path of technological change. There is neither knowledge-production sector in his model, nor an education sector.

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* The paths for emissions and concentrations proposed as optimal by Nordhaus differ markedly from the ones that we postulate

* He proposes as “optimal” a path where emissions and concentrations keep increasing past the end of the 21st century

* His proposed values for 2100 are about 11 GtC in emissions, with concentrations at 586.4 ppm at 2100 and at a peak of about 680 ppm in 2180

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Emissions

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Concentrations

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• In light of the recent climate science research, we view Nordhaus’s “optimal” emission and concentration figures as excessively high, likely to bring about irreversible changes in temperature and unavoidable negative impacts in the welfare of future generations

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The Stern Review (2007)

• It does not carry out a full optimization exercise. It is rather a Cost-Benefit (C-B) analysis arguing that the “costs of inaction are larger than costs of action.”

• Assuming a path of growth for the GDP, it compares a Business as Usual (laissez-faire) path of GHG emissions with paths entailistrong, early action on climate change

its benefits outweigh its costs.

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• Discount rates have different roles in Cost-Benefit Analysis and in discounted-utilitarianism optimization

• Discounted-utilitarianism optimization uses the pure time discount rate to weight the utilities of the various generations

• Cost-Benefit Analysis uses the consumption discount rate to evaluate the changes in future consumption streams due to a particular (marginal) investment project, relative to an exogenously reference path

• The Stern Review uses a pure time discount rate of = 0.001 (based on the survival justification), together with = 1 and (1.3 % per annum), yielding a consumption discount rate of 0.014.

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• Its commentators suggest higher consumption discount rates

• Because the Stern Review does not solve an optimization program, its recommendations are in principle open to the objection that the consumption discount rate should reflect the rates of return of the available investment alternatives:

• Even if, using a consumption discount rate of 0.014, carbon emission reductions pass the Cost-Benefit test, future generations could conceivably be better off if the current generation avoided incurring the costs of GHG reductions and invested instead in other intergenerational public goods.

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• Our approach is in a sense dual to Cost-Benefit analysis.

• The latter takes as given a path for the economic variables, and recommends a path for the environmental variables (based on a cost-benefit criterion in the spirit of discounted utilitarianism).

• We, on the contrary, take as given a path for the environmental variables, and recommend paths for the economic variables (based on the criteria of pure sustainability and of sustainable growth).

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Next (work in progress)• Our 2000 welfare levels are that of the

average US citizen • Justice requires designing paths which

allow those in the South to converge in welfare to the welfare enjoyed in the rich North in 2 or 3 generations

• What sacrifices this will require of the North is the topic of our current research

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Objective

• Max utility of first generation in South

subject to

convergence to ray in two generations

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Model

• Calibrated to US and China data

• Same production function, but North & South initially have different education levels and stocks of knowledge & physical capital

• Knowledge spills over from N to S

• Transfers of output allowed

Method

• Choose growth rate (say 1% per annum = 28% per generation)

• Constrain North to grow at rate

• Converge to ray, so that N & S reach same utility, and grow at rate for t > 3

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Hence, at the solution

• Recall that the utility levels of North’s and South’s Generations 0 are historically given

• For t > 3, both N & S are in steady state, their utilities equal and growing together at rate .

• Generations 2 and 3 in N & S are the transition generations

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And

• The utility level of N’s generation t is that of N’s generation 0 multiplied by (1+ )t

• The utility of South’s transition generations 1 & 2 is endogeneized via the optimization program

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Hence

• The worst off generation (t > 0) is South’s Generation 1

• The Maximin principle requires the maximization of its utility

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But then

• Generation 1 in South invests little in education, with imposes low utility in South’s Generation 2

• The resulting path implies, in the South, little growth from 1 to 2, together with a big jump from 2 to 3.

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On the other hand

• Maximizing the utility of Generation 2 in the South subject to Generation 1 growing at rate of at least gives a smoother path with little loss of utility for Generation 1

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Optimal paths for a 1% sustained annual growth(28% per generation)

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Summary

• Ethical Criteria– Sustainability and sustainable growth rather

than discounted utilitarianism

– Optimization rather than C-B analysis

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Optimistic Conclusions

• One million caveats– Purely normative analysis: what an external

ethical observer would recommend

(abstract from bargaining, mkt. equilibrium).– No uncertainty– Cobb-Douglas– Central role of education & knowledge

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