Maximin Utility with Renewable and Nonrenewable Resources

1
Maximin Utility with Renewable and Nonrenewable
Resources

• Jon M. Conrad

2
In the spirit of Rawls (1971)

• If you did not know the generation into which you
  would be born, what rules would you want society
  to adopt relating to consumption, the rate of
  harvest from a renewable resource, and the rate
  of extraction from a nonrenewable resource?
• You might want rates of consumption harvest, and
  extraction which maximized the utility of the
  least-well-off generation. This is known as the
  maximin criterion.
• In this paper, Conrad numerically explores the
  maximin criterion in a discrete-time model with a
  renewable and a nonrenewable resource and a
  finite number of non-overlapping generations.

3
The Rawlsian social contract

• Takes the form of a vector of four rates
• a the fraction of a manufactured good that is
  consumed
• ß the fraction of a renewable resource that is
  harvested
• ?F the fraction of a nonrenewable resource
  extracted for production of the manufactured
  good, and
• ?U the fraction of the nonrenewable resource
  extracted for direct consumption (utility).
• These rates are adopted by all generations.

4
The maximin social contract

• The maximin social contract does not lead to
  equal utility across a finite number of
  generations
• But it typically results in an equitable
  intergenerational distribution of utility, as
  measured by the Gini coefficient.
• We explore how the four rates in the maximin
  social contract depend on initial conditions and
  the number of generations.

5
Introduction

• Rawls (1971) asked what social contract one would
  want in place if
• "no one knows his place in society, his class
  position or social status, nor does anyone know
  his fortune in the distribution of natural assets
  and abilities, his intelligence, strength, and
  the like. The principles of justice are chosen
  behind a veil of ignorance." (Rawls, 1971, p.11.)
• To Rawls, the rational individual, not knowing
  his or her lot in life, would want rules in place
  that would maximize the utility of the
  least-well-off individual.
• This is now referred to as the maximin criterion.

6
II. The Model (a)

• Yt F(Kt ,QF, t )
• the output of the manufactured good by the tth
  generation.
• i.e. output is a function of the capital stock,
  Kt , and the rate of extraction from a
  nonrenewable resource, QF, t
• Population and work force are constant and
  unchanging through time.
• Ct aF(Kt ,QF, t )
• the consumption of manufactured goods, a gt 0
• Ht ßXt
• the harvest from a renewable resource of size Xt
  , 1 gt ß gt 0
• Ut U(Ct ,Ht ,QU, t )
• Utility of the tth generation depends on
  consumption of the manufactured good, harvest
  from the renewable resource, and QU, t
  (extraction from the nonrenewable resource
  consumed directly by the tth generation)

7

• QF, t ?FRt and QU, t ?URt
• denote the levels of extraction for production of
  the manufactured good and for direct consumption
  when the remaining reserves of the nonrenewable
  resource are Rt
• 1 gt ?F gt 0 , 1 gt ?U gt 0 , 1 gt (?F ?U ) gt
  0 ,
• Rt1 1 - (?F ?U) Rt
• the dynamics of remaining reserves,
• Kt1 Kt (1- a)F(Kt ,QF, t )
• the dynamics of the capital stock,

8

• Xt1 1- ß r(1- Xt/Xc )Xt
• the dynamics of the renewable resource
• r gt 0 is the intrinsic growth rate
• Xc gt 0 is the environmental carrying capacity
• and it is assumed that 1- r ß lt 1 (to avoid
  chaos)
• Let t 0,1,2,...,T be the number of generations
  under consideration, where t 0 is the current
  generation and T is the terminal generation
• a, ß, ?F , ?U denotes a social contract,
  invariant across the generations.

9
The optimization problem

• Seeks to find the social contract which will

10
Numerical analysis will require specification of

• The production function and the utility function
• and any parameters to those functions
• The parameter values for r , Xc ,
• The initial conditions, K0 , X0 , R0 , and T .
• Assume Cobb-Douglas forms for both the production
  function and utility function
• Yt F(Kt ,QF, t ) Kt?Q?F, t
• Ut U(Ct ,Ht ,QU, t ) CteHt?QfU, t
• where
• 1 gt ? gt ? gt 0
• 1 gt ? ?
• 1 gt e gt 0
• 1 gt ? gt 0 ,
• 1 gt f gt 0

11
Properties

• The production function and the utility
  function exhibit declining marginal product or
  utility, respectively, and both functions are
  homogeneous of degree less than one.
• These functions both have unitary elasticity of
  substitution between inputs (in the case of the
  production function) or commodities (in the
  case of the utility function).
• This will allow capital to substitute for
  extraction of the nonrenewable resource in
  production
• And for consumption or harvest to substitute for
  extraction from the nonrenewable resource in
  utility.
• In the social contract, a, ß, ?F , ?U, a
  positive, steady-state, level for Xt will be
  achieved at
• X Xc (r - ß)/r

12
III. The Gini Coefficient

• The Gini coefficient is a statistical measure of
  dispersion, commonly used to measure income
  inequality within a country.
• Consider the 45º line and the Lorenz Curve, L(X)
  , drawn in Figure 1.
• On the horizontal axis, moving from left to
  right, the population of a country is ordered
  from lowest income to highest income. The Lorenz
  Curve plots the cumulative share of income earned
  by the poorest X of the population.
• If earned income were equally distributed, the
  Lorenz Curve would be the 45º line.
• If the poorest X of the population earned Y of
  the income, where Y lt X, for X lt 1 100 , then
  the Lorenz Curve is convex and lies below the 45º
  line.
• The area below the 45º line and above the Lorenz
  Curve is the Gini coefficient divided by two
  (G/2).
• If one person earned all the income, the Lorenz
  Curve becomes the X and Y axes, the area below
  the 45º line is one-half, and G 1.
• If income were equally distributed, the Gini
  coefficient would be zero.
• Thus, higher Gini coefficients imply greater
  inequality.

13
(No Transcript)
14
More on Gini
15
IV. Maximin Social Contracts

• To start, we consider a simple maximin problem
  where T 10 , implying that 11 generations will
  be bound by the social contact a, ß, ?F , ?U.
  
• Table 1 shows an initial spreadsheet where the
  social contract has been arbitrarily set to a,
  ß, ?F , ?U 0.90, 0.20, 0.05, 0.05 , as
  indicated by the values in cells I8I11
• Other parameter values are shown in cells B3B13.
  

16
(No Transcript)
17
Simulation 1

• Given these parameter values, initial conditions,
  and the social contract, the spreadsheet in
  Table 1 simulates the values for consumption,
  harvest from the renewable resource, extraction
  for utility, extraction for production, the
  capital stock, remaining reserves, and utility in
  columns B through I.
• Column J indicates the rank of each utility level
  and column K computes the term (T 2 - RankUt
  )Ut , whose sum will be used in computing the
  Gini coefficient.
• In cells I28I30 we use the Excel functions
  MIN(I16I26) to return the minimum utility,
  MAX(I16I26) to return the maximum utility
  (just for comparison to the minimum), and compute
  the Gini coefficient for the utility vector using
  Equation (2).
• For this initial social contract we see utility
  monotonically declines from U0 3.28787 to U10
  2.86017 .
• The Gini coefficient registers G 0.02357 .
• In this initial spreadsheet, the steady-state
  stock for the renewable resource is X 80 ,
  which is reached in t 1.

18
Simulation 1 (Optimised)

• We now use Excels Solver to maximize the minimum
  utility in cell I28 by changing the social
  contract in cells I8I11.
• The maximin social contract and time paths for
  consumption, harvest, extraction, the capital
  stock, remaining reserves, and utility are given
  in Table 2. For this numerical example the
  maximin social contract is given by a, ß, ?F ,
  ?U 0.634, 0.5, 0.082, 0.087 . The maximized
  minimum utility is U2 3.87149 .
• Perhaps surprisingly, we see that the maximin
  social contract is Pareto superior to the initial
  social contact in that all utility levels have
  been increased.
• When the initial stock of the renewable resource
  exceeds the stock necessary to support maximum
  sustainable yield, XMSY , the maximin social
  contract will choose ß so that X XMSY Xc/2 .
  But, as noted above, X Xc (r - ß)/ r . With r
  1, equating these last two expressions for X
  implies ß 0.5.
• Under the maximin social contract, the
  nonrenewable resource has been depleted to a
  lower level than in the initial spreadsheet, R10
  15.84645 lt 34.86784 , while the terminal capital
  stock is higher, K10 15.39637 gt 2.97142 .
• Finally, the Gini coefficient is slightly lower
  under the maximin contract ( 0.02180 lt 0.02357 )
  indicating slightly more equitable time paths for
  consumption, harvest, and extraction.

19
(No Transcript)
20
Maximin Social Contract

• Why, in a finite-generation model, is it is not
  optimal to completely exhaust the nonrenewable
  resource. As Rt is reduced so too are the
  extraction levels, QF, t ?F Rt and Q U, t
  ?URt , leading to lower levels for QU, t and Ut .
  With all generations committed to the maximin
  social contract, compensating adjustments in
  consumption of the manufactured good or harvest
  of the renewable resource are more limited than
  if rates of consumption, harvest and extraction
  were allowed to vary over time. Complete
  exhaustion in T 10 would cause U10 0 . With
  the objective of maximizing the minimum utility,
  this is not optimal. Remaining reserves can only
  be depleted so far before the utility of later
  generations is reduced below the minimum utility
  under the maximin social contract.
• Solow (1974) noted the importance of initial
  conditions in the maximin problem.
• In the initial spreadsheet in Table 1, we
  arbitrarily set K(0) K0 1, R(0) R0 100 ,
  and X(0) X0 100 . We will refer to this
  initial condition as the less developed,
  resource-rich initial condition.
• For contrast, we consider two other initial
  conditions
• K(0) K0 1,R(0) R0 100 , and X(0) X0
  40 , the less developed, nonrenewable rich
  initial condition
• K(0) K0 100 , R(0) R0 40 , and X(0) X0
  40 ,the developed, resource-depleted initial
  condition.
• We compute the maximin social contract for these
  three initial conditions for T 10, 20, 50 . We
  are interested in how individual rates within the
  social contract change with a change in the
  initial condition and in the horizon length
  (number of generations). The results are shown in
  Table 3.

21
(No Transcript)
22
What can be gleaned from Table 3?

• Most consistent result an increase in T causes
  a reduction in both ?F and ?U . This makes sense,
  as the nonrenewable resource must be spread over
  a longer horizon for future generations to have
  positive production and utility.
• When X0 gt XMSY Xc/2 , the maximin social
  contract will always select a ß so that X Xc
  (r - ß)/ r XMSY Xc/2 . With r 1 this
  implies ß 0.5 . We see this rate of harvest for
  all T with the less-developed-resource-rich
  initial conditions (first row in Table 3) and,
  more surprisingly, in the developed-resource-depl
  eted initial conditions (the third row in Table
  3).
• Consumption of the manufactured good involves 1 gt
  a gt 0 in all cases except for the
  developed-resource-depleted case when T 10 . In
  this case the maximin social contract sets a
  1.05621, which results in each generation
  consuming more of the manufactured good (capital)
  than is produced, with the result that the stock
  of capital declines from its relatively abundant
  initial condition of K(0) K0 100 to K10
  91.67416 . With abundant capital and a relatively
  short horizon, this social contract makes sense.
  If you can achieve a higher minimum utility by
  eating your capital stock, do so. For longer
  horizons (T 20 and T 50 ) capital consumption
  in excess of production is no longer optimal in
  the maximin social contract.
• In the less-developed-resource-rich and in the
  less-developed-nonrenewable-rich cases (rows one
  and two) the Gini coefficient increases as T
  increases. As the horizon increases for these
  initial conditions the utility profile assumes a
  low-high-low pattern. With T 50 , the peak
  utility level occurs in t 17 for all three
  initial conditions. In the developed-resource-depl
  eted case (the third row) the Gini coefficient
  decreases as T increases. In this row, when T
  10 , the utility profile monotonically declines
  as earlier generations obtain higher utility from
  higher levels of consumption of the initially
  abundant capital good. Comparing the maximized
  minimum utilities in the three rows of Table 3,
  if one could choose their initial conditions, but
  not the value of T, (a partial veil of
  ignorance), one would prefer to gamble on T
  starting from a developed resource-depleted
  initial state.
• Finally, for the less-developed-nonrenewable-rich
  initial conditions, where R(0) R0 100 and
  X(0) X0 40 , we observe an increase in ß as T
  increases. This result may at first seem
  counter-intuitive. From a relatively low initial
  stock for the renewable resource one might think
  that the maximin contract would choose ß 0.5.
  Examination of the utility profiles provides the
  answer. In all three cases in row two of Table 3,
  the minimum utility is U0. The least costly way
  to increase the utility of this first generation
  is to bump up the initial harvest from the
  renewable resource which is only possible if X0 gt
  X Xc (r ß)/r . With X(0) X0 40 , this
  requires ß gt 0.6 .
• In all three cases in row two H0 gt Ht , for t
  1,2,...,T and this pattern of harvest is required
  to maximize the minimum utility, U0 .

23
Conclusions

• The model lead to an operational definition of
  the Rawlsian social contract. The social
  contract became a vector of rates of consumption,
  harvest, extraction for production of the
  manufactured good and extraction for direct
  utility. It was a social contract because it was
  adopted by all generations.
• The maximin social contract maximized the minimum
  generational utility over the horizon t
  0,1,...,T .
• While an analytic solution for the maximin social
  contract was not possible, the problem could be
  programmed on an Excel spreadsheet and easily
  solved for modest values of T .
• Is there anything to be learned from this simple
  model? Perhaps.
• When adding a renewable resource to the maximin
  problem, if there is no cost to harvest and if
  X0gt XMSY , the maximin social contract would move
  the resource stock to the level that supports
  maximum sustainable yield. This adjustment takes
  place in t 0 where H0 (X0 ! XMSY ) and then
  Xt XMSY and Ht HMSY for t ! 1 regardless of
  the value for T .
• If X 0 lt XMSY the harvest rate ß may be greater
  than the rate associated with maximum sustainable
  yield if it is the least-cost way to increase Min
  Ut U0 .
• Does the finite horizon in this computational
  model diminish it relevance. I think not. The
  difficulty with making sustainable choices when
  one cares about future generations is that we
  dont know the stocks that future generations
  will find essential or desirable. We dont know
  the preferences of future generations nor the
  technologies which will influence the value of
  manufactured or natural capital in the future.
• Uncertainty about preferences and technology make
  maximin over an infinite horizon an elegant
  exercise of questionable relevance. It may be
  better to build finite-horizon models with
  greater reality in their economic and resource
  structure and explore, computationally, maximin
  in greater detail.

24
References

• dAutume, A. and K. Schubert. 2008. Hartwicks
  Rule and Maximin Paths when the Exhaustible
  Resource has Amenity Value, Journal of
  Environmental Economics and Management, 56(3)
  260-274.
• Hartwick, J. M. 1977. Intergenerational Equity
  and the Investing of Rents from Exhaustible
  Resources, American Economic Review,
  66(5)972-974.
• Rawls, J. 1971. A Theory of Justice, Harvard
  University Press, Cambridge.
• Solow, R. M. 1974. Intergenerational Equity and
  Exhaustible Resources, Review of Economic
  Studies (Symposium on the Economics of
  Exhaustible Resources), 29-45.
• Withagen, C and G. B. Asheim. 1998.
  Characterizing Sustainability The Converse of
  Hartwicks Rule, Journal of Economic Dynamics
  and Control, 23159-165.