Figure 1: Environmental function and its price
Thomas Cool, June 1 & June 21 2000
Consultancy & Econometrics
Report TC-2000-06-01
JEL A00, Q00
If an economic system has a regime switch, then we could
assume that the preferences have remained basically the same, but that
only the state of information has changed. An alternative view is that
there has been a shift of preferences, as for example individual persons
can change their mind too. Economic theory needs the concept of a ‘meta
social welfare function’ (meta-SWF) in order to explain switches in preference
regimes. Hueting’s argument on the switch in environmental policy from
laissez faire to sustainability is an interesting example for this argument.
Cool (2000) presents the extended social welfare function
SWF(x ; I), where x is the allocation over agents, and where
the state of information I is included in the SWF, to express in
a shorthand fashion that society’s choice can depend upon the state of
information. The basic notion is that the SWF remains the same over the
regimes, but a condition can change. Information is basically just an example
for such a condition, since also another variable can cause the switch.
In a sense we could allow for time as the ‘explanatory’ variable. Basically,
of course, we can have a dynamic situation that gives the evolution of
the SWF over time, with perhaps a dramatic change at the switch point.
But simply designing a path of SWF(t) will not do, since economics has
to model the process of choice that is involved in the making of the change.
An alternative approach is to assume different social welfare functions
per regime, for example SWF(x) and SWF*(x) if there are two
regimes. But with different SWF’s we would need a meta-SWF to explain the
shift.
The issue actually holds for any regime switch. A useful
example is the issue of the choice in environmental policy between laissez
faire and sustainability.
The environment can be seen as generating various
functions that enable life and economic activity. In the past these functions
were free, and thus had no price attached to them. Nowadays, however, these
functions become scarce, and thus get to be priced. The (unmanaged) market
price - or ‘laissez faire’ price - of an environmental function can be
derived as the cost that an economic agent has to make if he or she wants
to enjoy the function. Alternatively, the government may impose controls
to influence that price (and we get a market with controls). Choosing a
correct price is important also for statistical purposes, since a figure
like ‘national income’ is calculated while using prices.
A government can have various objectives when choosing its controls. One important objective might be ‘sustainability’, i.e. that the environmental functions are used such that nature can run its course, and such that later generations are not overly hindered by current uses. Hueting presents the choice for sustainability as socially optimal. "In other words", social welfare should increase as a result of the choice for sustainability. The choice for sustainability would generally mean that people would use less resources, and ‘national income’ as currently measured might well be lower. By economic intuition we expect that a move to a better situation is reflected in the upward movement of at least some indicator. If ‘national income’ goes down, then at least social welfare has to go up. This paper hopes to clarify this issue.
We will show in particular:
It follows, therefor, that the discussion on sustainability
may be a bit more complex than originally thought.
Figure 1 is the, one might say, renowned Hueting graph
of the relation between an environmental function and its price. The upward
sloping curve gives the producer costs ('supply'), found by looking at
the costs of making the function available - such as water clean-up. The
downward sloping curve gives the laissez faire user costs ('demand'), based
upon such laissez faire prices. It could be constructed from the efforts
by the agents to compensate for the loss of function by choosing other
activities or using other resources. These costs should be added to give
total unit costs. The suggestion is that the observed choice is at the
minimum of this summed costs. Basically, though, a social welfare function
would select the observed point, by balancing the environmental costs with
other objectives (not shown). Anyway, statistically, we could observed
the implied price (total cost) at ‘observed preferences’. Alternatively,
society imposes a norm of higher availability, and then the intersection
of the vertical norm and the cost curves gives the normed price.
Figure 1: Environmental function and its price
Incidently, society’s norm will be derived from individual
preferences. It has been conjectured by some, in verbal discussions, that
Hueting would ‘impose’ the norm of sustainability. This however appears
to be a misunderstanding. The difference between the laissez faire situation
and the normed situation appears to derive from different considerations
- as holds regime switches in general.
It will be useful to model the problem. Let us consider
two non-overlapping generations who ‘share’ 100 units of oil and 100 units
of water. The first generation will make the decisive decision how much
to use itself, and it will bequeath the remainder to its descendants. To
do so, the first generation uses a social welfare function (SWF), which
function not only contains its own direct income yNow
but also the indirect welfare that it derives from the situation for the
descendants. This indirect welfare is based on the direct income yFuture
that the descendants are hypothesised to achieve. We follow Ramsey in a
lack of a rate of discount.
The SWF will here be a Constant Elasticity of Subsitution (CES) function that neglects the distribution of income. Next to an ‘egotistic’ base situation SWF, we regard the alternative SWF* in which society switches its preferences so that it becomes more understanding of the needs of future generations. The SWF* includes a bonus welfare injection that derives from making the switch:
The income of the generations is determined by production
functions that depend upon the allocations of the factors of oil and water.
With a constant technology, and i = Now, Future:
We solve the model by the program originally developed
by Noguchi (1993) and further developed by Cool (1999). In the plots, the
base ‘egotistic’ situation has continuous lines, and the alternative ‘sustainable’
situation has dashed lines. We use r = 2/3 (s=
3/5) and r = -2/3 (s= 3). We also assume
that the switch bonus = 0.
Figure 2 plots the production possibility curves and the
SWF indifference maps of the two situations. Clearly the alternative SWF
allows more consumption for the future generation.
Figure 2: Production Possibility Curves & Indifference Maps
Figure 3 plots the Edgeworth-Bowley diagram, with
Now in the lower left and the Future in the upper right. The movement
is downwards along the contract curve. Since the production functions are
the same, the contract curve is a straight line. Consequently, the percentage
that a generation takes of the resources is the same for all resources.
Figure 3: Edgeworth-Bowley Diagram
The following tables give the numerical outcomes of
the two regimes. The social optimum is found as in Table 1. The associated
allocations are in Table 2 - left and right side. Given our simple assumptions,
we also get a simple result. When you compare the two regimes, please note
that the prices are normalised per regime to a unit price for Now,
and thus are not comparable over regimes.
Table 1: Utility, production and national income for two regimes
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We arrive at a closer understanding by regarding two
other graphs. Let us choose water allocation at the optimal level, and
vary the allocation of oil. Figure 4 shows the SWF and SWF* graphs as functions
of the allocation of oil to the Now generation. Figure 5 plots the output
levels of the Now and Future generation. Output of the Now generation goes
up when it uses more and more oil. At the same time the resource for the
Future generation goes down, and hence income goes down. As the income
of the Future generation goes down, then this eventually affects the social
welfare of the Now generation as well.
We now can understand the Hueting graph a bit better. The switch from the laissez faire situation to the normed ‘sustainability’ situation can be a switch from one SWF to an alternative SWF*. The horizontal axes in Figure 4 and Figure 5 give depletion, which is the opposite of availability. Due to the higher preference for presumed future consumption, current output becomes less and future output goes up, and hence the availability of the resource goes up as well.
In this example, we see that the absolute value of social welfare is less in the new situation. Lesser consumption Now is compensated by a benefit to the future generations, but not fully. With ordinal welfare, this does not mean much of course. If we assign meaning to the numerical values, however, then a non-zero switch bonus is required. In this case the bonus should be at least 2.6 welfare function points (or output to that effect). In a general approach, we would need a meta-SWF to choose between the SWF and SWF* - i.e. to deal with that constant. Such a meta-SWF would automatically assign a value to the different welfare scales.
Alternatively, if we use the SWF(x; I) approach, then the regime switch could be the result of a change of the state of information from I = 0 to I = 1, and we would get SWF(x, 0) = SWF(x) and SWF(x; 1) = SWF*(x). By implication the two welfare scales are considered to be comparable, and the bonus would be the implied value of the information.
This discussion thus corroborates Huetings position, but adds a useful clarification. This analysis also shows that there is scope for research on how people’s choices are affected.
Hueting’s position might be interpreted as: If
society decides for sustainability, then apparently this is an (meta-SWF)
improvement. A problem with this position could be Hume’s gap between Is
and Ought. From observing a certain situation, we cannot conclude that
it apparently is optimal. However, Hueting’s position would be valid if
the emphasis is on ‘decides’. If society decides, i.e. aggregates
its preferences, to sustainability, then this by definition gives the new
social preference. But for the same reason, it is not obvious that society
would make this choice. It might as well think that sustainability has
no bonus. Note for example that the SWF* optimum certainly is lower
in terms of the original SWF, so some people who think in terms of the
original function will have a hard time to see the improvement.
We clarified that regime switches can be represented
by the information approach or by the meta-SWF approach to preference switches.
And we showed that these are to some extend equivalent.
Using this, we clarified the discussion on the policy choice on sustainability.
This analysis also shows that there is scope for research
on how people’s choices are affected.
Literature
Cool, Th. (2000), "Definition and Reality in the General Theory of Political Economy", Scheveningen, ISBN 90-802263-2-7
Hueting, R., c.s. (2000), "Prijs het milieu, niet het beleid", Economisch-Statistische Berichten 4244, February 25, p157-159 (Dutch)
Hueting, R. (1992), "The economic functions of the environment", in P. Ekins and M. Max-Neef (ed), "Real life economics", Routledge
Noguchi, Asahi (1993), "General Equilibrium models", in
Varian (ed), "Economic and financial modeling with Mathematica", Springer
Telos 1993 pp 104-123
The following is the Economics Pack program to produce above results.
Needs["Economics`Pack`"]
Economics[AGE]
(* SWF *)
SetFunction[case1, now] =
{Function ® ( 0.4 oilNow^rho + 0.6 waterNow^rho)^(1/rho),
CoefficientList ® {rho ® 1 - 1/3},
Factors ® {oilNow, waterNow} }
SetFunction[case1, future] =
{Function ® (c oilFuture^rhoF + (1-c) waterFuture^rhoF)^(1/rhoF),
CoefficientList ® {c ® 0.4, rhoF ® 1 - 1/3},
Factors ® {oilFuture, waterFuture} }
SetModel[NumberOfSectors ® 2, NumberOfFactors ® 2, Utility ® CES,
Production ® {Sector[1] ® SetFunction[case1, now],
Sector[2] ® SetFunction[case1, future] } ]
$Coefficients
Results[AGEmodel]
ownpars = {Utility ® {Scale[Utility] ® 1, RTS[Utility] ® 1, S[Utility] ® .6,
FactorE[1] ® 0.7, FactorE[2] ® 0.3},
Production ® {}, Resources ® {100, 100}}
asgn = Assign[ownpars]
eq = Equilibrium[ownpars]
AllocationTable[Allocation[ownpars]]
shares = (FactorPrices /. eq) * (Resources /. ownpars) / (YEq /. eq)
cpc23 = CPCDiagram[ownpars, AxesLabel ® {"Now", "Future"},
AspectRatio ® Automatic]
ploteq1 = EdgeworthBowley[ownpars, Factor ® {1, 2}, PlotPoints ® 50];
(* Analysis *)
swf1 = Utility /. Results[AGEmodel]
asgn = Join[ Assign /. Results[AGEmodel], {oilFuture ® 100 - oilNow ,
waterFuture ® 100 - waterNow}]
SWF[oilNow_, waterNow_] = swf1 /. asgn
swfplot = Plot[SWF[x, 63], {x, 0, 100}, AxesLabel ® {oilNow, "Social\nwelfare"}];
swfplot3D = Plot3D[otherSWF[x, y], {x, 0, 100}, {y, 0, 100}]
y1[oilNow_, waterNow_] = Sector[1] /. Results[AGEmodel] /. asgn
y2[oilNow_, waterNow_] = Sector[2] /. Results[AGEmodel] /. asgn
outputplot = Plot[{y1[x, 63], y2[x, 63]}, {x, 0, 100},
AxesLabel ® {oilNow, "Output"}, PlotRange ® All, AxesOrigin ® {0, 10}];
(* SWF 2 *)
otherpars = {Utility ® {Scale[Utility] ® 1, RTS[Utility] ® 1, S[Utility] ® .6,
FactorE[1] ® .5, FactorE[2] ® .5},
Production ® {}, Resources ® {100, 100}}
Assign[otherpars]
othereq = Equilibrium[otherpars]
AllocationTable[Allocation[otherpars]]
shares = (FactorPrices /. othereq) * (Resources /. otherpars) / (YEq /. othereq)
othercpc = CPCDiagram[otherpars, AxesLabel ® {"Now", "Future"},
AspectRatio ® Automatic, ContourStyle ® {Dashing[{0.02}]}];
cpcfin = Show[cpc23, othercpc];
Export["C:\dump\cpc.gif", cpcfin, "gif"]
othereb = EdgeworthBowley[otherpars, Factor ® {1, 2}, PlotPoints ® 50,
ContourStyle ® {Dashing[{0.02}]}];
ebfin = Show[ploteq1, othereb, FrameLabel ® {"Oil", "Water"}]
Export["C:\dump\eb.gif", ebfin, "gif"]
(* Analysis *)
otherasgn = Join[Assign /. Results[AGEmodel], {oilFuture ® 100 - oilNow ,
waterFuture ® 100 - waterNow}]
otherSWF[oilNow_, waterNow_] = swf1 /. otherasgn
otherswfplot = Plot[otherSWF[x, 50], {x, 0, 100},
AxesLabel ® {oilNow, "Social\nwelfare"},
PlotStyle ® {Dashing[{0.02}]}];
swffin = Show[swfplot, otherswfplot]
Export["C:\dump\swf.gif", swffin, "gif"]
othery1[oilNow_, waterNow_] = Sector[1] /. Results[AGEmodel] /. otherasgn
othery2[oilNow_, waterNow_] = Sector[2] /. Results[AGEmodel] /. otherasgn
otheroutputplot = Plot[{othery1[x, 50], othery2[x, 50]}, {x, 0, 100},
AxesLabel ® {oilNow, "Output"}, PlotRange ® All,
AxesOrigin ® {0, 10}, PlotStyle ® {Dashing[{0.02}]}];
oplt = Show[outputplot, otheroutputplot]
Export["C:\dump\output.gif", oplt, "gif"]
(* Hueting graph *)
supply[x_] := x^2 + .2
demand[x_] := .1 + 1/(x + .25)
huet1 = Plot[{supply[x], demand[x]}, {x, 0.2, 1.5}, Ticks ® None,
AxesOrigin ® {0, 0}, AxesLabel ® {"Availability of\nEnv. Function", "Price"} ];
as = Asymptot[{1.25, 3}, True, DashingValues ® {0, 0}];
poi = PointOfIntersection[demand, supply, 3]
as2 = Asymptot[poi, False]
as3 = Asymptot[{1.25, supply[1.25]}, False]
huetplot = Show[huet1, as, as2, as3]
Export["C:\dump\hueting.gif", huetplot, "gif"]