“Intergenerational Equity and Exhaustible Resources”
By Robert Solow
Arrow Team
*Handout is modified from 2011 presentation by Traviss Cassidy, Antonios Koumpias, Jianxing Liang,
and Yanxi Zhou. We hope there is some added value.
BACKGROUND AND INTRODUCTION
The vast majority of economic research works within a “utilitarian” framework. This means that
economists evaluate behavior and policies based on their consequences for total net utility—
aggregate utility of all individuals in a society or market minus aggregate disutility. For example,
an individual’s “optimal” behavior is that which results in the most utility for the individual.
Likewise, “socially optimal” outcomes maximize total net utility.
This approach assumes that utility can be measured, observed, compared among individuals, and
summed over all individuals. As an ethical system, utilitarianism is consequentialist—it holds
that actions should be judged according to their consequences. (In this case, the consequences for
total net utility are what matter.) In a strict interpretation, utilitarianism makes no explicit
argument for equity among individuals; an extremely unequal society may be desirable if total
utility is maximized in such an arrangement.1
An American philosopher named John Rawls strongly opposed the utilitarian framework, most
famously in A Theory of Justice (1971). As an alternative, he asked readers to imagine that we all
start our lives in an “original position,” in which we are completely unaware of the physical and
material endowments (e.g., wealth, race, gender, ethnicity, intelligence, strength, etc.) that we will
enjoy over the course of our lives. Behind this “veil of ignorance,” he argued, we would agree to
a social arrangement (or “contract”) that maximizes the welfare of the least well-off member of
society. This is known as the “max-min” principle and is expressed mathematically as:
W = min (U1, … , Un )
In terms of expected utility theory, Rawls implicitly assumes that individuals in the original
position are infinitely risk averse: they care only about the worst possible outcome.
Rawls was ambivalent about how to treat intergenerational justice, partly because exchange
occurs only in one direction: the current generation influences the well-being of the next
generation, but no generation can influence the welfare of previous generations. In this case the
max-min principle cannot be applied.
In the paper, Robert Solow applies the max-min principle to an optimal growth problem that
assumes a continuum of generations. (An infinite number of infinitesimally small generations.)
The model is based on Frank Ramsey’s 1928 article, “A Mathematical Theory of Saving,” though
1
Of course, an appeal to diminishing marginal utility could lead a utilitarian to choose an equitable societal
arrangement.
1
Solow’s optimization of the model is completely different. A basic Ramsey model entails the
following:
∫
( ( ))
(w.r.t. c(t) and subject to certain physical constraints that are specified later in the paper)
In Ramsey’s version of the model, we maximize the sum of utilities of all the generations
stretching to the infinite horizon. Maximization may require early generations to scarify
themselves by forgoing consumption in order to save and increase the utility of future generations.
Though, max-min principle has aspects that the standard additive-welfare approach does not.
However, Solow’s approach to Rawls’max-min principle is completely different, as will be seen.
1. THE MAX-MIN PRINCIPLE AND OPTIMAL ECONOMIC GROWTH
Solow gives the intergenerational problem a more direct treatment than Rawls did in A Theory of
Justice by making a few assumptions. For tractability, Solow assumes that every individual is
identical within and across generations. He also equates utility with consumption to further
simplify the model. Lastly, he assumes that at each moment in time all individuals consume an
equal share. This reduces the problem to one of equity between “instants in time.” Each instant in
time represents a new “generation.”
Solow argues that, with some exceptions, the max-min principle requires that every individual in
every generation consume the same amount. If one individual were to consume more than another,
then welfare W = min (U1, … , Un ) could be increased by reducing the first individual’s
consumption and increasing the other’s consumption until they are equalized. An exception to
this rule is the presence of a technical obstacle to the equalization of consumption over time,
which Solow ignores as trivial.
2. CONSTANT POPULATION, CONSTANT TECHNOLOGY, NO SCARCE NATURAL
RESOURCES
Q: net output
K: capital
K0: initial capital
L: labor
capital per person (assuming L is total labor and total population and that everyone
works)
2
Solow first considers the case of zero population growth or constant population, constant
technology, and zero scarcity. He assumes the production function is homogeneous of
degree one, which is why he can write
where ( )
(
). Equation 2 is given by
(
)
This can be re-written as ̇
, which says that the change in our stock of
capital is equal to the amount of output that we don’t consume, i.e. that we save. In
optimal control theory, K is our state variable, C is our control variable, and the rewritten Equation 2 is our equation of motion. K is a stock that accumulates over a period
of time, whereas C can take any nonnegative value (assuming perfect financial markets)
in any moment in time. This is why we relate C to ̇ rather than to K.
The largest aggregate consumption that can be maintained forever is obtained by
setting ̇
(
). This means each generation consumes all of the
output and neither saves nor dissaves, maintaining the capital stock at its original level.
Here, one important assumption is that capital does not depreciate.
3. EXPONENTIALLY GROWING POPULATION
Now Solow lets the labor force grow by setting
Substituting (1) into (2) and dividing by L gives
, where n is the growth rate.
where
is per-capita consumption. Substituting the new L into the definition of k
and taking the derivative with respect to time results in:
From (3) it is clear that, once we choose a time path c(t) for consumption, our time path
k(t) for per-capita capital accumulation is determined. This is because, in any moment in
time, any per-capita output f(k) that is not consumed or lost via population growth is used
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to accumulate capital. A time path c(t) is feasible if it results in a capital time path such
that ( )
—that is, as long as our path does not cause capital to disappear.
Because we are applying the max-min principle, maximization requires finding the largest
feasible constant c0 that equalizes consumption of all individuals across generations.
(
)
( )
Assuming that f(k) is increasing, concave, ( )
( )
guarantees that f(k) will intersect nk only once and that ( )
will
eventually disappear.
The shape of the curve in Figure 1 is due to the fact that nk increases at a constant rate
(linearly) whereas f(k) increases at a decreasing rate (it is concave).
( )
Solow argues that setting
̇
, so there is no capital accumulation and
is optimal by the following: at c*,
( ) . Thus c* is feasible. Any
is also feasible but is clearly not optimal, as it is not the largest feasible constant
c. Setting
gives ̇
by (3), so we will quickly lose capital until
, at
which point we will have nothing to consume because production is zero. Thus
is optimal.
The max-min rule requires that each generation invest only enough to provide capital for
the increase in population, i.e. setting ( )
Utilitarianism calls for earlier
generations to consume less so that later generations can consume more, but this fails the
Rawlsian criterion.
4. TECHNICAL PROGRESS
Solow now adds labor-augmenting technology:
4
where a is the growth rate of technology and
worker. Combining (1a) and (2) gives
is capital per “effective”
Solow chooses to keep
rather than writing
because the max-min
principle is only concerned with standard per-capita consumption. Taking the time
derivative of z and substituting gives
( ) (
)
This is the change in effective capital per worker. Setting
gives ̇ ( )
but ̇ ( )
because technology growth causes the final term in
(4) to decrease. Thus, if we start at then we soon begin to accumulate capital, allowing
future generations to consume more. Thus, we are allowing future generations to
consume more than present generations if we use this scheme. So, what we want to do is
set c higher in the present period, since later generations are able to maintain the same
level of consumption as the early generations thanks to technological progress. It follows
that the largest feasible c will give a time path ( )
and will satisfy the
( )
transversality condition
That is, under the optimal c, society
asymptotically consumes the entire capital stock, so no capital is left over at the end of
the planning period, but we use less and less because of the effect of technology.
5. SUMMARY SO FAR
The max-min principle calls for zero net saving with stationary technology, and for
negative net saving with growing technology. But, the max-min criterion is at the mercy
of the initial conditions: if initial capital is very small, no more will be accumulated and
the standard of living will be low forever.
To see the problem here, entertain the following thought experiment: assume two periods,
and assume utility is an exponential function of c. Then, in the standard (non-Rawlsian)
case, we would want to group all of our consumption in one period. But, in the Rawlsian
case, the two generations’ utility taken as a whole is lower so that they can both be equal.
The effect is compounded by the fact that we have technological augmentation of labor.
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6. EXHAUSTIBLE RESOURCES
Now we introduce exhaustible resources into the production function, which becomes
Q  F ( K , L, R) , where R is a rate of flow of a natural resource. For the problem to be
interesting, R should “essential,” which means R  0  Q  0 (otherwise initial stock of
R would be used up early to shore up consumption and accumulate capital, then the
problem reduces to the previous ones without R ), and the average product of R should
be unbounded (otherwise only a finite amount of output can ever produced, and the only
level of aggregate consumption maintainable for infinite time is zero because we need
intergenerational equity here).
For simplicity, consider the Cobb-Douglas production function,
Q  F ( K , L) R h , (0  h  1) , with F homogeneous of degree 1  h (HD 1  h ), so Q is
HD1. Therefore,
Q  emgt Lg Rh K 1 g h
…(6),
where mg is the rate of Hicks-neutral technical progress; equivalently, m is the rate of
labor-augmenting technical progress.
If you are familiar with production functions, move on. If not and if interested, keep
reading. First, a digression on the marginal rate of (technical) substitution (MRS). The
K MPL
MRS of capital to labor is defined as MRS K  L  
. That is, to maintain the

L MPK
same amount of production Q, reducing one unit of labor, we lose MPL , and to make up
for this, we need to increase MRS K  L unit of capital, so that we have MPK * MRS K  L unit
of production to make up for the labor loss. If MRS K  L is high, that means we need more
capital to compensate for one unit loss of labor.
The elasticity of substitution between capital and labor is defined as
d ( K / L) d ( MPL / MPK )
d ln( K / L)
, which is the ratio of [the percentage
K  L 
/

K/L
( MPL / MPK ) d ln(MPL / MPK )
change in the ratio of capital and labor] and [the percentage change in the ratio of MRS
of capital and labor]. Notice the latter is also the price ratio of labor and capital in a
competitive market. This measures how easy it is to shift between labor and capital. If
 K  L is low, that means a given percentage change in price ratios does not cause the stock
ratio of capital and labor to change much, which means the substitution is difficult.
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Now for a digression on Hicks-neutral technical progress. Define elasticity of output
Q / Q Q / K
Q / Q Q / L
E
MPK / MPL
, and EL 
. Define   K 
. If
EK 


K / K
Q/K
L / L
Q/L
EL
K/L
technological progress increases  , then it is capital-augmenting, because for a given
capital and labor ratio (the denominator), it helps capital to produce more relative to labor
(the numerator). If it reduces  , it is labor-augmenting. If it does not change  , it is
called neutral technological progress. If K/L remains constant over time, it is called Hicks
neutral. (Robert J. Barro, Xavier Sala-i-Martin, Economic Growth, page 52)
Define y  R / Lemt , z  K / Lemt , c  C / L , and use previous mathematical tricks. We get
z  z1 g h y h  (n  m) z  ceemt
….(7)
Given c and y (or C and R) and either z(0) or K(0), we can generate z(t) and therefore K(t)
with equation (7).
Formally the optimization problem is
max c0
s.t. z  z1 g  h y h  (n  m) z  ce  emt

L0  y (t )e( m  n )t dt  R
0
y (t )  0, t  0
z (t )  0, t  0
c(t )  c0 , t  0
What to note here is that we must find the largest constant consumption per head. This
means we end up using exactly all of our resources.
7. EXHAUSTIBLE RESOURCES: REFORMULATION
Basically, the previous optimization problem is reformulated into a trial and error
problem. Choose an arbitrary c0 from equation (7) and solve

min  y (t )e( m  n )t dt
0
s.t. z  z1 g  h y h  (n  m) z  ce  emt
y (t )  0, t  0
z (t )  0, t  0
7
If the minimized
minimized


0


0
y (t )e( m n )t dt is greater than R / L0 , choose a smaller c0 . If the
y (t )e( m n )t dt is smaller than R / L0 , choose a larger c0 . Repeat this until,
when minimized,


0
y(t )e( m n )t dt  R / L0 .
A necessary condition for a minimum of this modified problem is the existence of a
shadow-price (also as the efficiency price) of capital in terms of the natural resource,
p(t ) with properties
phz1 g h y h 1  1
p / p  (1  g  h) z
 g h
...(9a )
y
h
...(9b)
(9a) means that the resource should be drawn down in such a way that its marginal
product is kept equal to its own efficiency price. (9b) means that the rate of change of the
shadow price of resources should equal the sum of the rate of change of the shadow price
of a unit of capital and the own rate of return from using capital (optimally) to produce
itself; or, a rational investor, calculating with efficiency prices, should be at all times
indifferent at the margin between holding capital goods and holding mineral deposits as
earning assets. Generalizing beyond the Cobb-Douglas case, along an optimal path the
proportional rate of change of the marginal productivity of the resource should always
equal the level of the marginal product of reproducible capital.
Take the logarithm of (9a), differentiate with respect to time and use the result to
eliminate p / p from (9b), and we get
1 g  h
ce mt
y / y  (
)(m  n 
)
1 h
z
...(10).
But, this is a dead end because it contains time explicitly.
8. ZERO POPULATION GROWTH AND ZERO TECHNICAL PROGRESS AGAIN
In this section, Solow applied the Max-Min principle to the case of no population growth (ZPG)
and no technology progress (ZTG).
With the Cobb-Douglas function, output per unit of natural-resource input, Q/R, goes to infinity
as R goes towards zero. That is, the average product of the natural resource is unbounded.
Q
R K
 emgt ( )h ( )1 g h means that at any given time (so t is fixed), output per worker, Q/L, goes
L
L L
to zero as R goes to zero. Without technical progress, the only way to maintain constant
8
consumption is through fast enough capital accumulation, so that K/L goes to infinity as R/L
drops towards zero.
Stylized facts suggest that exponential population growth is an inappropriate idealization.
Therefore suppose the population does not grow, that is, n=0. So L0 can be normalized to 1.
Since there is no technical progress, then we have m=0. “For temporary notational reasons,” (I
think it is convenience reasons so that the following equations and solutions are neat) set
1  g  h  a and h  b . Then (7) and (10) become
z  z a yb  c
a
y
cy / z
1 b
...(7a ')
...(10a ')
Take for granted that 0  a, b  1, a  b  1 , given the properties of the Cobb-Douglas function.
Assume a  b , i.e., the elasticity of output with respect to reproducible capital
(
dQ / Q Q / K

 1  g  h ) exceeds that with respect to exhaustible resources (=h). “This
dK / K
Q/K
seems quite safe: from factor shares, a would be at least three times b.”
The reason for assuming a  b is to make the integral


0
y (t )dt converge and hence get a
solution for an maintainable constant level of consumption per head. (If you are interested why
a  b is admissible and b  a is not, please see Appendix B in the paper.) In other words, under
these assumptions (ZPG, ZTP, a  b ), and constraints ( y(t )  0, t  0 , z(t )  0, t  0 ,


0
y(t )e( m n )t dt  R / L0 ), we can get a solution for c0  u b (
a  b R b a b
) z0  u , and
b L
z (t )  z0  ut , y(t )  (c0  u)1/ b ( z0  ut ) a /b . For the integral to converge, y (t ) must drop to zero.
By equation (7a’), z (t ) must be unbounded, which means u  0 . Otherwise, as y (t ) drops to zero,
the RHS of (7a’) becomes and remains negative, and any bounded z (t ) will become negative.
The larger the R (less resource-constrained) and z0 (more endowment of capital), the larger the
range of feasible c0 , i.e., u b (
a  b R b a b
) z0  u , the maximum of c0 is larger.)
b L
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In a nutshell, this section tells us that, in the case of ZPG and ZTP, which we will continue to
discuss in the following sections, there is a solution for c0 (indefinitely-maintainable and
positive), and correspondingly, y(t )  0, z(t )   as t   .
9. ZPG AND ZTP, DETAILED SOLUTION
Equations (7 ) and (10 ) are necessary conditions for an optimal choice of y(t), given
. Since
they are autonomous equations, they can be analyzed in Figure 2. Any trajectory in Figure 2 is
time path of z and y.
The curve is defined by the equation:
…(11)
Substitute (11) to (7 ) and (10 ), we can get:
…(12)
(
)
(
)
…(13)
These provide a solution to (7 ) and (10 ) that lies on the curve (11) for all t≧0.
Lastly, we need to show that choices (12) and (13) do minimize the total resource use under a
constant
condition. That is to say, there exists a certain level of
that (12) and (13) each has
a nonnegative solution.
Set ∫
()
̅
and determine that largest feasible
. Routine computation gives:
10
̅
( )
(
(
)
)
(
)
(
).
10. ZPG AND ZTP: DISCUSSION
Assumptions:
1) The allowable
is a concave unbounded function of the initial capital stock per worker
with constant population L.
2) The elasticity of substitution between resources and labor-and-capital is at least one.
Then, in the situation with exhaustible resources, any level of consumption per worker c can be
maintained if only the initial capital per work k is large enough.
Under the Rawlsian criterion, the optimal program calls for K/L to grow from the very beginning
and for R/L to fall from the very beginning. If a > b, earlier generations would use up the
resources very fast, building up the capital stock in return.
From
(11), along the optimal path,
1) net output is constant.
2) (1-b) = (1-h) of output is consumed; b = h of output is net investment.
3) h is quite small since it is the elasticity of output with respect to resource input.
4) net investment (fraction b of output ) is enough to maintain output and consumption constant
with decreasing resource inputs.
5) from
proportional to (
in (9a), the shadow-price of capital in terms of resources is
(
)
)
. Therefore, the shadow-price of resources in terms of the
produced commodity rises ultimately like
where
.
Below is a brief comparison of the model in the paper and the Ramsey model, which will be
discussed in detail in the extension section.
Assuming constant population and Cobb-Douglas production, the Ramsey model yields:
1) consumption per head rises without limit over time.
2) It will use up the pool of natural resources more slowly than a Rawlsian society.
3) earlier generations will have less that the constant Rawlsian consumption per head c, and later
generations will lead a higher standard life.
11. EXPONENTIAL POPULATION GROWTH WITH LIMITED RESOURCES
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Solow argues that unlimited population growth is not plausible in a world of limited natural
resources. Nevertheless, he briefly touches on the case of an economy whose labor force grows
exponentially while no technical progress takes place.
if
then (7)
̇
(7c)
No positive constant consumption per worker exists for which there exists a function ( )
constrained by (8). If ( ) is bounded, then as ( )
,where
, and eventually
RHS of (7c) dominates the first,
which forces ̇
. If ( ) is unbounded then the second term,
, and again,
, on the
turns negative.
13. SUMMARY
There are 2 main takeaways from Solow’s analysis:
1. Max-Min Criterion is appropriate for intertemporal planning decisions
o Drawback #1: requires a big enough initial capital stock to induce capital
accumulation; otherwise, no capital will be accumulated and the standard
of living will be low forever.
o Reaches unreasonable results when the population is constant and
technical progress in unbounded
2. Incorporating exhaustible resources in the model does not offset the basic
results, insofar as the elasticity of substitution between natural resources and
reproducible capital/labor-and-capital goods is no less than unity.
Earlier generations are entitled to reduce the pool of natural resources if and only if they
add to the stock of reproducible capital (rapid resource-saving technical progress).
Essentially, the max-min criterion requires us to completely degrade the environment
while building up the capital stock, until we live on the Death Star.
Compared with Ramsey’s model, it is the total utilitarian that need to maximize.
The path is shown as follow: the c’=0 and k’=0 louses are shown in the graph. The intersection
point is the steady state. And the curve that passes the steady point is the unique steady saddle
path. Points on this path will converge to the steady point.
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Based on the phase diagram, the path of consumption per person is not constant. At the stead
state, the per capita consumption is increasing at the rate of population growth rate/technology
growth rate.
Variable
̅
( )
()
Interpretation
Initial labor
Initial net output
Bound on resources
Individual utilities
Initial consumption per worker
Initial capital per effective worker
Cobb-Douglas parameter for R
Time path for capital stock under maximum
maintainable consumption
Capital
labor
Net output
Rate of flow of natural resource, extracted
from a pre-existing pool
Social Welfare
Rate of technological augmentation of labor
(section 5)
Consumption per worker
Reduced form production function
Capital per worker
Rate of labor-augmenting technical progress
(sections 6+)
Rate of hicks-neutral technical progress
Rate of growth in labor force (equivalent to
13
( )
pop growth rate since labor supplied
inelastically)
Shadow price of capital
Time
Arbitrary constant in solution to equation 7a
Resource per effective worker
Capital per effective worker
14