## 1. Production Possibilities in Ramsey's Formulation




Ramsey's goal was practical: "How much of a nation's output should
it save for the future?" The demographic profile over time was taken by
him to be given, meaning that future numbers of people were seen as
exogenous and predictable. We were therefore to imagine that economic
policies have a negligible effect on reproductive behaviour (but see
Dasgupta, 1969, for a study of the joint population/saving problem,
using Classical Utilitarianism as the guiding principle). Parfit (1984)
christened choices involving the same demographic profile, "Same
Numbers Choices."




The ingredients of Ramsey's theory are individuals' lifetime
well-beings. Government House in his world maximizes the expected sum
of the lifetime well-beings of all who are here today and all who will
ever be born, subject to resource constraints. The optimum distribution
of lifetime well-beings across generations is derived from that
maximization exercise. Of course, the passage of time is not the same
as the advance of generations. An individual's lifetime well-being is
an aggregate of the flow of well-being she experiences, while
intergenerational well-being is an aggregate of the lifetime
well-beings of all who appear on the scene. It is doubtful that the two
aggregates should have the same functional form. On the other hand,
there is little evidence to suggest that we would be way off the mark
in assuming they do have the same form. As a matter of practical
ethics, it helps enormously to approximate by not distinguishing the
functional form of someone's well-being over time from that of
well-being across the generations. Ramsey adopted this short-cut.
People were also taken to be identical, so we may as well assume that
there is a single individual at each date. The move removes any
distinction between time and generations. An alternative interpretation
would have us imagine that the economy consists of a single dynasty,
where parents in each generation leave bequests for their children
(Meade 1966, adopted this interpretation). Ramsey also assumed,
probably because the mathematics is simpler, that time is a continuous
variable, not discrete.




Let \(t \ge 0\) denote time. In Ramsey's model there is no
uncertainty (but see Levhari and Srinivasan, 1969, for one of the first
of many extensions of the Ramsey model that incorporate uncertainty
about future possibilities). The economy is endowed with a *single,
non-depreciating* commodity that can be worked by labour to produce
output at each date (Gale 1967 and Brock 1973, were among the first
of many extensions of the Ramsey model that contain a heterogeneous
collection of capital goods). The economy is assumed to be closed to
international trade (opening the economy to trade involves only a minor
extension to Ramsey's model). That means some of the output can be
invested so as to add to the commodity's stock while the remainder can
be consumed immediately. We call the stock of the commodity that serves
to produce output, "capital." The problem is then to find the optimum
allocation of output at each date between consumption and
investment.




Ramsey assumed that work is unpleasant. But because including the
disutility of work in our account of his work here would add nothing of
substance, we suppose that labour supply is an exogenously given
constant (e.g., it is independent of the wages labour can demand). That
enables us to suppress the supply of labour in both production and the
factors affecting well-being.




If \(K\) is the stock of capital of the economy's one and only
commodity, output is taken to be \(F(K)\), where \(F(0) = 0\) (i.e.,
output is zero if there is no capital), \(dF(K)/dK \gt 0\) (i.e.,
the marginal product of capital is positive), and \(d^2 F(K)/dK^2
\le 0\) (i.e., the marginal product of \(K\) does not increase with
\(K\)). \(F(K)\) is a *flow* (production at a moment in time), in
contrast to \(K\), which is a *stock* (quantity of capital,
period). Notice also that output depends solely on the stock of
capital. No mention is made of possible improvements in the quality of
capital or labour. Thus, there is no prospect of technological
progress or accumulation of human capital in Ramsey's model (but
see Mirrlees 1967, for one of the first of many extensions of the
Ramsey model that include technological advances in production and
human capital formation); nor are there any natural resources in the
model (but see Dasgupta and Heal 1974, for one of the first of many
extensions of the Ramsey model that include natural capital in
production).




Let \(C(t)\) be consumption at \(t\). It is a
flow (units of consumption per moment). Similarly, we write
\(K(t)\) for the stock of capital at \(t\). As
\(dK(t)/dt\) is the rate of change in the
capital stock at \(t\), it is "net investment at \(t\),"
which too is a flow. And because the capital stock is assumed not to
depreciate, gross investment equals net investment.




In Ramsey's model anticipated output at each moment equals the sum
of intended investment and intended consumption. Intentions are always
realized. To put it in technical language, the economy is in
equilibrium at each moment, which is another way of saying that at each
moment intended saving equals intended investment. (The assumption
needs no explanation in a model with a single agent, but has real bite
in a world where savers are not the same agents as investors.) Capital
is assumed to be always fully deployed, and labour (which is hidden in
the production function \(F(K)\)) is taken to be fully
employed. Output at \(t\) is \(F(K(t))\).
It follows that the economy is driven by the dynamical equation



\[\tag{1}
\frac{dK(t)}{dt} = F(K(t)) - C(t)
\]


Equation (1) says that if consumption is \(C(t)\), investment is what
remains of output. So, Ramsey's problem can be cast equally as,
"How much of a nation's output should it consume?"
If consumption is less than output at \(t\) (i.e., \(C(t) \lt
F(K(t))\), investment is positive (i.e., \(dK(t)/dt \gt 0)\) and the
stock of capital increases; but if consumption exceeds output at
\(t\), investment is negative, which means capital is eaten into and
the stock declines (i.e., \(dK(t)/dt \lt 0).\) We now imagine that
Government House is advised by a "socially-concerned
citizen," the person being someone who is trying to determine
the right balance between the economy's consumption and
investment at each date. We shall call that person the *decision
maker*, or DM. Ramsey imagined that DM is a
Classical-Utilitarian.


## 2. The Classical-Utilitarian Calculus




Classical Utilitarianism identifies the good as the expected sum of
well-being over time and across generations. Here is Sidgwick (1907:
414) on the matter:



> 
> 
> It seems ... clear that the time at which a man exists cannot
> affect the value of his happiness from the universal point of view;
> and that the interests of posterity must concern a Utilitarian as much
> as those of his contemporaries, except in so far as the effect of his
> actions on posterity - *and even the existence of human
> beings to be affected* - must necessarily be more
> uncertain. (Italics added)




To formalize this, we consider an arbitrary date \(t\) at which DM is
deliberating. Let \(\tau\) denote dates not earlier than \(t\) (i.e.,
\(\tau \ge t)\). Ramsey considered a deterministic, infinitely lived
world (but see Yaari, 1965, for the first of many extensions of the
Ramsey model that incorporate the risk of individual or societal
extinction). Well-being is assumed to be a numerical quantity. Let
\(U(t)\) be well-being at \(t\), and let \(V(t)\) be an aggregate
measure of the flow of well-being across time and generations, as
evaluated at time \(t\). Ramsey followed Sidgwick in assuming that



\[\tag{2}
V(t) = \int^{\infty}\_t[U(\tau)]d\tau
\] 



\(V(t)\) is *intergenerational well-being* at \(t\). Because
Ramsey's world is deterministic, \(V(t)\) is also the expected
value of \(V(t)\). So Sidgwick's criterion is the \(V(t)\) in
equation (2).




Well-being at any given date is assumed to be a function solely of
consumption at that date. We therefore write \(U(t) =
U(C(t))\). Ramsey assumed that marginal well-being is positive (i.e.,
\(dU(C)/dC \gt 0)\) but diminishes with increasing consumption levels
(i.e., \(d^2 U(C)/dC^2 \lt 0)\). The latter property implies that
\(U(C)\) is a *strictly concave* function. (Edgeworth 1885,
had routinized the idea that marginal well-being declines with
increasing consumption.) Thus equation (2) can be written as



\[\tag{3}
V(t) = \int^{\infty}\_t [U(C(\tau))]d\tau
\]



Classical Utilitarianism, as reflected in equation (3), requires
that if \(U\) is a numerical measure of well-being, then so is
\(\alpha U+\beta\), where \(\alpha\) is a positive number and \(\beta\)
is a number of either sign. Formally, we say that \(U\) is unique
up to "positive affine transformations." We confirm presently that the
theory's recommendations are invariant under such transformations.


### 2.1 Zero Discounting of Future Well-Beings




In equation (3), future values of \(U\) are not discounted when
viewed from the present moment, \(t\). This particular move has
provoked more debate among economists and philosophers than any other
feature of Ramsey's theory of optimum saving. The debate has on
occasion been shriller than even we economists are used to (see in
particular Nordhaus 2007). At the risk of generalizing wildly,
economists have favoured the use of positive rates to discount future
well-beings (e.g., Arrow and Kurz 1970), whereas philosophers have
insisted that the well-being of future people should be given the same
weight as that of present people (e.g., Parfit 1984).




What would Classical Utilitarianism with positive discounting of
future well-beings look like? Let \(\delta \gt 0\) be the rate at which
it is deemed desirable to discount future well-beings (for simplicity
we take the discount rate to be constant). Then, in place of equations
(2)-(3), intergenerational well-being at \(t\), would read as




\[\tag{4}
\begin{align}
V(t) &= \int^{\infty}\_t [U(\tau)e^{-\delta(\tau -t)}]d\tau \\
 &= \int^{\infty}\_t [U(C(\tau))e^{-\delta(\tau -t)}]d\tau, t \ge 0 \\
\end{align}\]


In equation (4), \(\delta\) the "time discount rate" and
\(e^{-\delta}\) the resulting "time discount factor."



\(\delta \gt 0\) implies \(e^{-\delta} \lt 1\). That
means \(e^{-\delta(\tau -t)}\) tends to
zero exponentially as \(\tau\) tends to infinity. In the latter
part of his paper Ramsey (1928: 553-555) did use equation (4) to study
the problem of optimum saving, but he did not approve of the
formulation. Instead, he wrote (p. 543) that to discount later
\(U\)'s in comparison with earlier ones is "... ethically
indefensible and arises merely from the weakness of the imagination."
In a book that inaugurated the formal study of economic development,
Harrod (1948: 40) followed suit by calling the practice a "...
polite expression for rapacity and the conquest of reason by
passion."




Strong words, but to some economists, the Ramsey-Harrod stricture in
a deterministic world reads like a Sunday pronouncement. Solow (1974a:
9) expressed this feeling exactly when he wrote, "In solemn conclave
assembled, so to speak, we ought to act as if the [discount rate on
future well-beings] were zero."




But the matter cannot be settled without a study of production and
consumption possibilities open to an economy. Consider the following
tension between two sets of considerations:


1. Low rates of consumption by generations sufficiently far into
the future would not be seen to be a bad thing by the current DM if
future well-beings were discounted at a positive rate. So today's DM
would recommend high consumption rates for now and the near future even
if that meant generations in the distant future would live in penury.
But if such a policy were followed, the demands of a further moral
requirement to Classical Utilitarianism that DM may hold, namely,
"intergenerational equity," would not be met. Therefore we should
follow Ramsey and not discount future well-beings.
2. Write \(dF(K)/dK\) as \(F\_K\). From equation (1) it is simple to
deduce that \(F\_K\) is the rate of return on investment. In
Ramsey's economy \(F\_K \gt 0\), which means every unit of output
that is saved yields more than a unit of future consumption, other
things equal. For example, if DM were to reduce consumption at \(t\)
by a unit, the additional consumption that would be available in the
briefest of periods later - we write that as \(\Delta t\)
- without affecting consumption at any future date would be
\(1+[dF(K(t))/dK(t)]\Delta t\). The productivity of capital is thus
tied to the arrow of time, which creates a bias in favour of future
generations. This bias gives bite to the adage, "We can do
something for posterity, but what can posterity ever do for us?"
The thought inevitably arises that perhaps the bias should be
countered in DM's calculus if some attention were to be given by
her to intergenerational equity in realized well-being as a supplement
to Classical Utilitarianism. That in turn suggests that DM should
abandon Ramsey and discount future well-beings at a positive rate.




The force of each consideration has been demonstrated in the
economics literature. It has been shown in the context of a simple
model that if production requires produced capital and exhaustible
resources, then optimum consumption declines to zero in the long run if
future well-beings are discounted at a positive rate (Dasgupta and
Heal 1974), but increases indefinitely if we follow Ramsey in not
discounting future well-beings (Solow 1974b). The exercises tell us
that the long-run features of optimum saving policies depend on the
relative magnitudes of the rate at which future well-beings are
discounted and the long-term productivity of capital assets.




There is a more general point here, which was explored by Koopmans
(1960, 1965, 1967, 1972) in a remarkable set of publications on the
idea of economic development. In such complex exercises as those
involving consumption and investment over a long time horizon, it is
foolish to regard any ethical principle (e.g., Classical
Utilitarianism) as sacrosanct. One can never know in advance what it
may run up against. A more judicious tactic than Ramsey's would be to
to play off one set of ethical assumptions against another in
not-implausible worlds, see what their implications are for the
distribution of well-being across generations, and then appeal to our
intuitive senses before arguing over policy. Settling *ex ante*
whether to use a positive rate to discount future well-beings could be
a self-defeating
 move.[1]


## 3. The Problem of Optimum Saving




Ramsey considered a world with an indefinite future. This could
appear to be an odd move, but it has a strong rationale. Suppose DM
were to choose a horizon of \(T\) years. As she doesn't know when
our world will end, she will want to specify the resources that should
be left behind at \(T\) in case the world doesn't terminate then.
But to find a justification for the amount to leave behind at
\(T\), DM will need an assessment of the world beyond \(T\).
That would, however, amount to including the world beyond \(T\).
And so on.




Denote a consumption stream from the present \((t = 0)\) to infinity
as \(\{C(t)\}.\) \(K(0) \gt 0\) circumscribes the economy; it is the
quantity of capital that society has inherited from the
past. Mathematicians would call \(K(0)\) an "initial
condition." The problem Ramsey set himself was to determine the
consumption stream \(\{C(t)\}\) from 0 to infinity that DM would
select if she were a Classical Utilitarian.


### 3.1 Undiscounted Utilitarianism




Call a consumption stream \(\{C(t)\}\) *feasible*
if it satisfies equation (1) with initial condition \(K(0)\). In
Ramsey's deterministic world the Classical Utilitarian formulation of
the problem of optimum national saving at date \(t = 0\) is
thus:


 "From the set of all feasible consumption streams, find that
\(\{C(t)\}\) which maximizes



\[
V(0) = \int^{\infty}\_0 [U(C(t))]dt."
\]


We will call this optimization problem, *Ramsey Mark I*.




There is a serious difficulty with *Ramsey Mark I*: it is not
coherent. Infinite sums don't necessarily converge. For any
\(\{C(t)\}\) for which the infinite integral doesn't
converge, \(V(0)\) doesn't exist. If the integral is
non-convergent for every feasible consumption streams
\(\{C(t)\}\), the maximization problem is meaningless: One
cannot maximize something that appears to be a real-valued function
\(V(0)\) when in fact the function doesn't exist.




The force of this observation can be seen in



**Example 1** (attributed to David Gale)




Suppose as an extreme special case of the Ramsey economy,
\(F(K) = 0\) for all \(K \ge 0\). Then equation (1)
reduces to



\[\tag{5}
\frac{dK(t)}{dt} = - C(t)
\]



The economy described in equation (5) consists of a non-deteriorating
piece of cake, of size \(K(0) \gt 0\) at the initial date. It is
obvious that every consumption stream \(\{C(t)\}\) satisfying equation
(5) tends to zero in the long run. Formally, \(C(t) \rightarrow 0\) as
\(t \rightarrow \infty\).




Because the \(U\)-function is unique up to positive affine
transformations, we may without any loss of generality normalize it so
that \(U(0) \ne 0\). It is then obvious that for all feasible
\(\{C(t)\}\), \(V(0)\) in *Ramsey Mark I*
diverges to minus infinity if \(U(0) \lt 0\), but diverges to
plus infinity if \(U(0) \gt 0\). That an optimum policy does not
exist in the cake-eating model can be seen if we now recall that
\(U(C)\) has been assumed to be strictly concave. The
assumption implies that any non-egalitarian distribution of consumption
among the generations can be improved upon by a suitable
redistribution. The ideal distribution would be equal consumption for
all generations. The only consumption stream with the latter property
is \(C(t) = 0\) for all \(t\). But that's the worst
possible distribution. QED


### 3.2 Re-normalizing Undiscounted Utilitarianism




The question arises whether there are circumstances in which there
is a best consumption stream even though \(V(0)\) does not
converge for all consumption streams. Ramsey formulated the question by
altering the way the saving problem is posed.




Imagine that well-being is bounded above no matter how large
consumption happens to be. Let \(U\) be the numerical measure of
well-being that DM chooses to work with. (All positive affine
transformations of \(U\) would be equally legitimate measures of
well-being.) Let \(B\) be the lowest upper bound of \(U\). Ramsey
christened it "Bliss". Because the rate of return on
investment \((F\_K)\) in his model is positive, consumption would grow
indefinitely and tend to infinity in the long run if saving rates were
suitably chosen. That means there are possible paths of economic
development in which \(U(C(t))\) tend to \(B\) in the long run. But
that implies there are possible paths of economic development in which
the short-fall of \(U(C(t))\) from \(B\) tends to zero in the long
run. If the short-fall tends to zero fast enough, the undiscounted
integral of the difference between \(U(C(t))\) and \(B\) would exist,
and DM could seek to maximize the modified integral. So we
have *Ramsey Mark II*, which reads as




"From the set of all feasible consumption streams, find that
\(\{C(t)\}\) which maximizes



\[
V(0) = \int^{\infty}\_0 [U(C(t))-B]dt."
\]



Notice that *Mark II* is a transformation of *Mark I*.
The transformation amounts to re-normalizing the optimality criterion.
Not only was the move from *Mark I* to *Mark II* on
Ramsey's part ingenious, it also displayed his moral integrity. It
would have been easy enough for him to ask DM instead to discount
future consumption and expand the range of circumstances in which
Utilitarianism provides an answer to the problem DM is attempting to
solve. He chose not to do that.




Ramsey's intuition in moving from *Mark I* to *Mark
II* was powerful, but in a paper that initiated the modern
literature on the Ramsey problem, Chakravarty (1962) observed that to
rely exclusively on the condition Ramsey had identified as being
*necessary* for a consumption stream to be the optimum (see
below) can lead to absurd results (see below, Sect. 4). In effect
Chakravarty observed that infinite integrals, even when cast in the
re-normalized form in *Ramsey Mark II*, don't necessarily
converge to finite values.


### 3.3 The Overtaking Criterion




What was needed was to de-link the question whether infinite
well-being integrals converge from the question whether optimum
consumption streams exist. That insight was provided by Koopmans (1965)
and von Weizsacker (1965). The latter author's re-statement of
the problem of optimum saving was as follows:




We say that the feasible consumption stream \(\{C^\*(t)\}\)
is *superior* to a feasible consumption stream \(\{C(t)\}\) if
there exists \(T \gt 0\) such that for all \(t \ge T\),



\[\tag{6}
\int^t\_0 [U(C^\*(s))]ds \ge \int^t\_0 [U(C(s))]ds
\]



We call \(\{C^\*(t)\}\) *optimum* if it is superior
to all other feasible consumption streams.




The condition that is represented in inequality (6) is known as the
*Overtaking Criterion* (OC), for that is what it is. OC avoids
asking whether the integrals on either side of inequality (6) converge
as \(t \rightarrow \infty\). If they do, OC reduces to Classical
Utilitarianism. But OC is able to respond to Ramsey's saving problem in
a wider class of situations. In his work Koopmans (1965) identified a
canonical economic model in which the \(U\)-function is bounded
above and in which *Ramsey Mark II* is equivalent to an
optimization problem that is posed in terms of OC.




What are we to make of the ethics of discounting the well-beings of
future generations? Ramsey (1928) began by dismissing it but then
studied it at the tail end of his paper. DM could of course justify
discounting future well-being if there is a possibiliy of future
extinction. Sidgwick (1907) himself noted that in the passage quoted
earlier. If Classical Utilitarianism is taken to commend the expected
sum of well-beings, then the "hazard rate" at date
\(t\) (i.e., the probability of extinction at date \(t\)
conditional on society surviving until \(t)\) would appear in the
expression for expected well-being as a discount rate for well-being at
\(t\). The question remains whether Classical Utilitarianism would
insist on zero-discounting of future utilities in a deterministic
world.




In a remarkable pair of works Koopmans (1960, 1972) exposed internal
contradictions in ethical reasoning in a deterministic world in both
*Ramsey Mark I* and *Ramsey Mark II*. He (and
subsequently Diamond, 1965) showed that if relatively weak normative
requirements are imposed on the concept of intergenerational well-being
in a deterministic world, equal treatment of the \(U\)-function
across generations has to be abandoned. We turn to that now.


### 3.4 Discounted Utilitarianism




It transpires the mathematics is a lot simpler if, instead of
assuming time is continuous, time is taken to be discrete. Thus we now
assume that \(t = 0,1,2,\ldots\) . Assume also that
intergenerational well-being at \(t = 0\) can be measured in terms
of a numerical function \(V\). The idea is to require the
function, which is defined on infinite well-being streams, to satisfy
properties that reflect ethical directives.




Let \(\{U(t)\}\) be an infinite well-being stream, that is, \(\{U(t)\}
= (U(0),U(1),\ldots ,U(t),\ldots)\). We say \(V(\{U(t)\})\)
is *continuous* if in an appropriate mathematical sense the
values of \(V\) for well-being streams \(\{U(t)\}\) that don't
differ much in the space of \(\{U(t)\}\)s are close to one another. A
further condition on the \(V\)-function that is ethically attractive
is "monotonicity". To define the notion let us say a
well-being stream is "superior" to another if no
generation enjoys less well-being along the former than along the
latter and if there is at least one generation that enjoys greater
well-being in the former than it does in the latter. We say that \(V\)
is *monotonic* if \(V\) is larger for a well-being stream than
it is for another if the former is superior to the latter.




Both properties are attractive. Lexicographic orderings
notwithstanding, there are not convincing arguments against continuity.
Of course Rawls (1972) placed priority rules and the lexicographic
orderings on the objects of interest in his conception of justice that
come with them at the centre of his theory, but that has proved to have
been one of his most contentious moves. The richness and depth of his
analysis would not be lessened if small tradeoffs were admitted between
the objects of justice. And it's hard to find reasons against
monotonicity. Even Rawls, whose work was so pointed toward distributive
justice, insisted on monotonicity.




But it can be shown that any \(V\)-function that satisfies
continuity and monotonicity must have generation discounting built into
it. It would seem the real numbers are not rich enough to accommodate
infinite well-being streams in a manner that respects continuity and
monotonicity while awarding the well-beings of all generations equal
weight. Proof of the proposition is in Diamond (1965), and was
attributed by the author to Menahem Yaari. So we now introduce positive
well-being discounting in the \(V\)-function and formulate
*Ramsey Mark III*.




Return once again to the formulation where time is continuous. As
previously, we say a consumption stream \(\{C(t)\}\) is
*feasible* if it satisfies equation (1) with an initial capital
stock of \(K(0)\). *Ramsey Mark III* (Ramsey 1928,
553-555) is then:




"From the set of all feasible consumption streams, find that
\(\{C(t)\}\) which maximizes



\[
V(0) = \int^{\infty}\_0 [U(C(t))e^{-\delta t}]dt, \delta \gt 0."
\]


In *Mark III* the discount rate \(\delta\) is a positive
constant. That means the corresponding discount factor
\(e^{-\delta}\) is less than 1. The latter in turn can be
shown to mean that in a wide range of economic models
\(e^{-\delta t}\) tends to zero at so fast a rate
that *Mark III* has an answer.




Let \(\{C^\*(t)\}\) be the solution of *Ramsey Mark
III*. Heuristically it is useful to imagine that there is a DM at
each date. The measure of intergenerational well-being for the DM at
date \(t\) is the \(V(t)\) of equation (4). Notice that the ethical
views of the successive DMs are congruent with one another. There is
thus no need for the DMs to draw up an "intergenerational
contract". The DM at every date will want to choose the level of
consumption it deems to be optimum, aware that succeeding DMs will
choose in accordance with what she had planned for them. In modern
game theoretic parlance, Ramsey's optimum consumption stream
\(\{C^\*(t)\}\) is a "non-cooperative" (Nash) equilibrium
among the DMs.


## 4. The Ramsey Rule and Its Ramifications




We now construct an informal version of the variational argument
Ramsey used for determining \(\{C^\*(t)\}\) in *Mark
III*. Loosely speaking, the DMs require the marginal rate of
ethically indifferent substitution between consumption at any two brief
periods of time to equal the marginal rate at which consumption can be
transformed between those same pair of brief periods of time. Their
equality (i.e., the right balance from among the
"desirables" and the "feasibles") is a
*necessary* property of an optimum consumption stream.




Ramsey constructed a mathematical expression of the property, but
did not look for conditions that, taken together, are both necessary
and *sufficient*. We will use a simple example, which is also in
his paper, to show how a sufficient condition can be obtained.


### 4.1 The Variational Argument




Write \(dU/dC = U\_C\) and \(d^2 U/dC^2 = U\_{CC}.\) Let \(\{C(t)\}\) be
a feasible consumption stream. We first deduce a formal expression for
the marginal rate of ethically indifferent substitution between
consumption at any two brief periods of time. Suppose the intention is
to reduce consumption at some future date \(t\) by a small quantity
\(\Delta C(t)\) and raise consumption at a nearby date \(t+\Delta t\)
while keeping consumption at all other dates the same as in
\(\{C(t)\}\). The loss in well-being that would follow from the move
is \(e^{-\delta t}U\_{C(t)}\Delta C(t)\). We now seek to
determine the percentage increase in consumption that would be
required at \(t+\Delta t\) if \(V(0)\) is to remain unchanged; because
that's the marginal rate of ethically indifferent substitution
between consumption at \(t\) and consumption at \(t+\Delta t\). Denote
that rate by \(\varrho(t)\). Then \(\varrho(t)\) must be the
percentage rate at which discounted marginal well-being declines at
\(t\). It also follows that \(\varrho(t)\) is the rate the DM at \(t =
0\) would use to discount a unit of consumption at \(t\) so as to
bring it to the present (because that's what is meant by the
percentage rate at which discounted marginal well-being declines at
\(t\) - for a formal demonstration, see Dasgupta, 2008). Some
economists call \(\varrho(t)\) the *consumption rate of
interest* (Little and Mirrlees 1974), others call it
the *social rate of discount* (Arrow and Kurz
1970). \(\varrho(t)\) is a fundamental object in social cost-benefit
analysis.




Let \(\Delta\) be vanishingly small. Then, by definition



\[\tag{7}
\varrho(t) = - [d(e^{-\delta t}U\_{C(t)})/dt]/e^{-\delta t}U\_{C(t)}
\]



So as to simplify the notation let \(g(C(t))\) denote the percentage
rate of growth in \(C(t)\) (i.e. \(g(C(t)) = [dC(t)/dt]/C(t)\), which
can be negative), and let \(\sigma(C)\) denote the elasticity of
marginal well-being (i.e., \(\sigma(C) = -CU\_{CC}/U\_C \gt
0)\). Equation (7) then simplifies to



\[\tag{8}
\varrho(t) = \delta + \sigma(C(t))g(C(t))
\]



Because \(\{C^\*(t)\}\) is by assumption the optimum, no feasible
deviation from \(\{C^\*(t)\}\) can increase \(V(0)\). That means the
consumption rate of interest \((\varrho(t))\) must equal the social
rate of return on investment \((F\_{K(t)})\) at every \(t\). To see
why, suppose in some vanishingly small interval of time \(F\_{K(t)} \gt
\varrho(t)\). Then \(V(0)\) could be increased by consuming a unit
less at \(t\) and enjoying the return of \((1+F\_{K(t)})\) soon after.
Alternatively, if \(F\_{K(t)} \lt \varrho(t), V(0)\) could be increased
by consuming a unit more at \(t\) and reducing consumption soon after
by an amount equal to the return \((1+F\_{K(t)})\). But that means
the consumption rate of interest \(\varrho(t)\) equals the social rate
of return \(F\_{K(t)}\) along \(\{C^\*(t)\}\) at every date. Using
equation (8) we have,



\[\tag{9}
\delta + \sigma(C(t))g(C(t)) = F\_{K(t)}
\]



Equation (9) is the Ramsey Rule. It is a necessary condition for
optimality in *Ramsey Mark III* and is unarguably the most
famous equation in intertemporal welfare economics. The rule is a
formal statement of the requirement of \(\{C^\*(t)\}\), that
the marginal rate of substitution between consumption at two nearby
dates (the left-hand-side of eq. 9) equals the marginal rate of
transformation between consumption at those same pair of nearby dates
(the right-hand-side of eq. (9). It is simple to confirm that equation
(9) is invariant under positive affine transformations of the
\(U\)-function.


### 4.2 Incompleteness in Ramsey's Analysis




Presently we will specify a \(U\)-function for which \(\sigma\) is
independent of \(C\). For the moment we merely suppose that
\(\sigma\) is constant. In that case the Ramsey Rule reads as



\[\tag{10}
\delta + \sigma g(C(t)) = F\_{K(t)}
\]



In *Ramsey Mark III*, \(K(0)\) is given as an inheritance from
the past. That means \(F\_{K(0)}\) is given as an initial condition, it
is not a choice for the DM at \(t = 0\). Moreover \(\delta\) and
\(\sigma\) are parameters, both reflecting ethical values. The DM can
therefore determine \(g(C(0))\) from equation (10). But that's
the optimum percentage rate of growth in consumption at the initial
date. The Ramsey Rule gives the DM an equation for determining the
initial growth rate of consumption, but it does not say what the
initial *level* of consumption ought to be. Below we show by
way of an example that there are an infinity of feasible consumption
paths satisfying the Ramsey Rule. It follows that the DM at \(t = 0\)
needs a further condition to determine \(C^\*(0)\).


**Example 2** (the linear economy)




Assume



\[\begin{align}
\tag{11a} F(K) &= \mu K, \mu \gt 0 \\
\tag{11b} U(C) &= - C^{-(\sigma -1)}, \sigma \gt 1
\end{align}\]



From equation (11a) it follows that \(F\_K = \mu\), which means the
rate of return on investment is constant. From equation (11b) it
follows that \(\sigma\) is the elasticity of marginal well-being.
Notice also that \(U(C) \rightarrow -\infty\) as \(C \rightarrow 0\)
and that, under the chosen normaliztion of the \(U\)-function, \(U(C)
\rightarrow 0\) as \(C \rightarrow \infty\). Using equation (11a) in
equation (1) yields,



\[\tag{12}
\frac{dK(t)}{dt} = \mu K(t) - C(t)
\]



Write \(m = (\mu -\delta)/\sigma\). Applying equations (11a-b)
to equation (10) reduces the Ramsey Rule to



\[\tag{13}
\frac{dC(t)}{dt} = [(\mu - \delta)/\sigma]C(t) = mC(t)
\]



Equation (13) says that if \(\mu \lt \delta , C(t)\) declines to 0 at
an exponential rate. Empirically, the pausible case to consider is
\(\mu \gt \delta\), which is what we shall do here. It means that the
rate of return on investment \((\mu)\) exceeds the rate at which time
is discounted \((\delta)\). And that in turn means \(m \gt 0\).
Integrating equation (13) yields



\[\tag{14}
C(t) = C(0)e^{mt}
\]



Equation (14) says \(C(t)\) grows exponentially at the
rate \(m\). We reconfirm a point that was made previously, that
although equation (14) reveals the rate of growth optimum consumption
at the initial date (i.e., \(t = 0)\), it doesn't reveal the
initial level of consumption (i.e., \(C(0)\)). That's the
indeterminacy in the Ramsey Rule.




The simplest way to determine the optimum initial consumption,
\(C^\*(0)\), is to observe from equation (14) that if
\(C^\*(t)\) grows indefinitely at the rate \(m\), so
should \(K(t)\) be required to grow at that same rate.
The reason is that if the growth rate of \(K(t)\) were to
be less than \(m\), capital would be eaten into, which means the
stock would be exhausted in finite time. The economy would then cease
to exist \((V(0)\) would be minus infinity if the future
trajectory of the economy were to be thus.) If on the other hand the
growth rate of \(K(t)\) were to exceed \(m\), there
would be over-accumulation of capital, in the sense that consumption
would be lower at every date than it needs be. The situation would
resemble one where DM throws away a part of the initial capital stock
\(K(0)\) and then settles on a saving behaviour that satisfies the
Ramsey Rule.




Exponential growth in our linear economy (eq. 11a) tells us that
the saving rate should be constant. Let us define the *saving
rate*, \(s\), as the proportion of output (GDP) that is
invested at each instant. Then equation (1) can be re-written as



\[\tag{15}
\frac{dK(t)}{dt} = s\mu K(t)
\]



Equation (15) says that intended saving equals intended investment.
Integrating equation (15) yields



\[\tag{16}
K(t) = K(0)e^{s\mu t}
\]



But we are insisting that both \(K(t)\) and \(C(t)\) should grow at
the same rate. Equations (14) and (16) therefore imply



\[\tag{17}
m = \frac{\mu -\delta}{\sigma} = s\mu
\]



The saving rate in equation (17) is the optimum. So we write it as
\(s^\*\). Thus



\[\tag{18}
s^\* = \frac{m}{\mu} = \frac{\mu -\delta}{\sigma\mu} \lt 1
\]



Equations (16)-(18) tell us that the optimum rate of growth of
consumption, \(g^\*\), is




\[\tag{19}
g^\* = \frac{\mu -\delta}{\sigma} \gt 0
\]





Notice also that if \(\delta = 0\), equation (18) reduces to



\[\tag{20}
s^\* = \frac{1}{\sigma}
\]



Equation (20) offers as elegant a simplified answer as there could
be to the question with which Ramsey started his paper.


### 4.3 The Transversality Condition




The linear technology (eq. 11a) and the iso-elastic
\(U\)-function (eq. 11b) allowed us to recognise immediately
that if a consumption stream satisfying the Ramsey Rule is to be the
optimum, both capital and consumption should grow at the same
exponential rate, \(m\). Identifying a sufficient condition for
optimality in more general models is a lot more difficult. What we need
is a condition on the long-run features of a consumption stream
satisfying the Ramsey Rule that can ensure it is the optimum. von
Weizsacker (1965) showed that the required condition relates to the
long-run behaviour of the social value of capital associated with that
consumption stream. We now formalize the condition.




Let \(U\) be the unit of account. Consider a consumption stream
\(\{C(t)\}\). It follows that \(U\_{C(t)}\) is the social worth of a
marginal unit of consumption. Write \(P(t)\) for \(U\_{C(t)}. P(t)\)
is called the (spot) *accounting price* of consumption. Because
\(e^{-\delta t}P(t)\) is the discounted value of \(P(t)\), it is
called the present-value accounting price of consumption. If
\(\{C(t)\}\) satisfies the Ramsey Rule in *Mark III*,
\(e^{-\delta t}P(t)\) is also the present-value accounting price of a
unit of capital stock. von Weizsacker (1965) showed that a sufficient
condition for the optimality of \(\{C(t)\}\) is \(e^{-\delta
t}P(t)K(t) \rightarrow A\) as t \(\rightarrow \infty\), where \(A\) is
a (finite) non-negative number. In words, a necessary and sufficient
condition for \(\{C(t)\}\) to be the optimum is (i) that it satisfies
the Ramsey Rule, and (ii) that the present-value of the
economy's stock of capital is finite. Condition (ii), which is
widely known as the "transversality condition," eliminates
those feasible consumption streams that satisfy the Ramsey Rule but
along which there is excessive saving. A simple calculation confirms
that in Example 2 the transversality condition is satisfied if the
saving rate is \(s^\*\) (eq. 18).


### 4.4 Numerical Estimates of the Optimum Rate of Saving




Equation (18) says that \(s^\*\) is an increasing function of the
return on investment \((\mu)\), a decreasing function of the time rate of
discount \((\delta)\), and a decreasing function of the elasticity of
marginal well-being \((\sigma)\). Each of these properties is intuitively
obvious:




(1) The higher is the rate of return on investment \((\mu)\), the
greater is the gain to future generations from a marginal increase in
saving by initial generations. That says the optimum rate of saving
should be an increasing function of \(\mu\), other things equal.




(2) The larger is the value of the time discount rate \((\delta)\)
chosen by DM, the lower is the weight that she awards to the well-being
of future generations. That implies higher optimum consumption levels
for early generations (Sect. 2.1), which in turn implies that the
optimum rate of saving is lower, other things equal.




(3) As the return on investment is positive \((\mu \gt 0)\), the arrow
of time displays a bias in favour of future generations (Sect. 2.1).
But the larger is the chosen value of \(\sigma\), the more DM displays
concerns over equity in consumption across the generations. Therefore,
the larger is that concern, the higher is the optimum rate of
consumption to be enjoyed by initial generations. So we should expect
the optimum rate of saving to be a decreasing function of \(\sigma\),
other things equal.




It is instructive to consider stylized figures for the parameters on
the right-hand-sides of equations (18) and (19), respectively. Although
stylized, they are figures for the pair of ethical parameters \(\sigma\)
and \(\delta\) that economists who have written on the economics of
climate change have assumed in their work. To be sure, the welfare
economics of climate change has demanded more complicated models than
the model that is represented in equations (1) and (11a), but as we
confirm below, it has not offered any additional *theoretical*
insights. In what follows we take a year to be the unit of time and
assume that \(\mu = 0.05\) (i.e., 5% a year). Along the optimum, the
consumption rate of interest equals the rate of return on investment
(the Ramsey Rule), which means that the optimum consumption rate of
interest equals a constant 5% a year.




A figure of 5% a year for \(\mu\) implies a capital-output ratio
\((1/\mu)\) of 20 years, which is far higher than the estimates of
capital-output ratios from inter-industry studies that economists in
various parts of the world have arrived at (Behrman 2001); a
representative figure for 1/\(\mu\) in that literature is 3 years. But
their estimates have been based on a definition of
"capital" that is confined to "produced"
capital, such as factories, roads, ports, and buildings. Human capital
(education, health, knowledge) is missing from them, as is natural
capital (ecosystems, sub-soil resources). Ramsey's model, as
encapsulated in equation (11a), embraces all forms of capital goods. No
doubt his formulation requires a heroic (read, impossible!) feat of
aggregation, but when all capital goods that enter production are taken
into account, we should expect an aggregate capital-output ratio (which
we should call the (inclusive) wealth-output ratio), to be a lot higher
than 3 years; perhaps even higher than 20 years (Arrow et al., 2012,
2013). Large categories of capital goods are absent from the national
economic accounts that inform we economists' understanding of
production and consumption possibilities (Dasgupta, 2019). It would
thus seem there is still a long way to go before we can reach a good
approximation of what we should bequeath to our descendants.



**Example 3** (taken from the economics of climate
change)



We now turn our attention to the values of the two ethical parameters
in equation (11b) that were chosen by three economists in their study
of the economics of climate change.



\[\begin{align}
\tag\*{Cline (1992)} \sigma = 1.5 \quad &\text{and} \quad \delta = 0 \\
\tag\*{Nordhaus (1994)} \sigma = 1 \quad &\text{and} \quad \delta = 0.03 \text{ (3% a year)} \\
\tag\*{Stern (2007)} \sigma = 1 \quad &\text{and} \quad \delta = 0.001 \text{ (0.1% a year)} 
\end{align}\]


(NB: \(\sigma = 1\) corresponds to the logarithmic well-being function,
that is \(U(C) =\) log\(C\), and can be obtained as
a limit of the functional form of \(U(C)\) in equation
(11b) as \(\sigma \rightarrow 1.)\)




We impose those parameter values to find that the optimum saving
rate \(s^\*\) (eq. 18) and the optimum rate of growth of consumption
(eq. 19) are, in turn:



\[\begin{align}
\tag{21a} s^\* = 67\% \quad &\text{and} \quad g^\* = 3.3\% \text{ a year (Cline)} \\
\tag{21b} s^\* = 40\% \quad &\text{and} \quad g^\* = 2.0\% \text{ a year (Nordhaus)} \\
\tag{21c} s^\* = 98\% \quad &\text{and} \quad g^\* = 4.9\% \text{ a year (Stern)} 
\end{align}\]

### 4.5 Commentary




A national saving rate of 40% (eq. 21b) is no doubt high by the
standards of contemporary western economies, but there are countries
that in recent years have achieved 40-45% saving rates (China is
a prominent example). A figure of 67% for \(s^\*\) (eq. 21a) is higher
than the saving rate in any country, but is not beyond belief. The
truly outlandish figures is 98% (eq. 21c). It is outlandish
especially because the figure is the optimum saving rate no matter how
small \(K(0)\) happens to be. Admittedly, the model here
(eqs. 11a-b) is phenomenally stylized, but it does bring out
sharply the observation of Koopmans (1965), that it is foolish to
assume \(\delta = 0\) (or close to 0) without first checking its
possible consequences for the distribution of well-being across the
generations.




Equation (19) has shown that the optimum growth rate of consumption
is bounded above by \(\mu\), which explains why \(g^\*\) is less than
5% a year for each of the three parametric specifications we have
considered. The specifications come from three studies in the
welfare economics of global climate change, in which the authors worked
with models that are a lot more complex than Ramsey's. And yet
their findings are exactly what his formulation would point to
(Dasgupta 2008), namely, that other things equal, the lower is the
chosen value of \(\delta\) and/or the larger is the damage to future
well-being that is expected to be caused by global climate change, the
greater is the investment level DM should recommend to avert climate
change or soften the effects of that change on human well-being. The
often shrill debate (e.g., Nordhaus 2007) on the extent to which
global investment should be directed at reducing the deletarious
effects of climate change was spurred by differences in model
specification among climate-change economists.




The linear technology (eq. 11a) and the iso-elastic
\(U\)-function (eq. 11b), when taken together, have offered
deep insights even though we have restricted the discussion to
pen-and-paper calculations here. The functional forms are not
believable; nevertheless, Ramsey made use of them. His paper showed
that unbelievably simplified models, provided their construction is
backed by strong intuition, can illuminate questions that are
seemingly impossible to frame, let alone to answer
quantitatively. That has been Ramsey's enduring gift to
theoretical economics.