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\begin{center}\textbf{\Large MATHEMATICAL APPENDIX} \\ to \\
Principles of Economics \\ by \\ Alfred Marshall
\\ (http://www.econlib.org/library/Marshall/marPContents.html)\end{center} N${\scshape OTE}$
I. (p. 93). The law of diminution of marginal utility may be
expressed thus:---If $u$ be the total utility of an amount $x$ of
a commodity to a given person at a given time, then marginal
utility is measured by $\displaystyle\frac{du}{dx} \cdot \delta
x$; while $\displaystyle\frac{du}{dx}$ measures the \it marginal
degree \rm$\;$of utility. Jevons and some other writers use
``Final utility" to indicate what Jevons elsewhere calls Final
degree of utility. There is room for doubt as to which mode of
expression is the more convenient: no question of principle is
involved in the decision. Subject to the qualifications mentioned
in the text $\displaystyle \frac{d^2u}{dx^2}$ is always negative.
\vskip7pt N${\scshape OTE}$ II. (p. 96). If $m$ is the amount of
money or general purchasing power at a person's disposal at any
time, and $\mu$ represents its total utility to him, then
$\displaystyle{\frac{d\mu}{dm}}$ represents the marginal degree of
utility of money to him.
If $p$ is the price which he is just willing to pay for an amount $x$ of the commodity which gives him
a total pleasure $u$, then $$\frac{d\mu}{dm} \Delta p = \Delta
u; {\rm and} \frac{d\mu}{dm} \frac{dp}{dx}=\frac{du}{dx}.$$
If $p'$ is the price which he is just willing to pay for an amount
$x'$ of another commodity, which affords him a total pleasure
$u'$, then $$\frac{d\mu}{dm} \cdot \frac{dp'}{dx'} =
\frac{du'}{dx'};$$ and therefore \hspace{55pt}
$\displaystyle{\frac{dp}{dx}:\frac{dp'}{dx'} =
\frac{du}{dx}:\frac{du'}{dx'}.}$
\noindent (Compare Jevons' chapter on the {\it Theory of
Exchange}, p. 151.)
Every increase in his means diminishes the
marginal degree of utility of money to him; that is,
$\displaystyle \frac{d^2\mu}{dm^2}$ is always negative.
Therefore, the marginal utility to him of an amount $x$ of a
commodity remaining unchanged, an increase in his means increases
$\displaystyle \frac{du}{dx} \div \frac{d\mu}{dm}$; i.e. it
increases $\displaystyle \frac{dp}{dx}$, that is, the rate at
which he is willing to pay for further supplies of it. We may
regard $\displaystyle \frac{dp}{dx}$ as a function of $m, u$, and
$x$; and then we have $\displaystyle \frac{d^2p}{dm\,dx}$ always
positive. Of course $\displaystyle \frac{d^2p}{du\,dx}$ is always
positive.
\vskip7pt N${\scshape OTE}$ III. (p. 103). Let $P, P'$ be
consecutive points on the
\begin{wrapfigure}[10]{r}{275pt}{\epsfig{file=MrshMathAppg1.eps,height=178pt}}\end{wrapfigure}
demand curve; let $PRM$ be drawn perpendicular to $Ox$, and let
$PP'$ cut $Ox$ and $Oy$ in $T$ and $t$ respectively; so that $P'R$
is that increment in the amount demanded which corresponds to a
diminution $PR$ in the price per unit of the commodity.
Then the elasticity of demand at $P$ is measured by
\begin{eqnarray*}\displaystyle{\frac{P'R}{OM} \div \frac{PR}{PM},} \quad &\displaystyle{{\rm
i.e.\;by} \; \frac{P'R}{PR} \times \frac{PM}{OM};} \cr\cr
&\displaystyle{{\rm i.e.\;by} \; \frac{TM}{PM} \times
\frac{PM}{OM},} \cr\cr &\hspace{17pt}\displaystyle{{\rm i.e.\;by}
\; \frac{TM}{OM} \; {\rm or\;by} \; \frac{PT}{Pt}.}
\end{eqnarray*}
\noindent When the distance between $P$ and $P'$ is diminished
indefinitely, $PP'$ becomes the tangent; and thus the proposition
stated on p. 103 is proved.
It is obvious {\it \`a priori} that the measure of
elasticity cannot be altered by altering relatively to one another
the scales on which distances parallel to $Ox$ and $Oy$ are
measured. But a geometrical proof of this result can be got easily
by the method of projections: while analytically it is clear that
$\displaystyle \frac{dx}{x} \div \frac{-dy}{y}$, which is the
analytical expression for the measure of elasticity, does not
change its value if the curve $y=f(x)$ be drawn to new scales, so
that its equation becomes $qy=f(px)$; where $p$ and $q$ are
constants.
If the elasticity of demand be equal to unity for all
prices of the commodity, any fall in price will cause a
proportionate increase in the amount bought, and therefore will
make no change in the total outlay which purchasers make for the
commodity. Such a demand may therefore be called a {\it constant
outlay demand}. The curve which represents it, a {\it constant
outlay curve}, as it may be called, is a rectangular hyperbola
with $Ox$ and $Oy$ as asymptotes; and a series of such curves are
represented by the dotted curves in the following figure.
There is some advantage in accustoming the eye to the shape of
these curves; so that when looking at a demand curve one can tell
at once whether it is inclined to the vertical at any point at a
greater or less angle than the part of a constant outlay curve,
which would pass through that point. Greater accuracy may be
obtained by tracing constant outlay curves on thin paper, and then
laying the paper over the demand curve. By this means it may, for
instance, be seen at once that the demand curve in the figure
represents at each of the points $A,B,C$ and $D$ an elasticity
about equal to one: between $A$ and $B$, and again between $C$ and
$D$, it represents an elasticity greater than one; while between
$B$ and $C$ it represents an elasticity less than one. It will be
found that practice of this kind makes it easy to detect the
nature of the
assumptions with regard to the character of the demand for a
commodity, which are implicitly made in drawing a demand curve of
any particular shape; and is a safeguard against the unconscious
introduction of improbable assumptions.
\begin{center}\epsfig{file=Mrshg2aApp.eps,width=200pt,height=200pt}\end{center}
The
general equation to demand curves representing at every point an
elasticity equal to $n$ is $\displaystyle \frac{dx}{x} + n
\frac{dy}{y} =0$, i.e. $xy^{n}=C$.
It is worth noting that in
such a
curve $\displaystyle \frac{dx}{dy} = - \frac{C}{y^{n+1}}$; that
is, the proportion in which the amount demanded increases in
consequence of a small fall in the price varies inversely as the
$(n+1)^{\rm th}$ power of the price. In the case of the constant
outlay curves it varies inversely as the square of the price; or,
which is the same thing in this case, directly as the square of
the amount.
\begin{wrapfigure}[11]{r}{180pt}{\epsfig{file=MrshMathAppg3.eps,height=175pt}}\end{wrapfigure}
\vskip7pt N${\scshape OTE}$ IV. (p. 110). The lapse of
time being measured downwards along $Oy$; and the amounts, of
which record is being made, being measured by distances from $Oy$;
then $P'$ and $P$ being adjacent points on the curve which traces
the growth of the amount, the rate of increase in a small unit of
time $N'N$ is $$\frac{PH}{P'N'} = \frac{PH}{P'H} \cdot \frac{P'H}{P' N'}
= \frac{PN}{Nt} \cdot \frac{P'H}{P'N'} = \frac{P'H}{Nt};$$
\noindent since $PN$ and $P'N'$ are equal in the limit.
If
we take a year as the unit of time we find the annual rate of
increase represented by the inverse of the number of years in
$Nt$.
If $Nt$ were equal to $c$, a constant for all points of the curve,
then the rate of increase would be constant and equal to
$\displaystyle \frac{1}{c}$. In this case $\displaystyle -x
\frac{dy}{dx} =c$ for all values of $x$; that is, the equation to
the curve is $y = a - c \log x$
\vskip7pt N${\scshape OTE}$ V. (p.
123). We have seen in the text that the rate at which future
pleasures are discounted varies greatly from one individual to
another. Let $r$ be the rate of interest per annum, which must be
added to a present pleasure in order to make it just balance a
future pleasure, that will be of equal amount to its recipient,
when it comes; then $r$ may be 50 or even 200 per cent. to one
person, while for his neighbour it is a negative quantity.
Moreover some pleasures are more urgent than others; and it is
conceivable even that a person may discount future pleasures in an
irregular random way; he may be almost as willing to postpone a
pleasure for two years as for one; or, on the other hand, he may
object very strongly indeed to a long postponement, but scarcely
at all to a short one. There is some difference of opinion as to
whether such irregularities are frequent; and the question cannot
easily be decided; for since the estimate of a pleasure is purely
subjective, it would be difficult to detect them if they did
occur. In a case in which there are no such irregularities, the
rate of discount will be the same for each element of time; or, to
state the same thing in other words, it will obey the exponential
law. And if $h$ be the future amount of a pleasure of which the
probability is $p$, and which will occur, if at all, at time $t$;
and if $R = 1 + r$; then the present value of the pleasure is
$phR^{-t}$. It must, however, be borne in mind that this result
belongs to Hedonics, and not properly to Economics.
Arguing still
on the same hypothesis we may say that, if $\varpi$ be the
probability that a person will derive an element of happiness,
$\Delta h$, from the possession of, say, a piano in the element of
time $\Delta t$, then the present value of the piano to him is
$\displaystyle{\int^{T}_{0} \varpi R^{-t} \frac{dh}{dt} dt}$. If
we are to include all the happiness that results from the event at
whatever distance of time we must take $T = \infty$. If the source
of pleasure is in Bentham's phrase ``impure,'' $\displaystyle \frac{dh}{dt}$
will probably be negative for some values of $t$; and
of course the whole value of the integral may be negative.
\vskip7pt N${\scshape OTE}$ VI. (pp. 132, 3). If $y$ be the
price at which an amount $x$ of a commodity can find purchasers in
a given market, and $y = f(x)$ be the equation to the demand
curve, then the total utility of the commodity is measured by
$\int^{a}_0 f(x)dx$, where $a$ is the amount consumed.
If however
an amount $b$ of the commodity is necessary for existence, $f(z)$
will be infinite, or at least indefinitely great, for values of
$x$ less than $b$. We must therefore take life for granted, and
estimate separately the total utility of that part of the supply
of the commodity which is in excess of absolute necessaries: it is
of course $\int^{a}_b f(x)dx$.
If there are several commodities
which will satisfy the same imperative want, as e.g. water and
milk, either of which will quench thirst, we shall find that,
under the ordinary conditions of life, no great error is
introduced by adopting the simple plan of assuming that the
necessary supply comes exclusively from that one which is
cheapest.
It should be noted that, in the discussion of consumers'
surplus, we assume that the marginal utility of money to the
individual purchaser is the same throughout. Strictly speaking we
ought to take account of the fact that if he spent less on tea,
the marginal utility of money to him would be less than it is, and
he would get an element of consumer's surplus from buying other
things at prices which now yield him no such rent. But these
changes of consumers' rent (being of the second order of
smallness) may be neglected, on the assumption, which underlies
our whole reasoning, that his expenditure on any one thing, as,
for instance, tea, is only a small part of his whole expenditure.
(Compare Book V. ch. ${\scshape II}$. $\S 3$.) If, for any reason,
it be desirable to take account of the influence which his
expenditure on tea exerts on the value of money to him, it is only
necessary to multiply $f(x)$ within the integral given above by
that function of $xf(x)$ (i.e. of the amount which he has already
spent on tea) which represents the marginal utility to him of
money when his stock of it has been diminished by that amount.
\vskip7pt N${\scshape OTE}$ VII. (p. 134). Thus if $a_1, a_2,
a_3, \ldots$ be the amounts consumed of the several commodities of
which $b_1, b_2, b_3, \ldots$ are necessary for existence, if $y =
f_1 (x), y = f_2 (x), y = f_3 (x), \ldots$ be the equations to
their demand curves and if we may neglect all inequalities in the
distribution of wealth; then the total utility of income,
subsistence being taken for granted, might be represented by
$\Sigma\int_b^a f(x)dx$, if we could find a plan for grouping
together in one common demand curve all those things which satisfy
the same wants, and are rivals; and also for every group of things
of which the services are complementary (see Book V. ch.
${\scshape VI}$). But we cannot do this: and therefore the formula
remains a mere general expression, having no practical
application. See footnote on pp. 131, 2; also the latter part of
Note XIV.
\vskip7pt N${\scshape OTE}$ VIII. (p. 135). If $y$ be
the happiness which a person derives from an income $x$; and if,
after Bernoulli, we assume that the increased happiness which he
derives from the addition of one per cent. to his income is the
same whatever his income be, we have $\displaystyle x \frac{dy}{dx}
=K$, and $ \therefore$ $y = K \log x +C$ when $K$ and $C$
are constants. Further with Bernoulli let us assume that, $a$
being the income which affords the {\it barest} necessaries of
life, pain exceeds pleasure when the income is less than $a$, and
balances it when the income equals $a$; then our equation becomes
$\displaystyle y = K \log \frac{x}{a}$. Of course both $K$ and
$a$ vary with the temperament, the health, the habits, and the
social surroundings of each individual. Laplace gives to $x$ the
name {\it fortune physique}, and to $y$ the name {\it fortune
morale}.
Bernoulli himself seems to have thought of $x$ and $a$ as
representing certain amounts of property rather than of income;
but we cannot estimate the property necessary for life without
some understanding as to the length of time during which it is to
support life, that is, without really treating it as income.
Perhaps the guess which has attracted most attention after
Ber\-noulli's is Cramer's suggestion that the pleasure afforded by
wealth may be taken to vary as the square root of its amount.
\vskip7pt N${\scshape OTE}$ IX. (p. 135). The argument that fair
gambling is an economic blunder is generally based on Bernoulli's
or some other definite hypothesis. But it requires no further
assumption than that, firstly the pleasure of gambling may be
neglected; and, secondly $\phi{''} (x)$ is negative for all values
of $x$, where $\phi (x)$ is the pleasure derived from wealth equal
to $x$.
For suppose that the chance that a particular event will happen is
$p$, and a man makes a fair bet of $py$ against $(1-p)y$ that it
will happen. By so doing he changes his expectation of happiness
from $$\phi (x) \ \hbox{to} \ p \phi \{x + (1 - p) y \} + (1 - p)
\phi (x - py).$$ \noindent This when expanded by Taylor's Theorem
becomes \begin{eqnarray*}\phi (x) &&+ \frac{1}{2} p (1 - p)^2 y^2
\phi{''} \{x + \theta (1 - p) y \} \\ \\ &&+ \frac{1}{2} p^2 (1 -
p) y^2 \phi{''} ( x - \Theta py);\end{eqnarray*} \noindent
assuming $\phi{''} (x)$ to be negative for all values of $x$, this
is always less than $\phi (x)$.
It is true that this loss of probable
happiness need not be greater than the pleasure derived from the
excitement of gambling, and we are then thrown back upon the
induction that pleasures of gambling are in Bentham's phrase
``impure''; since experience shows that they are likely to
engender a restless, feverish character, unsuited for steady work
as well as for the higher and more solid pleasures of life.
\vskip7pt N${\scshape OTE}$ X. (p. 142). Following on the same
lines as in Note I., let us take $v$ to represent the disutility
or discommodity of an amount of labour $l$, then
$\displaystyle \frac{dv}{dl}$ measures the marginal degree of
disutility of labour; and, subject to the qualifications mentioned
in the text, $\displaystyle \frac{d^{2}v}{dl^2}$ is positive.
\vskip7pt Let $m$ be the amount of money or general purchasing
power at a person's disposal, $\mu$ its total utility to him, and
therefore $\displaystyle \frac{d\mu}{dm}$ its marginal utility.
Thus if $\Delta w$ be the wages that must be paid him to induce
him to do labour $\Delta l$, then $\displaystyle \Delta w \frac{d \mu}{dm}
= \Delta v$, and $\displaystyle \frac{d w}{dl} \cdot
\frac{d \mu}{dm} = \frac{d v}{ dl}$.
\vskip7pt If we assume that
his dislike to labour is not a fixed, but a fluctuating quantity,
we may regard $\displaystyle \frac{dw}{dl}$ as a function of $m,
v$, and $l$; and then both $\displaystyle \frac{d^{2}w}{dm\;dl},
\frac{d^{2}w}{dv\;dl}$ are always positive.
\vskip7pt N${\scshape
OTE}$ XI. (p. 248). If members of any species of bird begin to
adopt aquatic habits, every increase in the webs between the
toes---whether coming about gradually by the operation of natural
selection, or suddenly as a sport,---will cause them to find their
advantage more in aquatic life, and will make their chance of
leaving offspring depend more on the increase of the web. So that,
if $f(t)$ be the average area of the web at time $t$, then the
rate of increase of the web increases (within certain limits) with
every increase in the web, and therefore $f{''} (t)$ is positive.
Now we know by Taylor's Theorem that $$f(t + h) = f (t) + hf' (t)
+ \frac{h^2}{1.2} f{''} (t + \theta h);$$ \noindent and if $h$ be
large, so that $h^2$ is very large, then $f (t + h)$ will be much
greater than $f(t)$ even though $f'(t)$ be small and $f{''} (t)$
is never large. There is more than a superficial connection
between the advance made by the applications of the differential
calculus to physics at the end of the eighteenth century and the
beginning of the nineteenth, and the rise of the theory of
evolution. In sociology as well as in biology we are learning to
watch the accumulated effects of forces which, though weak at
first, get greater strength from the growth of their own effects;
and the universal form, of which every such fact is a special
embodiment, is Taylor's Theorem; or, if the action of more than
one cause at a time is to be taken account of, the corresponding
expression of a function of several variables. This conclusion
will remain valid even if further investigation confirms the
suggestion, made by some Mendelians, that gradual changes in the
race are originated by large divergences of individuals from the
prevailing type. For economics is a study of mankind of particular
nations, of particular social strata; and it is only indirectly
concerned with the lives of men of exceptional genius or
exceptional wickedness and violence.
\vskip7pt N${\scshape OTE}$ XII. (p. 331). If, as in Note X., $v$
be the discommodity of the amount of labour which a person has to
exert in order to obtain an amount $x$ of a commodity from which
he derives a pleasure $u$, then the pleasure of having further
supplies will be equal to the pain of getting them when
$\displaystyle \frac{d u}{dx} = \frac{dv}{dx}$.
If the pain of labour be regarded as a negative
pleasure; and we write $U \equiv -v$; then $\displaystyle \frac{du}{dx} +
\frac{du}{dx} =0$, i.e. $ u + U = {\rm a}$ maximum at
the point at which his labour ceases.
\vskip7pt N${\scshape OTE}$ XII. $bis$ (p. 793). In an article in
the {\it Giornale degli Economisti} for February, 1891, Prof.
Edgeworth draws the
\begin{wrapfigure}[9]{r}{200pt}{\epsfig{file=MrshMathAppg4.eps,height=145pt}}\end{wrapfigure} adjoining
diagram, which represents the cases of barter of apples for nuts
described on pp. 414--6. Apples are measured along $Ox$, and nuts
along $Oy$; $Op = 4, \ pa = 40$; and $a$ represents the
termination of the first bargain in which 4 apples have been
exchanged for 40 nuts, in the case in which $A$ gets the advantage
at starting: $b$ represents the second, and $c$ the final stage of
that case. On the other hand, $a'$ represents the first, and $b',
c', d'$ the second, third, and final stages of the set of bargains
in which $B$ gets the advantage at starting. $QP$, the locus on
which $c$ and $d'$ must both necessarily lie, is called by Prof.
Edgeworth the {\it Contract Curve}.
Following a method adopted in
his {\it Mathematical Psychics} (1881), he takes $U$ to represent the
total utility to $A$ of apples and nuts when he has given up $x$
apples and received $y$ nuts, $V$ the total utility to $B$ of
apples and nuts when he has received $x$ apples and given up $y$
nuts. If an additional $\Delta x$ apples are exchanged for $\Delta
y$ nuts, the exchange will be indifferent to $A$ if $$\frac{dU}{dx}
\Delta x + \frac{dU}{dy} \Delta y = 0;$$ \noindent and it will
be indifferent to $B$ if $\displaystyle \frac{dV}{dx} \Delta x +
\frac{dV}{dy} \Delta y = 0$. These, therefore, are the equations
to the indifference curves $OP$ and $OQ$ of the figure
respectively; and the contract curve which is the locus of points
at which the terms of exchange that are indifferent to $A$ are
also indifferent to $B$ has the elegant equation
$\displaystyle \frac{dU}{dx} \div \frac{dU}{dy} = \frac{dV}{dx}
\div \frac{dV}{dy}$.
\vskip7pt If the marginal utility of nuts be constant for $A$ and
also for $B$, $\displaystyle \frac{dU}{dy}$ and $\displaystyle
\frac{dV}{dy}$ become constant; $U$ becomes $\Phi (a - x) + \alpha
y$, and $V$ becomes $\Psi (a - x) +\beta y$; and the contract
curve becomes $F(x) =0$; or $x = C$; that is, it is a straight
line parallel to $Oy$, and the value of $\Delta y \colon \Delta x$
given by either of the indifference curves, a function of $C$;
thus showing that by whatever route the barter may have started,
equilibrium will have been found at a point at which $C$ apples
have been exchanged, and the final rate of exchange is a function
of $C$; that is, it is a constant also. This last application of
Prof. Edgeworth's mathematical version of the theory of barter, to
confirm the results reached in the text, was first made by Mr
Berry, and is published in the {\it Giornale degli Economisti} for
June, 1891.
Prof. Edgeworth's plan of
representing $U$ and $V$ as general functions of $x$ and $y$ has
great attractions to the mathematician; but it seems less adapted
to express the every-day facts of economic life than that of
regarding, as Jevons did, the marginal utilities of apples as
functions of $x$ simply. In that case, if $A$ had no nuts at
starting, as is assumed in the particular case under discussion,
$U$ takes the form $$\int^x_0 \phi_1 (a - x) dx + \int^y_0 \psi_1
(y) dy;$$ \noindent similarly for $V$. And then the equation to
the contract curve is of the form $$\phi_1 (a - x) \div \psi_1 (y)
= \phi_2 (x) \div \psi_2 (b - y);$$ \noindent which is one of the
Equations of Exchange in Jevon's {\it Theory}, 2nd Edition, p.
108.
\vskip7pt N${\scshape OTE}$ XIII. (p. 354). Using the same
notation as in Note V., let us take our starting-point as regards
time at the date of beginning to build the house, and let $T'$ be
the time occupied in building it. Then the present value of the
pleasures, which he expects to derive from the house, is $$H =
\int^T_{T'} \varpi R^{-t} \frac{dh}{dt} dt.$$
Let $\Delta v$ be
the element of effort that will be incurred by him in building the
house in the interval of time $\Delta t$ (between the time $t$ and
the time $t + \Delta t$), then the present value of the aggregate
of effort is $$V = \int^{T'}_0 R^{-t} \frac{dv}{dt} dt.$$
If there
is any uncertainty as to the labour that will be required, every
possible element must be counted in, multiplied by the
probability, $\varpi^{'}$, of its being required; and then $V$
becomes $\displaystyle \int^{T'}_0 \varpi R^{-t} \frac{dv}{dt}
dt$.
\vskip7pt If we transfer the starting-point to the date of the
completion of the house, we have $$H = \int^{T_1}_0 \varpi R^{-t}
\frac{dh}{dt} dt \quad \hbox{and} \quad V = \int^{T'}_0 \varpi R^t
\frac{dv}{dt} dt,$$ \noindent where $T_1 = T - T'$; and this
starting-point, though perhaps the less natural from the
mathematical point of view, is the more natural from the point of
view of ordinary business. Adopting it, we see $V$ as the
aggregate of estimated pains incurred; each bearing on its back,
as it were, the accumulated burden of the waitings between the
time of its being incurred and the time when it begins to bear
fruit.
Jevons' discussion of the investment of capital is somewhat
injured by the unnecessary assumption that the function
representing it is an expression of the first order; which is the
more remarkable as he had himself, when describing Gossen's work,
pointed out the objections to the plan followed by him (and
Whewell) of substituting straight lines for the multiform curves
that represent the true characters of the variations of economic
quantities.
\vskip7pt N${\scshape OTE}$ XIV. (p. 357). Let
$\alpha_1, \alpha_2, \alpha_3, \ldots$ be the several amounts of
different kinds of labour, as, for instance, wood-cutting,
stone-carrying, earth-digging, etc., on the part of the man in
question that would be used in building the house on any given
plan; and $\beta, \beta', \beta{''}$, etc., the several amounts of
accommodation of different kinds such as sitting-rooms, bed-rooms,
offices, etc. which the house would afford on that plan. Then,
using $V$ and $H$ as in the previous note, $V, \beta, \beta',
\beta{''}$ are all functions of $\alpha_1, \alpha_2, \alpha_3,
\ldots$, and $H$ being a function of $\beta, \beta', \beta{''},
\ldots$ is a function also of $\alpha_1, \alpha_2, \alpha_3,
\ldots$ We have, then, to find the marginal investments of each
kind of labour for each kind of use.
\begin{eqnarray*}
\frac{dV}{da_1} &=& \displaystyle{\frac{d H}{d \beta} \frac{d
\beta}{da_1} = \frac{d H}{d \beta'} \frac{d \beta'}{da_1} =
\frac{d H}{d \beta{''}} \frac{d \beta{''}}{da_1} = - \cdots} \cr
\cr \frac{dV}{ da_2} &=& \displaystyle{\frac{d H}{d \beta} \frac{d
\beta}{da_2} = \frac{d H}{d \beta'} \frac{d \beta'}{da_2} =
\frac{d H}{d \beta{''}} \frac{d \beta{''}}{da_2} = \cdots \cdot}
\end{eqnarray*}
These equations represent a balance of effort and benefit. The
real cost to him of a little extra labour spent on cutting timber
and working it up is just balanced by the benefit of the extra
sitting-room or bed-room accommodation that he could get by so
doing. If, however, instead of doing the work himself, he pays
carpenters to do it, we must take $V$ to represent, not his total
effort, but his total outlay of general purchasing power. Then the
rate of pay which he is willing to give to carpenters for further
labour, his marginal demand price for their labour, is measured by
$\displaystyle \frac{d V}{da}$; while $\displaystyle \frac{d H} {d
\beta}, \frac{d H}{d \beta'}$ are the money measures to him of the
marginal utilities of extra sitting-room and bed-room
accommodation respectively, that is, his marginal demand prices
for them; and $\displaystyle \frac{d \beta}{da}, \frac{d \beta'}
{da}$ are the marginal efficiencies of carpenters' labour in
providing those accommodations. The equations then state that the
demand price for carpenters' labour tends to be equal to the
demand price for extra sitting-room accommodation, and also for
extra bed-room accommodation and so on, multiplied in each case by
the marginal efficiency of the work of carpenters in providing
that extra accommodation, proper units being chosen for each
element.
When this statement is generalized, so as to cover
all the varied demand in a market for carpenters' labour, it
becomes:---the (marginal) demand price for carpenters' labour is
the (marginal) efficiency of carpenters' labour in increasing the
supply of any product, multiplied by the (marginal) demand price
for that product. Or, to put the same thing in other words, the
wages of a unit of carpenters' labour tends to be equal to the
value of such part of any of the products, to producing which
their labour contributes, as represents the marginal efficiency of
a unit of carpenters' labour with regard to that product; or, to
use a phrase, with which we shall be much occupied in Book VI ch.
${\scshape I}$., it tends to be equal to the value of the ``net
product'' of their labour. This proposition is very important and
contains within itself the kernel of the demand side of the theory
of distribution.
Let us then suppose a master builder to have it
in mind to erect certain buildings, and to be considering what
different accommodation he shall provide; as, for instance,
dwelling-houses, warehouses, factories, and retail shop-room.
There will be two classes of questions for him to decide: how much
accommodation of each kind he shall provide, and by what means he
shall provide it. Thus, besides deciding whether to erect villa
residences, offering a certain amount of accommodation, he has to
decide what agents of production he will use, and in what
proportions: whether e.g. he will use tile or slate; how much
stone he will use; whether he will use steam power for making
mortar etc. or only for crane work; and, if he is in a large town,
whether he will have his scaffolding put up by men who make that
work a speciality or by ordinary labourers; and so on.
Let him
then decide to provide an amount $\beta$ of villa accommodation,
an amount $\beta'$ of warehouse, an amount $\beta{''}$ of factory
accommodation, and so on, each of a certain class. But, instead of
supposing him to hire simply $\alpha_1, \alpha_2, \ldots$
quantities of different kinds of labour, as before, let us class
his expenditure, under the three heads of (1) wages, (2) prices of
raw material, and (3) interest on capital: while the value of his
own work and enterprise makes a fourth head.
Thus let $x_1, x_2,
\ldots$ be the amounts of different classes of labour, including
that of superintendence, which he hires; the amount of each kind
of labour being made up of its duration and its intensity.
Let $y_1, y_2, \ldots$ be amounts of various kinds of raw materials,
which are used up and embodied in the buildings; which may be
supposed to be sold freehold. In that case, the pieces of land on
which they are severally built are merely particular forms of raw
material from the present point of view, which is that of the
individual undertaker.
Next let $z$ be the amount of locking up,
or appropriation of the employment, of capital for the several
purposes. Here we must reckon in all forms of capital reduced to a
common money measure, including advances for wages, for the
purchase of raw material; also the uses, allowing for
wear-and-tear etc. of his plant of all kinds: his workshops
themselves and the ground on which they are built being reckoned
on the same footing. The periods, during which the various
lockings up run, will vary; but they must be reduced, on a
``compound rate,'' i.e. according to geometrical progression, to a
standard unit, say a year.
Fourthly, let $u$ represent the money
equivalent of his own labour, worry, anxiety, wear-and-tear etc.
involved in the several undertakings.
In addition, there are
several elements, which might have been entered under separate
heads; but which we may suppose combined with those already
mentioned. Thus the allowance to be made for risk may be shared
between the last two heads. A proper share of the general expenses
of working the business (``supplementary costs,'' see p. 360) will
be distributed among the four heads of wages, raw materials,
interest on the capital value of the organization of the business
(its goodwill etc.) regarded as a going concern, and remuneration
of the builder's own work, enterprise and anxiety.
Under these
circumstances $V$ represents his total outlay, and $H$ his total
receipts; and his efforts are directed to making $H - V$ a
maximum. On this plan, we have equations similar to those already
given, viz.:---
\begin{eqnarray*}
&\displaystyle{\frac{d V}{dx_1} = \frac{d H}{d \beta} \cdot
\frac{d \beta}{d x_1} = \frac{d H}{d \beta'} \cdot \frac{d
\beta'}{d x_1} = \ldots} \cr\cr &\displaystyle{\frac{d V}{dx_2} =
\frac{d H}{d \beta} \cdot \frac{d \beta}{d x_2} = \frac{d H}{d
\beta'} \cdot \frac{d \beta'}{d x_2} = \ldots}\cr\cr
&\cdots\cdots\hspace{12em} \cr\cr &\displaystyle{\frac{d V}{ dy_1}
= \frac{d H}{d \beta} \cdot \frac{d \beta}{d y_1} = \frac{d H}{d
\beta'} \cdot \frac{d \beta'}{d y_1} = \ldots} \cr\cr
&\cdots\cdots\hspace{12em}\cr\cr &\displaystyle{\frac{d V}{dz} =
\frac{d H}{d \beta} \cdot \frac{d \beta}{d z} = \frac{d H}{d
\beta'} \cdot \frac{d \beta'}{ d z} = \ldots} \cr\cr
&\displaystyle{\frac{d V}{du} = \frac{d H}{d \beta} \cdot \frac{d
\beta}{d u} = \frac{d H}{d \beta'} \cdot \frac{d \beta'}{d u} =
\ldots .}
\end{eqnarray*}
That is to say, the marginal outlay
which the builder is willing to make for an additional small
supply, $\delta x_1$, of the first class of labour, viz.
$\displaystyle \frac{d V}{dx_1} \delta x_1$, is equal to
$\displaystyle \frac{dH}{d \beta} \cdot \frac{d \beta}{d x_1}
\delta x_1$; i.e. to that increment in his total receipts $H$,
which he will obtain by the increase in the villa accommodation
provided by him that will result from the extra small supply of
the first class of labour: this will equal a similar sum with
regard to warehouse accommodation, and so on. Thus he will have
distributed his resources between various uses in such a way that
he would gain nothing by diverting any part of any agent of
production---labour, raw material, the use of capital---nor his
own labour and enterprise from one class of building to another:
also he would gain nothing by substituting one agent for another
in any branch of his enterprise, nor indeed by any increase or
diminution of his use of any agent. From this point of view our
equations have a drift very similar to the argument of Book III.
ch. ${\scshape V}$. as to the choice between the different uses of
the same thing. (Compare one of the most interesting notes $(f)$
attached to Prof. Edgeworth's brilliant address to the British
Association in 1889.)
There is more to be said (see V. ${\scshape
XI}$. 1, and VI. ${\scshape I}$. 8) on the difficulty of
interpreting the phrase the ``net product" of any agent of
production, whether a particular kind of labour or any other
agent; and perhaps the rest of this note, though akin to what has
gone before, may more conveniently be read at a later stage. The
builder paid $\displaystyle \frac{dV}{dx_1}\delta x_1$ for the
last element of the labour of the first group because that was its
net product; and, if directed to building villas, it brought him
in $\displaystyle \frac{dH}{d\beta}\cdot \frac{d\beta}{d
x_1}\delta x_1$, as special receipts. Now if $p$ be the price per
unit, which he receives for an amount $\beta$ of villa
accommodation, and therefore $p\beta$ the price which he receives
for the whole amount $\beta$; and if we put for shortness
$\Delta\beta$ in place of $\displaystyle \frac{d\beta}{d
x_1}\delta x_1$, the increase of villa accommodation due to the
additional element of labour $\delta x_1$; then the net product we
are seeking is not $p\Delta \beta$, but $p \Delta \beta + \beta
\Delta p$; where $\Delta p$ is a negative quantity, and is the
fall in demand price caused by the increase in the amount of villa
accommodation offered by the builder. We have to make some study
of the relative magnitudes of these two elements $p\Delta \beta$
and $\beta \Delta p$.
If the builder monopolized the supply of
villas, $\beta$ would represent the total supply of them: and, if
it happened that the elasticity of the demand for them was less
than unity, when the amount $\beta$ was offered, then, by
increasing his supply, he would diminish his total receipts; and
$p \Delta \beta + \beta \Delta p$ would be a negative quantity.
But of course he would not have allowed the production to go just
up to an amount at which the demand would be thus inelastic. The
margin which he chose for his production would certainly be one
for which the negative quantity $\beta \Delta p$ is less than
$p\Delta \beta$, but not necessarily so much less that it may be
neglected in comparison. This is a dominant fact in the theory of
monopolies discussed in Book V. chapter ${\scshape XIV}$.
It is dominant also in the case of any producer who has a limited trade
connection which he cannot quickly enlarge. If his customers have
already as much of his wares as they care for, so that the
elasticity of their demand is temporarily less than unity, he
might lose by putting on an additional man to work for him, even
though that man would work for nothing. This fear of temporarily
spoiling a man's special market is a leading influence in many
problems of value relating to short periods (see Book V. chs.
${\scshape V.\ VII.\ XI.}$); and especially in those periods of
commercial depression, and in those regulations of trade
associations, formal and informal, which we shall have to study in
the second volume. There is an allied difficulty in the case of
commodities of which the expenses of production diminish rapidly
with every increase in the amount produced: but here the causes
that govern the limits of production are so complex that it seems
hardly worth while to attempt to translate them into mathematical
language. See V. ${\scshape XII.}$ 2.
When however we are studying
the action of an individual undertaker with a view of illustrating
the normal action of the causes which govern the general demand
for the several agents of production, it seems clear that we
should avoid cases of this kind. We should leave their peculiar
features to be analysed separately in special discussions, and
take our normal illustration from a case in which the individual
is only one of many who have efficient, if indirect, access to the
market. If $\beta \Delta p$ be numerically equal to $p\Delta
\beta$, where $\beta$ is the whole production in a large market;
and an individual undertaker produced $\beta^\prime$, a thousandth
part of $\beta$; then the increased receipt from putting on an
additional man is $p \Delta \beta^\prime$, which is the same as
$p\Delta \beta$; and the deduction to be made from it is only
$\beta^\prime \Delta p$, which is a thousandth part of $\beta
\Delta p$ and may be neglected. For the purpose therefore of
illustrating a part of the general action of the laws of
distribution we are justified in speaking of the value of the net
product of the marginal work of any agent of production as the
amount of that net product taken at the normal selling value of
the product, that is, as $p\Delta\beta$.
It may be noticed that
none of these difficulties are dependent upon the system of
division of labour and work for payment; though they are brought
into prominence by the habit of measuring efforts and
satisfactions by price, which is associated with it. Robinson
Crusoe erecting a building for himself would not find that an
addition of a thousandth part to his previous accommodation
increased his comfort by a thousandth part. What he added might be
of the same character with the rest; but if one counted it in at
the same rate of real value to him, one would have to reckon for
the fact that the new part made the old of somewhat less urgent
need, of somewhat lower real value to him (see note 1 on p. 417).
On the other hand, the law of increasing return might render it
very difficult for him to assign its true net product to a given
half-hour's work. For instance, suppose that some small herbs,
grateful as condiment, and easily portable, grow in a part of his
island, which it takes half a day to visit; and he has gone there
to get small batches at a time. Afterwards he gives a whole day,
having no important use to which he can put less than half a day,
and comes back with ten times as great a load as before. We cannot
then separate the return of the last half-hour from the rest; our
only plan is to take the whole day as a unit, and compare its
return of satisfaction with those of days spent in other ways; and
in the modern system of industry we have the similar, but more
difficult task of taking, for some purposes, the whole of a
process of production as a single unit.
It would be possible to
extend the scope of such systems of equations as we have been
considering, and to increase their detail, until they embraced
within themselves the whole of the demand side of the problem of
distribution. But while a mathematical illustration of the mode of
action of a definite set of causes may be complete in itself, and
strictly accurate within its clearly defined limits, it is
otherwise with any attempt to grasp the whole of a complex problem
of real life, or even any considerable part of it, in a series of
equations. For many important considerations, especially those
connected with the manifold influences of the element of time, do
not lend themselves easily to mathematical expression: they must
either be omitted altogether, or clipped and pruned till they
resemble the conventional birds and animals of decorative art. And
hence arises a tendency towards assigning wrong proportions to
economic forces; those elements being most emphasized which lend
themselves most easily to analytical methods. No doubt this danger
is inherent in every application not only of mathematical
analysis, but of analysis of any kind, to the problems of real
life. It is a danger which more than any other the economist must
have in mind at every turn. But to avoid it altogether, would be
to abandon the chief means of scientific progress: and in
discussions written specially for mathematical readers it is no
doubt right to be very bold in the search for wide
generalizations.
In such discussions it may be right, for
instance, to regard {\it H} as the sum total of satisfactions, and
{\it V} as the sum total of dissatisfactions (efforts, sacrifices
etc.) which accrue to a community from economic causes; to
simplify the notion of the action of these causes by assumptions
similar to those which are involved, more or less consciously, in
the various forms of the doctrine that the constant drift of these
causes is towards the attainment of the {\it Maximum
Satisfaction}, in the net aggregate for the community (see above
pp. 470--5); or, in other words, that there is a constant tendency
to make $H-V$ a maximum for society as a whole. On this plan the
resulting differential equations of the same class as those which
we have been discussing, will be interpreted to represent value as
governed in every field of economics by the balancing of groups of
utilities against groups of disutilities, of groups of
satisfactions against groups of real costs. Such discussions have
their place: but it is not in a treatise such as the present, in
which mathematics are used only to express in terse and more
precise language those methods of analysis and reasoning which
ordinary people adopt, more or less consciously, in the affairs of
every-day life.
It may indeed be admitted that such discussions
have some points of resemblance to the method of analysis applied
in Book III. to the total utility of particular commodities. The
difference between the two cases is mainly one of degree: but it
is of a degree so great as practically to amount to a difference
of kind. For in the former case we take each commodity by itself
and with reference to a particular market; and we take careful
account of the circumstances of the consumers at the time and
place under consideration. Thus we follow, though perhaps with
more careful precautions, the practice of ministers of finance,
and of the common man when discussing financial policy. We note
that a few commodities are consumed mainly by the rich; and that
in consequence their real total utilities are less than is
suggested by the money measures of those utilities. But we assume,
with the rest of the world, that as a rule, and in the absence of
special causes to the contrary, the real total utilities of two
commodities that are mainly consumed by the rich stand to one
another in about the same relation as their money measures do: and
that the same is true of commodities the consumption of which is
divided out among rich and middle classes and poor in similar
proportions. Such estimates are but rough approximations; but each
particular difficulty, each source of possible error, is pushed
into prominence by the definiteness of our phrases: we introduce
no assumptions that are not latent in the practice of ordinary
life; while we attempt no task that is not grappled with in a
rougher fashion, but yet to good purpose, in the practice of
ordinary life: we introduce no new assumptions, and we bring into
clear light those which cannot be avoided. But though this is
possible when dealing with particular commodities with reference
to particular markets, it does not seem possible with regard to
the innumerable economic elements that come within the
all-embracing net of the doctrine of Maximum Satisfaction. The
forces of supply are especially heterogeneous and complex: they
include an infinite variety of efforts and sacrifices, direct and
indirect, on the part of people in all varieties of industrial
grades: and if there were no other hindrance to giving a concrete
interpretation to the doctrine, a fatal obstacle would be found in
its latent assumption that the cost of rearing children and
preparing them for their work can be measured in the same way as
the cost of erecting a machine.
For reasons similar to those given
in this typical case, our mathematical notes will cover less and
less ground as the complexity of the subjects discussed in the
text increases. A few of those that follow relate to monopolies,
which present some sides singularly open to direct analytical
treatment. But the majority of the remainder will be occupied with
illustrations of joint and composite demand and supply which have
much in common with the substance of this note: while the last of
that series Note XXI. goes a little way towards a general survey
of the problem of distribution and exchange (without reference to
the element of time), but only so far as to make sure that the
mathematical illustrations used point towards a system of
equations, which are neither more nor less in number than the
unknowns introduced into them.
\vskip7pt N${\scshape OTE}$ XIV.
{\it bis} (p. 384). In the diagrams of this chapter (V. ${\scshape
VI.}$) the supply curves are all inclined positively; and in our
mathematical versions of them we shall suppose the marginal
expenses of production to be determined with a definiteness that
does not exist in real life: we shall take no account of the time
required for developing a representative business with the
internal and external economies of production on a large scale;
and we shall ignore all those difficulties connected with the law
of increasing return which are discussed in Book V. ch. ${\scshape
XII}$. To adopt any other course would lead us to mathematical
complexities, which though perhaps not without their use, would be
unsuitable for a treatise of this kind. The discussions therefore
in this and the following notes must be regarded as sketches
rather than complete studies.
Let the factors of production of a
commodity {\it A} be $a_1,a_2$, etc.; and let their supply
equations be $y=\phi_1(x), \ y=\phi_2 (x)$, etc. Let the number of
units of them required for the production of $x$ units of {\it A}
be $m_1 x, m_2x, \ldots$ respectively; where $m_1,m_2, \ldots$ are
generally not constants but functions of $x$. Then the supply
equation of {\it A} is $$y=\Phi(x)=m_1\phi_1(m_1 x)+m_2\phi_2(m_2
x)+\cdots \equiv \Sigma \{ m\phi (mx)\}.$$
Let $y=F(x)$ be the
demand equation for the finished commodity, then the derived
demand equation for $a_r$ the $r^{\rm th}$ factor is $$y=F(x)-\{
\phi(x)-m_r \phi_r(m_r x)\}.$$
But in this equation $y$ is the
price, not of one unit of the factor but of {\it m} units; and to
get an equation expressed in terms of fixed units let $\eta$ be
the price of one unit, and let $\xi =m_r x$, then
$\displaystyle n=\frac{1}{m_r}\cdot y$ and the equation becomes
$$\eta=f_r (\xi)= \frac{1}{m_r}\left[F\left(\frac{1}{
m_r}\xi\right) -\left\{ \phi\left(\frac{1}{m_r}\xi\right)-m_r
\phi_r (\xi) \right\} \right]. $$
If $m_r$ is a function of $x$,
say $=\psi_r(x)$; then $x$ must be determined in terms of $\xi$ by
the equation $\xi=x \psi_r(x)$, so that $m_r$ can be written
$\chi_r(\xi)$; substituting this we have $\eta$ expressed as a
function of $\xi$. The supply equation for $a_r$ is simply
$\eta=\phi_r(\xi)$.
\vskip7pt N${\scshape OTE}$ XV. (p. 386). Let the demand equation
for knives be
\begin{eqnarray*}
&y=F(x).................\mathrm{(1),} \cr \mathrm{let \ the \
supply \ equation \ for \ knives \ be} &y=\Phi
(x).................\mathrm{(2),} \cr \mathrm{let \ that \ for \
handles \ be \ }\hspace{90pt} &y=\phi_1
(x)\,................\mathrm{(3),} \cr \mathrm{and \ that \ for \
blades \ be \ }\hspace{91pt} &y=\phi_2
(x)\,................\mathrm{(4),}
\end{eqnarray*} \noindent then the demand equation for handles is
$$\hspace{187pt}y=f_1(x)=F(x)-\phi_2(x)\,......\mathrm{(5).}$$
The measure of elasticity for (5) is $\displaystyle
-\left\{\frac{xf_1^\prime (x)}{f_1 (x)} \right\}^{-1}$, that is,
$$-\left\{ \frac{xF^\prime (x) -x\phi_2^\prime (x)}{f_1(x)}
\right\}^{- 1}; $$ \noindent that is, \hspace{55pt}
$\displaystyle{\left\{ - \frac{xF^\prime (x)}{F(x)}\cdot
\frac{F(x)}{ f_1 (x)} + \frac{x\phi_2^\prime (x)}{f_1(x)}
\right\}^{- 1}.}$
This will be the smaller the more fully the following conditions
are satisfied: (i) that $\displaystyle -\frac{xF^\prime (x)}{
F(x)}$, which is necessarily positive, be large, i.e. that the
elasticity of the demand for knives be small; (ii) that
$\phi_2^\prime (x)$ be positive and large, i.e. that the supply
price for blades should increase rapidly with an increase, and
diminish rapidly with a diminution of the amount supplied; and
(iii) that $\displaystyle \frac{F(x)}{f_1(x)}$ should be large;
that is, that the price of handles should be but a small part of
the price of knives.
A similar, but more complex inquiry, leads to
substantially the same results, when the units of the factors of
production are not fixed, but vary as in the preceding note.
\vskip7pt N${\scshape OTE}$ XVI. (p. 387). Suppose that {\it m}
bushels of hops are used in making a gallon of ale of a certain
kind, of which in equilibrium $x^\prime$ gallons are sold at a
price $y^\prime = F(x^\prime)$. Let $m$ be changed into $m+\Delta
m$; and, as a result, when $x^\prime$ gallons are still offered
for sale let them find purchasers at a price $y^\prime +\Delta
y^\prime$; then $\displaystyle \frac{\Delta y^\prime}{\Delta m}$
represents the marginal demand price for hops: if it is greater
than their supply price, it will be to the interest of the brewers
to put more hops into the ale. Or, to put the case more generally,
let $y=F(x,m),\ y=\Phi (x,m)$ be the demand and supply equations
for beer, $x$ being the number of gallons and $m$ the number of
bushels of hops in each gallon. Then $F(x, m)-\Phi (x, m)=$ excess
of demand over supply price. In equilibrium this is of course
zero: but if it were possible to make it a positive sum by varying
$m$ the change would be effected: therefore (assuming that there
is no perceptible change in the expense of making the beer, other
than what results from the increased amount of hops)
$\displaystyle \frac{dF}{dm}=\frac{d\Phi}{dm} $: the first
represents the marginal demand price, and the second the marginal
supply price of hops; and these two are therefore equal.
This method is of course capable of being extended to cases in which
there are concurrent variations in two or more factors of
production.
\vskip7pt N${\scshape OTE}$ XVII. (p. 388). Suppose
that a thing, whether a finished commodity or a factor of
production, is distributed between two uses, so that of the total
amount $x$ the part devoted to the first use is $x_1$, and that
devoted to the second use is $x_2$. Let $y=\phi(x)$ be the total
supply equation; $y=f_1(x_1)$ and $y=f_2 (x_2)$ be the demand
equations for its first and second uses. Then in equilibrium the
three unknowns $x$, $x_1$, and $x_2$ are determined by the three
equations $f_1(x_1)=f_2(x_2)=\phi(x);\;x_1+x_2=x$.
Next suppose
that it is desired to obtain separately the relations of demand
and supply of the thing in its first use, on the supposition that,
whatever perturbations there may be in its first use, its demand
and supply for the second use remains in equilibrium; i.e. that
its demand price for the second use is equal to its supply price
for the total amount that is actually produced, i.e.
$f_2(x_2)=\phi(x_1+x_2)$ always. From this equation we can
determine $x_2$ in terms of $x_1$, and therefore $x$ in terms of
$x_1$; and therefore we can write $\phi (x)=\psi(x_1)$. Thus the
supply equation for the thing in its first use becomes $y=\psi
(x_1)$; and this with the already known equation $y=f_1(x_1)$
gives the relations required.
\vskip7pt N${\scshape OTE}$ XVIII.
(p. 389). Let $a_1,a_2, \ldots$ be joint products, $m_1x, m_2x,
\ldots$ of them severally being produced as the result of $x$
units of their joint process of production, for which the supply
equation is $y=\phi (x)$. Let $$y=f_1(x), \quad y=f_2(x), \ldots
$$ \noindent be their respective demand equations. Then in
equilibrium $$m_1f_1 (m_1x)+m_2f_2(m_2x)+\cdots=\phi(x). $$
Let $x^\prime$ be the value of $x$ determined from this equation; then
$f_1(m_1x^\prime)$, $f_2(m_2 x^\prime)$ etc. are the equilibrium
prices of the several joint products. Of course $m_1,m_2$ are
expressed if necessary in terms of $x^\prime$.
\vskip7pt N${\scshape OTE}$ XIX. (p. 390). This case corresponds, {\it
mutatis mutandis}, to that discussed in Note XVI. If in
equilibrium $x^\prime$ oxen annually are supplied and sold at a
price $y^\prime=\phi (x^\prime)$; and each ox yields $m$ units of
beef: and if breeders find that by modifying the breeding and
feeding of oxen they can increase their meat-yielding properties
to the extent of $\Delta m$ units of beef (the hides and other
joint products being, on the balance, unaltered), and that the
extra expense of doing this is $\Delta y^\prime$, then
$\displaystyle \frac{\Delta y^\prime}{\Delta m}$ represents the
marginal supply price of beef: if this price were less than the
selling price, it would be to the interest of breeders to make the
change.
\vskip7pt N${\scshape OTE}$ XX. (p. 391). Let $a_1, a_2,
\ldots$ be things which are fitted to subserve exactly the same
function. Let their units be so chosen that a unit of any one of
them is equivalent to a unit of any others. Let their several
supply equations be $y_1=\phi_1 (x_1),\ y_2=\phi_2 (x_2),
\ldots\;$.
In these equations let the variable be changed, and let
them be written $x_1=\psi_1 (y_1),\ x_2=\psi_2 (y_2), \ldots$. Let
$y=f(x)$ be the demand equation for the service for which all of
them are fitted. Then in equilibrium $x$ and $y$ are determined by
the equations $y=f(x);\ x=x_1+x_2+\cdots, y_1=y_2=\cdots =y$. (The
equations must be such that none of the quantities $x_1, x_2,
\ldots$ can have a negative value. When $y_1$ has fallen to a
certain level $x_1$ becomes zero; and for lower values $x_1$
remains zero; it does not become negative.) As was observed in the
text, it must be assumed that the supply equations all conform to
the law of diminishing return; i.e. that $\phi_1^\prime (x),
\phi_2^\prime (x), \ldots$ are always positive.
\vskip7pt N${\scshape OTE}$ XXI. (p. 393). We may now take a bird's-eye
view of the problems of joint demand, composite demand, joint
supply and composite supply when they all arise together, with the
object of making sure that our abstract theory has just as many
equations as it has unknowns, neither more nor less.
In a problem of joint demand we may suppose that there are $n$
commodities $A_1, A_2, \ldots A_n$. Let $A_1$ have $a_1$ factors
of production, let $A_2$ have $a_2$ factors, and so on, so that
the total number of factors of production is $a_1+a_2+a_3+\cdots +
a_n$: let this $=m$.
First, suppose that all the factors are different, so that there
is no composite demand; that each factor has a separate process of
production, so that there are no joint products; and lastly, that
no two factors subserve the same use, so that there is no
composite supply. We then have $2n+2m$ unknowns, viz. the amounts
and prices of $n$ commodities and of $m$ factors; and to determine
them we have $2m+2n$ equations, viz.---(i) $n$ demand equations,
each of which connects the price and amount of a commodity; (ii)
$n$ equations, each of which equates the supply price for any
amount of a commodity to the sum of the prices of corresponding
amounts of its factors; (iii) $m$ supply equations, each of which
connects the price of a factor with its amount; and lastly, (iv)
$m$ equations, each of which states the amount of a factor which
is used in the production of a given amount of the commodity.
Next, let us take account not only of joint demand but also of
composite demand. Let $\beta_1$ of the factors of production
consist of the same thing, say carpenters' work of a certain
efficiency; in other words, let carpenters' work be one of the
factors of production of $\beta_1$ of the $n$ commodities $A_1,
A_2,\ldots \;$. Then since the carpenters' work is taken to have
the same price in whatever production it is used, there is only
one price for each of these factors of production, and the number
of unknowns is diminished by $\beta_1-1$; also the number of
supply equations is diminished by $\beta_1-1$: and so on for other
cases.
Next, let us in addition take account of joint supply. Let
$\gamma_1$ of the things used in producing the commodities be
joint products of one and the same process. Then the number of
unknowns is not altered; but the number of supply equations is
reduced by $(\gamma_1 -1)$: this deficiency is however made up by
a new set of $(\gamma_1-1)$ equations connecting the amounts of
these joint products: and so on.
Lastly, let one of the things
used have a composite supply made up from $\delta_1$ rival
sources: then, reserving the old supply equations for the first of
these rivals, we have $2 (\delta_1-1)$ additional unknowns,
consisting of the prices and amounts of the remaining
$(\delta_1-1)$ rivals. These are covered by $(\delta_1-1)$ supply
equations for the rivals and $(\delta_1-1)$ equations between the
prices of the $\delta_1$ rivals.
Thus, however complex the problem
may become, we can see that it is theoretically determinate,
because the number of unknowns is always exactly equal to the
number of the equations which we obtain.
\vskip7pt N${\scshape
OTE}$ XXII. (p. 480). If $y=f_1(x),\ y=f_2(x)$ be the equations to
the demand and supply curves respectively, the amount of
production which affords the maximum monopoly revenue is found by
making $\{ xf_1(x)- xf_2(x)\}$ a maximum; that is, it is the root,
or one of the roots of the equation $$\frac{d}{dx}\{ xf_1(x)-
xf_2(x) \}=0. $$
The supply function is represented here by
$f_2(x)$ instead of as before by $\phi(x)$, partly to emphasize
the fact that supply price does not mean exactly the same thing
here as it did in the previous notes, partly to fall in with that
system of numbering the curves which is wanted to prevent
confusion now that their number is being increased.
\vskip7pt
N${\scshape OTE}$ XXIII. (p. 482). If a tax be imposed of which
the aggregate amount is $F(x)$, then, in order to find the value
of $x$ which makes the monopoly revenue a maximum, we have
$\displaystyle \frac{d}{dx}\{ xf_1(x)- xf_2(x)-F(x)\}=0$; and it
is clear that if $F(x)$ is either constant, as in the case of a
license duty, or varies as $xf_1(x)-xf_2(x)$, as in the case of an
income-tax, this equation has the same roots as it would have if
$F(x)$ were zero.
Treating the problems geometrically, we notice that, if a fixed
burden be imposed on a monopoly sufficiently to make the
mo\-nop\-oly revenue curve fall altogether below $Ox$, and
$q^\prime$ be the point on the new curve vertically below $L$ in
fig. 35, then the new curve at $q^\prime$ will touch one of a
series of rectangular hyperbolas drawn with $yO$ produced
downwards for one asymptote and $Ox$ for the other. These curves
may be called Constant Loss curves.
Again, a tax proportionate to
the monopoly revenue, and say $m$ times that revenue ($m$ being
less than 1), will substitute for $QQ^\prime$ a curve each
ordinate of which is $(1-m)\times$ the ordinate of the
corresponding point on $QQ^\prime$; i.e. the point which has the
same abscissa. The tangents to corresponding points on the old and
new positions of $QQ^\prime$ will cut $Ox$ in the same point, as
is obvious by the method of projections. But it is a law of
rectangular hyperbolas which have the same asymptotes that, if a
line be drawn parallel to one asymptote to cut the hyperbolas, and
tangents be drawn to them at its points of intersection, they will
all cut the other asymptote in the same point. Therefore if
$q_3^\prime$ be the point on the new position of $QQ^\prime$
corresponding to $q_3$, and if we call $G$ the point in which the
common tangent to the hyperbola and $QQ^\prime$ cuts $Ox,\
Gq_3^\prime$ will be a tangent to the hyperbola which passes
through $q_3^\prime$; that is, $q_3^\prime$ is a point of maximum
revenue on the new curve.
The geometrical and analytical methods
of this note can be applied to cases, such as are discussed in the
latter part of $\S 4$ in the text, in which the tax is levied on
the produce of the monopoly.
\vskip7pt N${\scshape OTE}$ XXIII. {\it bis} (p. 489). These
results have easy geometrical proofs by Newton's method, and by
the use of well-known properties of the rectangular hyperbola.
They may also be proved analytically. As before let $y=f_1(x)$ be
the equation to the demand curve; $y=f_2(x)$ that to the supply
curve; and that to the monopoly revenue curve is $y=f_3(x)$, where
$f_3(x)=f_1(x)-f_2(x)$ the equation to the consumers' surplus
curve $y=f_4 (x)$; where $$f_4(x)=\frac{1}{x} \int_0^x f_1(a)
da-f_1(x).$$ \noindent That to the total benefit curve is
$y=f_5(x)$; where $$f_5(x)=f_3(x)+f_4(x)=\frac{1}{x}\int_0^x
f_1(a) da-f_2(x);$$ \noindent a result which may of course be
obtained directly. That to the compromise benefit curve is
$y=f_6(x)$; where $f_6(x)=f_3(x)+nf_4(x)$; consumers' surplus
being reckoned in by the monopolist at $n$ times its actual value.
To find {\it OL} (fig. 37), that is, the amount the sale of which
will afford the maximum monopoly revenue, we have the equation
$$\frac{d}{dx}\{xf_3 (x)\}=0; \ \hbox{i.e. }
f_1(x)-f_2(x)=x\{f_2^\prime (x)-f_1^\prime (x)\};$$ \noindent the
left-hand side of this equation is necessarily positive, and
therefore so is the right-hand side, which shows, what is
otherwise obvious, that if $Lq_3$ be produced to cut the supply
and demand curves in $q_2$ and $q_1$ respectively, the supply
curve at $q_2$ (if included negatively) must make a greater angle
with the vertical than is made by the demand curve at $q_1$.
To find {\it OW}, that is, the amount the sale of which will
afford the maximum total benefit, we have $$ \frac{d}{dx}\{{xf_5
(x)}\}=0;\; \hbox{i.e. } f_1(x)-f_2(x)-xf_2^\prime (x)=0. $$
\noindent To find {\it OY}, that is, the amount the sale of which
will afford the maximum compromise benefit, we have
\begin{eqnarray*}
&\displaystyle{\frac{d}{dx} \{ x f_{6}(x) \}} = 0; \cr \hbox{i.e.
} &\displaystyle{\frac{d}{dx} \left\{
(1-n)xf_1(x)-xf_2(x)+n\int_0^x f_{1}(a) da \right\}} = 0; \cr
\hbox{i.e. } &\displaystyle{(1-n)
xf_1^\prime(x)+f_1(x)-f_2(x)-xf_2^\prime (x)} = 0.
\end{eqnarray*}
\noindent If $OL=c$, the condition that {\it OY} should be greater
than {\it ON} is that $\displaystyle \frac{d}{dx}\{xf_6(x) \}$ be
positive when $c$ is written for $x$ in it; i.e. since
$\displaystyle \frac{d}{dx}\{xf_3(x)\}=0$ when $x=c$, that
$\displaystyle \frac{d}{dx}\{xf_4(x)\}$ be positive when $x=c$;
i.e. that $f_1^\prime (c)$ be negative. But this condition is
satisfied whatever be the value of $c$. This proves the first of
the two results given at the end of V. ${\scshape XIV.}$ 7; and
the proof of the second is similar. (The working of these results
and of their proofs tacitly assumes that there is only one point
of maximum monopoly revenue.)
One more result may be added to those in the text. Let
us write $OH=a$, then the condition that $OY$ should be
greater than {\it OH} is that $\displaystyle \frac{d}{
dx}\{nf_6 (x)\}$ be positive when $a$ is written for $x$: that
is, since $f_1(a)=f_2(a)$, that
$(1-n)f_1^\prime(a)-f_2^\prime (a)$ be positive. Now
$f_1^\prime (a)$ is always negative, and therefore the
condition becomes that $f_2^\prime (x)$ be negative, i.e. that the
supply obey the law of increasing return and that $\tan \phi$
be numerically greater than $(1-n) \tan \theta$, where
$\theta$ and $\phi$ are the angles which tangents at {\it A}
to the demand and supply curves respectively make with $Ox$.
When $n=1$, the sole condition is that $\tan \phi$ be
negative: that is, $OW$ is greater than {\it OH} provided the
supply curve at {\it A} be inclined negatively. In other words, if
the monopolist regards the interest of consumers as identical
with his own, he will carry his production further than the
point at which the supply price (in the special sense in
which we are here using the term) is equal to the demand
price, provided the supply in the neighbourhood of that point
obeys the law of increasing return: but he will carry it less
far if the supply obeys the law of diminishing return.
\vskip7pt N${\scshape OTE}$ XXIV. (p. 565). Let $\Delta x$ be the
probable amount of his production of wealth in time $\Delta t$,
and $\Delta y$ the probable amount of his consumption. Then the
discounted value of his future services is $\displaystyle \int_0^T
R^{-t} \left( \frac{dx}{dt}-\frac{dy}{ dt} \right)dt;$ where $T$
is the maximum possible duration of his life. On the like plan the
past cost of his rearing and training is $\displaystyle
\int_{-T^\prime}^0 R^{-t} \left( \frac{dy}{dt}-\frac{dx}{dt}
\right)dt$, where $T^\prime$ is the date of his birth. If we were
to assume that he would neither add to nor take from the material
wellbeing of a country in which he stayed all his life, we should
have $\displaystyle{\int_{-T}^T R^{-t} \left(
\frac{dx}{dt}-\frac{dy}{ dt} \right)dt=0}$; or, taking the
starting-point of time at his birth, and $l=T^\prime +T=$ the
maximum possible length of his life, this assumes the simpler
form, $\displaystyle{\int_0^l R^{-t} \left(
\frac{dx}{dt}-\frac{dy}{dt} \right)dt=0.}$
To say that $\Delta x$ is the probable amount of his production in
time $\Delta t$, is to put shortly what may be more accurately
expressed thus:---let $p_1,p_2, \ldots$ be the chances that in
time $\Delta t$ he will produce elements of wealth $\Delta_1 x,
\Delta_2 x, \ldots$, where $p_1+p_2+\cdots =1$; and one or more of
the series $\Delta_1 x, \Delta x, \ldots$ may be zero; then $$
\Delta x=p_1\Delta_1 x+p_2 \Delta_2 x+\cdots \cdot $$
\begin{center} \textbf{The End} \end{center}
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