Outline:

1st Introduction

2nd
The most important representatives

3rd
The Walras's system of equations

4th
The Walrasian auctioneer

5th
Assumptions about the utility

6th
The system of indifference curves

7th
The budget equilibrium

8th
Present goods and future goods

9th
Leisure utility versus consumer utility

5th
Assumptions about the utility

In the
following, we will turn to the works of Vilfredo Pareto, but we will limit
ourselves to Pareto's utility-theoretical considerations. As is well known,
Pareto argued against the marginal utility approach of the Vienna School;that utility units can only
be measured ordinally, but not cardinally.

We
always speak of an only ordinal measure of utility when it can be stated that
the utility of two goods A and B can either be classified as equally high or
when the utility of one good A is clearly higher or also lower than the utility
of good B. A cardinal measure of utility, on the other hand, is present when it
can be stated by how many multiples the utility of good A is greater or also
smaller than the utility of good B. With the possibility of a cardinal measure
of utility it could be established, for example, that good A provides three
times as much utility as good B, whereas if we limit ourselves to an ordinal
measure we could only say that good A provides a higher utility than good B.

If we
follow Vilfredo Pareto, the assumptions of the marginal utility approach are
wrong. We have seen in the previous chapter on the Vienna school that it is
calculated by using a utility function and marginal utility function, where the
quantity of the good is drawn on the abscissa and the total utility or the
marginal utility that is obtained with alternative quantities of goods can be
read on the ordinate. If we now assume that a point D, which is twice as far
from the coordinate origin as a point C, also indicates that the utility at
point D is therefore also twice as high as the utility at point C, then we are
in fact applying a cardinal measure.

Pareto's
thesis that units of utility can only be measured ordinarily, but not
cardinally, is not accepted by all economists; since famous economic theorists,
such as A. P. Lerner or Jan Tinbergen, were still of the firm conviction that
units of utility can indeed be measured cardinally. They have also developed
certain methods of how it was possible to measure utility cardinally. But the
vast majority of economists, especially those scientists who deal with welfare
theory topics, have adopted Pareto's position, and also in this context the
modern welfare theory is in the most general sense called Paretian welfare
theory.

This
dictum now affects primarily Gossen’s first Law. Within the framework of the
Gossen’s first Law, one can still reasonably say that an increased consumption
of a good leads to a decrease in marginal utility. We can even remain with the
usual diagram to illustrate the course of utility and draw a total utility
curve as well as a marginal utility curve into this diagram. The statement of
such a curve is then limited to the statement that two points which have a
different distance from the coordinate origin have a different level of
utility; that the point which is further away from the coordinate origin also
signals a higher utility, whereby it remains unclear how strong these
differences are in detail.

Vilfredo
Pareto has furthermore supported the thesis that units of utility can neither
be compared with each other interpersonally. Utility was a subjective quantity;
it was only possible for an individual to compare the utility of different
goods that it consumes itself. However, it was not possible to directly compare
the utility expectations of different persons with each other.

This
was in fact a deadly blow to the older welfare theory, which sought to prove,
among other things, that the reduction of the income differentiation increased
the welfare of the whole society. This thesis was justified on the basis that
the rich, whose income was taxed with a monetary unit, would suffer a smaller
loss of utility, because of the law of diminishing marginal utility of income,
than the poor, to whom these sums of money were allocated and who thus
experienced an increase in utility. A redistribution from rich to poor would
increase the utility for the whole of society as long as there was a
differentiation of income.

At this
point, we do not want to deal with the multitude of questionable assumptions
that have to be made to reach this conclusion. At this point, it is sufficient
to show that this conclusion is invalid already because, in Pareto's opinion,
there is no way to compare the utility changes between the rich and poor.

6th
The system of indifference curves

To be
able to deal with the influence of utility concepts on household demand for
consumer goods without applying a cardinal utility measure, Pareto introduces a
system of indifference curves that reflects the demand structure of a person (a
household). We want to present this system here, albeit using the presentation
of Edgeworth, who adopted and refined this instrument and whose presentation
has been adopted by most textbooks in use today.

We
want to assume a diagram on whose axes the quantities of two consumer goods x_{1} and x_{2} are
plotted. This limitation to two consumer goods serves solely to simplify the
representation. In reality we always assume that a household demands a variety
of consumer goods with its income. But if one wants to apply the graphical
presentation, only two goods can be viewed at the same time. At best, one could
distinguish three types of goods in a three-dimensional space, although here
one would have to accept that the representation already exceeds the
imagination of many. Theoretically, it is of course possible to represent every
finite number of goods in an n-dimensional space with a limitation to an
analytical analysis. We want to limit ourselves here to a two-dimensional space
and thus ensure the highest possible comprehensibility of Pareto's ideas.

Now
we consider any point U_{1} within this diagram. This point U_{1} is asigned a certain quantity of goods of
the good X_{1}, namely X_{11}, as well as of the good X_{2},
namely X_{21}. If our individual chooses the point U_{1}, it
can consume the quantity X_{11} of good X_{1} and the quantity X_{21} of good X_{2}. In this way, it reaches a certain level
of utility attributed to the point U_{1}, which Pareto calls the degree of
ophelimity in order to distinguish himself from the language use of the school
of marginal utility. However, there is no reason why we should not continue to
speak of a level of utility and therefore we want to maintain this language use
of the school of marginal utility.

Now
we want to appropriate the insight that our demand structure would allow an
individual to consume the goods to be consumed in a different relationship than
previously assumed. Here, we speak of the possibility of substitution. We can
subtract one unit from the one good X_{1}; in accordance with the law of diminishing
marginal utility, we experience a loss of utility. We can now compensate this
loss of utility by consuming more of the other good X_{2}. Here, we speak of substituting good X_{1} (the
diminished good) with good X_{2} (the increased good).

Now
we want to assume that our individual consumes so much more of good X_{2} that
he just reaches a level of utility that corresponds to the starting point U_{1}.
We connect both points with each other. In the same way, we continue to replace
good X_{1} with good X_{2} and connect the newly created points with
each other. In this way, we obtain an indifference curve with the utility level
U_{1}, which is characterised by the fact that all combinations
(points) lying on this curve enable an equally high utility level, namely U_{1}.

One of the most important
basic statements of the subjective value theory in Pareto's version is that
this indifference curve shows a convex curvature (in relation to the coordinate
origin). What does this mean materially? If we continue with the substitution
of the one good X_{1} by the respectively other good X_{2}, we
must add more and more units of the good X_{2} to compensate for the
loss of a unit of the good X_{1}.

If we change from a difference
consideration (one unit of the good X_{1} is always subtracted) to an
differential consideration (the respectively subtracted quantities of the good
X_{1} become smaller and smaller up to the infinitely small quantity),
then we can attach a tangent to the indifference curve. The angle of this
tangent corresponds to the differential quotient dX_{1}/dX_{2}.
If we now continue with the substitution, then the convex curvature of the
indifference curve has the effect that the marginal rate of the substitution
(the tangent angle) decreases more and more. Vilfredo Pareto refers in this
context to the law of the decreasing marginal rate of substitution.

It can now be shown easily
that this law of decreasing marginal rate of substitution can be derived from
the law of decreasing marginal utility, thus referring to the same regularity.
Because if we remove one unit from good X_{1}, then this leads to an
increase of the marginal utility according to the law of decreasing marginal
utility. If we now add as many units of X_{2} simultaneously until the
previous level of utility is reached again, then again, a loss of utility per
unit of good takes place due to the law of diminishing marginal utility. Since
the substitution process combines the changes in both types of goods, it is
necessary to consume more and more of good X_{2}, because with
progressive substitution, the loss of utility at good X_{1} becomes
greater and greater and the gain in utility of a unit at good X_{2}
becomes smaller and smaller.

We continue our analysis
starting from point U_{1} by consuming one additional unit of good X_{1}.
Since we assume that each increase in the consumption of a good causes a
certain increase in utility, this change in consumption means that the new
point U_{2} expresses a higher level of utility U_{2} in
comparison to the starting point. Starting from this point, we can, again by
substitution, search for other combinations of both goods (i.e. points), which
have the property that they all guarantee the same level of utility. If we
connect these points with each other, then we get a second indifference curve U_{2}
correspondingly, whereby this second curve differs from the first curve in the
way that each point of the new curve has a higher utility level than the points
on the originally developed indifference curve. However, by how much higher
this second utility level is than the first one, cannot be said.

We can now continue with this procedure and assign an indifference curve to any arbitrary point within our diagram. In this way we obtain a dense set of indifference curves, which means that such an indifference curve passes through any point in the diagram. Since we want to assume that substitution processes are possible starting from any utility level and that also the law of diminishing marginal rate of substitution applies to any utility level, all indifference curves arising in this way show a convex curvature in the direction of the coordinate origin.

For logical reasons, the indifference curves can not intersect. Indeed, an intersection would mean, that one and the same combination of goods causes two different levels of utility at the same time, which would be contradictory.

However, this does not say anything about whether all indifference curves have the same curvature, or whether the marginal rate of substitution changes with an increasing utility level. The curvature of the individual indifference curves naturally depends on the coefficients of the utility function. In the previous chapter on the Vienna School, I had ascertained that sometimes, in analogy to the law of diminishing returns at Cobb-Douglas production functions, also in household theory it is assumed that the law of diminishing marginal utility applies only to a partial variation of a single consumer good, but that if, due to an income increase, all goods are increasingly consumed in the same ratio, the marginal utility would remain constant. In this context it was referred to as the law of the constant marginal utility level product.

7th
The budget equilibrium

In a further step, we now want to deal with the question how the statements formulated in Gossen's second law must be modified if we assume with Pareto only an ordinal utility measure. We remember from the previous chapter: If we speak of Gossen's first law, it is about the question on which determinants the level of utility or marginal utility depends. The second Gossen's law, on the other hand, formulates an equilibrium condition; an answer is expected to the question under which conditions a household maximises its utility.

Here, the formulation of Gossen's second law is based on a given income and an arbitrary distribution of the income to the individual consumer goods and it is analysed whether a change in the distribution of the selected quantities of goods leads to an increase in utility. According to Gossen, all possibilities of increasing utility by means of a change in the composition of the consumed bundle of goods are only exhausted if the marginal utility of the income (i.e. of the last income unit) has the same level for all uses.

This question can only be
answered if we know both the income and the prices of the consumer goods. For,
if I consume one unit less of good X_{1}, I only know how much income I
save by this, if I know the price of good 1. And vice versa it is also true
that I only know how many units of the other good X_{2} I can buy
additionally from the saved income, if I also know the price of good X_{2}.

With the help of the given
income and the prices for good X_{1} and good X_{2}, which are
also assumed to be given and constant, a so-called budget line (or as it is
sometimes called, a revenue line or consumption line) can be constructed. We
start from our diagram above, but for reasons of simplification we consider
initially only one single indifference curve.

Let us assume two extreme cases:
The examined household could spend its entire income e only on the purchase of
the good X_{1}. It could then buy exactly e/p_{1 }units of good
X_{1}. If, for example, the income is 100 monetary units (GE) and the
price for a unit of good X_{1} is 5, the household could purchase 100/5
= 20 units of good X_{1}, since the value of the income obviously
corresponds to the value of the purchased goods:

However, our household could
also use its income to purchase the good X_{2} alone and could then
purchase a total of e/p_{2 }units of this good. Again, the equation
applies:

Now we must assume in reality,
that a household normally acquires both (all available) goods, that it is only
a question of which part of the income is spent on good X_{1} and which
remaining part is spent on good X_{2}. We can now assume that the
exchange ratio is determined by the given and constant prices and therefore
remains constant, too.

This means, however, that all
factually possible combinations of the two goods must necessarily lie on a
straight line, assuming constant income and prices. But if we know two points
of a straight line (in our case the two extreme points e/p_{2 }and e/p_{1}),
then we can connect these two points with each other and thus obtain the budget
line, which informs us which combinations of goods are actually possible at
all.

This budget line now touches an indifference curve in any case. Since we have shown that there is one (and only one) indifference curve running through each point of the diagram, it is also ensured that there is always one and only one point on the budget line which is tangent to an indifference curve. This tangent point then also corresponds to a very specific level of utility.

Now we can show that this tangent point necessarily indicates the highest possible level of utility. For this purpose, we will now go back to a diagram in which several indifference curves are drawn:

Now let us check with the help
of this diagram whether the tangential point of an indifference curve with the
budget line guarantees - as claimed - the highest possible level of utility.
This tangential point corresponds to the point P_{1} in our diagram. A
clearly higher utility level would have been guaranteed by point P_{2},
since it lies on an indifference curve which is further away from the
coordinate origin than the indifference curve which touches the budget line.
However, since no point of this indifference curve coincides with the budget
line, there is no possibility for the household to achieve this higher level of
utility under the given circumstances.

Point P_{3} and point
P_{4}, by contrast, lie on the budget line and can therefore be
realised very well under the given circumstances, but they intersect an
indifference curve which is closer to the coordinate origin than the
indifference curve with the tangential point and thus guarantees a lower level
of utility than at the starting point P_{1}. Thus, it is proven that a
household maximises its utility exactly when it chooses the combination of
goods corresponding to the tangential point.

The statement that a household maximises its utility exactly if and only if it chooses a combination of goods in which the budget line is tangent to an indifference curve is thus completely in line with Gossen's second law. According to this law, utility maximisation is achieved when the marginal utility of the incomes is equally high in all types of use. This requirement is met by Pareto's statement that the marginal rate of substitution must correspond to the angle of the budget.

The marginal rate of
substitution expresses, however, the __subjective__ exchange ratio that a
household chooses in a substitution. The price ratio, by contrast, indicates
the __objective__ ratios at which goods are exchanged on the market. This
means that utility is maximised exactly when the subjective and objective
exchange ratios correspond. The subjective exchange ratios are determined by
the marginal rate of substitution, while the objective exchange ratios are
determined by the ratio of the prices to each other.

8th
Present goods and future goods

The indifference curve system developed by Pareto and Edgeworth can now also be used to examine how a household divides its current income between present and future consumption desires. Instead of two consumer goods, we therefore plot the present goods in demand on one axis (the ordinate) and the future goods in demand on the other axis (the abscissa). We therefore turn here to a problem that was primarily investigated by Eugen von Böhm-Bawerk in the context of his theory of capital and interest. On the coordinate axes, monetary amounts are drawn which must be raised for the demand for the goods and not - as in the previous scheme - quantities of goods.

This way out by means of monetary amounts is necessary since the demand for present goods as well as the demand for future goods usually consists of several types of goods. This is true for the present goods as well as especially for the future goods, whereby it has not only to be considered that in a future period several types of goods will be demanded, but furthermore that the amount of money saved today can be split up to several periods in the future.

The set of indifference curves is developed here following the same method that was used for the demand structure regarding different types of goods. The position and curvature of these indifference curves and thus the course of the marginal rate of substitution is determined by a multitude of facts. Here also those behaviours are included, which Eugen von Böhm-Bawerk described as a principle of positive time preference. But also the question of which risk someone takes when he invests his savings in interest-bearing investments and furthermore how prepared to take risks the individual is in each case, ultimately determines the course of these indifference curves.

Furthermore, let us ask ourselves how we can develop the budget line in this case. Let us proceed here according to the same method as already mentioned above regarding the structure of the amounts of consumer goods in demand. In a first extreme case, we use the total income available for both periods for present goods, whereby ex definitione just an amount of money is spent for the present goods that corresponds to the income: e. We plot this amount on the ordinate.

However, we must assume that every individual must have a subsistence level in every period, including the present, to survive at all. Therefore, it makes sense when we consider the total available income only as far as it is above a subsistence level. The value e then corresponds to the difference between the actual income and the subsistence level.

In the other extreme case, the entire disposable income would be reserved for the purchase of future goods. We want to assume here that the household invests the saved income parts at the current interest rate, so that it can assume that in the next future period the savings sum multiplied by the interest factor (1 + interest rate, e.g. 1.03 with an interest rate of 3%) will be available. If the entire current income would be saved, then after one year a sum of money of e * 1.03 monetary units (GE) would be available under the assumptions made. We would then have to plot this amount on the abscissa.

However, we must now consider that it is not 100% certain that we can actually dispose of this amount after one year. We have always assumed that investments are more or less risky, and it is also unknown whether the interest rate has not changed in the meantime. These two factors make it necessary for us to weight the relevant amount of money with the amount of risk taken. For example, if we had to fear that the risk would be 50% to 50%, which means that only in half of the cases we could expect to get back 100% of our invested capital, the amount of money saved would have to be multiplied by 0.5.

We are now in a position to construct the budget line for the savings scheme. Again, we want to assume a constant exchange ratio between a monetary unit in the current period and the next future period, in other words, a constant interest rate. We can therefore again draw the budget curve as a straight line, whereby the budget line in this case indicates the amounts of money that may be available in the two periods with different divisions between present and future.

Again, the tangent point of this budget line with one of the indifference curves indicates the savings rate, which guarantees maximum utility over both periods.

Finally, we must clarify to what extent the two models discussed - the model of income distribution among the individual quantities of consumer goods and the model for determining the savings amount - harmonise with each other. The first model could be modified in such a way that the budget line does not refer to the total income, but only to the part of the income left over from a previous decision on the savings amount for consumption purposes. The budget line would then correspond to the amount of money reserved for consumption purposes.

However, such a procedure would certainly not guarantee an optimal decision. Because in order to indicate, with which saving sum the total income is maximised over time, it requires also the knowledge of all prices, also the prices of the individual consumer goods. In this respect the approach of Walras was surely correct, at which all factors of production, thus also the capital supply or the planned savings sum are to be seen principally in dependence of the prices of all goods as well as all factors of production. Thus, we remain with the fact that the construction of models with only 2 types of usage only serves the better illustration, but that in reality all decisions must be made uno actu, i.e. at the same time.

9th
Leisure utility versus consumer utility

The indifference curve scheme
developed by Pareto can finally also be used to clarify how an individual uses
his total time in hours (ST) for the gainful work time ST_{A} as well
as for leisure time ST_{F}. In our diagram, we plot the amount of
leisure time spent in hours per day on the ordinate and on the abscissa the
amount of consumer goods that our individual can buy if it performs an economic
activity.

Regarding the set of indifference curves, in principle the same applies as for the two previous models (determination of consumer goods, determination of the savings sum). We can assume that these indifference curves are also run convexly to the origin of the coordinates. Also here, it can be assumed that leisure time and working time are in a certain substitutive ratio to each other, e.g. that one gives up a little more leisure time in order to receive a higher income during working time. Ultimately, this is a question of the utility that the individual derives from leisure time and the utility that the individual experiences by buying and consuming consumer goods from the income earned during working time. The assumed curvature of the individual indifference curves is linked to the fact that also here the marginal rate of substitution from leisure time to working time diminishees with increasing substitution.

How do we get to a budget line in this scheme? First of all, we have to clarify what total time we assume per day. A day has 24 hours. However, we cannot really assume that these 24 hours per day are available, i.e. that our individual is free to decide how to divide this period between leisure time and working time. Every human being has a minimum of free time, which he cannot fall below, because otherwise his existence would be endangered. So we subtract from the 24 hours per day the leisure time minimum, which - so may be assumed - would be 8 hours and get from the difference: 24 - 8 = 16 a total time of 16 hours, which the individual can dispose of. We want to disregard the possibility that the individual is additionally hindered in this decision by institutional regulations, since it is not primarily a question of which alternatives the individual actually faces, but rather which alternatives would be possible if the individual were free to determine the working time.

Here, too, we can spend all of
our time, i.e. the 16 hours per day in leisure time. In this case we plot the
value 16 (hours) on the ordinate. Or we use the entire freely available time
for economic work. Here we receive an income that we spend on consumer goods.
It is the amount of consumer goods we can afford with the respective income and
this amount is determined by the product working hours (ST_{A})
multiplied by the wage rate per hour (l) and divided by the price of the
consumer goods for the consumer goods bundle (p):

Since the wage-price ratio for the household is given and constant, we can create the budget line again by connecting the two corner points.

Again, it applies that where this budget line touches an indifference curve, there is the optimal distribution of the available total time between leisure time and working time.