# 8th Lausanne school Part II

Outline:

1st Introduction

2nd The most important representatives

3rd The Walras's system of equations

4th The Walrasian auctioneer

6th The system of indifference curves

7th The budget equilibrium

8th Present goods and future goods

9th Leisure utility versus consumer 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 x1 and x2 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 U1 within this diagram. This point U1 is asigned a certain quantity of goods of the good X1, namely X11, as well as of the good X2, namely X21. If our individual chooses the point U1, it can consume the quantity X11 of good X1 and the quantity X21 of good X2. In this way, it reaches a certain level of utility attributed to the point U1, 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 X1; 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 X2. Here, we speak of substituting good X1 (the diminished good) with good X2 (the increased good).

Now we want to assume that our individual consumes so much more of good X2 that he just reaches a level of utility that corresponds to the starting point U1. We connect both points with each other. In the same way, we continue to replace good X1 with good X2 and connect the newly created points with each other. In this way, we obtain an indifference curve with the utility level U1, which is characterised by the fact that all combinations (points) lying on this curve enable an equally high utility level, namely U1.

# 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 X1 by the respectively other good X2, we must add more and more units of the good X2 to compensate for the loss of a unit of the good X1.

If we change from a difference consideration (one unit of the good X1 is always subtracted) to an differential consideration (the respectively subtracted quantities of the good X1 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 dX1/dX2. 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 X1, 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 X2 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 X2, because with progressive substitution, the loss of utility at good X1 becomes greater and greater and the gain in utility of a unit at good X2 becomes smaller and smaller.

We continue our analysis starting from point U1 by consuming one additional unit of good X1. 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 U2 expresses a higher level of utility U2 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 U2 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 X1, 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 X2 I can buy additionally from the saved income, if I also know the price of good X2.

With the help of the given income and the prices for good X1 and good X2, 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 X1. It could then buy exactly e/p1 units of good X1. If, for example, the income is 100 monetary units (GE) and the price for a unit of good X1 is 5, the household could purchase 100/5 = 20 units of good X1, since the value of the income obviously corresponds to the value of the purchased goods:

# e = X1 * p1  è   X1 = e/p1

However, our household could also use its income to purchase the good X2 alone and could then purchase a total of e/p2 units of this good. Again, the equation applies:

# e = X2 * p2  è   X2 = e/p2

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 X1 and which remaining part is spent on good X2. We can now assume that the exchange ratio is determined by the given and constant prices and therefore remains constant, too.

# (e/p1)/ (e/p2) = p2/p1

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/p2 and e/p1), 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 P1 in our diagram. A clearly higher utility level would have been guaranteed by point P2, 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 P3 and point P4, 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 P1. 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 STA as well as for leisure time STF. 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 (STA) multiplied by the wage rate per hour (l) and divided by the price of the consumer goods for the consumer goods bundle (p):

# STA* l /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.