Comments on the Rate of Capitalisation in Valuation Models in an Islamic Economy

In this paper the author tries to analyse the rate of capitalization in Islamic economies. He contends that it is not a profit/ loss-sharing rate, but is the sum of two quantities — the marginal efficiency of investment on inter-temporal allocation of income and the price for riskbearing. It appears from the author’s discussion that he believes risktaking is to be encouraged by Islam. Some errors of mathematics and logic in this analysis, are pointed out. Rather than considering how the analysis should be corrected, I have discussed the question of “risk” in Islamic economics. On the basis of this discussion some suggestions regarding Islamic economics are made. These suggestions do not only have relevance to the question of “interest” but also to Zakah.

It is difficult to discuss matters relating to definitions. If a term defined in one context is extended beyond that context it becomes very difficult to decide which extension should be accepted. To remove confusion in terminology one would generally tend to accept the first extension and avoid any other extensions to avoid confusion in the terminology. For this reason I fail to see the advantage of taking the author’s definition in place of that of other Islamic economists. As such the main motivation for the paper under discussion seems very weak to me. However, that is not a very good reason for not discussing the paper further. Before going onto that discussion, I would like to state what I believe to a very important, if somewhat obvious, principle, “Economic realities are not altered by the terminology used to discuss them”. Thus the terminology should be chosen for the convenience of discussions and can not be imposed on us by any other considerations.

The first point in the paper on which I would like to comment is an economic one. The author assumes that in a working Islamic economy, the usual, capitalistic “exchange basis of value” will no longer be applicable, but people will consider “Islamic benefits”. As I understand it, the author says that the basis of value for the individual would be some sort of a “social benefit value”. If such a situation could be brought about, there would hardly be any need of laws to regulate social and economic behaviour of individuals. Certainly, in such a Utopian world most of the normal problems of economics would no longer be relevant. Economics is mainly concerned with a conflict of interests. There are no conflicts appearing in the case considered by the author as everyone is trying to maximize the same objective function. Clearly, the questions raised in this “economics” are entirely different from those arising in normal economic theory. Thus, these new questions need new concepts and can not be discussed in terms of the earlier concepts. The whole idea of "capital” and "the rate of capitalization” etc. need to be reassessed for this new "economics”. To appreciate the tremendous shift of viewpoint acquired here it is only necessary to see what a difference was made by Marx’s change from the capitalistic basis of value to a "labour basis of value”. Even though the fundamental point of dealing with conflicting interests was preserved, and only the means of measuring value was altered, a complete change of the basic concepts was involved. Here, when we remove the very basis of all economic discussion, it is no longer clear that we are still discussing economics. It seems to me that the use of the previous methods and terminology is entirely unjustified with this basic assumption. For further discussion, therefore, we shall disregard this assumption. This approach appears to be consistent with Islam, which assumes that humans are selfishly motivated, and provides a code to regulate their selfish behaviour. Thus the individual Muslim is not to be regarded as trying to maximize "social benefit” but rather to maximize his individual utility function subject to the constraints imposed by the Islamic state in which he is living — a situation which can be dealt with in usual economic terms.

The above-mentioned point leads the author to disagree with the other workers in Islamic economics. The difference essentially boils down to a definition of “Islamic economics”. The earlier definitions, though still not quite precise, are very clearly recognisable as economics. The author’s "definition”, also vague, would be better termed on “Utopian Islamic economics”, where all ideals have already been achieved. One would expect such a perfect system to be static and hence there should be no rates appearing in it. One would expect, therefore, that the rates appearing in the work, result from some error of analysis which needs to be pinpointed. This argument is certainly not rigorous, nor is it meant to be. It is merely, expressed to indicate that an error was expected by me. The need for rigour comes in the analysis of the author’s argument, which follows.

*We are informed that the “profit function”, n, is given by

tt = R — C + u             (1)

R being the annual revenue, C the annual cost and u a stochastic (or random) variable satisfying the requirement

E(u)=0,E(u2) = a2              (2)

where E(u) represents the expectation value of the stochastic variable u. What this implies is that the variable u is already known throughout the range and its average is zero. From Equations (1) and (2) it becomes obvious that

o2 = E(tt2) - [E(tt)]2          (3)

where E(tt) is just the expectation value of the profits, being R—C, i.e. the difference between the revenue and the costs. The author confuses this expectation value of profits, i.e. the average profits, with the expected profits. The expected profit is the guessed amount of profit that the entrepreneur is expecting to get, and not the expectation value of the random variable “profit”. In fact “u” is not an observable variable and only “tt” can be actually measured. Thus a2 is defined only in terms of it, i.e. by Equation (3). This equation only applies with complete knowledge of profits before hand and does not refer to the guesses of the entrepreneur. Not assuming any foreknowledge all that can be reasonably said is that the investor’s utility function is defined by

au         au

U = U(tt, a),      >         0          <0        (4)

dir        da

where n is the expected (guessed) profit and a the expected risk. The
earlier Equations (1)—(3), can not be meaningfully used.

The author talks about an undefined constant ‘a’. It is claimed that it has been “shown” to be some quantity, but there is no reference as to where it has been shown. This lack of definiteness is typical of the whole analysis. Here it is particularly distressing as it is basic to the author’s “definition” of the rate of capitalization, which becomes meaningless as it is expressed in terms of an undefined constant by the equation

p = a + w a2     (5)

w being some other, unspecified, constant. Equation (5) does not have any economic content beyond saying that the rate of capitalization is linearly related to a2, which would be defined by Equation (5) if that equation were meaningful. Even accepting a more meaningful definition by Equation (3) with a2 being the measure of the entrepreneur’s expected risk, we would be left with the task of trying to understand the significance of a and w. Supposing that this had been done to our satisfaction, we would still have to understand the significance of p as a whole as that has not been expounded by the author. Later, the author returns to "the rate of capitalization” and provides his own "definition” of the quantity as the sum of two quantities: “the required rate of return” and “the investor’s premium for risk-bearing”. There is no attempt made to justify this definition, or to relate it to Equation (5). As the title seems to indicate that this concept is the main theme of the paper, and there is confusion regarding it, one has to say that the subject has not been adequately dealt withf.

So far no basic errors have been committed in the mathematics. The errors dealt with have been of a deeper nature. The paper provides basic errors of mathematics and economics. The situation discussed is very odd in itself. There are supposed to be two entrepreneurs not dependent on anything else in the economy, one of whom is called "the Central Treasury” while the other is called “the investor”. In spite of these names they are taken to behave as “entrepreneur 1 ” and “entrepreneur 2”. There is nothing in the utility function, or its corresponding constraints, to distinguish the entrepreneur from the Central Treasury. This is the basic error of economics referred to. We now come to the basic mathematical error. A maximand is written down by the author

U* = \ J ] { n v i r 2 ; o 2 v o 2 2 ) + \ [ U 2 { n v n 2 ; o 2 v o 2 2 ) -  U * j  (6)

where U1 is supposed to be the utility function of “the investor”, and U2 of “the Central Treasury”; 7r1 and 7t2 are their respective expected profits, and a21 and a22 their respective risks; X is an unspecified lagrange multiplier presumably to be fixed by optimization requirements; and is an undefinable utility function. There is no way that, within the framework used, U1 should depend on ir2 or a22. Again, the way it is presented, U| could be defined, Equation (6) is devoid of content.

To continue with an analysis of the mathematical errors, the author writes down two further conditions,

2 2 2
IT = 7T1 + 7T2; 0=0j + O 2    (7)

where n is the total profit and o2 the total risk. Whereas the profits will add, the risks will not. To see this fact most simply consider a partnership in a pure gamble, i.e. o2 = Zi (in a probabilistic measure) clearly each partner also runs a risk of Z* in that case, and not a quarter. Somehow the author would like to consider a2 as “the amount of money involved in the risk” and not “the risk” itself. Without an exact definition of what o2 is actually supposed to be, [remember that it could not be defined by Equations (2) or (3)] there is no question of being able to accept Equation (7).

At this stage, the author performs some completely meaningless steps. He differentiates Equation (6), partially, relative to 7r1 and o2 and on the right hand side relative to two independent variables together. Nothing here makes sense. Then he requires that these partial derivatives be zero, which is a very odd optimization constraint. The net result is that if the calculations are performed correctly otherwise, the author is committing the fallacy of dividing by zero. He then claims, that his (fallacious) steps easily give

da22/da2] = —1    (8)

which is also untrue. Had the claim even been true, Equation (8) would normally be considered meaningless as it appears to relate two unrelated (independent) variables. Whereas the two independent variables could be added, no restriction on their sum is valid, as they are then no longer independent variables. In fact what the author does to require that o2 in Equation (7) be constant. This requirement clearly yields Equation (8). Thus he is actually thinking of o2^ and o22 as being parametrized by some parameter s, and o2 being constant for all values of a in our domain of interest. This assumption can not be economically justified at all.

The above-mentioned error is repeated for expected profits to yield

d7r2/d7r1 = —1   (9)

by assuming that the expected profits ir are constant. As the meaning of constancy is not for a given enterprise but for any enterprise this assumption is clearly false. In the case of a2^, since the economic significance of the term was not clear, it was not possible to make such an emphatic statement. Apart from these errors there is an inconsistency in two of the assumptions made by the author. Since any intangible “Islamic returns” will add for the two partners and will not cancel, the profit could not remain constant anyhow. Hence this assumption (of constant profits) does not allow n to include “Islamic returns”.

The errors committed have already become too numerous to be amenable to correction. Thus, for example, the author writes down the following:

9U. / 97T,           (9U1 / 97r0) (97r « , / 9ir1)        9U, Pa,

                        1          =                                  I           I           L =       L= _ (io)

9U2/9tt2            (9U2/9tt2)          9U2       Po2

Here the first step is incorrect. Consider the case that U1 is a function of 7r1 only, then 9U1 /97r2 = 0. Thus, even purely mathematically, the first step is fallacious. Again, it is not meaningful to take the partial derivative of one utility relative to another, as utilities are functions of the independent variables and not themselves some basis of independent variables. Even had that error not been present, the derivative of one independent variable relative to another can never be partial, but may be definable as a total derivative. Even supposing that a total derivative is what was meant (or, disregarding other problems, the partial derivative could be regarded as correct) still

9U1/9tt2            9U.,

             *          (11)

9U2/9tt2            9U2

in general. Of course, there is also the error of Equation (9) which is used in the second step of Equation (10). It is not possible to continue to discuss the errors meaningful. If one wanted to pursue this somewhat unpromising approach, the whole analysis would have to be reworked from scratch. A rough sketch of the attempt that can be made is outlined below.

Consider a situation of n entrepreneurs with expected profits n. and expected risks a., and one Central Treasury with expected profits no

and expected o2q. The economically reasonable utility functions are

Vo2o.°21 ff2n>   (12)

ui = UjOi,; o2j) (i = 1,..,n)

with the requirements

au./a^x), au./902.<o         (13)

The optimal value will be for maximum profits and minimum risks for the entrepreneurs. However, the Central Treasury has a much more complicated requirement since it maximizes other profits than its own and it minimizes other risks as well as its own. It would be the task of Islamic economists to state reasonable economic requirements for U within an Islamic framework. Thus, for example, we would probably require that

aUj/aTr, > o, au./aa2. < o, (i = o, i, 2,... n)           (14)

particularly simple model may be

Uo=j, WiUi+W0U0   (15)

where Wj are some appropriate weighting factors and Un =0n (tt , a2)

O O ' o’ o'

satisfies Equation (13). Thus the utility function of the Central Treasury would be a suitably weighted average of the (n+1) entrepreneurs, with the clear requirement that WQ < W.. The Wj would be fixed by conditions such as the national needs and allocative considerations, being time dependent in general [A geometric mean would also bemused in place of Equation (15)].

The “interest” forbidden at the time was easily understood as money was well defined over fairly long periods of time. The use of this "money” to obtain more “money" at the expense of other people was definitely forbidden in Islam, in fact, it could be argued that Islam disapproved of more “money" being obtained without any work done for the extra “money”. The conjunction was to the exhorbitant “interest” charged by private money-lenders. Clearly that “interest” was the result of the very high risk taken by the money-lender. In no way can it be argued that this risky investment was encouraged in Islam while the comparatively safe investment of a shopkeeper in goods for sale was discouraged. The element of risk was a social evil and to be discouraged as such. However, as we know even in the physical sciences today, nothing is certain and so some element of risk is inevitable. We can only reduce the risk but it will never be possible to eliminate it.

Let us now turn to “interest" in present-day terms. “Money” is no longer a well-defined quantity in that its value changes over short periods of time. Its value decreases daily more than it changed in decades in the early Islamic times. Also, the “interest” given by banks is obviously not a means of exploiting the needy people. Even the “interest” taken by a bank bears only a superficial resemblance to the “interest” forbidden by Islam. The disallowed “interest” must be that of early Islamic times and not the modern banking “interest”. It is certainly a perverted view of Islam which disallows safe investment and encourages the originally disallowed interest under another name. Remembering that one of the purposes of Zakah was as a disincentive to hoarding unutilized wealth, there should be a general disincentive for keeping “money” instead of investing it in a way that is beneficial for the economy. These points must be taken into account along with inflation when discussing the question of “interest" in an Islamic economy. Rather than getting bogged down with terminology on which there may be disagreement, I shall avoid terms like “profit” and “interest” and only talk of “returns" on an investment.

I would like to suggest that instead of trying to avoid inflation and its related problems it should be used to achieve certain goals of an Islamic economy. In particular it could be used as a means of collecting a wealth tax on unutilized “money” holdings, as their value will decrease in real terms. Ideally “money” should be invested in the Central Treasury to give returns greater than the inflationary loss. The Central Treasury would give loans on the basis of national needs at a rate of returns to the treasury in excess of the returns given by the bank. The purpose of these conditions is to keep a positive cost of capital so as to avoid overspending. In this way we have a minimum risk investment for everybody. The risk is taken by the whole economy. The investors gain collectively or lose collectively, in keeping with basic Islamic requirements.

To sum up, numerous errors of logic, economics and mathematics were found in the analysis of the paper under discussion. As the main aim of the paper is to analyse rather than to suggest policies, it does not seem as if much can be gained from it. In particular this paper, along with many others, assumes that Islam encourages risky ventures and discourages safe investments. In fact the opposite in true — Islam discourages gambling and private money-lending while encouraging investments which are safe, like shopkeeping. As such there should not be a question of “profit-sharing” against “interest”, but of the “private money-lending” against “collective investment”, i.e. an exploitative “interest” against a modern collective “interest”. Replacing these emotive terms by the word “returns” we see that we require safe investment in a Central Treasury which gives returns at a higher rate than the rate of inflation. The treasury will loan money for investment at a higher rate than the rate of inflation. The treasury will loan money for investment at a higher rate than the rate it gives. The money would be used in ventures that fulfil national needs and not merely to maximize profits. This method enables us to collect Zakah on an utilised money holdings by setting aside as much money as would have been lost due to inflation by all the people who have not invested in the Central Treasury.

Source: Fiscal Policy and Resource Allocation in Islam, Ziauddin Ahmed, Munawar Iqbal and M. Fahim Khan. Republished with permission.