Abstract; JEL reference A12, B13, B21, D11 and D61

The neo-classic revolution of the XIXth century seventies applied the Newtonian Physics Algebra to economics and in so-doing was soon confronted with the ever-present problem of the value of money. From these days onwards several solutions to circumvent rather than to solve the issue were devised. The utility indifference and the functional approaches are shown not to be more realistic than the original Marshallian one. The most known solution to the problem is the “Slutsky Matrix”, obtained through a device called “compensated demand function”. In the present paper the compensation operation is shown – in analytic and experimental terms – to have no scientific support; the problem is restated in basic differential calculus language equating the central role of the value of money.

Acknowledgements

The author is thankful to Isabel Coutinho – engineer by training – for showing how this question can be treated by standard Algebra as well as to the Physicist Professor Nuno Sá for his support. Both of them bare no responsibility to the insufficiencies of the text fully assumed by the author.

*PAULO CASACA,

Paulo Casaca, Av. Paul Hymans, 124/5 B-1200 Brussels, Belgium, Ph. +32474813815, E-Mail [email protected]

** 1. Introduction**

Stanley Jevons applied the Physics Newtonian model to the economics behaviour, in what was known as the marginalist revolution. The application presented a major problem, referred by Marshall (1890) – most likely the main publicist of the new school of thought – as the problem of the variation of the marginal utility of money causing the non-integrability of ordinary demand functions.

The value of money has always been the most intractable problem of economics – or chrematistics if we are to use the original and logic Aristotelian concept– and it is remarkable that this fundamental philosophical question surfaced again under an Algebraic form in the context of such a new revolutionary approach.

As the Marshallian solution of assuming the constancy of the value of money seemed to fly in the face of evidence (not least to Marshall himself) Vilfred Pareto (1909) followed a general equilibrium alternative where utility is kept constant. This is a no-less stringent assumption and, as we shall see, is equivalent to consider the constancy of the value of money.

Some other authors thought of particular utility functions where the variation in the value of money would not constitute a problem. Paul Samuelson was the author that better analysed the issue and showed that the underlying assumptions of these utility functions were not more realistic than the alternative assumptions of utility indifference or of fixed value for money.

The most famous solution came from Eugene Slutsky, as a sort of device that could slice from the relevant Jacobian the intractable side, which would disappear if “compensated” – the so-called income effect – and the well-behaved residual variation, that would guarantee the full integrability of demand functions, and therefore, bypassing the problem.

There is not a unique classical formulation for this solution, since after Slutsky (1915) formulated his matrix, Hicks (1934 and 1956) made at least two different approaches to the problem and never presented a complete analytical treatment of the subject. Allen (1932, 1933, 1934-a) is the best source from this point of view. The plethora of articles written on the subject addressed specific issues but never made a consistent analytical formulation of the compensation device.

Samuelson (1938-b and 1948) introduced revealed preferences as an alternative approach to the classical differential calculus demand theory formulation, and this approach became dominant in the last decades in scientific papers, whereas graphical charts are widely used in more didactic material.

In spite of the huge popularity of the Slutsky construction, it is virtually impossible to find complete presentations of the subject using the same scientific and appropriate language all along the demonstration. Most demonstrations in manuals jump from differential calculus analysis to graphical charts or relational algebra and or refer to authors who did it. For the sake of clarity and consistency, we will therefore stick to the classical differential calculus formulations.

The Slutsky matrix is a differential calculus construction. The inconsequential use of differential calculus analysis, graphical charts and relational algebra that is widespread in modern manuals (Varian (1992) and Kreps (1991) are the most typical) is of poor use when a thorough assessment of the problem is needed.

Even the “Handbook of Mathematical Economics” (Intriligator, 1982, Vol. I, p.83) makes no analytic treatment of the subject, inscribing a subscript “comp” under a differential with no other explanation than “Here “comp” refers to a compensated change in price, where income is compensated so as to keep utility constant”.

To our knowledge, the only attempt to build a consistent and complete differential calculus presentation of the “Slutsky Matrix” was made by Henderson and Quandt (1958, pp. 25 – 27), and therefore we will follow it.

Otherwise, contrarily to an implicit assumption of the Economics community, there is nothing peculiar in the mathematics of the formulation of the Utility Potential, which is the same as used for potentials in exact sciences. Therefore, we will also follow closely the differential calculus of vector fields of Apostol (1962, II volume, chapters 8 and 10) for the presentation of the vector fields expenditure and ordinary demand functions.

Another reason why this issue became unnecessarily complex and difficult to understand is that most of the discussion on the fundamental issues is done on different areas of application of economics that do not intercommunicate properly.

Otherwise, this essential issue is dealt with in the context of ever more challenging assumptions turning it ever more impossible to understand. This complexity hides the real nature of the problem instead of clarifying it. Therefore, we will keep our discussion in the context of a pure “Robinson Crusoe” economy with a pure maximising/minimising behaviour, assuming that the solution to the more complex issues demands a solution to the more simple issue, and not the reverse.

Furthermore, although it is easy to test the correction of the compensation theory on ordinary demand functions, no one ever did it. In the end of the present paper we will use an example to show that how the well-known Slutsky symmetry has no relation whatsoever with the compensation device.

** 2. The Vector Field Expenditure Demand Function **

We assume that a consumer engages in a constrained minimising behaviour where an expenditure function E = p_{i}x_{i} is minimised subject to an utility constraint U = U(x_{i}) – where the utility function is a strictly quasi-concave function – defined on the set composed of n+1 elements; n commodities (x_{i}) and 1 Lagrange multiplier (µ) variables.

Z = p_{i}x_{i} + µ[U – U(x_{i})] (2.1)

with the following first order extremum conditions:

p_{i} = µU_{i} (2.2)

U = U(x_{i})

If we express these first order conditions for a constrained minimum (2.2) as a function of prices and utility we will obtain a vector field continuously differentiable demand function f*, named “Expenditure Demand Function”, defined on the n+1 element set S composed of n prices (p_{i}) and 1 utility (U) variables.

This function is different from the so-called Hicksian or compensated demand function because it does not assume that utility is constant, nor that income is compensated. Furthermore, it considers the whole set of m equations resulting from (2.2) and not only the first n equations. The standard practice in contemporary Economics manuals of considering money as having a constant value was shown to be unacceptable by Marshall already in the nineteenth century.

x_{i} = f*_{i} (p_{i}, U) (2.3)

µ = f*_{m} (p_{i}, U)

The total differential of the m equations set (2.2) on the Ω + S space is an m differential equation set whose array of coefficients is the same as the bordered Hessian matrix:

∑µU_{ij }+ U_{i}dµ = dp_{i } (2.4)

∑U_{j}dx_{j} = dU

Denoting the Hessian determinant by D* and the cofactor of the element in the first row and first column by D*_{11}, the cofactor of the element in the first row and second column by D*_{12}, etc., the solution of (2.4) by Cramer’s rule is:

dx_{i} = (∑D*_{ji}dp_{j} + D*_{mi}dU)/D* (2.5)

dµ = (∑D*_{jm}dp_{j} + D*_{mm}dU)/D*

If we divide each of these equations, successively by dpj, dU, while assuming the other n variables to remain constant we will obtain the total differential of the vector field f*, the Jacobian matrix F*:

f*^{i}_{j}, (i, j = 1,…m) (2.6)

Each term f*^{i}_{j} is of the general form:

f*^{i}_{j} = D*^{j}_{i}/D* (2.7)

The Jacobian matrix F*, as a consequence of the Young’s Theorem, is symmetric and, as the set S is defined on the positive orthant space (which is convex) this ensures that the demand function f* is a gradient, and so its line integral is independent of the specific path of price and utility variations:

_{i=1}^{m} _{C}f*_{i}dk_{i} (k_{i} = p_{i}, for i = 1,2,…,n, k_{m} = U) (2.8)

We shall name this line integral “Expenditure Consumer’s Surplus” (ECS).

** 3. The Ordinary Demand Function. **

We assume that a consumer engages in a constrained maximising behaviour where the utility function U = U(x_{i}) – defined the same way as in section 2 – is maximised subject to an expenditure constraint p_{i}x_{i} = E and defined on the commodity – Lagrange multiplier space, , composed of n commodities (x_{1}, x_{2},…,x_{n}) and 1 Lagrange multiplier (λ).

V = U(x_{i}) + λ(E – p_{i}x_{i} ) (3.1)

The first order extremum conditions are:

U_{i} = λp_{i} (3.2)

p_{i}x_{i} = E

If we express these first order conditions for a constrained maximum (3.2) as a function of prices and expenditure we will obtain a function f, we will name “Ordinary Demand Function” defined on the budget set composed of n prices (p_{i}) and 1 expenditure (E) variable. Here we will not follow the standard practice of ignoring the m^{th} equation of this demand function.

x_{i} = f_{i} (p_{i, }E) (3.3)

λ = f_{m }(p_{i, }E)

The total differential of the m equations set (3.2) on the + space is an m differential equation set whose array of coefficients is the same as the bordered Hessian matrix:

∑U_{ij}dx_{j} – p_{i}dλ = λdp_{i} (3.4)

∑p_{j}dx_{j }= dE – x_{i}dp_{i}

Denoting the Hessian determinant by D and the cofactor of the element in the first row and first column by D_{11}, the cofactor of the element in the first row and second column by D_{12}, etc., the solution of (3.4) by Cramer’s rule is:

dx_{i} = ∑λD_{ji} dp_{j} + + D_{mi}(-dE +_{i=1}^{n}x_{i}dp_{i})]/D

dλ = ∑λD_{jm}dp_{1} + D_{mm}(-dE +_{i=1}^{n}x_{i}dp_{i})]/D (3.5)

If we divide each of these equations, successively by dpj, dE while assuming the other n variables to remain constant we will obtain the total differential of the vector field f, the Jacobian matrix F:

f^{i}_{j}, (for i, j, = 1, 2, …n, m) (3.6)

The elements of the n first columns of matrix (3.6) are:

f^{i}_{j} = λD_{ji}/D + x_{j}D_{mi}/D (3.7)

(i=1, 2, …, n, m; j=1, 2, …, n)

While the elements of the m^{th} column are:

f^{i}_{m} = -D_{mi}/D (3.8)

(i=1, 2, …, n, m)

Substituting (3.8) into (3.7), we have

f^{i}_{j} = λD_{ji}/D – x_{j}f^{i}_{m} (3.9)

The Jacobian matrix F is clearly not symmetric. The line integral of the vector field ordinary demand function along an unspecified path of price and income variations follows the equation:

_{i=1}^{m} _{C}f_{i}dk_{i} (k_{i} = p_{i}, for i = 1,2,…,n, k_{m} = E) (3.10)

This is a non-significant integral, as it adds up variables expressed in money units and a variable expressed in utility units. Furthermore, it is dependent on the path of integration. If we are to consider only the first n elements in the integral, we will have the “Marshallian Consumer’s Surplus” (MCS).

Its values, in general, have no relation (either “cardinal” or “ordinal”) with utility and, therefore, cannot be considered welfare indicators.

In next section we will see how this issue may be restated in a coherent issue with a meaningful answer.

** 4. The recoverability of Expenditure and Utility Functions **

From (2.2) and (3.2), keeping in consideration basic principles of duality we can obtain (see Coto-Millán, 2003, chapter 5; Hillier et al, 2005, chapter 6)

λ = µ^{-1}; (4.1)

Considering the n first equations of (2.2), the last equation of (2.4) divided by λ and the last equation of (3.4), we will have:

λ^{-1}dU = p_{i}dx_{i} = dE – x_{i}dp_{i} (for i = 1, 2,…n) (4.2)

From equality (4.2) we can obtain two equations:

dE = x_{i}dp_{i} + λ^{-1}dU (4.3)

dU = -λx_{i}dp_{i + }λdE (4.4)

Equation (4.3) means that the vector field expenditure demand function is the gradient of the expenditure function. The “Expenditure Consumer’s Surplus” (ECS) represented in the equation (2.8), therefore, measures exactly the expenditure difference from any particular price and utility situation to another, consequently recovering exactly the underlying indirect expenditure function. The plethora of Hicksian variations can therefore be seen as simple particular numeric cases of the same integral.

E^{1} – E^{0} = ECS (4.5)

Equation (4.4) shows us that the differential equation of utility is not the ordinary demand function and so, regardless of the dependency of the path of integration, we should not expect the Marshallian Consumer’s Surplus to give us utility. Equation (4.4) shows us also that, provided that it is found to be exact, there is a demand function that is the gradient of the indirect utility function.

To show that equation (4.4) is an exact differential, we will first consider the problem for the n first elements of the n first equations of its Jacobian.

We must demonstrate that:

(λx_{i})/p_{j} = (λx_{j})/p_{i} (4.6)

Developing the left member of (4.6) we have:

λx_{i}/p_{j} + x_{i}_{λ}/p_{j}, (4.7)

From (3.7):

λ/p_{j} = λD_{jm}/D + x_{j}D_{mm}/D (4.8)

From (3.8):

λ/E = -D_{mm}/D (4.9)

x_{j}/E = -D_{mj}/D

From the Young’s theorem we know that:

D_{jm} = D_{mj} (4.10)

And so we can replace (4.8) by the following expression:

λ/p_{j} = -λx_{j}/E -x_{j}λ/E (4.11)

Transforming (4.7) into

λx_{i}/p_{j} + x_{i}(-λx_{j}/E -x_{j}λ/E) (4.12)

If we follow the same steps for the right side of equation (4.6) we will obtain:

λx_{j}/p_{i} + x_{j}(-λx_{i}/E -x_{i}λ/E) (4.13)

So equality (4.6) can be presented as:

λx_{i}/p_{j} +x_{i}(-λx_{j}/E -x_{j}λ/E) = λx_{j}/p_{i} +x_{j}(-λx_{i}/E -x_{i}λ/E) (4.14)

and changed into:

λx_{i}/p_{j} -λx_{i}x_{j}/E -x_{i}x_{j}λ/E = λx_{j}/p_{i} -λx_{j}x_{i}/E -x_{i}x_{j}λ/E (4.15)

And into:

λx_{i}/p_{j} -λx_{i}x_{j}/E = λx_{j}/p_{i} -λx_{j}x_{i}/E (4.16)

Or, still, to:

x_{i}/p_{j} + x_{j}x_{i}/E = x_{j}/p_{i} +x_{i}x_{j}/E (4.17)

This is the familiar Slutsky reversibility law, or a general term of the Slutsky symmetric matrix.

Regarding the m^{th} row and column of the present Jacobian, we must show that:

-λx_{i}/E = λ/p_{i} (4.18)

From (3.7) we know that:

f^{m}_{i} = λD_{im}/D + x_{i}D_{mm}/D (4.19)

Developing the left side of equation (4.18) we have:

-λx_{i}/E = -x_{i}λ/E – λx_{i}/E (4.20)

And, according to (3.8), (4.20) may be transformed in the following way:

-x_{i}λ/E – λx_{i}/E = x_{i}D_{mm}/D + λD_{im}/D (4.21)

And so (4.18) is also proved to be true, and therefore the differential equation (4.4) is found to be exact. Another way of stating this fact is that the multiplication of the first n equations of the vector field ordinary demand function by -λ produces a gradient of the Indirect Utility Function.

g_{i} = -λx_{i} = -λf_{i} (p_{i}, E) (4.22)

g_{m} = λ = f_{m} (p_{i}, E)

We shall call this vector field “Utility Demand Function”. Its Jacobian will take the general form:

g^{i}_{j} = g^{j}_{i} = -λx_{i}/p_{j} – x_{i}λ/p_{j} (for i,j = 1,2,…,n)

g^{i}_{m} = g^{m}_{i} = -x_{i}λ/E – λx_{i}/E (for i = 1, 2,…, n, m) (4.23)

As the Jacobian matrix G is symmetric the line integral of the vector field utility demand function is independent of the specific path of prices and income variations. It follows the equation:

_{i=1}^{m} _{C}g_{i}dk_{i} (k_{i} = p_{i}, for i = 1,2,…,n, k_{m} = E) (4.24)

We shall name this line integral “Utility Consumer’s Surplus” (UCS); it measures exactly the utility difference from any price and expenditure situation to another. It recovers the indirect utility function.

U^{1} – U^{0} = UCS (4.25)

As we reach this point it becomes clear that the problem of the integrability of demand function, equated at the beginning of the century as a path dependency of integration of the Marshallian demand function, can be better equated in terms of two fundamental remarks:

(IV.1) The observable (Marshallian) demand function (3.3) is not a Gradient of the Utility Function.

(IV.2) The expenditure (2.3) and the utility (4.22) demand functions are indeed the Gradients of the underlying Expenditure and Utility constrained Functions but they are expressed in terms of non-observable variables, marginal cost of utility or Utility and marginal utility of money (expenditure).

** 5. The indifference approach **

The indifference approach was due to Edgeworth, as Pareto himself points out (p.169, 1909). Contrarily to the standard practice of demand theory, we can use it only in the context of expenditure minimisation and not in the context of utility maximisation, that is to say, we can apply this restriction when utility is in the domain of the function as in equation (4.3) and not when it is in the range of the function, as in equation (4.4).

As we have seen, the main problem of the expenditure demand function is that utility and its associated Lagrange multiplier are not observable variables. A solution for this problem is to place it on the utility indifference hyperspace, that is to say, to establish the condition dU = 0. Keeping in mind equation (4.3) we can see that this restriction is equivalent to consider the marginal utility of money as a constant. This restriction makes the integration of the Lagrange multiplier unnecessary.

Since the commodity expenditure demand vectors are also dependent on utility, and as in this case utility is supposed to be an unknown constant, the integration of the vector field expenditure demand function does not produce a unique cardinal determination of money expenditure.

However, this procedure fully maintains the ordinal expenditure determination properties, since the Jacobian (2.6) symmetry guarantees that f*^{i}_{j} = f*^{j}_{i} (for i, j = 1, 2,…, n, m), regardless of the value assumed for utility.

In computational terms this Paretian or ordinal technique allows us, therefore, to find a unique and ordinal consumer’s surplus, that is to say, to build expenditure demand functions with constant utility.

Although this solution is computationally conceivable, and it fully overcomes remark IV.1, it does not help us from the point of view of remark IV.2, since we cannot reasonably expect any consumer to obey this utility constancy rule, which ultimately cannot be observed. This ordinalist solution is therefore unable to solve the problem of testing by experiment the minimising behaviour.

** 6. The search for an integrating factor **

An alternative way of looking at the integrability issue is that an integrating factor capable of transforming the ordinary demand function into a gradient is found or is proved to exist. This is the standard method applied to this kind of situations in Physics (see Apostol, 1961).

If we determine the integrating factor that transforms the ordinary demand function in such a way that each term f^{i}_{j} is transformed into a symmetric f**^{i}_{j}, we produce a “Slutsky Substitution Terms Matrix” that solves the problem of the path of integration of the vector field ordinary demand function at least for the first n rows and columns of Jacobian (3.6). This alternative is the standard calculus method to attain the results the compensation approach tried to achieve.

The “existence theorem” of such integrating factors loosely mentioned by Samuelson (1950, p.380) was formally presented by Hurwicz and Uzawa (1971) in a more elaborate form.

However, this “existence theorem” is redundant. If we just consider the first n equations of (3.3) as Hurwicz and Uzawa do, -λ is the integrating factor of the ordinary demand function, as we have seen in section 4, and it is the solution for the problem they equated, that does not exist as such.

Still, the relevant Jacobian here will not be the Slutsky Matrix but the matrix (4.23). As we have seen, this emphasises the importance of the recoverability problem (remark IV.2) and emphasises the inadequacy of the Slutsky Matrix to answer the relevant question.

** 7. The functional approach**

We should also consider the utility functions whose special properties guarantee the symmetry or quasi-symmetry of Jacobians of ordinary demand functions.

Samuelson (1942, 1965 and 1974) and Chipman and Moore (1976) defined two partial cases: homothetic utility functions with constant income and the so-called “vertical [or horizontal] parallel preferences” with constant price of the “money good”. The most common homothetic function is the Cobb-Douglas function, widely used on the offer side of the equilibrium model.

Both these types of utility functions require such unnatural restrictions that, so far, they are mere curiosities with no practical use, Samuelson (op. cit.) being the author who better explained why.

These utility functions are nothing other than a formal expression of the original Marshallian assumption of the constant value of money, so they might be viewed as a restatement of the original Marshallian approach. The gradients of these functions, or of the first n elements of these functions, produce Jacobians F that are the same as Jacobians F*.

In spite of these well-known limitations of homothetic utility functions, there are manuals that use them as a “proof” of the soundness of the compensation operation (e.g. Varian, 1992, even uses the term “Cobb-Douglas Slutsky equation”). Even more awkwardly, the very same authors who consider homothetic utility functions to be unrealistic, do often use them to exemplify their findings (e.g. Deaton and Muellbauer (1980-a; 1980-b) and Kim (1997)).

** 8. The “Slutsky Matrix” construction**

Slutsky (1915), realised that the first n rows and columns of (3.6) presented as in (3.9) would become a symmetric matrix if we transform them in the following way:

f**^{i}_{j} = f^{i}_{j} + x_{j}f^{i}_{m} = λD_{ji}/D = f**^{j}_{i} (8.1)

Slutsky called each of these terms “residual variations” and called equality (8.1) “the law of reversibility of residual variations”, presented in the following form, identical with equation 4.17 (Slutsky, 1915, p. 43):

x_{j}/p_{i} + x_{i}(x_{j}/s) = x_{i}/p_{j} + x_{j}(x_{i}/s) (8.2)

(s represents income)

Keeping in mind that the multiplication of a matrix array by λ increases the value of the determinant by the same multiple and using equalities (4.1) and (4.2) we can express the equality:

f*^{i}_{j} = f**^{i}_{j} (for i, j = 1, 2, …, n) (8.3)

This equality shows us that the Slutsky Matrix is simply the Jacobian of the Expenditure Demand Function (2.6).

However, instead of realising this fact, Slutsky thought it would be the consequence of a sort of compensation virtual operation: “the increment dp_{i} of price (“apparent loss”) accompanied by an increment of income equal to the apparent loss, can be said to be the compensated variation of price”. He further thought that this double operation could indeed transform each term f^{i}_{j} into f**^{i}_{j}. He supplied no analytical support to his claim, which, as we will see is fully erroneous.

The demand theory revolution of the thirties gave new clout and notoriety to the original Slutsky “compensation” approach. Slutsky (1915) was introduced in Anglo-Saxon Economics by Schultz (1935) and Allen (1936), only in the aftermath of the rediscovery of the “compensation” idea by Hicks (1934) and Allen (1934-b). Therefore, the latter is to a large extent independent of the former.

The main novelty brought forward by the Hicksian “compensation” was that it no longer assumed the constancy of the purchased quantities of all goods that had formerly been bought, but just the utility derived from those goods, maintaining consumers at specific “utility levels”.

The starting point of the Slutsky or the Hicksian versions of the compensation operation is equality (4.2):

λ^{-1}dU = p_{i}dx_{i} + p_{j}dx_{j} + …. = dE – x_{i}dp_{i} – x_{j}dp_{j}… (4.2)

On the Slutsky version the restriction imposed is:

p_{i}dx_{i} = 0 (for i=1, 2,….n) (8.4)

On the Hicksian version the restriction imposed is:

dU = 0 (8.5)

If we have a movement in one of the prices p_{j}, simultaneously with a movement in expenditure and with no further movements in the remaining variables, both the conditions (8.4) and (8.5) yield the following equality:

dE = x_{j}dp_{j} (8.6)

If we now consider (3.9) as a commodity differential:

dx_{i} = (λD_{ji}/D)dp_{j} -(x_{j}f^{i}_{m})dp_{j} (8.7)

And replace x_{j}dp_{j},

dx_{i} = (λD_{ji}/D)dp_{j} -f^{i}_{m}dE (8.8)

If now we consider the “compensation” to be translated as f^{i}_{m}dE and add this element to equality (8.7), we would obtain:

dx_{i} = (λD_{ji}/D)dp_{j} (8.9)

This is supposed to be the “compensated” or residual variation terms, from which “Slutsky Matrices” should be formed.

There are endless variations to this procedure, none less erroneous than the original Slutsky one. Some authors force the commodity x_{i} to vanish, some others consider the differentials dp_{j} (in the second term) and dE to disappear, more commonly authors present some obscure charts from where the Slutsky equation could be understood, etc.

As Slutsky himself did not give a great deal of importance to his compensation operation, the whole of the affair would be rather secondary if the standard demand theory, following the papers of Allen (1934-b) and Hicks (1934), had not transformed this operation into a central element of the theory.

The so-called Slutsky matrix is nothing other than a trivial identity. However, it has no relation whatsoever with “compensation” operators, utility indifference, or with Ordinary Demand Functions, contrarily to what standard Demand Theory assumed from the thirties onwards. We can see this through the following numerical application.

** 9. A Numerical Application **

Designating the differentials x_{1}dp_{1}, x_{2}dp_{2}, µdU; x_{1}dp_{1}, x_{2}dp_{2} λdE and -λx_{1}dp_{1}, -λx_{2}dp_{2}, λdE [respectively for the expenditure, ordinary and utility demand functions] by 1, 2 and 3, we will note the paths of integration by superscripts according to the following rule: path number (1); order of integration; [1,2,3]; (2) – [1,3,2]; (3) – [2,1,3]; (4) – [2,3,1]; (5) – [3,1,2]; (6) – [3,2,1].

The branches of each path will be noted by the subscripts 1, 2 or 3 according to the differential under integration. We shall only consider paths formed by the different ordering of the total integration of each differential term as an infinite number of paths would exist otherwise.

We will naturally exclude here homothetic and other special functions dealt with in section 7. Let’s consider the quasi-concave utility function U = x_{1}x_{2} + x_{2}.

For the constrained minimising behaviour, the first order minimum conditions are:

p_{1} = µx_{2} (9.1)

p_{2} = µ(x_{1} + 1)

U = x_{1}x_{2} + x_{2}

The expenditure demand function f* is:

x_{1} = U^{1/2}p_{1}^{-1/2}p_{2}^{1/2} – 1 (9.2)

x_{2} = U^{1/2}p_{2}^{-1/2}p_{1}^{1/2}

µ = U^{-1/2}p_{1}^{1/2}p_{2}^{1/2}

The commodity x_{1} assumes nonnegative values only when the restriction Up_{2} p_{1} is imposed on the domain of the expenditure demand function.

The Jacobian matrix F* takes the form:

-2^{-1}U^{1/2}p_{1}^{-3/2}p_{2}^{1/2} 2^{-1}U^{1/2}p_{1}^{-1/2}p_{2}^{-1/2} 2^{-1}U^{-1/2}p_{1}^{-1/2}p_{2}^{1/2} (9.3)

2^{-1}U^{1/2}p_{1}^{-1/2}p_{2}^{-1/2} -2^{-1}U^{1/2}p_{1}^{1/2}p_{2}^{-3/2} 2^{-1}U^{-1/2}p_{1}^{1/2}p_{2}^{-1/2}

2^{-1}U^{-1/2}p_{1}^{-1/2}p_{2}^{1/2} 2^{-1}U^{-1/2}p_{1}^{1/2}p_{2}^{-1/2} -2^{-1}U^{-3/2}p_{1}^{1/2}p_{2}^{1/2}

The matrix (9.3) is symmetric in agreement with the results presented on section 2 and thus ensuring the vector field expenditure demand function to be a gradient. A price variation will yield the same value for the expenditure independently of the path of the variation.

If we consider the following movements in the space S: p_{1}^{0} = p_{2}^{0} = p_{2}^{1 }= 1, U^{0} = 2,25 p_{1}^{1} = 1/2,; U^{1} = 2; (test 1) the function f will yield the result:

x_{1}^{0} = 0.5, x_{2}^{0} = 1.5, µ^{0} = 2/3; x_{1}^{1} = x_{2}^{1} = 1; µ^{1} = ½; E^{0} = 2; E^{1} = 1,5.

And:

ECS^{i} = -.5 (for i, j = 1, 2,…,6)

Which is the expenditure value that shall be added to E^{0} to obtain E^{1}.

As we can see, the Hessian symmetry and the resulting independence of the path of integration have no relation with utility indifference or with any “compensation” operator. As we have stressed, these assumption and device are totally meaningless for the matters in discussion.

For the constrained maximising behaviour the first order maximum conditions are:

x_{2} = λp_{1} (9.4)

x_{1} + 1 = λp_{2}

E = x_{1}p_{1} + x_{2}p_{2}

The ordinary demand function f is:

x_{1} = 2^{-1}Ep_{1}^{-1} – 2^{-1} (9.5)

x_{2} = 2^{-1}p_{1}p_{2}^{-1} + 2^{-1}Ep_{2}^{-1}

λ = 2^{-1}Ep_{1}^{-1}p_{2}^{-1} + 2^{-1}p_{2}^{-1}

For the commodity x_{1} to assume only nonnegative values the restriction E p_{1} must exist on the domain of the ordinary demand function.

The Jacobian matrix F takes the form:

-2^{-1}Ep_{1}^{-2} 0 2^{-1}p_{1}^{-1} (9.6)

2^{-1}p_{2}^{-1} -2^{-1}p_{2}^{-2}(p_{1} + E) 2^{-1}p_{2}^{-1}

-2^{-1}Ep_{1}^{-2}p_{2}^{-1} -2^{-1}p_{2}^{-2}(1 + Ep_{1}^{-1}) 2^{-1}p_{1}^{-1}p_{2}^{-1}

This matrix is not symmetric, so the integration of the vector field ordinary demand function (MCS) will present different results according to the path of integration followed, as we have seen in section 3.

Once again, we can verify that the results of the integration of this function will be meaningless, regardless of what happens with the values of utility and the “compensation” operator.

We can test the “compensation” operator with a variation of one of the prices and the expenditure such that the resulting utility will not change, which is the standard definition of the “compensation” operator (test 2).

Let’s then find an E^{1 }such that the following conditions are met:

p_{1}^{0} = p_{2}^{0} = p_{1}^{1 }=1; E^{0} = 2; p_{2}^{1 }=2; U^{0} = U^{1} = 2.25

We can obey this condition if we consider the first two equations of (9.2) and the last equation of (9.4).

x_{1} = U^{1/2}p_{1}^{-1/2}p_{2}^{1/2} – 1 (9.7)

x_{2} = U^{1/2}p_{2}^{-1/2}p_{1}^{1/2}

E = x_{1}p_{1} + x_{2}p_{2}

From this we will obtain:

x_{1}^{1} = 1.5*2^(1/2) – 1; x_{2}^{1} = 1.5*2^(-1/2), and E^{1} = 6*2^(-1/2) – 1

If we now calculate the MCS, we will obtain MCS^{1} = MCS^{3} = MCS^{4} = 1.74 and for the remaining three paths we will obtain MCS^{2} = MCS^{5} = MCS^{6} = 2.48, which are different and have no relation with the associated utility variation (which, as we know, is zero).

The fact that in this experiment MCS_{1} was zero created two groups of three paths, each of them showing the same results. If a variation in p_{1 }is considered, six different results will be found.

This numerical example shows that utility indifference and the “compensation” operator are absolutely irrelevant in obtaining a symmetric Jacobian and so to the solution of the path dependency of integration, contradicting directly the statements of the “compensation” theory, and therefore, confirming what we stated in section 8.

Let’s consider now the utility demand function g derived from the previously considered function f:

-λx_{1} = -2^{-2}E^{2}p_{1}^{-2}p_{2}^{-1} + 2^{-2}p_{2}^{-1} (9.8)

-λx_{2} = -2^{-2}p_{1}p_{2}^{-2} – 2^{-1}Ep_{2}^{-2} – 2^{-2}E^{2}p_{1}^{-1}p_{2}^{-2}

λ = 2^{-1}Ep_{1}^{-1}p_{2}^{-1} + 2^{-1}p_{2}^{-1}

The Jacobian matrix G is:

2^{-1}E^{2}p_{1}^{-3}p_{2}^{-1} 2^{-2}p_{2}^{-2}(E^{2}p_{1}^{-2}-1) -2^{-1}p_{1}^{-2}p_{2}^{-1}E (9.9)

2^{-2}p_{2}^{-2}(E^{2}p_{1}^{-2}-1) p_{2}^{-3} (2^{-1}p_{1}E + E + 2^{-1}p_{1}^{-1}E^{2}_{)} -(2^{-1}p^{-2}) (1+Ep_{1}^{-1})

-2^{-1}p_{1}^{-2}p_{2}^{-1}E -(2^{-1}p^{-2}) (1+Ep_{1}^{-1}) 2^{-1}p_{1}^{-1}p_{2}^{-1}

This matrix is symmetric, so this demand function is a gradient and according to section 4 its integration (UCS) will give us the exact utility variations.

We can now consider the previous tests, or indeed, any other variation, and we will see that UCS is not dependent of the path of integration and it recovers exactly the considered utility variation.

For test 1, the values for UCS^{i} = (i, = 1, 2,…6) are equal to U^{1} – U^{0} = -0.25. For test 2, UCS^{i} = 0, (for i = 1, 2,…6).

** 10. Conclusions**

The so-called “Slutsky Matrix” is nothing else than the Jacobian of the “Expenditure Demand Function”, which is a correct Jacobian of this function, but certainly not of the Ordinary Demand Function. There is no “compensation”, regardless of the way it is defined, that can ever transform the Jacobian of the later into the Jacobian of the former.

Apparently, the Demand Theory of the thirties, confronted with a function dependent of unobservable variables but which was a gradient, and a function that was independent of unobservable variables but that was not a gradient, thought that it would be possible to take the best out of the two functions, and this is what the compensation operation is all about. However, as it is shown in sections 4 and 8, this duality does not work that way and the compensation device is baseless.

The last decades of last century have been dominated by a wide range of theoretical, mathematical or statistical solutions designed to bridge the gap between the Marshallian Consumer’s Surplus and the “Exact Consumer’s Surplus”. Among others we can quote: Burns, (1973); Foster and Neuberg, (1974); Bergson, (1975); Willig, (1976); Bruce, (1977); Dixit and Weller, (1979); Zajac, (1979) Chipman and Moore, (1980); Hausman, (1981); Vartia, (1983); Stahl, (1984); Coursey and Schulze, (1987); Vives, (1987); Weitzman, (1988) Lewbel (1989), Ebert (1995) and Kim (1997). None of these solutions tested conveniently the validity of the “Slutsky Matrix”.

In this paper we proved in mathematical and experimental terms that there is no such a thing as a “Slutsky Substitution Terms Matrix” that can be related to an Ordinary Demand Function by “compensation”, integrating factor, utility constancy or any known mathematical tool.

Our reformulation of the question of integrability confirms the importance given in the beginning of the century to the observability of the marginal utility of expenditure (λ).

As long as no scientific process is found to replace λ, or to determine to a sufficient extent its variation pattern, the utility potential theory will continue in the same deadlock as it was at the beginning of last century.

References

Allen, R. D., 1932, The Foundations of a Mathematical Theory of Exchange, Economica 12, 197-226.

– 1933, On the Marginal Utility of Money and its Applications, Economica 13, 186-209.

– 1934-a, The Nature of Indifference Curves, Review of Economic Studies 1, 110-121.

– 1934-b, A Reconsideration of the Theory of Value, Part II, Economica NS, 2, 196-219.

– 1936, Slutsky Theory of Consumer’s Choice, Review of Economic Studies 3, 120-129.

Apostol, T. M., 1961 [1968]. Calculus, II (J. Wiley & Sons, New York).

Aristotle, [2008], Les Politiques, (Flammarion, Paris)

Bergson, A., 1975, A Note on Consumer’s Surplus, Journal of Economic Literature13, 38-44.

Bruce, N., 1977, A Note on Consumer’s Surplus, the Divisa Index and the Measurement of Welfare Changes, Econometrica 45, 1033-1038.

Burns, M. E., 1973, A Note on the Concept and Measure of Consumer’s Surplus, American Economic Review 63, 335- 344.

Chipman, J. S. and J. C. Moore, 1976, The Scope of Consumer’s Surplus Arguments, in: Tang et al., eds., Evolution, Welfare and Time in Economics: Essays in Honour of Nicholas Georgescu – Roegen (Lexington) 69-123.

– 1980, Compensation Variation, Consumer’s Surplus and Welfare, American Economic Review 70, 933-949.

Coto-Millán, 2003, Contributions to Economics, Utility and Production, theory and applications (Springer)

Coursey and Schulze, 1987, The Disparity Between Willingness to Accept and Willingness to Pay Measures of Value, Quarterly Journal of Economics, CII, 679-690.

Deaton, A. and J. Muellbauer (1980-a), Economics and Consumer Behaviour (Cambridge University Press, Cambridge).

– 1980-b, An Almost Ideal Demand System, American Economic Review, June, 312-26.

Ebert, U., 1995, Consumer’s Surplus: Simple Solutions to an Old Problem, Bulletin of Economic Research 47, 285-294

Foster and Neuberg, 1974, The Ambiguity of the Consumer’s Surplus Measure of Welfare Change, Oxford Economic Papers 26, 66- 77.

Hausman, J. A., 1981, Exact Consumer’s Surplus and Dead-weight Loss, American Economic Review 71, 662-676.

Henderson J. and R. Quandt, 1958 [1985], Microeconomic Theory; a Mathematical Approach, Mc Graw-Hill.

Hicks, J. R., 1934, A Reconsideration of the Theory of Value, Part I, Economica NS, 2, 52-76.

– 1956 [1958], A Revision of Demand Theory, (Oxford University Press, London). Spanish translation: Revision de la Teoria de la Demanda (Fondo de Cultura Económica, México).

Hillier, Frederick and Lieberman, Gerald, 2005, Introduction to Operations Research, McGraw-Hill

Hurwicz, L. and H. Uzawa, 1971, On the Integrability of Demand Functions, in: John Chipman et al. eds., Preferences, Utility and Demand, (Harcourt Brace Jovanovich, Inc, New York) 114-148.

Kim, H. Y., 1997, Inverse Demand Systems and Welfare Measurement in Quantity Space, Southern Economic Journal 63, 3663-679.

Kreps, D., 1991, A Course in Microeconomic Theory, (Harvester Wheatsheaf, New York)

Lewbel, A., 1989, A Path-independent Divisa-like Index for PIGLOG Preferences, Economica 56, 121-123.

Marshall, A., 1890 [1964], Principles of Economics; an Introductory Volume, (Macmillan and Co Ltd, London).

Morey, E. R., 1984, Confuser Surplus, American Economic Review 74, 163-172.

Pareto, V.,1909, [1927]. Manuel d’Economie Politique (Marcel Giard, Paris).

Popper, Karl R., 1959 [1990]The Logic of Scientific Discovery. (Unwin Hyman Ltd. London)

Samuelson, P. A., 1938-a, A Note on the Pure Theory of Consumer’s Behaviour, Economica, August, 61-71.

– 1938-b, The Numerical Representation of Ordered Classifications and the Concept of Utility, Review of Economic Studies 6, 65-70.

– 1942, Constancy of the Marginal Utility of Income, in: Oscar Lange et al. eds., Studies in Mathematical Economics and Econometrics, in Memory of Henry Schultz (University of Chicago Press, Chicago), 75-91.

– 1947, Foundations of Economic Analysis (Harvard University Press, Cambridge Mass.)

– 1948, Consumption Theory in Terms of Revealed Preference, Economica, 243-253.

– 1950, The Problem of Integrability in Utility Theory, Economica 27, 355-385.

– 1965, Using Full Duality to Show that Simultaneously Additive Direct and Indirect Utilities Imply Unitary Price Elasticity of Demand, Econometrica, 33, 781-801.

– 1974, Complementarity – An Essay on the 40th Anniversary of the Hicks-Allen Revolution in Demand Theory, Journal of Economic Literature 12, 1225-1289.

Samuelson, P. A. and S. Swamy, 1974, Invariant Economic Index Numbers and Canonical Duality: Survey and Synthesis, American Economic Review, 64, 566-593.

Silberberg, E., 1972, Duality and the Many Consumer’s Surpluses, American Economic Review 62, 942-952.

Slutsky, E., 1915 [1952] Sula Teoria del Bilancio del Consumatore, Giornale degli Economisti e Rivista di Statistica, 51, 1-26. English translation: On the Theory of the Budget of the Consumer, in: Stigler, G. J. and K. Boulding, Readings in Price Theory (George Allen and Unwin Ltd., London) 27-56.

Stahl, D., 1984, Monotonic Variations of Consumers Surplus and Comparative Performance Results, Southern Economic Review 51, 503-520.

Uzawa, H., 1971, Preference and Rational Choice in Theory of Consumption in Chipman, J. et al. eds., Preferences, Utility and Demand (Harcourt Brace Jovanovich, Inc, New York), 7-28.

Varian, H. R.,1992, Microeconomic Analysis (W.W. Norton Company, New York)

Vartia, Y. O., 1983, Efficient Methods of measuring Welfare Change and Compensated Income in Terms of Ordinary Demand Curves, Econometrica 51, 79-98.

Vives, X., 1987, Small Income Effects: A Marshallian Theory of Consumer’s Surplus and Downward Sloping Demand, Review of Economic Studies, LIV, 87-103.

Weitzman, M., 1988, Consumer’s Surplus as an Exact Approximation when Prices are Appropriately Deflated, Quarterly Journal of Economics, CIII, 543-553.

Zajac, E. 1979, Dupuit-Marshall Consumer’s Surplus, Utility and Revealed Preference, Journal of Economic Theory 20, 260-270.

Original version written in Lisbon, 1989

Last edited in Brussels, 2015-10-06

___________________________

(José Paulo Martins Casaca)

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