Toward a Difference Approach: A Discrete Meat Demand System

While the differential approach to economic analysis is useful, the difference approach is indispensable as almost all economic data are discrete, rather than continuous. Thus, we must investigate the integration of the differential with difference approaches. We show a difference quotient corresponding to a differential quotient, which is generally called a derivative, and a partial difference quotient corresponding to a partial differential quotient, which is generally called a partial derivative. From these, the difference approach produces a discrete demand system with logarithmic mean elasticities as parameters that corresponds to a continuous demand system with point elasticities as parameters produced by the differential approach. These systems should satisfy each budget constraint: the former for finite-change variables and the latter for infinitesimal-change variables. Based on these, we consider a discrete meat demand system, apply it to monthly demand for fresh meat in Japan, and estimate it using a weighted RAS method. The estimated demand system has two desirable properties: each estimated demand (theoretical value) of the conditional demand function coincides with each observed demand, and this system satisfies the difference budget constraint.

differential values to their corresponding difference values, which creates new problems (see Section 4 and Appendix B). Thus, we must also investigate a difference approach to avoid these problems.
In this study, we derive and estimate a difference demand system whose parameters have to possess some properties, which we call a discrete demand system. When we estimate the parameters of each conditional demand function, we assume that our discrete demand system has two specific features: all independent and dependent variables (prices, total expenditures, and demands) are pre-determined (i.e., measurable, observable, or exogenously given), and all parameters (price and income elasticities, and residuals) of the system are post-determined. (As the ith demand is not a dependent variable whenever we estimate this demand system, it would be better for us to use another term instead of the function. However, we use the function for convenience). We can regard all pre-determined variables, variables calculated from these, and real numbers (1 and 0 will be used later) as real data (i.e., real-world data).
In response to this assumption, we develop a new method, which we call the weighted RAS (WRAS) method. The WRAS method can estimate the parameters using only two points data (i.e., initial and terminal data). All the parameters that are estimated using our WRAS display the most desirable properties: each theoretical demand of the conditional demand function calculated using estimated parameters and independent variables coincides with its real value (observable demand), and these parameters satisfy the budget conditions. Thus, we can say that the parameters are consistent with real data (see Sections 5 and 6).
In the below, all variables are assumed to be economic data, and thus they are positive and discrete except for events that assume differentiability for their description. (Wherever we have to consider these events, all variables are assumed to be continuous and these functions are either differentiable or totally differentiable). For simplicity, they are usually not unity when we need to take their logarithms and only the natural logarithm is used. Any economic datum x, which may be called a variable, is commonly given as x t at point t (e.g., day, month, year). Then, we have two differences such as Δx ≡ x 1 x 0 and Δlogx ≡ logx 1logx 0 = log(x 1 /x 0 ), where the subscript 0 represents an initial point and 1 represents a terminal point. The two differences are always those at these points, except in Subsection 4.3, and Sections 5 and 6. These two differences are also assumed to be non-zero to obtain interesting results, unless we set Δx = 0 to define a partial difference quotient. Naturally, we consider the limit: lim ∆ → 0.
For the differential approach, the two above-mentioned differences (i.e., the differentials) are ordinarily written as dx and dlogx. As these differentials cannot be observed or measured, they cannot be used as real data (note that a derivative or a partial derivative is theoretically given by its definition). Thus, there is no differential demand system of which parameters are consistent with real data.
Here, we show the most useful correspondences between the differential and difference approaches (Tsuchida, 2018). From the definition (1), we obtain ∆ log = ∆ ( ). ⁄ In contrast, we know the familiar relationship: From Eqs. (4) and (5), we can find the following correspondences: infinitesimal − change variables { log ↔ ∆ log ↔ ∆ ↔ ( ) } finite − change variables, infinitesimal-change variables and the latter for finite-change variables. While the former conditions are well-known, the latter conditions have never been known. A specific demand system embeds the budget constraint within itself. Hence, the differential budget constraint and difference budget constrain are embedded in the differential and difference versions of the demand system, respectively. In Section 5, we apply our discrete meat demand system to monthly demand for fresh meat in Japan, and the parameters are estimated using the WRAS method. The WRAS method can derive the discrete meat demand system whose parameters are consistent with real data. In Section 6, we present some concluding remarks, wherein we illustrate the consistency with real data using matrix algebra again.

Elasticities: Derivations and Definitions
We begin to represent the elasticities produced by the differential and difference approaches using various functions. Our difference approach can derive the difference versions corresponding to the elasticities produced by the differential approach. We use terms such as differential quotient, which is generally called a derivative, in Subsection 2.1 (see also Tsuchida, 2018) and partial differential quotient, which is generally called a partial derivative, in Subsection 2.2 (see, e.g., Takayama, 1974;Berck & Sydsӕter, 1991;Bronshtein et al., 2015). In this section and the next two sections, first we explain the differential approach under the expression [Inf-Change], and then we explain the difference approach under the expression [Fin-Change]. When two functions are written on the same line, the first is for the differential approach and the second is for the difference approach in which the subscript t represents a point.

1) = and = ( ) ( is a constant)
[Inf-Change] The differential approach leads to: = −1 . From this, the differential quotient is obtained as follows: A point elasticity e * is given by * ≡ = log log = .
[Fin-Change] The difference approach leads to: An arc elasticity is defined as wherein A(x) = (x 1 + x 0 )/2 is the arithmetic mean. Our difference approach induces a new elasticity, called a log-mean elasticity, which is defined as www.scholink.org/ojs/index.php/rem Research in Economics and Management Vol. 5, No. 3, 2020 Published by SCHOLINK INC. * ≡ If A(x) ≈ L(x) (x is X and Y) is assumed, the arc elasticity (8) approaches the log-mean elasticity (9).
Comparing Eq. (7) with Eq. (9), we can find close correspondences between the differential and difference approaches. We emphasize these as follows: The correspondence shown in (10) is easily understood from the following definition of a derivative: If ∆ → X → 0, we can employ the above correspondences. Similar correspondences will be found below.
It must be noted that the point and log-mean elasticities usually depend on a point and two points (i.e., an initial point and a terminal point), respectively. Thus, the point elasticity is point-dependent and the log-mean elasticity is two-points-dependent (hereafter twop-dependent). See Subsection 2.2.
2) = and = ( is a positive constant) The differential quotient and point elasticity are = and * = = = 1. Note that A(Y) = A(cX) = cA(X) for a constant c.

Functions of Two Variables: A Partial Differential Quotient and a Partial Difference Quotient
1) = and = ( ) ( ) ( a double logarithmic fuction; and and are constants) which is the function obtained by differentiating Y with respect to X, treating Z as a constant. The total From (13), we obtain the following: which is a differential quotient dY/dX under the condition: dZ = 0 (Note 1) and essentially equivalent to Eq. (12). We designate the middle term in Eq. (14) the partial differential quotient of Y with respect to X. The partial differential quotient of Y with respect to Z is defined in a similar manner.
Thus, the point elasticity of Y with respect to X, e X , is The point elasticity with respect to Z is similarly defined. These elasticities for the double-log (double logarithmic) function become independent of the point.
[Fin-Change] The difference approach leads to Our partial difference quotient of Y with respect X is defined as which is the difference quotient ΔY/ΔX under the condition: ΔZ = 0 (Note 2) and corresponds to Eq.
(14). The partial difference quotient of Y with respect to Z is defined in a similar manner.
Our log-mean elasticity of Y with respect to X, , is While the log-mean elasticity is usually dependent on two points, this elasticity is not.
The partial difference quotient and log-mean elasticity are very effective concepts and play key roles in our discrete demand system. This elasticity also corresponds to Eq. (15). These correspondences are shown as follows, and will be utilized latter if ∆ → X → 0 and ∆Z = Z = 0: The two (differential and difference) approaches to the double-log function produce the same elasticity, that is, the point elasticities at all points between the initial and terminal points equal the log-mean elasticity. (As explained above, the point and log-mean elasticities for other functions are usually point-dependent and twop-dependent, respectively). The log-mean elasticity of Y with respect to Z is analogously defined.
2) = 1 + 2 X and = 1 ( ) ( ) + 2 ( 1 and 2 are positive constants; and and are constants) Thus, the point elasticity of Y with respect to X, e X , is This point elasticity is dependent on a specific point because the variables X, Z, and Y are those values at that point. The point elasticity of Y with respect to Z is analogously defined.
The log-mean elasticity of Y with respect to Z is similarly defined.

Continuous and Discrete Log-Change Demand Functions and Their Elasticities
In this section, we discuss two specific demand functions and their elasticities. We also use two approaches: differential and difference approaches. To simplify our equations, there are two commodities, i and j. Our fundamental ith demand functions are = ( , , )and = ( , , ), wherein p, q, and y represent price, quantity, and income (i.e., total expenditure), respectively, and the subscripts i and j represent the commodities. In this section, the budget constraints are discarded. in which e ij is the point price elasticity of q i with respect to price p j and h i is the point income elasticity of q i with respect to income y. For simplicity, we call these the point elasticities. Thus, the log-change demand function can be rewritten as We call this the continuous log-change demand function. We use continuous and discrete to stress the demand function or system applied to continuous and discrete data, respectively.
[Fin-Change] The difference approach leads to the following log-change demand function and elasticities: in which ε ij is the log-mean price elasticity of q i with respect to p j and η i is the log-mean income elasticity of q i with respect to y. We also call these the log-mean elasticities. The demand function, which is called the discrete log-change demand function, can be rewritten as 2) The ith demand functions: = c + + + y and = + + + ( , , , and are constants) [Inf-Change] The differential approach leads to: = + + . Thus, we have: The point elasticities are = ( = and )and = .
Therefore, the continuous log-change demand function for this case can also be given by (22).

Continuous Demand System and Discrete Demand System
From the explanations in Section 3, it is inferred that we can derive a general differential demand system from Eq. (22) and a difference demand system from Eq. (23). In this section, we provide more details about these systems.
[Inf-Change] The pedantic derivation of the general differential case is easy. Our fundamental demand system is: Where p = {p 1 , p 2 , …, p n }, = Σ , and n are the price vector, income (total expenditure), and the number of all commodities in this system, respectively. The summation ∑ or ∑ is always made over all values of i. Here q i is implicitly derived using constrained utility maximization. The total differentiation of this function yields: Thus, we have the general differential demand system as follows: Recall that e ij denotes the point price elasticity and h i denotes the point income elasticity. These elasticities possess various properties produced using a utility function and should satisfy the two conditions derived by the differential budget constraint (30) below (see also Subsection 4.3 and Appendix A). If we substitute the Slutsky equation and another budget constraint (see Appendix B) into (24), we obtain a model similar to the Rotterdam model or its variant (see, e.g., Barten, 1993;Matsuda, 2005;Clements & Gao, 2014). We do not employ the Slutsky equation, so our price elasticities are always Marshallian.
We define a continuous demand system as shown in Eq. (25), based on Eq. (24): where the final term u i is the residual and exhibits all of the effects induced by the other factors that are not employed as explanatory variables (e.g., weather and consumer sentiment). The elasticities and residuals should satisfy the above conditions and the residual condition that is explained later, respectively.
[Fin-Change] The pedantic derivation of a general difference demand system is impracticable, because the difference approach is only applied to a specific demand system. Considering the derivation process of Eq. (23) and the correspondences between the differential and difference approaches in (6), (10), (11), (18), and (19), we define a difference demand system: wherein and are the log-mean price and income elasticities that should satisfy the two conditions derived using the difference budget constraint discussed below. If we consider numerous demand systems based on microeconomic theory, we can derive this system from most of them. For example, we can derive such a system from the linear expenditure system (LES) as shown in Appendix A (Note 3). Recall that the LES is far removed from Eq. (26). See also Appendix C. If we assume a certain demand system derived using constrained utility maximization, the characteristics of the utility function regulate these elasticities. If ∆ → → 0 and ∆ → → 0 (for all ); and ∆ → → 0, the difference demand system (26) approaches the general differential demand system (24).
We use this to define a discrete demand system: where the final term is the residual corresponding to u i above. The conditions that their elasticities should satisfy are the same as those in Eq. (26) and the condition regarding the residual is discussed below.
We may derive the following approximation formed from Eqs. (24) and (26): This log-change demand system is similar to the Rotterdam model or its variants (e.g., Barten, 1964;Theil, 1965Theil, , 1975Clements & Gao, 2014). If we use log-change values Δ log , Δ log , and ∆ log as in (28), we have to employ the budget constraint for finite-change variables shown below.
Therefore, the approximation in (28) may not satisfy this difference budget constraint because this constraint only applies to the log-mean elasticities. See Appendix B for further details.

Conditions for Elasticities and Residuals
Frisch (1959) showed the conditions that the point elasticities of a demand system have to satisfy (see also Deaton & Muellbauer, 1980b;Barten, 1993). We follow some of these conditions and derive some new conditions for log-mean elasticities.

Homogeneity Conditions Derived from the Homogeneous Function
The homogeneity condition may be meaningful for infinitesimal-change variables, but not necessarily meaningful for finite-change variables.
[Inf-Change] The demand function should be homogeneous of degree zero in prices and income. Thus, we have = ( , ) = ( , ), ( = 1, 2, … , ), wherein γ is a positive constant. This is known as the homogeneity restriction. Since we have the following from Euler's theorem: thus, the homogeneity condition for the point elasticities is given by [Fin-Change] If the demand function is homogeneous of degree zero, we have the following formal equation: wherein p 1 = γp 0, p 0 = {p 0i } (price vector), y 1 = γy 0 , 0 = Σ 0 0 , and γ is a positive constant. We also call this the homogeneity restriction. As the restriction requires the differences of the demands to become null (i. e., ∆ = 0) for all commodities, our difference quotient and partial difference quotient are inactive. Therefore, we cannot define the homogeneity condition for the log-mean elasticities.

Conditions Derived from the Budget Constraints
The budget constraints are as follows: For finite-change variables, our difference approach must utilize the transformation m ti or w ti as defined above (see also Tsuchida, 2018). We call the former Transformation-M and the latter Transformation-W. [ Equation (30) is the differential budget constraint, from which we produce two conditions: the Engel condition on the income elasticities and Cournot conditions on the price elasticities (Note 4): ∑ = 1( = 1, 2, … , ), (Engel condition); ∑ = − ( and = 1, 2, … , ), (Cournot conditions).
These conditions hold only for the point elasticities. The general differential demand system (24) should satisfy these conditions. The important point here is that the differential demand system based on a specific demand system automatically satisfies these conditions because the budget constraint (30) is embedded within itself (see Subsection 4.3 and Appendix A). The same is true for the difference case presented below. As a continuous demand system needs a condition regarding the residuals, we derive this condition.
Multiplying both sides of Eq. (25) by w i and summing for i, we get: Using the two conditions and the budget constraint (30), we have the following Residual condition: Theil's approximation (Theil, 1975/76, Eq. (2.4) in Chap. 2) to the budget constraint (30) is wherein A(w i ) = (w 1i + w 0i )/2 is the arithmetic mean of the two budget shares. This approximation produces different conditions to those outlined above. For a more detailed discussion, see Appendix B.
[Fin-Change] Using Transformation-M, we derive two types of conditions: the Engel condition on the income elasticities and Cournot conditions on the price elasticities (Similar conditions derived using Transformation-W are shown in Appendix A). To obtain these conditions, we utilize the following difference budget constraint: We call these conditions the M-Engel and M-Cournot conditions. If the difference demand system (26) is not based on a specific demand system, these conditions must be satisfied.
Furthermore, the residuals of the discrete demand system need a new condition. Multiplying both sides of (27) by ( ) ( ) ⁄ and summing for i, we have wherein we use Eqs. (32) and (33). Thus, we produce the M-Residual condition as follows: Here, we explain the advantage of using the above three conditions as constraints over the parameters (elasticities and residuals). Since our discrete demand system is not derived from a specific demand www.scholink.org/ojs/index.php/rem Research in Economics and Management Vol. 5, No. 3, 2020 system, this should satisfy these conditions. Thus, we can exploit these conditions. This demand system is composed of n demand functions, and these parameters are exhibited as an × ( + 2) matrix, wherein the columns relate to the three conditions and each row relates to each demand function in (27).
We employ these features to estimate the parameters. A more detailed explanation is provided in Subsection 5.2 and Section 6.
It is also worth noting that the weights of the aggregation, ( ) ( ), ⁄ are those of an ideal log-change index. We have known two ideal log-change indices: the Montgomery index and the Vartia-Sato index (Vartia, 1976;Sato, 1976;Balk, 2008;Tsuchida, 2014). From our budget constraint (31), we can identify the Montgomery index. (For the Vartia-Sato index, see Appendix A). The first and second terms on the right-hand side of Eq. (31) are the ideal log-change price and quantity indices, respectively. We call these weights the Montgomery weights, which leads to the well-known inequality: The ingenious proof of the inequality (35) was discussed by Balk (2008, p. 87), who used Jensen's inequality for a convex (or concave) function. As our data are assumed to be discrete, we should seek further proof, which is presented in Appendix D.

Elasticities That Satisfy the Budget Conditions
It should be noticed that the point elasticities and log-mean elasticities are usually point-dependent and twop-dependent, respectively. To clearly illustrate this, first we explain the difference case for Transformation-W in detail and then shortly explain the differential case.
To estimate the parameters of each difference demand system (26) using yearly data from 2000 to 2015, we use the difference demand system (36), its budget constraint (37), and its budget conditions (Engel (38) and Cournot (39) conditions) in years t and s = t -1 as follows: The superscripts and subscripts t and s represent the year; and subscripts i, j, and k represent commodities. Each difference is expressed as follows: ∆ log = log − log , etc. The superscript and subscript t represents the terminal year, whereas s represents the initial year. We find the Vartia-Sato index from the budget constraint (37), and the weights therein are Vartia-Sato's, which sum to 1. This budget constraint produces the two conditions (38) and (39)  If all elasticities are independent of all points, we have the following equation (40) instead of (36): To satisfy the budget constraint (37) for every pair of years, we have the following for all t: = 2001, 2002, … , and all and ).
In (41), we rewrite the Vartia-Sato weight as = ( , ) ∑ ( , ) ⁄ . From these equations, we can obtain the elasticities to satisfy Eq. (37). Without loss of generality, we assume three commodities (i, j, and k= 1, 2, 3) and any three years (t = r, u, v: and r ≠ u, u ≠ v, etc.) We can use Eq. (42)  Next, we solve the Engel condition. This is given by From (43), we obtain = 1 for all i.
Multiplying both sides of the above equation by L(w ti ,w si ) and summing for i, we obtain the budget constraint (37). Hence, this system satisfies this constraint.
The general differential demand system (24) should satisfy the differential budget constraint (30) and its budget conditions (the Engel and Cournot conditions) in every year; that is, all elasticities must be point-dependent (see the differential version of the LES shown in Appendix A). If all elasticities are independent of all points, we can obtain these elasticities using similar procedures to the above-mentioned difference case. Therefore, we have the following primitive differential demand system: log = − log + log , for all (2000, … , 2015) and all , Where d log q ti , d log p ti , and d log y t are the differentials in year t. This system also satisfies the differential budget constraint (30). www.scholink.org/ojs/index.php/rem Research in Economics and Management Vol. 5, No. 3, 2020 An example of yearly demand system for the above primitive difference and differential demand systems is log = − log + log + log(1⁄ ) , ( = 1, 2, … , ).
Thus, all budget shares take the same value, that is, = ( )⁄ = 1⁄ for all t and i. This demand system is derived from the utility function: = ∑ log (see Phlips, 1974, pp. 65-66). (By maximizing the Lagrangean with respect to q ti and minimizing it with respect to a Lagrangean multiplier λ, we have 1⁄ = λ and = ∑ . From the two equations, we obtain λ = ⁄ ).
Generally, we have 16 sets of point elasticities for the differential demand system and 15 sets of log-mean elasticities for the difference demand system. This means that most of the point elasticities and log-mean elasticities are point-dependent and twop-dependent, respectively (Note 6).

Estimating Method
We showed in Section 4 that the discrete demand system is given by (27) and their parameters (elasticities and residuals) should satisfy the M-Engel, M-Cournot, and M-Residual conditions. In this section, we estimate these elasticities and residuals using monthly expenditure data for fresh (or raw) meat purchases over the previous year in Japan. An explanation of the data used is provided in Appendix E.
We do not suppose a specific demand system. Fresh meat is composed of four commodities: beef (i or j = 1), pork (i or j = 2), chicken (i or j = 3), and others (i.e., other meats, i or j = 4). The variables q i , p i , and are, respectively, the ith demand, price, and residual, whereas y is the total expenditure on these types of meat. The residual includes the contributions of other factors excluding p i (I = 1, 2, 3, 4) and y.
Most of those contributions may be induced by substitutes and complements for fresh meat (e.g., ham, sausage, and cooked foods such as croquette and Hamburg steak). Implicitly, assuming Transformation-M, all elasticities in (44) are M-elasticities and should satisfy the three above-mentioned conditions. We call Eq. (44) the M-Demand equation conditions in the below. (When we employ a continuous meat demand system such as that in (25), it worth mentioning the homogeneity condition (29). Each conditional demand function may not satisfy this condition since the prices of the substitutes and complements for fresh meat are not contained in the function.) We employ a two-stage method to estimate the parameters in (44 using the WRAS method. Our main aim in this section is not to evaluate these final results, but rather to explain the new method.

[First Stage]
First, we calculated the shares of average monthly expenditure on the various types of fresh meat from 2014 to 2016 as shown in Table 1. It can be seen that in Japan, pork has the largest share, followed by beef, except for December. For December, beef has the largest share, which may stem from a seasonal effect (in particular, the preparation of sukiyaki for dinner).  Because the consumption tax rate was increased in April 2014, the data that we used to estimate the parameters were from April 2015/2014 to December 2016/2015. Thus, each data type has 21 samples.
The parameters estimated using OLS are shown in Table 2. All own price elasticities and income (total expenditure) elasticities have proper signs and high t-values.

[Second Stage]
In this stage, we estimate parameters in a specific month. We selected the months of October and November after considering the change in the consumption tax rate and seasonal variations (e.g., ceremonial usage and the rainy season, see also Table 1). Below, we explain the method that we use to estimate the parameters for October. The same method is also applied for November. Data used were the monthly changes from 2014 to 2015 and from 2015 to 2016, which are shown in Tables 3, 4, and 5.
We assume that our parameters are the same in both periods to get steady results. Based on this assumption, the following Montgomery weights needed to be used. Given that the subscripts 4, 5, and 6

+.
Our control variables are the three averages of the log-change values and the above-mentioned Montgomery weights, which are simply redefined as in which the subscripts t and s may be deleted.  First, we settled the rough elements of matrix A, as shown in Table 6. Each = −1.3, = 0.1 or − 0.1 ( ≠ ), = 1, and = 0 ( and = 1, 2, 3, 4). The signs of the price and income elasticities are the same as those in Table 2. All the initial values employed by our WRAS method must be positive. So, we modified the values presented in Table 6  We use our WRAS method to constrain the initial values sufficiently to satisfy the following four To calculate the row and column totals for the matrices, we employ the following weights: Row weights: ∆ log 1 , ∆ log 2 , ∆ log 3 , ∆ log 4 , ∆ log , 1; Column weights: 1 , 2 , 3 , 4 .
Our weighted column totals were also changed, and the left-hand sides of the Thus, the right-hand side of this condition, which is the sixth weighted column total, is 6 # = 1.1 ∑ 4 =1 .
Using the above-mentioned weights and the control totals, we apply the WRAS method to the initial values of matrix A0 and obtain the elements of matrix A1 and matrix A2 below. The above-mentioned values such as ∆ log # and # are always used as the numerators to obtain the coefficients of the row and column constraints below.

1.2) Column constraints
Using the M-Cournot conditions and the elements of matrix A1, we obtained the coefficient of the jth column constraint s j as follows: Using the M-Engel condition and the elements of matrix A1, we obtained the coefficient of the fifth column constraint s 5 as follows: Similarly, we obtained the coefficient of the sixth column constraint s 6 from the M-Residual condition and the elements of matrix A1 as follows: If this constraint is redundant, we always set 6 = 1. For more details, see the final results presented below.
All coefficients are positive because all a1 ij > 0. Using these coefficients and the elements of matrix A1, we determined the new matrix A2 as follows: ,.

2) Second round and matrix AX
Using the elements of matrix A2 and the coefficients of the row and column constraints calculated using the procedures outlined above, we determined matrices A3 and A4. Repeating these computations, we obtained matrices A5, A6,..., AX, wherein X = 2x (x = 4, 5, …).We repeated these computations until all the elements of matrix AX were nearly equal to those of matrices AX -2 and AX -1 (X -2 = X -2, X -1 = X -1). This scenario occurred at about X = 100. To get stable results, we employed the matrix with X = 800 (i.e., A800). Below, we call these values the convergent elements.  We adopted the convergent elements without the sixth column constraint as the final results.
Subtracting each additive value from the corresponding element in Table 8, we got the elasticities and the residuals, which are shown in Table 9.  Tables 2 and 6.
Hence, we can say that our estimated parameters are consistent with real data, which has the following two implications (also see the next section). Substituting these parameters and real data such as p i and y on the right-hand side of Eq. (44), we obtain the theoretical demand for each commodity that equals the real (or actual) value. Substituting these parameters and the Montgomery weights on the left-hand sides of Eqs. (32), (33), and (34), we get the corresponding values that equal the real values on the right-hand sides, respectively (Besides, we get the budget constraint (31) from Eqs. (44), (32), (33), and (34); that is, these parameters satisfy the budget constraint).
Similarly, we adopted the convergent elements without the sixth column constraint as the final results for November and got their elasticities and residuals. The results are shown in Table 10, and all values also satisfy the four above-mentioned conditions. Thus, the estimated parameters for November are also consistent with real data. All the estimated elasticities have the same signs as those shown in Table   9. www.scholink.org/ojs/index.php/rem Research in Economics and Management Vol. 5, No. 3, 2020 Published by SCHOLINK INC.
Additionally, each estimated elasticity in October is similar to that in November, but the residuals differ (e.g., _11 = −0.8532 in Oct. vs. _11 = −0.8409 in" . ", _1 = −0.0459 in" . . " _1 = −0.0181 " . ). " To gain more accurate results, the procedures that we use to obtain the initial values in matrix A are crucial. (Strictly speaking, we cannot assess whether these estimated elasticities are reasonable and have the proper signs because we have never tried to estimate these parameters). It may be helpful to compare our results with those obtained using the differential approach (e.g., Fousekis & Revell, 2000; Okrent & Alston, 2012 (Appendix -Table A.4)) and other approaches (e.g., Hayes, Wahl, & Williams, 1990).

Concluding Remarks
Some differential demand systems or their variants, such as the Rotterdam model, are well-known.
However, we cannot derive a differential demand system that is consistent with real data, as the differentials such as dx and dlogx (x is any economic datum) are unable to be observed or measured.
Thus, we must develop an alternative approach, that is, the difference demand system.
Various correspondences between the differential and difference approaches (or calculi) are presented by Tsuchida (2018), and this study extends and evolves these correspondences. Specifically, we have shown a difference quotient corresponding to a differential quotient, which is generally called a derivative, and a partial difference quotient corresponding to a partial differential quotient, which is generally called a partial derivative. From these, we have derived a continuous (i.e., differential) log-change demand function with point elasticities as parameters and a discrete (i.e., difference) log-change demand function with logarithmic mean elasticities as parameters. Based on these results, we have defined continuous and discrete demand systems that should satisfy each budget constraint.
We can also apply these demand systems to any group of commodities (e.g., a meat demand system).
Our discrete meat demand system was applied to monthly demand for fresh meat (beef, pork, chicken, and other meats) in Japan, and its parameters (elasticities and residuals) were estimated using the weighted RAS (WRAS) method, which is handy and practical. Whereas 24 parameters must be estimated in a given month, 13 control variables are used in our method. Nevertheless, our WRAS method can derive a discrete meat demand system in which the estimated parameters are consistent with real data. This implies that each theoretical value of the conditional demand functions calculated using estimated parameters and independent variables coincides with its real value (observed demand), and each set of these parameters satisfies the Engel, Cournot, and Residual conditions.
As the difference approach to the demand system and its estimating method have scarcely been studied, we offer several remarks (see also the difference version of the AIDS in Appendix A). 1) We begin by reconsidering the level of consistency with real data, on which we place particular emphasis. For an alternative explanation, we use matrix algebra. Our estimated parameters in October shown in Table 9 are given as a 4×6 matrix B = {b ij }. For example, 12 = 12 and 45 = 4 . Our real data are given by vectors. For example, any vector x and its transpose x t are given by www.scholink.org/ojs/index.php/rem Research in Economics and Management Vol. 5, No. 3, 2020 90 Published by SCHOLINK INC.
2) While the normal and continuous demand systems are based on economic theory (see, e.g., Barten, 1977;Piggott & Marsh, 2011), the discrete demand system is not. Our discrete demand system can only be derived from a specific demand system based on economic theory. Thus, we have to investigate how to combine economic theory with the discrete demand system. First, we define a utility function and derive a demand system based on the relevant theory. Next, we derive the discrete demand system from this system and estimate its parameters using the WRAS method.
3) Additionally, we have to investigate how to identify a utility function whose parameters have coherent properties to those produced using our These elasticities satisfy the Homogeneity, Engel, and Cournot conditions explained in Subsection 4.2.
These proofs are easy, and thus are omitted. From the initial assumptions, -1 < e ii < 0 and e ij < 0 (j ≠ i).
These elasticities are point-dependent. Assuming that (A1) was estimated using yearly data from 1996 to 2015, we have 20 sets of elasticities from the above relationships. Note that p i , q i , and y are yearly data; and a i and b i are constants.
We have two ideal log-change indices that are derived using Transformation-M ( = ) and Transformation-W( = / ). We can apply these transformations to the LES, and thus two difference versions are obtained. First, we utilize Transformation-M.
Substituting m ti and k ti = p ti b i into the above LES, we have: If we assume that the discrete meat demand system (44) is based on the LES, we have the following discrete meat demand system: ∆ log = ∑ ∆ log + ∆ log + .
The price and income elasticities are the same as those outlined above and the residuals satisfy the M-Residual condition (34). These 19 sets of elasticities and residuals can also be estimated using the WRAS method, and should coincide with the above M-elasticities calculated using a i , b i , and various log-means. Hence, we can determine whether our assumption is plausible. Applying Transformation-W in the below, we can similarly do.
From this, we obtain This is another difference version of the LES, whose own and cross price elasticities and income elasticities, which are called W-elasticities, are These W-elasticities differ from the M-elasticities.
Based on the other budget constraint using Transformation-W (1 = ∑ ), we obtain 0 = ∑ ∆ = ∑ ( )∆ log = ∑ ( ) (∆ log + ∆ log − log ), From Eq. (A3), we find the Vartia-Sato index. The first and second terms of the right-hand side are, respectively, the ideal log-change price and quantity indices (Note 10). This budget constraint (A3) produces the two conditions related to W-elasticities as follows:  The W-elasticities also satisfy Eqs. (A4) and (A5). These proofs are also easy. Note that these aggregation weights add up to 1and differ from those of the M-elasticities.
If the ratio of two positive variables, x 1 and x 0 , is close to unity (i.e., x 1 /x 0 ≈ 1), we have a good relationship (Tsuchida, 2014(Tsuchida, , 2018: wherein G(x) = (x 0 x 1 ) 0.5 is the geometric mean and A(x) is, as mentioned above, the arithmetic mean. If all variables below satisfy these assumptions, we obtain the following approximations: Thus, the weights derived using Transformation-M approach those derived using Transformation-W.
Note that the right-hand side of (A10) is the price term of the differential budget constraint (30).
The above (A12) corresponds to (A9) and (A13) corresponds to (A10), though the right-hand side of (A13) is the price term of the difference budget constraint (A3). This system is produced using Transformation-W. (If we use Transformation-M, we have to multiply both sides of (A6) by y, whereby we have numerous non-linear terms.) Deaton and Muellbauer (1980a) use the following approximation (A14) instead of (A13): This may be regarded as an approximation of (A10), but is neither the price term of the difference budget constraint (A3) nor that of Theil's approximation (B2) in Appendix B. They also show another difference version (Deaton & Muellbauer, 1980a, Eq. (21)).
Given that Eq. (A6) leads to our meat demand system, its discrete meat demand system is given by ∆ log = ∑ ∆log + Δ log + , ( and = 1, 2, 3, 4), wherein the parameters are Here, is the Kronecker delta, which takes 1 if i = j, and 0 otherwise; and is the residual and includes an error that is accompanied by the approximation (A13). From the budget constraint (A3), we have the Engel condition (A4) and Cournot conditions (A5); and the Residual condition that is given by Hence, we can estimate these parameters in Eq. (A15) using our WRAS method. Based on those results and some log-means, we get the parameters and in (A6). Thus, we can check whether some estimated parameters have the properties coherent to those of its utility function.
wherein the two elasticities are M-elasticities (they may also be W-elasticities). Note that A(w i ) and the elasticities are dependent on two points. The two elasticities in (B1) will approximately satisfy the two conditions (B3) and (B4), if we assume that the elasticities e ij and h i in (B1) are close to the log-mean elasticities and in (B4) and (B3). Given that a double-log function produces this continuous demand system, the assumptions are fulfilled. Recall that all double-log functions produce point elasticities that are always equal to the log-mean elasticities discussed in Subsection 2.2.

Appendix C: Differential and Difference Approaches for a Composite Function
There are some functions for which we cannot always derive the correspondence between the differential and difference approaches (see Tsuchida, 2018). An example is presented below.
The differential approach quickly derives: = log + log = log log + log = log + ( log ⁄ ) log . (C1) Note that all variables are positive and not 1 when we take their logarithms.
The correspondences between (C1) and (C2) can be found only when Z > 1 and Z t > 1. Hence, we cannot always derive the difference version for a demand function that has a composite explanatory variable such as that shown above. www.scholink.org/ojs/index.php/rem Research in Economics and Management Vol. 5, No. 3, 2020 98 Published by SCHOLINK INC.
If 1 = 0 for all i, the inequality turns out to be the identity. The log-mean has the following property: ( 1 , 0 ) = wherein z = 2 ( = 3, 4, 5, … ).The last equation is obtained by expanding the first equality of (D2) repeatedly and the same as the Corollary 2 in Carlson (1972). Moreover, we have two relationships: ( Hence, we can apply Hölder's inequality (Berck & Sydsӕter, 1991) given by wherein a i and b i are positive; and p > 1, q > 1, and 1/p + 1/q = 1. Using the inequality, we produce the following inequalities: Using these inequalities and comparing each term in the numerator of the above equations with the corresponding term in the denominator, we find the inequality (D1). Whereas we only use the values composed of the first and second terms of the last expansion of Eq. (D2), we can apply Hölder's inequality to those terms including the third and higher terms, for which comparisons we use the bellow.