Lu ZHANG
Beijing Institute of Technology Zhuhai
lu.zhang@cgt.bitzh.edu.cn
Abstract
Given a relative lack of knowledge about Japanese consumer preferences for fish, Japanese fish inverse demand is modeled by the inverse Lewbel demand system and the inverse almost ideal demand system. Results indicate that both inverse demand systems fit analyzing the Japanese fish consumption.
Key words: inverse Lewbel demand system, inverse almost ideal demand system.
1. Introduction
A basic question in applied economic analysis is to find the appropriate method to specify a demand system. That is, should prices be taken as given with quantities endogenous, or should quantities be assumed to be fixed and price depends on the existing supplies. Fresh fish, a typical quickly perishable good which has been established as a respectable, challenging subject in demand analysis, its quantity can’t adjust in the short run. Thus, the producers are virtually price takers. Obviously, inverse demand system should be applied in the fishes’ demand system analysis. This result is proved by Eales, Durham and Wessells’s study in 1997[ James Eales, Catherine Durham, and Cathy R. Wessells. “Generalized Models of Japanese Demand for Fish” American Agricultural Economics Association.79(Nov 1997):1153-1163.].
An inverse demand system, which expresses the relative or normalized prices paid as a function of total real expenditure and the quantities available of goods, has been accepted by more and more economists till now. Based on the direct translog demand system which was developed by Christen, Jorgenson and Lau(1975)[ Christensen, L.R., D.W. Jorgenson and L.J Lau, “ Transcendental logarithmic utility functions” American Economic Review(1975) 65:367-383.], many inverse demand system developed. In 1994, just as the coincide of the Almost Ideal Demand System(AIDS) which was developed by Deaton and Muellbauer (1980)[ Deaton, A and J. Muellbauer, “An Almost Ideal Demand System” American Economic review (1980b) 70:312-326.], Eales and Unnevehr (1994)[ James S.Eales and Laurian J. Unnevehr, “The Inverse Almost Ideal Demand System” European Economic Review 38(1994) :101-115. ]brought forward the inverse demand system for AIDS which was called IAIDS and used it to analyze the U.S. meat demand. It soon becamed very popular due to its convenience in empirical analysis. The PIGLOG preference is convenient for welfare analysis, at the same time, homogeneity and symmetry restriction depend only on estimated parameters and so are easily imposed and test. In the same year, Eales(1994)[ James S. Eales “ The Inverse Lewbel Demand System” Joural of Agricultural and Resource Economics(1994),19(1):173-182.] suggested a new inverse demand system called Inverse Lewbel Demand System(ILDS) that maintained flexible function forms. It is aimed to solve the hypothesis problem which always occurs in parametric analysis. That is, if the functional form employed is inappropriate, the measurement of elasticity may be bias and consumer preference may be unsuitable. Thus, the more flexible the function form of the system is the better performance of the demand system. The ILDS nests the Direct Translog Demand System (DTDS) and the Inverse Almost Ideal Demand System (IAIDS). It allows demand analysts to test whether consumer preference is modeled with appropriate function forms. We will choose ILDS and IAIDS in our estimation of Japanese fish consumption.
According to Eales, Durham and Wessells’s(1997) [ James Eales, Catherine Durham, and Cathy R. Wessells. “Generalized Models of Japanese Demand for Fish” American Agricultural Economics Association.79(Nov 1997):1153-1163.]paper, Japanese spent 13.3% of their average monthly food budget on fish and seafood products in 1994, compared to the 9% on meat. Given the importance of fisheries products in Japanese ordinary lives, it is not surprising that we are not the first to model Japanese consumers’ demand for fish. However, it still very interesting to study the separable market in Japanese fish consumption. Thus, we will choose tuna, scad, sardine and bonito as four kind of representative fishing product to construct a separable market of Japanese fish consumption.
This paper is organized as follows: Section 2 describes the Inverse Lewbel Demand System (ILDS) and Inverse Almost Ideal Demand System (IAIDS). Section 3 conducts the quantitative analysis. Section 4 makes a short conclusion.
2.The Inverse Lewbel Demand System and Inverse Almost Ideal Demand System
The Inverse Lewbel Demand System can be derived from the following direct and logarithmic utility function:
(1)
Where:
(2)
Here denotes the quantity of commodity, andmeans the total utility.
By using the Hotelling-Wold Identity, it is easy to derive the budget share equation for commodity , which is denoted as .
(3)
By employing the distance functioncorresponding to the utility function given in equation (1), we can attain
(4) Exponentiating and collecting terms gives:
(5)
Or
(6)
Then, we can add up restriction on these equations, or more precisely, add up restriction on these equations’ parameters. Those restrictions can be listed as follows,
(7)
Imposing these restrictions on equation (6), then we can get this log distance function:
(8)
So, the compensated inverse demands will be:
=
.
(9)
However, at the optimum,, then , so the last term vanishes. We substituting the unobservable utility from equation (1) in the first term will lead the equation (9) becomes:
(10)
Since IAIDS model is nested in the ILDS model, if we add one more restriction , then ILDS will reduce to IAIDS:
(11)
3. Quantities Analysis
3.1 Data Description
Data were gathered from Japan Statistical Bureau. The data set consist of monthly data ranks from January, 2000 to February, 2013. The categories contain one high-value fresh fish tuna and three medium value fresh fish as scad, sardine and bonito. Although these are not the only fishes sold commercially in Japan, they are representative ones. The quantity and nominal per month expenditure for these four fishes are applied directly to this analysis. The reason can be listed as follows: According to Eales, Durham and Wessells(1997)[ James Eales, Catherine Durham, and Cathy R. Wessells. “Generalized Models of Japanese Demand for Fish” American Agricultural Economics Association.79(Nov 1997):1153-1163.], Seasonality is very impressive in Japanese fish consumption, driven by two demand factors. First, the Japanese receive large bonuses in December, which averaged 4% per capita over the period covered by the data. Second, there are two gift-giving seasons in Japan, one in July and a very important one in December. This leads to December peaks in both prices and quantities for some highly value fish products. However, they use twelfth differences (instead of first) with a correction for first order autocorrelation (AR(1)) in the errors and no seasonal dummy variables , allowing for monthly shifts in the differential demands, and substituting quarterly for monthly dummy variables. None of these had any substantial effect on the results. As a matter of course, we will not take data-processing and use the nominal price index expenditure. The following table shows the statistical descriptions of the data set.
Table 1: Descriptive Statistics 2001:1-2013:2
Stat.Des Mean Median Maximum Minimum Std.Dev
W1 0.613445 0.611225 0.819672 0.476692 0.080275
W2 0.151707 0.152847 0.198354 0.082341 0.022350
W3 0.067793 0.066788 0.142324 0.029334 0.017583
W4 0.167055 0.168802 0.276278 0.058328 0.060486
X1 238.8219 234.5000 367.0000 160.0000 46.29809
X2 139.4795 132.0000 243.0000 59.00000 39.98935
X3 74.51370 72.50000 175.0000 25.00000 23.53581
X4 101.1849 91.50000 229.0000 30.00000 51.61996
Y 90495.08 89105.94 133150.0 50897.42 19812.31
Notes: category1=tuna, category2=scad, category3=sardine, category4=bonito.
From the Table 1, we see that tuna is associated with the highest average share in total expenditure, at 61.34%. Bonito and scad rank second and third, respectively, at 16.71% and 15.17%. Sardine is the lowest at 6.78%. But as further revealed in Table 1 there is substantial variation in the share values in the data set. Average total selling by species, in 100 grams, are also reported in table 1. Again, tuna is associated with the highest selling over the sample period, bonito and scad occupy the second and third rank. There is apparently considerable variation in the data. In all, there appears to be sufficient variation in the data so that it would be possible to determine what systematic relationships exist among the demands for these various fishes.
3.2 Empirical results and analysis
3.2.1 The basic models without any control of serial correlation
Our ultimate goal is the direct comparison of the two generalized demand systems, which motivated our reparameterization of these models. Before these, we estimate the ILDS and IAIDS without any control of serial correlation; Table 2 shows the result of the following systems:
Table 2: Empirical results for ILDS and IAIDS
ILDS IAIDS
0.310766*** 0.583655***
0.157538*** 0.253501***
0.090102*** 0.071352***
0441593*** 0.091492***
0.241436*** 0.232429***
-0.076922*** -0.080621***
-0.039836*** -0.47205***
-0.090056*** -0.104603***
-0.076922*** -0.080621***
0.084326*** 0.078341***
-.718349E-03 0.415188E-03
0.590660E-02 0.186563E-02
-0.039836*** -0.047205***
-0.718349E-03 0.415188E-03
0.058870*** 0.059118***
-0.012220*** -0.012328***
-0.124678*** -0.104603***
-0.668507E-02 0.186563E-02
-0.018316*** -0.012328***
0.096369*** 0.115066***
-0.414284E-03*** 0.031951***
-0.142029E-03 0.010184***
0.152689E-04 -0.010125**
0.541044E-03*** -0.032009***
ILDS IAIDS
D-W Statistics D-W Statistics
Tuna 0.952779 1.95967 0.953052 1.97938
Scad 0.852135 2.37041 0.854080 2.38443
Sardine 0.796593 1.53034 0.796321 1.54498
Bonito 0.971966 1.53266 0.969159 1.55118
Note :1)*: the value is statistically significant at 90% level.
2)**: the value is statistically significant at 95% level.
3)***: the value is statistically significant at 99% level.
4) category1=tuna, category2=scad, category3=sardine, category4=bonito.
5) The results are calculated by deleting the equation.
Table 2 shows that most of the parameters in ILDS and IAIDS are significant at 99% level. And more interesting thing is that many of the estimated parameters in ILDS and IAIDS are similar to each other. For instance, the estimated value of , , and are very close to each other. At the same time, single-equation values for each model, defined as the square of the simple correlation between actual and fitted shares, are reported in Table 2. Each model apparently fits the (share) data reasonably well. With the sardine category generally displaying the lowest explanatory power ( is near 0.8) and the bonito category the highest explanatory power ( is near 0.97). D-W Statistics of two models both behave well and have similar value. However, the D-W Statistics of sardine and bonito are all much smaller than 2, which all close to 1.5, means that serial correlation problem may exist in our estimation. In all, two systems still behave the same. The reason why they behave like this may be traced back to the special setting of the ILDS. IAIDS is nested in ILDS. There is just one difference in their parameter restriction. In some cases, the difference between their estimated results will be beneath discussion. Thus, we can conclude that both ILDS and IAIDS represent the demand system very well.
3.2.2 The test of serial correlation
From the data description before, we have already known that the data for this paper is time series that untreated. Then, we may face serial correlation in our estimation of the demand system. And some of the categories’ D-W Statistics are very different form 2. So the test of serial correlation for these models is necessary. In order to test whether the serial correlation problem exists or not, we will construct two new equation which add AR(1) into the ILDS model and IAIDS model. After adding the AR(1), the budget share equations for ILDS and IAIDS will be:
(12)
(13)
Define is the coefficient of AR(1), we set the hypothesis as follows:
The regression results from equation (12) and (13) are shown in the table 3:
Table 3: The empirical result for ILDS and IAIDS for AR(1) model
ILDS IAIDS
-0.042200** -0.022084*
Note: 1)*: the value is statistically significant at 90% level;
2)**: the value is statistically significant at 95% level;
3)***: the value is statistically significant at 99% level;
From table 3, we can easily get the conclusion that we can reject the null hypothesis at 90% significant level, which means that the ILDS and IAIDS do have the serial correlation problem.
3.2.3 The model considering the serial correlation problem
According to Yen and Chern’s (1992)[ Steven T. Y and Wen.S.Chern. “Flexible Demand Systems with Serially Correlated Errors: Fat and Oil Consumption in the United States” American Agricultural Economics Association (1992) 74 (3): 689-697.] study, we are able to estimate the model with serial correlation by adding up the error term. The error term can be represented as follow forms:
(14)
By adding equation (14) into the estimated budget share, the budget share equation will become:
(15)
After substituted the method into equation (10) and (11), the two equation will be transformed as:
(16)
(17)
Then we will estimate these two equations above based on our data. The results will be listed as follows:
Table 4: Empirical Results for ILDS and IAIDS (with serial correlation control)
ILDS IAIDS
0.286826*** 0.450392***
0.158942*** 0.211027***
0.073762*** 0.115362***
0.480471*** 0.223219***
0.245695*** 0.227261***
-0.079576*** -0.085793***
-0.038490*** -0.044944***
-0.088346*** -0.096523 ***
-0.079576*** -0.085793***
0.086796*** 0.083795***
-0.178855E-03 0.453173E-03
0.618646E-02 0.154470E-02
-0.038490*** -0.044944***
-0.178855E-03 0.453173E-03
0.060070*** 0.058720***
-0.010552*** -0.014229***
-0.127630*** -0.096523***
-0.704202E-02 0.154470E-02
-0.021402*** -0.014229***
0.092711*** 0.109207***
-0.414922E-03*** 0.182952E-03***
-0.159983E-03*** 0.569122E-04 ***
-0.243498E-04 -0.491707E-04**
0.599255E-03*** -0.190693E-03***
0.153778*** 0.043128**
ILDS IAIDS
D-W Statistics D-W Statistics
Tuna 0.952405 2.30184 0.952722 2.04771
Scad 0.839221 2.62373 0.848933 2.45046
Sardine 0.804030 1.72929 0.793089 1.57550
Bonito 0.973950 1.87320 0.971292 1.63431
Note: 1)*: the value is statistically significant at 90% level.
2)**: the value is statistically significant at 95% level.
3)***: the value is statistically significant at 99% level.
4) category1=tuna, category2=scad, category3=sardine, category4=bonito.
5) The results are calculated by deleting the equation.
Compare table 3 and table 4, we can find the parameters which are significant at 99% level in the ILDS with serial correlation control is one more than the ILDS without serial correlation control. However, the parameters which are significant at 99% level in the IAIDS with serial correlation are still the same as the IAIDS without serial correlation. The estimated values are different form each other due to the functional form change. For both ILDS and IAIDS, the do not change a lot. Thevalues of scad increase in two models. Furthermore, the D-W Statistic of sardine and bonito are increase and become closer to 2 in ILDS with serial correlation control, while they have no change in IAIDS with serial correlation control compare to the old one. It means that the serial correlation problem hasn’t been solved in IAIDS with serial correlation control.
From the result of table 4, Yen and Chern’s method do help us to reduce the errors due to the serial correlation. Even though many problems still exist, the result we obtained is just passable and more reliable and valid for the analysis of the Japanese fish consumption.
3.2.4 The Elasticity Estimation
Next, we compare elasticity results with two inverse demand systems for Japanese fishes’ consumption. ILDS and IAIDS both represent the same picture of demand sensitivities. We will estimate and analyze the Marshallian elasticity, the scale elasticity and the Hicksian elasticity. The two approaches we employ, inverse demand system with serial correlation control and without serial correlation control, will be shown at the same time.
3.2.4.1 The Marshallian quantity elasticity estimation
The Marshallian quantity elasticity tells you how much price of the commodity must change in order to induce the consumer to absorb marginally more of commodity. The equation for computing the Marshallian quantity elasticity will be deduced as follows. Let denotes the Marshallian inverse demand functions obtain from a direct utility function.
For ILDS models:
(18)
(19)
where
(20)
If , then the ILDS model will reduce to IAIDS:
where
(21)
From equation (20) and (21), if , the goods are complements, if, the goods are substitutes. The result of the Marshallian quantity elasticity can be listed as follows:
Table 5: Estimation of Marshallian elasticity for ILDS
Without serial correlation control With serial correlation control
M11 -0.64152 -0.63887
M12 -0.16013 -0.16924
M13 -0.099602 -0.10213
M14 -0.18148 -0.18348
M21 -0.54233 -0.56397
M22 -0.47891 -0.46795
M23 -0.039419 -0.040537
M24 0.0042374 0.0013740
M31 -0.62214 -0.60884
M32 -0.045180 -0.041993
M33 -0.16614 -0.15080
M34 -0.21488 -0.19538
M41 -0.77876 -0.79827
M42 -0.074076 -0.080664
M43 -0.14405 -0.16671
M44 -0.45748 -0.48593
Note: category1=tuna, category2=scad, category3=sardine, category4=bonito.
Table 6: Estimation of Marshallian elasticity for IAIDS
Without serial correlation control With serial correlation control
M11 -0.66214 -0.62939
M12 -0.14222 -0.14003
M13 -0.077603 -0.073349
M14 -0.17012 -0.15753
M21 -0.58430 -0.56515
M22 -0.49753 -0.44830
M23 0.0018952 0.0029619
M24 0.012809 0.010122
M31 -0.57864 -0.66429
M32 0.037083 0.0068206
M33 -0.12609 -0.13145
M34 -0.18299 -0.21035
M41 -0.47523 -0.57508
M42 0.050887 0.0093992
M43 -0.071399 -0.084814
M44 -0.31264 -0.34837
Note: category1=tuna, category2=scad, category3=sardine, category4=bonito.
From table 5, we can find that there is no big difference in Marshallian elasticity between ILDS with serial correlation control and the model without serial correlation control. And the own quantity elasticity of four categories are all negative, which keep in line with the economic intuition that increasing supply will decrease the own price. There is only one positive item in Marshallian elasticity. The elasticity M24 between scad and bonito that is bigger than zero. However, the absolute value is very small. From the opposite view, the elasticity M42 between bonito and scad is still negative. Thus, it is hard to define they are complement commodities.
From table 6, we can find the values of elasticity still are similar to each other. The own quantity elasticity of four categories are still be negative. However, more positive elasticity present. M23, M24, M32, M42 are all become positive. It seems that scad and sardine, scad and bonito are complement commodities. However these relationships are very weak due to the absolute value of elasticity are still very small.
From the two tables above, we can find something interesting in common. For tuna, its own Marshallian quantity elasticity is the biggest one in the four categories while scad has got the smallest one. For tuna, the other three kind of fishes change has no big effect on its consumption which represent by the very small absolute value of Marshallian quantity. If we come back to the mean budget share of tuna, it will not surprise that it occupy the biggest one, which means that it plays as the most important consumption fish in Japanese daily life. For sardine, most of its Marshallian quantity elasticity’s absolute value is very small, which meet its role in the budget share.
3.2.4.2 The scale elasticity
Interpretation of result for inverse demand system is not as widely agreed upon as direct demand system. Instead of income elasticity, Anderson[ Anderson,R.W. “Some Theory of inverse Demand for Applied Demand Analysis” Eur.Econ.Rev.14(1980):281-90. ](1980)introduce scale elasticity, which can be characterized as the percentage change in the marginal value of commodity as the scale of consumption is expanded in one percent. Thus, for necessities, they have scale elasticity less than -1 while luxury who own the scale elasticity bigger than -1. The equation for computing the scale elasticity can be drive from a Marshallian inverse demand function:
(22)
Where denotes the scale, and the scale elasticity of good is defined as:
(23)
Since:
(24)
Thus:
(25)
For ILDS:
(26)
For IAIDS, by adding up the restriction that, it would become:
(27)
If , the commodity is luxury, while, the commodity becomes necessity. The estimated result can be found in table 7 and table 8.
Table 7 : Estimation of scale elasticity for ILDS
Without serial correlation control With serial correlation control
-1.08273 -1.09372
-1.05642 -1.07109
-1.04833 -0.99701
-1.45437 -1.53157
Note: category1=tuna, category2=scad, category3=sardine, category4=bonito.
Table 8: Estimation of scale elasticity for IAIDS
Without serial correlation control With serial correlation control
-1.05208 -1.00030
-1.06713 -1.00037
-0.85064 -0.99927
-0.80838 -0.99886
Note: category1=tuna, category2=scad, category3=sardine, category4=bonito.
For ILDS, if we don’t take care of the serial correlation, we can find that all the scale elasticity are smaller than , it means that all kinds of fish we take into account that are necessity. If we don’t ignore the serial correlation, we can find that except sardine, all the other fishes’ scale elasticity are smaller than negative one. Even for the sardine, the value of its scale elasticity still very close to negative one, which is -0.99701.
For IAIDS, if we ignore the serial correlation, we can find that the first two fishes’ scale elasticity are smaller than . And the last two fishes’ scale elasticity is close to negative one as well. For IAIDS with serial correlation control, the results are still the same.
Thus, we can make a conclusion that tuna and scad are necessity in Japanese daily life, and sardine and bonito still have high probability to be necessity if we take the estimation error into account.
3.2.4.3 The Hicksian elasticity
Hicksian elasticity, similar to the Marshallian elasticity, condition on the unchanged welfare level, aims at investigating the change of the commodity if the price of one good changes as well. It can be deduced by employing the Antonelli equation,
(28)
From equation (28), if , the goods are complements, and if , the goods are substitutes. The following tables illustrated the Hicksian elasticity in the ILDS and IAIDS model.
Table 9: Estimation of Hicksian elasticity for ILDS
Without serial correlation control With serial correlation control
H11 0.022693 0.031469
H12 0.0041191 -0.0031352
H13 -0.026208 -0.028188
H14 -0.00060340 -0.00014616
H21 0.10575 0.092498
H22 -0.31865 -0.30529
H23 0.032192 0.031877
H24 0.18072 0.18091
H31 0.020974 0.0022302
H32 0.11385 0.10942
H33 -0.095075 -0.083394
H34 -0.039750 -0.028259
H41 0.11344 0.14043
H42 0.14655 0.15194
H43 -0.045467 -0.063160
H44 -0.21452 -0.22920
Note: category1=tuna, category2=scad, category3=sardine, category4=bonito.
Table 10: Estimation of Hicksian elasticity for IAIDS
Without serial correlation control With serial correlation control
H11 -0.016742 -0.016318
H12 0.017393 0.011886
H13 -0.0062788 -0.0057190
H14 0.0056285 0.010152
H21 0.070328 0.047966
H22 -0.33564 -0.29638
H23 0.074239 0.070597
H24 0.19107 0.17781
H31 -0.056817 -0.051842
H32 0.16613 0.15858
H33 -0.068425 -0.063892
H34 -0.040893 -0.042847
H41 0.020670 0.037116
H42 0.17353 0.16110
H43 -0.016596 -0.047247
H44 -0.17760 -0.18093
Note: category1=tuna, category2=scad, category3=sardine, category4=bonito.
Several observations are in order regarding results presented in table 9 and 10. First, almost all fishes’ own Hichsian elasticity are negative except the tuna category in ILDS. This result still follows the normal economic rules that as the quantity of supply is bigger, the price of the commodity will be smaller. Second, there is tendency for the Hicksian own-quantity elasticity to become smaller, in absolute terms, by comparing to the Marshallian own-quantity elasticity. This result is consistent with those of Holt and Bishop (2002)[ Matthew T.Holt and Richard C. Bishop, “A Semiflexible Normalized Quadratic Inverse DemandSystem : An Application to The Price Formation of Fish” Empirical Economics (2002)27:23-47],who find that own-quantity elasticity tend to decline monotonically (in absolute terms) with the rank of Antonelli matrix is reduced. Third, all the estimated results come out to be same between ILDS and IAIDS. The difference between estimated values is very small.
3.2.5 Invariance Property
The existence of serial correlation impel us to delete one equation (which is the equation for the previous sections) to do the estimations. According to the econometric knowledge, it is easy and institutional to find that no matter which equation is deleted, the estimation result should be the same with each other. This section aims to test whether results for ILDS and IAIDS are invariance. Instead of deleting the equation, we will delete the equation for the following estimations. We will define the new estimation results as ILDS2 and IAIDS2, and the previous result which we estimated with serial correlation control will be redefined as ILDS1 and IAIDS1[ ILDS1 and IAIDS1 are the results from table 4.].
Table 11: Comparison of the results of ILDS
ILDS1 ILDS2
0.286826*** 0.286826***
0.158942*** 0.158942***
0.073762*** 0.073762***
0.480471*** 0.480471***
0.245695*** 0.245695***
-0.079576*** -0.079576***
-0.038490*** -0.038490***
-0.088346*** -0.088346***
-0.079576*** -0.079576***
0.086796*** 0.086796***
-0.178855E-03 -0.178854E-03
0.618646E-02 0.618646E-02*
-0.038490*** -0.038490***
-0.178855E-03 -0.178854E-03
0.060070*** 0.060070***
-0.010552*** -0.010552***
-0.127630*** -0.127630***
-0.704202E-02 -0.704203E-02
-0.021402*** -0.021402***
0.092711*** 0.092711***
-0.414922E-03*** -0.414922E-03***
-0.159983E-03*** -0.159983E-03***
-0.243498E-04 -0.243500E-04
0.599255E-03*** 0.599255E-03***
0.153778*** 0.153778***
ILDS1 ILDS2
D-W Statistics D-W Statistics
Tuna 0.952405 2.30184 .952405 2.30184
Scad 0.839221 2.62373 .839221 2.62373
Sardine 0.804030 1.72929 .806134 1.73294
Bonito 0.973950 1.87320 .973950 1.87320
Note: 1)*: the value is statistically significant at 90% level.
2)**: the value is statistically significant at 95% level.
3)***: the value is statistically significant at 99% level.
4) category1=tuna, category2=scad, category3=sardine, category4=bonito.
5) The results are calculated by deleting the equation.
Compare the results in table 11, we can find that the estimated values are almost the same, and theand D-W statistics are also similar to each other. There is no big difference between ILDS1 and ILDS2, the ILDS has the property of invariance.
Table 12:Comparison of the result of IAIDS
IAIDS1 IAIDS2
0.450392*** 0.450392***
0.211027*** 0.211027***
0.115362*** 0.115362***
0.223219*** 0.223219***
0.227261*** 0.227261***
-0.085793*** -0.085793***
-0.044944*** -0.044944***
-0.096523 *** -0.096523***
-0.085793*** -0.085793***
0.083795*** 0.083795***
0.453173E-03 0.453174E-03
0.154470E-02 0.154470E-02
-0.044944*** -0.044944***
0.453173E-03 0.453174E-03
0.058720*** 0.058720***
-0.014229*** -0.014229***
-0.096523*** -0.096523***
0.154470E-02 0.154470E-02
-0.014229*** -0.014229***
0.109207*** 0.109207***
0.182952E-03*** 0.182952E-03***
0.569122E-04 *** 0.569122E-04***
-0.491707E-04** -0.491706E-04***
-0.190693E-03*** -0.190694E-03
0.043128** 0.043128**
IAIDS1 IAIDS2
D-W Statistics D-W Statistics
Tuna 0.952722 2.04771 0.952722 2.04771
Scad 0.848933 2.45046 0.848933 2.45046
Sardine 0.793089 1.57550 0.870688 1.74081
Bonito 0.971292 1.63431 0.971292 1.63431
Note: 1)*: the value is statistically significant at 90% level.
2)**: the value is statistically significant at 95% level.
3)***: the value is statistically significant at 99% level.
4) category1=tuna, category2=scad, category3=sardine, category4=bonito.
5) The results are calculated by deleting the equation.
Compare the results in table 12, we can find that the estimated values of IAIDS and IAIDS2 are almost the same, and theand D-W statistics are also similar to each other. There is no big difference between IAIDS1 and IAIDS2, the IAIDS has the property of invariance as well.
4. Conclusion remarks
In recent years there has been growing interest in systems of inverse demand equations. In the effort of many brilliant scholars, a lot of interesting and powerful inverse demand systems were brought forward to us. In this paper, by employing the Inverse Lewbel Demand System (ILDS) and the Inverse Almost Ideal Demand System (IAIDS), we analyze the demand system of four kinds of fishes in Japan. Both ILDS and IAIDS suffer from serial correlation. In the model with serial correlation control, we can find that D-W statistic improve much for ILDS while there is no big change for IAIDS. However, two models’ coefficient estimation results are still very significant. As a matter of course, we still employ them to finish the elasticity analysis. The elasticity analysis includes the Marshallian Elasticity, Scale Elasticity and Hichsian Elasticity. In Japan, the main food fishes, including tuna, scad, sardine and bonito are all necessity, which matches the reality that fishes occupy a very important position in Japanese daily life. Finally, on balance, both ILDS and IAIDS appear promising, and may prove beneficial for the demand analysis for other commodities. However, the serial correlation problem in IAIDS is not well solved in this paper. Further studies are needed to improve the analysis.
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