Tohoku University Research Center for Policy Design Discussion Paper

TUPD-2023-004

A Bayesian analysis of vegetableproduction in Japan

Akira Hibiki
Graduate School of Economics and Management, Tohoku University

Koji Miyawaki
School of Economics, Kwansei Gakuin University

Graduate School of Economics and Management, Tohoku University

May 2023

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A Bayesian analysis of vegetableproduction in Japan
Akira Hibiki
Research Center for Policy Design, Graduate School of Economics and Management,
Tohoku University

Koji Miyawaki
School of Economics, Kwansei Gakuin University
andResearch Center for Policy Design, Graduate School of Economics and Management,
Tohoku University

May 8, 2023

Abstract
Microeconomic theory often assumes that a producer maximizes
its proﬁt. As a consequence, under perfect competition, the optimal
production amount is either zero or positive, where the latter satisﬁes
the condition that the price is equal to the cost for the additional
production amount (the marginal cost). This paper proposes two
statistical models directly derived from this relationship and develops
a Bayesian estimation method for the parameters included in thisrelationship. The models are applied to analyze vegetable productionin Japan.
Keywords: producer theory; Bayesian approach; Jeﬀreys’ prior.

1

Introduction

Microeconomic theory for producers starts from perfect competition, where
each producer chooses its production amount given the price by maximiz1

ing its proﬁt. When this situation as well as the single-output technology
assumption holds, the optimal production amount is derived by solving the
proﬁt maximization condition and either is zero or has a positive value thatsatisﬁes the equation between the price and the production amount, where
the price is equal to the change in cost from the additional production, i.e.,the marginal cost. When the marginal cost is high relative to the price, aproducer decides that no production amount is optimal. This paper proposes
two statistical models that describe such behavior, develops an estimation
methodology, and applies the models to the analysis of vegetable productionin Japan.
The statistical models that are directly derived from the proﬁt maximization condition are diﬀerent from the typical regression model in two aspects.
First, no production amount is explicitly included in the model. This isjustiﬁed not only from a theoretical point of view, as explained above, butalso from an empirical point of view. No production occurs for a substantial
number of observations, as shown in the next section.
Second, the conditional distribution will be speciﬁed for the variable ofthe marginal cost function. We are able to transform the model into its typical form. In the econometrics literature, the former is called the structuralmodel, while the latter is called the reduced form (see, e.g., Cameron andTrivedi (2005)). Although the parameters included in these two forms are
consistent with each other, the least squares estimation does not retain thisconsistency. That is, the least squares estimates for the former do not yield
the estimates for the latter when we use the relationship suggested by these
two forms. To guarantee such a parametrization invariance as well as toaddress the corner solution problem, this paper takes the Bayesian approach
using the Jeﬀreys prior.

2

The Jeﬀreys prior was proposed by Jeﬀreys (1946) in search for parametrization invariance and is widely known in Bayesian statistics for its desirable
properties, including this invariance and the ignorance of knowledge (seeKass and Wasserman (1996)). Although this prior possesses these plausible
properties, the resulting posterior distribution in our work is diﬃcult to tract
analytically, and we take the Markov chain Monte Carlo (MCMC) method
to approximately infer the posterior distribution (see, e.g., Gamerman and
Lopes (2006) for this method).
The proposed framework is applied to analyze vegetable production in
Japan. There are 38 vegetables in our dataset, and none are produced in
all prefectures of Japan. Such a production pattern is possibly due to climatic conditions (temperature and rainfall), in addition to market conditions.
Then, if the marginal cost became low relative to the price due to global
warming, it would be possible for producers to plant vegetables that are notproduced before, as suggested by the proﬁt maximization condition. This
aspect was ﬁrst examined by Mendelsohn et al. (1994) to show the impact of
global warming on agriculture in the United States. It is also measured by
Deschênes and Greenstone (2007).
This paper is organized as follows. After the motivating dataset is examined in Section 2, Section 3 proposes two statistical models that are directly
derived from the proﬁt maximization condition, and Section 4 describes theBayesian estimation method. The proposed models and method are applied
to analyze vegetable production in Japan in Section 5. Section 6 concludes
this paper.

32Vegetable production in JapanThe motivating data are the vegetable production data in Japan aggregatedat the prefecture level in 2018, which will be analyzed in Section 5. The data
are collected by the Ministry of Agriculture, Forestry and Fisheries, Japan,
and consist of 47 prefectures, each of which contains about ﬁve thousand
to about forty-ﬁve thousand agricultural units, about twenty thousand units
on average, according to the 2020 Census of Agriculture and Forestry. Eachprefecture records the production amount of 38 vegetables. This amount is
diﬀerent from the one sold in the market. Of 1, 786 (= 47 × 38) observations,
900 reports no production amount. Table 1 summarizes their distribution.
From this table, no production is observed in about 24 prefectures on average
Table 1: Distribution observations of no production amount
Minimum First quartile

Mean

Third quartile

Maximum

By vegetables

2

18.75

23.68

32.75

40

By prefectures

4

15

19.15

24

31

for each vegetable.
Such observations of no production occur partly because of the producer’s
response to climatic and market conditions. If the price were suﬃciently
high after taking account of these costs, all types of vegetables would be
produced in positive amounts. This aspect is incorporated by resorting to the
producer’s proﬁt maximization problem derived from microeconomic theory.
However, its direct application causes a statistical invariance problem, as
discussed in the following two sections.

4

3

Proﬁt maximization condition and statistical models

Two statistical models will be described in this section. Both of them arederived from the proﬁt maximization condition but based on diﬀerent speciﬁcations of the cost function. These models are relatively simple and theyignore several issues. They will be discussed later in this section.
The ﬁrst model is given bypi + λi = β0 + x′i β + ui ,if yi = 0,pi = αyi + β0 + x′ β + ui , if yi > 0,i(1)where subscript i denotes the i-th observational unit (i = 1, . . . , n). In theempirical analysis given in Section 5, the observational unit is the prefectureand n = 47 given each vegetable. In this model, yi is the production amount,pi is the price, and xi is the k-dimensional covariate vector. The error term
ui is assumed to follow a normal distribution with mean 0 and variance σ 2 ,i.e., N (0, σ 2 ).
The model parameters are (α, β0 , β, σ 2 , {λi }i∈C0 ), where C0 is a set ofobservational units whose yi is zero, i.e., C0 = {i | i = 1, . . . , n and yi = 0}.
The parameter λi is the premium additive to the price for a producer to start
production. The parameter σ 2 is the nuisance parameter, while the otherparameters are the parameters of interest, i.e., the structural parameters.
We assume both α and λi to be positive. This assumption arises naturally
from the theory related to our statistical model, which is explained later, andis incorporated as prior knowledge.
This statistical model is structural because it is directly derived from microeconomic theory. In the theory, under perfect competition, given the price5

p, a producer with a single-output technology chooses the production amounty that maximizes its proﬁt subject to the following nonnegative constraint:
max p · y − C(y),subject to y ≥ 0,ywhere C(y) is the cost function. Then, its ﬁrst-order condition is given byp + λ = C ′ , if y = 0,(2)p = C ′ ,if y > 0,
where C ′ is the derivative with respect to y (the so-called marginal costin economics) and λ > 0 is the Lagrange multiplier. This is the direct
result of the Karush-Kuhn-Tucker theorem. While Hall (1988) empiricallysuggests that this condition does not hold in the U.S. industry, no structural
alternative seems to be plausible for our motivating dataset. Thus, we takeit as the foundation of our analysis.
In this condition, it is the production amount that is decided. The Lagrange multiplier is sometimes called the shadow price in microeconomic theory. Under some regularity conditions, it is equal to the inﬁnitesimal change
in the nonnegative constraint. See, e.g., Mas-Colell et al. (1995) for detailed
explanations of the shadow price and the producer’s theory. By assumingthat the marginal cost function (C ′ ) is linear in the production amount andis observed with an additive normal error, we have statistical model (1) as anatural consequence of the microeconomic theory for producers.
The relationship between the microeconomic theory provided above andmodel (1) implies the four following points. First, because the right-hand sideof (1) is the marginal cost function, yi and xi are variables that determine
the marginal cost. Second, the positive constraints on α and λi are reasonable because the marginal cost usually increases as the production amount6

increases and because no production amount occurs when the marginal costis more than the price, respectively.
Third, the error term represents variations in the omitted variables andthe deviation from the ﬁrst-order approximation of the marginal cost function(i.e., the second- and higher-order terms). It would be more rigorous totruncate the distribution because it is the diﬀerence between the price andthe speciﬁed marginal cost function. Regarding this point, we note that thenext model (4) addresses this issue, and we provide a more detailed discussion
below.
Fourth, the dependent variable is not pi but yi . This aspect requires
special care when we estimate the model parameters. A similar and familiar
form is the Tobit regression model, where yi is on its left hand, which is givenbyyi = max{yi∗ , 0},(3)yi∗ = δ0 + γpi + x′i δ + vi ,where yi∗ is the latent production amount and the error term vi is assumed tofollow a normal distribution with mean 0 and variance τ 2 (see Chib (1992)
for its Bayesian estimation method). Both model (1) and model (3) should
be consistent with their theoretical origin, model (2). That is, the model
parameters (α, β0 , β, σ 2 , {λi }) and (γ, δ0 , δ, τ 2 , {yi∗ }) are consistent with eachother. Such consistency is easier to implement in the Bayesian estimation.
Section 4 discusses this aspect further.
The second model is given bylog pi + λi = β0 + x′i β + ui ,if yi = 0,log pi = α log yi + β0 + x′ β + ui , if yi > 1.
i

(4)

Because this model is essentially the same from a statistical point of view,
we reuse the parameters. However, their interpretation is diﬀerent, e.g., λi is
7not the additive premium but the logarithm of the multiplicative premium tostart production. In addition, the production amount jumps from yi = 0 toyi = 1 as the price rises. Then, the term on the left-hand side, β0 +x′i β +ui if
yi = 0, becomes the logarithm of the so-called shutdown price in economics.
A possible source of this jump is the nonsunk ﬁxed cost to produce theamount of the ﬁrst unit (see Mas-Colell et al. (1995)). From the perspectiveof microeconomic theory, the log model (4) is more reasonable in agriculture
than the linear model (1) because agriculture usually requires the rental anddepreciation costs regardless of the production amount.
These two models have advantages and weak points. The linear model
is a direct derivation from the proﬁt maximization condition and puts noconstraints on the measurement unit. However, it requires a truncation,such as I(pi − β0 − β ′ xi > ui ) for yi > 0, as discussed above. A similar
truncation occurs under the reduced form, which leads to the complicatedJeﬀreys prior, as noted in Chib (1992) and Amemiya (1973). Further, Angrist(2001) discusses that the marginal eﬀect without truncation on the dependent
variable is estimated to be similar to the marginal eﬀect with truncation.
Therefore, we do not incorporate this truncation in the linear model to avoid
such a complication.
The log model has a jump, as described above, and requires no truncationof the error term. However, it requires a constraint on the measurement unit.
The unit should be selected as long as it satisﬁes yi > 1 for all observations
with a positive production amount. This holds in our empirical dataset, theminimum of which is 47.
The linear and log models are diﬀerent with respect to the cost function that they are based on.

When the cost function is speciﬁed as aquadratic function of the production amount, we obtain the linear model.

8When log C(y) = a + b log(y), the marginal cost function is C ′ = bC(y)/y,leading tolog C ′ = a + log(b) + (b − 1) log(y).
Thus, it is straightforward to obtain the log model (4) with this speciﬁccost function. Model selection between these two models by the devianceinformation criterion (DIC) proposed by Spiegelhalter et al. (2002) will be
conducted in Subsection 5.2 with the empirical dataset.
Four issues related to the proposed statistical models are discussed. First,
the decision on the production amount is usually made before the actual equilibrium is observed, and it might be more reasonable to use the expectationof the price instead of its current values (the price contemporaneous with the
production amount), provided there is no uncertainty about the production
amount. A possible explanation for this situation is the rational expectation
assumption where expected values are observed.
This paper assumes the rational expectation, mainly because Cooley and
DeCanio (1977) and Goodwin and Sheﬀrin (1982) empirically showed that
this assumption is consistent with agricultural data. In addition, Subsection
5.4 uses the previous price instead of the current one and shows that thischange has little inﬂuence on the inferences about the parameters of interest.
The decision under uncertainty would be another aspect to consider.
Sandmo (1971) proposes its analytical framework, while Pope and Chavas(1994), Pope and Just (1996), and Moschini (2001) provide its empirical
speciﬁcations. However, to keep the models as simple as possible and to
avoid misspecifying the structure based on expectation, we do not pursue
this aspect in this paper.
Second, model parameter estimates can suﬀer from the endogeneity bias.
As an example, consider a factor that relates to the economic environment. It
9may also relate to the producer’s technology, so that the error term correlateswith the factor that relates to the economic environment if such a factor
is excluded from the model. On the other hand, this factor also aﬀects
the demand, so that the price is likely to correlate with this factor. Then,
this factor omitted from the model causes the endogeneity bias on model
parameter estimates. When we assume that the technology is independentof such a factor, it is not a problem. The short-run analysis is a case where
this assumption holds because the technology does not change so quickly.
Third, the marginal cost function is speciﬁed as linear in the production
amount, even after taking logarithms. An alternative is its nonparametricestimation proposed by Hall and Yatchew (2007). Our speciﬁcation can beconsidered to be its ﬁrst-order approximation. Estimation with higher-orderterms complicates the estimation methodology, and we leave it for a future
work.
Finally, the substitution between vegetables is ignored in this model.
Thus, as the theoretical foundation, we assume a producer with a singleoutput technology. Depending on the application context, the substitution
would be of interest. A theoretical framework for this case is the one proposedby Pfouts (1961), where a ﬁrm produces many products. Such a situation isobserved in the manufacturing industry, and is examined by Bernard et al.
(2010) with a continuum of products. A possible empirical speciﬁcation is
proposed by Just et al. (1983), useful for the inner solution.
In our model, such an aspect can be incorporated when the land is included in the proﬁt maximization condition. We do not include it by assuming that the producer has the unlimited access to the land. As in ourempirical analysis, this assumption is reasonable when the production unit
is the prefecture, which can be considered to have enough land.

10

4

Bayesian estimation

This section describes the Bayesian estimation of the linear model (1). The
estimation of the log model (4) can be implemented in a similar way, which
will not be described in this paper (see Remark 1 below).
The likelihood function under the linear model (1) is proportional to−n/2
f {yi }ni=1 | α, β0 , β, σ 2 , {λi }i∈C0 , {pi , xi }ni=1 ∝ |α|n1 σ 2




X
X
2
2 
1
,
× exp − 2
pi + λi − β0 − x′i β +pi − αyi − β0 − x′i β

2σ 
i∈C0

i∈C1

where C1 is the complement of C0 , i.e., C1 = {i | i = 1, . . . , n and yi > 0}
and n1 = |C1 |. The design matrices Z, X1 , and X2 are assumed to be offull column rank, where Z has (1, x′i ) for its i-th row (i = 1, . . . , n), X1 has(pi , yi , 1, x′i ) for its i-th row (i ∈ C1 ), and X2 has (pi , 1, x′i ) for its i-th row(i ∈ C0 ).
Remark 1. For the log model, change (pi , yi ) to (log pi , log yi ) and modify
C1 = {i | i = 1, . . . , n and yi > 1} to have the likelihood with respect to{log yi }ni=1 . Then, the following discussion holds similarly.
The prior density function is assumed to be 11
π α, β0 , β, σ 2 , {λi }i∈C0 ∝ I (α > 0)I σ2 > 0(k+4+n)/20α(σ 2 )
)1/2
(
Y
1
q12

I (λi > 0) ,×+
σ 2 2n1 1 − nn1
i∈C

(5)

0

where n0 = |C0 | and
q12 =

X
i∈C1

p2i −

X
i∈C1

zi′ piX
i∈C1!−1zi zi′

Xzi p i .

i∈C1

We use I(A) as the indicator function, where I(A) = 1 if A is true andI(A) = 0 otherwise. This prior is a product of the prior knowledge of the sign
11

constraint and the so-called Jeﬀreys prior derived from the structural model(1). Appendix A provides its derivation. The Jeﬀreys prior is often used in
the objective Bayes literature and considered to be one of the noninformative
priors (see, e.g., Kass and Wasserman (1996)). Because a noninformative
prior does not necessarily lead to a proper posterior distribution, the proofof its properness is given in Appendix B.
One main advantage of the Jeﬀreys prior is its invariance to parametertransformation. The proposed statistical model has another parametrization,
where yi is on the left-hand side (3). Both parametrizations should lead toposterior distributions that are consistent with each other after the appropriate parameter transformation. Otherwise, the prior distribution under one
model introduces prior knowledge that is diﬀerent from that under the othermodel. Such invariance is a desirable property for the statistical inference ofour models. This paper focuses on the parametrization of model (1) and its
Jeﬀreys prior because its statistical inferences are easier to conduct.
Our prior speciﬁcation has a side eﬀect. Eaton and Sudderth (2010) showthat the Jeﬀreys’ prior corresponds to the left-invariant Haar measure in theinvariance situation (see their Theorem 3.7). This implies that the resulting
estimator is invariant according to the linear unit transformation (from ton
to kilogram, for example). Because the choice of unit is arbitrary in ourempirical application, this property is plausible.
To conduct inferences, we apply the Gibbs sampler (one of the Markov
chain Monte Carlo methods) to approximately infer the posterior distribution. It is implemented in the following six-step algorithm.
Step 1. Initialize the model parameters (α, β0 , β, σ 2 , {λi }i∈C0 ).
Step 2. Generate α conditional on β0 , β, σ 2 , {λi }i∈C0 .

12

Step 3. Generate (β0 , β) conditional on α, σ 2 , {λi }i∈C0
Step 4. Generate σ 2 conditional on α, β0 , β, {λi }i∈C0 .
Step 5. Generate λi conditional on α, β0 , β, σ 2 for i ∈ C0 .
Step 6. Repeat Step 2 through Step 5.
Because the conditional densities used in Steps 2 and 4 are nonstandard, we
use the Metropolis-Hastings algorithm to generate a random sample fromthem. For Step 2, the Laplace approximation based on the Taylor series
expansion of its logarithm around the mode is applied (see Tierney and
Kadane (1986)). For Step 4, a mixture of inverse gamma distributions isused as the proposal distribution. For details of the full conditional posterior
density functions used in this Gibbs sampler, see Appendix C.

5

Empirical analysis of Japanese vegetableproduction

This section analyzes the prefecture-level vegetable production data in Japan
in 2018. As stated in Section 2, 38 vegetables are included in the analysis,
and they are separately analyzed by applying the models described in Section
3.

5.1

Data description

The variables to be used for the analysis are the production amount, theprice, and climate-related variables. See also Section 2 for the source of thedataset. The measurement unit for the production amount and the priceare ton and Japanese yen per kilogram, respectively. The price variable is13the one reported at the wholesale market. When a prefecture has more than
one market, we average the prices by using their sales amount at markets as
weight. For the climate-related variables, three variables are selected, i.e, theaverage temperature x1 , the average rainfall x2 , and the average number ofhours of sunlight x3 , collected by the Japan Meteorological Agency. Althoughthese variables vary even within a prefecture, we use the ones reported at
the meteorological observatory closest to the prefectural capital.
Wage and the rental cost of capital would be reasonable candidate variables for the marginal cost function. However, they are not available in Japan
partly because most agricultural units are family-based (about 96.4% of themare owned by an individual according to the 2020 Census of Agriculture andForestry), so that wage and the rental cost of capital are diﬃcult to measurefor such a unit.
Because the log model (4) is preferred to the linear model (1) in termsof DIC, as shown later, the logarithm of the production amount and priceare summarized in the form of a boxplot by each vegetable in Figures 1
and 2. The box gives the interquartile range (IQR) between the ﬁrst andthird quartiles with a line at the median. The two whiskers are the 1.5IQR length, and observations outside the whiskers are given by dots. Whenobservations are within the 1.5-IQR length, the whiskers are stretched at the
maximum/minimum.
Summary statistics of the climate-related variables are given in Table 2.

5.2

MCMC settings and model selection

For all vegetable data, we estimate the model parameters by the MCMC
method described in the previous section. Discarding the ﬁrst 10,000 samples, the subsequent 100,000 samples are generated, and every 5-th sample
14

log production amount [log(ton)]3
asparagus
broccoliburdock
cabbagecarrotcauliflower
celerych. cabbage
ch. leek
chingensai
cucumber
eggplant
garlicgingerhonewort
ja. radishkidney bean
komatsunalettucelotus rootmelon
mizunaonion
pea
potatoshungiku
soybeanspinachsquashstrawberry
sweet cornsweet pepper
tarotomatoturnipwatermelon
welsh onionyam

0

asparagus
broccoliburdock
cabbagecarrotcauliflower
celerych. cabbage
ch. leek
chingensai
cucumber
eggplant
garlicgingerhonewort
ja. radishkidney bean
komatsunalettucelotus rootmelon
mizunaonion
pea
potatoshungiku
soybeanspinachsquashstrawberry
sweet cornsweet pepper
tarotomatoturnipwatermelon
welsh onionyam

log price [log(yen/kg)]
15

10

5

Figure 1: Log production amount by vegetables.

9

8

7

6

5

4

Figure 2: Log price by vegetables.

is selected to have 20,000 samples. They are used to conduct the follow-

ing statistical inferences. The ineﬃciency factor is used to check whether thenumber of MCMC samples is high enough (see Chib (2004) for this measure).

15

Table 2: Summary statistics of the climate-related variables
Unit

Minimum First quartile Mean Third quartile Maximum

average temperature (x1 )

◦

C

9.55

15.38

16.10

17.31

23.51

average rainfall (x2 )

mm

2.27

3.74

4.87

5.47

8.68

average hours of sunlight (x3 ) hour

4.18

5.20

5.67

6.13

6.55

All ineﬃciency factors are less than 50.
We compare the model ﬁtting between the linear and log models by DIC.
We note that the likelihood with respect to yi is used for DIC, so that it
is comparable between both models. After the MCMC samples are divided
into ten equally sized groups, DIC is estimated for each group. Their average and standard deviation are an estimate of DIC and its standard error,
respectively. Judging from the estimated DIC, the log model is preferred,because the minimum diﬀerence is about 18, with a standard error of about
0.43, and the other standard errors are less than 1.

5.3

Analysis of the model parameters

Figure 3 shows boxplots of the (marginal) posterior distributions of the avP
erage of exp(λi ), that is, λ̄ = n10 i∈C0 exp(λi ), for all vegetables. Each box
represents the ﬁrst and third quartiles with a line at the mean, and bothends of the whiskers show the 95% credible interval.
As the log model is used, this average can be interpreted as the average(multiplicative) premium price to start production scaled by the price. The
premium diﬀers across vegetables, reﬂecting their economic and agriculturalenvironment. Most premium prices are around 1.2, and they are less thanabout 1.8.
Figure 4 shows the marginal eﬀect of the average temperature on the
16

−0.2

17

yam

welsh onion

watermelon

turniptomatotaro

sweet pepper

sweet cornstrawberrysquashspinachsoybeanshungiku

potatopea

onion

mizunamelon

lotus root

lettucekomatsunakidney bean

ja. radish

honewort

gingergarliceggplant

cucumberchingensai

ch. leek

ch. cabbageyamwelsh onion

watermelon

turniptomatotaro

sweet pepper

sweet cornstrawberrysquashspinachsoybeanshungiku

potatopea

onion

mizunamelon

lotus root

lettucekomatsunakidney bean

ja. radish

honewort

gingergarliceggplant

cucumberchingensai

ch. leek

ch. cabbage

celerycauliflower

carrotcabbageburdock

broccoli

asparagus

1.0

celerycauliflower

carrotcabbageburdock

broccoli

β1

asparagus

1.8

1.6

λ 1.4

1.2

Figure 3: Premium.

marginal cost. Boxplots are drawn in the same manner as the ones in Fig-

0.2

0.1

0.0

−0.1

Figure 4: Marginal eﬀect of the average temperature.

ure 3. This ﬁgure suggests that the association between the marginal cost

and the average temperature is nonuniform among vegetables. In its Sixth

Assessment Report by Working Group II, the Intergovernmental Panel on
Climate Change reports that the average temperature will rise by about 1.5
to 5 ◦ C in 2100 compared with the one in 1850–1900 (see Pörtner et al.
(2022)). Our result suggests that its inﬂuence on vegetable production varies
according to the type of vegetable.
To see this more clearly, we divide the vegetables into two groups according to their marginal posterior probabilities of β1 . For each vegetable,
we estimate its posterior probability that its β1 is positive/negative, i.e.,Pr(β1 > 0 | data) or Pr(β1 < 0 | data). When the probability is morethan 0.95, it is credible that the average temperature has a positive/negative
association with the marginal cost. The results are given in Table 3.
Table 3: Marginal posterior probability of β2

Positive

Negativeburdock, cucumber, potato, welsh

asparagus, chingensai, ginger, kid-

onion, onion, sweet corn, soybean,

ney bean, komatsuna, lettuce, lotusyamroot, pea, shungiku, taro

The posterior summary for the remaining model parameters is provided
in Appendix D.

5.4

Examination of the rational expectation assumption

The result above relies on the rational expectation assumption, as discussed
in Section 3. To examine whether this assumption is reasonable, we use
the price in 2017 instead of 2018 and see how this assumption aﬀects the18result. The same prior and MCMC settings as for the log model are used togenerate 20,000 samples. Figure 5 shows the comparison of posterior means
with the datasets in 2017 and 2018, overlaying the 45-degree line, for allmodel parameters. Judging from the posterior mean, the two results aremostly similar because most points gather around the 45-degree line and nosystematic deviation from the line exists. Therefore, we conclude that relying
on the rational expectation assumption would be reasonable, at least for thisdataset.

6

Discussion

This paper proposed two statistical models that are derived from the producer’s proﬁt maximization condition. To estimate the model parameters, wetake the Bayesian approach using the Jeﬀreys prior to guarantee parametrization invariance between the structural and reduced forms. The proposedmodels and estimation method are applied to analyze vegetable productionin Japan. The analysis estimates the premium price to start production andthe impact of the average temperature rise on the production amount. The
latter shows a nonuniform inﬂuence of global warming on vegetable production.
To analyze the impact of global warming more accurately, we need otherinformation, such as the demand structure. In addition, as discussed in theprevious section, the substitution between agricultural products is anotherfactor to incorporate when we analyze the impact, which can be addressedby the panel dataset. Finally, relaxing the assumption of perfect competition
would be useful when our framework is applied to another industry. Implementing the above points requires a complication of our methodology and

19

9

0.25

0.15

0.10

2017 price2017 price

0.15

β0

7

6

5

0.05

0.10

0.15

0.20

0.25

3

2018 price

0.00

6

7

8

−0.15 −0.10 −0.05 0.00

9

β2

0.1
0.0
−0.1
−0.2

0.05

0.10

0.15

0.20

2018 price−0.4 −0.3 −0.2 −0.1 0.0

0.1

0.2

0.3

2018 price

0.15

0.5

0.6

σ

0.4

0.3

0.2

0.4

0.0

0.1

λi

4

3

2

1
2

3

2018 price

Figure 5: Comparison.

20

0.2

0.3

0.4

2018 price

5

1

0.10

0.1

−0.3
0.00

0.05

2018 price

0.5

β3

0.2

−0.05

−0.10 −0.05

−0.05

0.6

0.3

2017 price

0.05

0.00

2018 price2017 price2017 price

0.10

5

0.4

0.20
0.15

4

2017 price

0.05

0.05

−0.10

4

0.00

β1

0.10

2017 price

α

0.20

8

4

5

will be addressed in future works.

A

Jeﬀreys’ prior

The Jeﬀreys prior is proportional to the square root of the determinant of theFisher information matrix. To this end, the following formula for the determinant of a partitioned matrix is applied: |A| = |A11 ||A22 − A21 A−1
11 A12 | =−1|A22 ||A11 − A12 A22
A21 |, where


A=


A11 A12.

A21 A22
See, e.g., Abadir and Magnus (2005) for a proof of this formula.
The Fisher information matrix for the linear model is given by
F = −E∂ 2 log f ∂ζ∂ζ ′∂ 2 log f∂λ∂ζ ′∂ 2 log f∂ζ∂λ′ ∂ 2 log f∂λ∂λ′,where ζ = (α, σ 2 , β0 , β ′ )′ and λ = (λi )i∈C0 . The expectation is over theconditional distribution of yi (i = 1, . . . , n). This is a partitioned matrix,
where Fij is its (i, j) block element for i, j = 1, 2. After some calculations,
we haveP

2n1+ α21σ2 α2F11 = 


n1
− ασ
2

1
ασ 2

F22 =

n1
2
i∈C1 di − ασ 2Pi∈C1 zi di1In ,σ2 0 


0

′



0

′


′
F12 = F21=n2(σ 2 )2

− σ12 zi

0




,


21

1
ασ 2

P

′
i∈C1 zi di

0
P
1
σ2

′
i zi z i




,


where zi′ = (1, x′i ), In0 is the n0 -dimensional unit matrix, the index i in F12
is over i ∈ C0 , and

di =



pi + λi − β0 − x′i β, if i ∈ C0 ,pi − β0 − x′ β,(6)

if i ∈ C1 .

i

Then,


2n1
+ α21σ2 α2−1F11 − F12 F22
F21 = 


1
ασ 2

P

2
i∈C1 di

n1
− ασ
2
P

n1
− ασ
2

1
ασ 2

n
2(σ 2 )2i∈C1 zi di

0

1
σ2

P
P



′
i∈C1 zi di 0′i∈C1 zi zi.
After combining all the expressions above and picking up the terms that
include the model parameters, we have


n1 
1
2n1 
1
−
+ 2 2 q12×
|F | =
P
2
1
′
α
n
α σ
0
i∈C1 zi zi
σ2
(
)
2
1
q
1
 .
∝
σ2 +
2n1 1 − nn1
α2 (σ 2 )k+4+n0n2(σ 2 )20

Therefore, we have the Jeﬀreys prior found in Equation (5).

22



B

Properness of the posterior distribution

This section will give a proof that the posterior distribution that uses Prior(5) is proper. The posterior density is proportional to

π α, β0 , β, σ 2 , {λi }i∈C0 ∝|α|n1 −1 σ
2 −(k+n0 +n+4)/2

(

)

n
1 X

exp − 2(di − αyi )2
σ i=1
)1/2
(
2

1
q
1

× I (α > 0) I σ 2 > 0+
σ 2 2n1 1 − nn1
Y×
I (λi > 0)
i∈C0

(

1 X
(di − αyi )2<|α|n1 −1 σ
exp − 2
σ i=1


 1
q1

× I (α > 0) I σ 2 > 0  + q
σn1
2n1 1 − nY×I (λi > 0) .

2 −(k+n0 +n+4)/2

n

)

i∈C0

Our proof shows that the integral of the function on the most right-hand side
over its parameter space is ﬁnite.
With the rank condition, two normalizing constants from the inverse
gamma distribution and the Arellano-Valle and Bolfarine generalized t distribution are ﬁnite. Then, we haveZ
π α, β0 , β, σ 2 , {λi }i∈C0 dσ 2 dβ0 dβ <(some ﬁnite constant)
× |α|n−1 (e′ M e)−(n0 +n+1)/2,

where M = In − Z(Z ′ Z)−1 Z ′ , e is the n-dimensional vector whose i-th row
is deﬁned as
ei =



pi + λi − αyi , if i ∈ C0 ,pi − αyi ,23

if i ∈ C1 .

(7)

Let
f (α, {λi }i∈C0 ) = α2

(n−1)/2(e′ M e)−(n0 +n+1)/2.

This function is integrable on any compact subset A of the nonnegative
orthant because of the following three reasons: (i) |α|(n−1)/2 is continuousand closed on A, (ii) e′ M e is so as well, and (iii) e′ M e > 0 because of therank condition.
Further, letg (α, {λi }i∈C0 ) = ∥ξ∥2

(n−1)/2(e′ M e)−(n0 +n+1)/2,

where ξ = (α, {λi }i∈C0 ) and ∥ξ∥ is the Euclidean norm of ξ. It is clear thatf ≤ g for any possible value of ξ. This function is integrable on A.
We will show that g = O(∥ξ∥−(n0 +2) ), where the order notation denotes|g| ≤ M ∥ξ∥−(n0 +2) for some constant M . The polar coordinate representationof ξ leads to
dr2
∥ξ∥2
=
,
e′ M ea + br + cr2
where a, b, c, d are some functions of triangular functions and r is the radial
distance from the origin. We note that cr2 = (e − p)′ M (e − p), wherep = (p1 , . . . , pn )′ . By the rank condition, cr2 > 0. So c ̸= 0 as long as r > 0.
Then, the left-hand side converges to 1/c as ∥ξ∥ goes to inﬁnity. Becauseg × ∥ξ∥(n0 +2)




=

∥ξ∥2
e′ M e(n0 +n+1)/2,

we have the result.
Because the order result shows that g is integrable over the nonnegative
orthant, f is so as well, which completes the proof.

24

C

Full conditional posterior density functionsFirst, the full conditional posterior density for α is proportional to
π α | β0 , β, σ 2 , {λi }i∈C0 ∝ |α|n1 −1"

where Q = σ −2

(
!
!)#
n
n
X
X
1
× exp − 2 α2yi2 − 2αyi d i
2σ
i=1
i=1


R
= |α|n1 −1 exp − α2 + Qα ,
2

Pn

−2
i=1 yi di (see Equation (6) for di ) and R = σPn

2i=1 yi .

This conditional density is nonstandard, and we apply the Metropolis-Hastings(MH) algorithm to draw a sample from it.
The proposal used in this step is derived as follows. The mode of thisconditional density is
m=

Q+

p
Q2 + 4R (n1 − 1),2R

and the second derivative of the log conditional density is given byH=−

n1 − 1− R.
m2

Then, the Taylor series expansion of the log conditional density around themode gives the proposal T N(0,∞) (m, −H −1 ), where T NS (µ, τ 2 ) denotes thetruncated normal distribution with mean µ and variance τ 2 on support S.
Let α(−1) and α̃ be the sample recorded in the previous Markov chain andthe candidate drawn from the proposal density, respectively. The candidate
is accepted with probabilityg (α̃)
min 1,,g (α(−1) )whereg (α) = |α|

n1 −1



1
2
exp − (R + H) α + (Q + mH) α .
225

Next, because the full conditional density for σ 2 is also nonstandard, theMH algorithm is applied as well. The proposal for this generation is a mixtureof the inverse gamma distributions IG(r1 /2, S1 /2) and IG((r1 + 1)/2, S1 /2)
with the respective weights w and 1 − w, where r1 = k + 2 + n + n0 , S1 =Pn
2
i=1 (di − αyi ) , and
w=

q
q 1
.
q1 + 2n1 1 − nn1

The acceptance probability for a candidate σ̃ 2 in terms of the previous sample
σ 2,(−1) is given by2

min 1,

g (σ̃ ),g (σ 2,(−1) )where g(σ 2 ) =q(1 − w)2 + w2 σ 2
1 − w + wσ.

The remaining full conditionals used in the Gibbs sampler are all standard. That is,′(β0 , β ′ ) | α, σ 2 , {λi }i∈C0 ∼N (b1 , B1 ) ,
λi | α, β0 , β, σ 2 ∼T N(0,∞) −pi + αyi + β0 + x′i β, σ 2 ,where zi′ = (1, x′i ), B1−1 = σ −2Pn′
i=1 zi zi , and b1= σ −2 B1Pn

i=1 ei zi(see

Equation (7) for ei ).

D

Analysis of other parameters

The regression coeﬃcients except for the intercept are summarized in Figure
6. The broader credible interval for α is attributed to the larger number ofprefectures of no production. The intercept and standard error of regression
are summarized in Figure 7.

26

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29

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30

−0.1

−0.2

0.6

0.4

0.2

β3

0.0

−0.2

−0.4

−0.6

−0.8
celery31

Figure 6: Posterior summary I.
sweet cornsweet pepper
tarotomatosweet cornsweet pepper

tarotomatoyam

strawberry

strawberry

welsh onion

squashsquashyamspinachspinachwelsh onion

turnipsoybeansoybeanturnipshungiku

shungiku

watermelon

taro
tomatopotatopotatoturnipsweet cornsweet pepper

pea

pea

watermelon

strawberry

onion

onion

yam

welsh onion

watermelon

squashspinachsoybeanshungiku

potatopea

onion

mizunamelon
mizunamelon

lotus root

lettucekomatsunakidney bean

ja. radish

honewort

gingergarliceggplant

cucumberchingensai

ch. leek

ch. cabbage

melon

lettucelotus root

lettucelotus root

carrotcauliflower

mizunakomatsunagarlicgarlickidney bean

eggplant

eggplant

komatsunacucumbercucumberkidney bean

chingensaichingensai

ja. radish

ch. leek

ch. leek

ja. radish

ch. cabbage

ch. cabbage

gingerceleryceleryhonewort

cauliflower

cauliflower

gingercarrotcarrot0.0

honewort

burdock
cabbageburdock

cabbagebroccoli

asparagus

cabbage

0.0
burdock

β2

broccoli

0.1

broccoli

0.2

asparagus

0.3

asparagus

0.6

0.5

0.4

α 0.3

0.2

0.1

0.00

32

Figure 7: Posterior summary II.
squashstrawberrysquashstrawberryturnipwatermelon
welsh onion
yamturnipwatermelon

welsh onion

yam

taro
tomatotaro

tomatosweet corn

spinachspinachsweet pepper

soybeansoybeansweet cornshungiku

shungiku

sweet pepper

pea
potatopotato

onion

onion

pea

melon

lotus root

mizunalettucelotus root

lettucemelonkomatsunakomatsunamizunaja. radishkidney beankidney bean

honewort

honewort

ja. radishgarlicgingereggplant

eggplant

gingercucumbercucumbergarlicch. leek
chingensai

ch. leek

chingensai

celery0.25
ch. cabbage

0.50

ch. cabbage

σ 0.75carrot1.00
cauliflower

1.25

celery1.50

cauliflower

cabbageburdock

broccoli

asparagus−2.5

carrotcabbageburdock

broccoli

asparagus

10.0

7.5

5.0

β0
2.5

0.0