A Bayesian Analysis of Vegetable Production in Japan

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 this relationship. The models are applied to analyze vegetable production in Japan.

Appendices

Appendix A: Jeffreys’ Prior

The Jeffreys’ prior is proportional to the square root of the determinant of the Fisher information matrix. To this end, the following formula for the determinant of a partitioned matrix is applied: $| \varvec{A} | = | \varvec{A}_{11} | | \varvec{A}_{22} - \varvec{A}_{21} \varvec{A}_{11}^{-1} \varvec{A}_{12} | = | \varvec{A}_{22} | | \varvec{A}_{11} - \varvec{A}_{12} \varvec{A}_{22}^{-1} \varvec{A}_{21} |$, where

$$\begin{aligned} \varvec{A} = \begin{pmatrix} \varvec{A}_{11} &{} \varvec{A}_{12} \\ \varvec{A}_{21} &{} \varvec{A}_{22} \end{pmatrix}. \end{aligned}$$

See, e.g., Abadir and Magnus (2005) for a proof of this formula.

The Fisher information matrix for the linear model is given by

$$\begin{aligned} \varvec{F} = -E \begin{pmatrix} \frac{ \partial ^{2} \log f }{ \partial \varvec{\zeta } \partial \varvec{\zeta }^{\prime } } &{} \frac{ \partial ^{2} \log f }{ \partial \varvec{\zeta } \partial \varvec{\lambda }^{A \prime } } \\ \frac{ \partial ^{2} \log f }{ \partial \varvec{\lambda }^{A} \partial \varvec{\zeta }^{\prime } } &{} \frac{ \partial ^{2} \log f }{ \partial \varvec{\lambda }^{A} \partial \varvec{\lambda }^{A \prime } } \end{pmatrix}, \end{aligned}$$

where $\varvec{\zeta } = (\alpha , \sigma ^{2}, \beta _{0}, \varvec{\beta }^{\prime })^{\prime }$ and $\varvec{\lambda }^{A} = (\lambda _{i}^{A})_{i \in {\mathcal {C}}_{0}}$. The expectation is over the conditional distribution of $y_{i}$ ($i = 1, \dots , n$). This is a partitioned matrix, where $\varvec{F}_{ij}$ is its (i, j) block element for $i, j = 1, 2$. After some calculations, we have

$$\begin{aligned} \varvec{F}_{11}&= \begin{pmatrix} \frac{2 n_{1}}{\alpha ^{2}} + \frac{1}{ \alpha ^{2} \sigma ^{2} } \sum _{i \in {\mathcal {C}}_{1}} d_{i}^{2} &{} -\frac{n_{1}}{ \alpha \sigma ^{2} } &{} \frac{1}{ \alpha \sigma ^{2} } \sum _{i \in {\mathcal {C}}_{1}} \varvec{z}_{i}^{\prime } d_{i} \\ -\frac{n_{1}}{ \alpha \sigma ^{2} } &{} \frac{n}{2 (\sigma ^{2})^{2}} &{} 0 \\ \frac{1}{ \alpha \sigma ^{2} } \sum _{i \in {\mathcal {C}}_{1}} \varvec{z}_{i} d_{i} &{} 0 &{} \frac{1}{\sigma ^{2}} \sum _{i} \varvec{z}_{i} \varvec{z}_{i}^{\prime } \end{pmatrix}, \\ \varvec{F}_{22}&= \frac{1}{\sigma ^{2}} \varvec{I}_{n_{0}}, \\ \varvec{F}_{12}&= \varvec{F}_{21}^{\prime } = \begin{pmatrix} \varvec{0}^{\prime } \\ \varvec{0}^{\prime } \\ -\frac{1}{\sigma ^{2}} \varvec{z}_{i} \end{pmatrix}, \end{aligned}$$

where $\varvec{z}_{i}^{\prime } = (1, \varvec{x}_{i}^{\prime })$, $\varvec{I}_{n_{0}}$ is the $n_{0}$-dimensional unit matrix, the index i in $\varvec{F}_{12}$ is over $i \in {\mathcal {C}}_{0}$, and

$$\begin{aligned} d_{i}&= {\left\{ \begin{array}{ll} p_{i} + \lambda _{i}^{A} - \beta _{0} - \varvec{x}_{i}^{\prime } \varvec{\beta }, &{}\text {if } i \in {\mathcal {C}}_{0}, \\ p_{i} - \beta _{0} - \varvec{x}_{i}^{\prime } \varvec{\beta }, &{}\text {if } i \in {\mathcal {C}}_{1}. \end{array}\right. } \end{aligned}$$

(7)

Then,

$$\begin{aligned} \varvec{F}_{11} - \varvec{F}_{12} \varvec{F}_{22}^{-1} \varvec{F}_{21} = \begin{pmatrix} \frac{2 n_{1}}{\alpha ^{2}} + \frac{1}{ \alpha ^{2} \sigma ^{2} } \sum _{i \in {\mathcal {C}}_{1}} d_{i}^{2} &{} -\frac{n_{1}}{ \alpha \sigma ^{2} } &{} \frac{1}{ \alpha \sigma ^{2} } \sum _{i \in {\mathcal {C}}_{1}} \varvec{z}_{i}^{\prime } d_{i} \\ -\frac{n_{1}}{ \alpha \sigma ^{2} } &{} \frac{n}{2 (\sigma ^{2})^{2}} &{} 0 \\ \frac{1}{ \alpha \sigma ^{2} } \sum _{i \in {\mathcal {C}}_{1}} \varvec{z}_{i} d_{i} &{} 0 &{} \frac{1}{\sigma ^{2}} \sum _{i \in {\mathcal {C}}_{1}} \varvec{z}_{i} \varvec{z}_{i}^{\prime } \end{pmatrix}. \end{aligned}$$

After combining all the expressions above and picking up the terms that include the model parameters, we have

$$\begin{aligned} | \varvec{F} |&= \begin{vmatrix} \frac{n}{2 (\sigma ^{2})^{2}}&0 \\ 0&\frac{1}{\sigma ^{2} } \sum _{i \in {\mathcal {C}}_{1}} \varvec{z}_{i} \varvec{z}_{i}^{\prime } \end{vmatrix} \times \left\{ \frac{2 n_{1}}{\alpha ^{2}} \left( 1 - \frac{n_{1}}{n} \right) + \frac{1}{ \alpha ^{2} \sigma ^{2} } q_{1}^{2} \right\} \\&\propto \frac{1}{ \alpha ^{2} \left( \sigma ^{2} \right) ^{k+4+n_{0}} } \left\{ \sigma ^{2} + \frac{ q_{1}^{2} }{ 2 n_{1} \left( 1 - \frac{n_{1}}{n} \right) } \right\} . \end{aligned}$$

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

Appendix B: Propriety 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

$$\begin{aligned} \pi \left( \alpha , \beta _{0}, \varvec{\beta }, \sigma ^{2}, \{ \lambda _{i}^{A} \}_{i \in {\mathcal {C}}_{0}} \right) \propto&| \alpha |^{n_{1}-1} \left( \sigma ^{2} \right) ^{-(k + n_{0} + n + 4) / 2} \exp \left\{ -\frac{1}{\sigma ^{2}} \sum _{i = 1}^{n} \left( d_{i} - \alpha y_{i} \right) ^{2} \right\} \\&\times I \left( \alpha> 0 \right) I \left( \sigma ^{2}> 0 \right) \left\{ \frac{1}{ \sigma ^{2} } + \frac{ q_{1}^{2} }{ 2 n_{1} \left( 1 - \frac{n_{1}}{n} \right) } \right\} ^{1/2} \\&\times \prod _{i \in {\mathcal {C}}_{0}} I \left( \lambda _{i}^{A}> 0 \right) \\ <&| \alpha |^{n_{1}-1} \left( \sigma ^{2} \right) ^{-(k + n_{0} + n + 4) / 2} \exp \left\{ -\frac{1}{\sigma ^{2}} \sum _{i = 1}^{n} \left( d_{i} - \alpha y_{i} \right) ^{2} \right\} \\&\times I \left( \alpha> 0 \right) I \left( \sigma ^{2}> 0 \right) \left( \frac{1}{\sigma } + \frac{q_{1}}{ \sqrt{2 n_{1} \left( 1 - \frac{n_{1}}{n} \right) } } \right) \\&\times \prod _{i \in {\mathcal {C}}_{0}} I \left( \lambda _{i}^{A} > 0 \right) . \end{aligned}$$

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 have

$$\begin{aligned} \int \pi \left( \alpha , \beta _{0}, \varvec{\beta }, \sigma ^{2}, \{ \lambda _{i}^{A} \}_{i \in {\mathcal {C}}_{0}} \right) d \sigma ^{2} d \beta _{0} d \varvec{\beta } <&(\text {some finite constant}) \\&\times | \alpha |^{n-1} \left( \varvec{e}^{\prime } \varvec{M} \varvec{e} \right) ^{-(n_{0} + n + 1) / 2}, \end{aligned}$$

where $\varvec{M} = \varvec{I}_{n} - \varvec{Z} ( \varvec{Z}^{\prime } \varvec{Z} )^{-1} \varvec{Z}^{\prime }$, $\varvec{e}$ is the n-dimensional vector whose i-th row is defined as

$$\begin{aligned} e_{i} = {\left\{ \begin{array}{ll} p_{i} + \lambda _{i}^{A} - \alpha y_{i}, &{}\text {if } i \in {\mathcal {C}}_{0}, \\ p_{i} - \alpha y_{i}, &{}\text {if } i \in {\mathcal {C}}_{1}. \end{array}\right. } \end{aligned}$$

(8)

Let

$$\begin{aligned} f \left( \alpha , \{ \lambda _{i}^{A} \}_{i \in {\mathcal {C}}_{0}} \right) = \left( \alpha ^{2} \right) ^{(n-1) / 2} \left( \varvec{e}^{\prime } \varvec{M} \varvec{e} \right) ^{-(n_{0} + n + 1) / 2}. \end{aligned}$$

This function is integrable on any compact subset A of the nonnegative orthant because of the following three reasons: (i) $| \alpha |^{(n-1)/2}$ is continuous and closed on A, (ii) $\varvec{e}^{\prime } \varvec{M} \varvec{e}$ is so as well, and (iii) $\varvec{e}^{\prime } \varvec{M} \varvec{e} > 0$ because of the rank condition.

Further, let

$$\begin{aligned} g \left( \alpha , \{ \lambda _{i}^{A} \}_{i \in {\mathcal {C}}_{0}} \right) = \left( \Vert \varvec{\xi } \Vert ^{2} \right) ^{(n-1) / 2} \left( \varvec{e}^{\prime } \varvec{M} \varvec{e} \right) ^{-(n_{0} + n + 1) / 2}, \end{aligned}$$

where $\varvec{\xi } = (\alpha , \{ \lambda _{i}^{A} \}_{i \in {\mathcal {C}}_{0}})$ and $\Vert \varvec{\xi } \Vert$ is the Euclidean norm of $\varvec{\xi }$. It is clear that $f \le g$ for any possible value of $\varvec{\xi }$. This function is integrable on A.

We will show that $g = O ( \Vert \varvec{\xi } \Vert ^{-(n_{0} + 2)})$, where the order notation denotes $|g| \le M \Vert \varvec{\xi } \Vert ^{-(n_{0} + 2)}$ for some constant M. The polar coordinate representation of $\varvec{\xi }$ leads to

$$\begin{aligned} \frac{ \Vert \varvec{\xi } \Vert ^{2} }{ \varvec{e}^{\prime } \varvec{M} \varvec{e} } = \frac{ d r^{2} }{ a + b r + c r^{2} }, \end{aligned}$$

where a, b, c, d are some functions of triangular functions and r is the radial distance from the origin. We note that $c r^{2} = ( \varvec{e} - \varvec{p} )^{\prime } \varvec{M} ( \varvec{e} - \varvec{p} )$, where $\varvec{p} = (p_{1}, \dots , p_{n})^{\prime }$. By the rank condition, $c r^{2} > 0$. So $c \ne 0$ as long as $r > 0$. Then, the left-hand side converges to 1/c as $\Vert \varvec{\xi } \Vert$ goes to infinity. Because

$$\begin{aligned} g \times \left( \Vert \varvec{\xi } \Vert ^{(n_{0} + 2)} \right) = \left( \frac{ \Vert \varvec{\xi } \Vert ^{2} }{ \varvec{e}^{\prime } \varvec{M} \varvec{e} } \right) ^{(n_{0} + n + 1) / 2}, \end{aligned}$$

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.

Appendix C: Gibbs Sampler

This section describes the Gibbs sampler for the Bayesian estimation of the linear model. It is implemented in the following six-step algorithm.

Step 1. Initialize the model parameters $(\alpha , \beta _{0}, \varvec{\beta }, \sigma ^{2}, \{ \lambda _{i}^{A} \}_{i \in {\mathcal {C}}_{0}})$.

Step 2. Generate $\alpha$ conditional on $\beta _{0}, \varvec{\beta }, \sigma ^{2}, \{ \lambda _{i}^{A} \}_{i \in {\mathcal {C}}_{0}}$.

The full conditional posterior density for $\alpha$ is proportional to

$$\begin{aligned} \pi \left( \alpha \mid \beta _{0}, \varvec{\beta }, \sigma ^{2}, \{ \lambda _{i}^{A} \}_{i \in {\mathcal {C}}_{0}} \right) \propto&\left| \alpha \right| ^{n_{1}-1} \\&\times \exp \left[ -\frac{1}{2 \sigma ^{2}} \left\{ \alpha ^{2} \left( \sum _{i = 1}^{n} y_{i}^{2} \right) - 2 \alpha \left( \sum _{i = 1}^{n} y_{i} d_{i} \right) \right\} \right] \\ =&\left| \alpha \right| ^{n_{1}-1} \exp \left( -\frac{R}{2} \alpha ^{2} + Q \alpha \right) , \end{aligned}$$

where $Q = \sigma ^{-2} \sum _{i = 1}^{n} y_{i} d_{i}$ (see Eq. (7) for $d_{i}$) and $R = \sigma ^{-2} \sum _{i = 1}^{n} y_{i}^{2}$. 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 this conditional density is

$$\begin{aligned} m&= \frac{ Q + \sqrt{ Q^{2} + 4 R \left( n_{1} - 1 \right) } }{ 2 R }, \end{aligned}$$

and the second derivative of the log conditional density evaluated at the mode m is given by

$$\begin{aligned} H&= -\frac{n_{1}-1}{m^{2}} - R. \end{aligned}$$

Then, the Taylor series expansion of the log conditional density around the mode gives the proposal $TN_{(0, \infty )} (m, -H^{-1})$. Let $\alpha ^{(-1)}$ and ${\tilde{\alpha }}$ be the sample recorded in the previous Markov chain and the candidate drawn from the proposal density, respectively. The candidate is accepted with probability

$$\begin{aligned}&\min \left[ 1, \frac{ g \left( {\tilde{\alpha }} \right) }{ g \left( \alpha ^{(-1)} \right) } \right] , \end{aligned}$$

where

$$\begin{aligned}&g \left( \alpha \right) = \left| \alpha \right| ^{n_{1}-1} \exp \left\{ -\frac{1}{2} \left( R + H \right) \alpha ^{2} + \left( Q + m H \right) \alpha \right\} . \end{aligned}$$

Step 3. Generate $(\beta _{0}, \varvec{\beta })$ conditional on $\alpha , \sigma ^{2}, \{ \lambda _{i}^{A} \}_{i \in {\mathcal {C}}_{0}}$.

The full conditional distribution for $(\beta _{0}, \varvec{\beta })$ is the multivariate normal distribution, which is given by

$$\begin{aligned} \left( \beta _{0}, \varvec{\beta }^{\prime } \right) ^{\prime } \mid \alpha , \sigma ^{2}, \{ \lambda _{i}^{A} \}_{i \in {\mathcal {C}}_{0}} \sim&N \left( \varvec{b}_{1}, \varvec{B}_{1} \right) , \end{aligned}$$

where $\varvec{z}_{i}^{\prime } = (1, \varvec{x}_{i}^{\prime })$, $\varvec{B}_{1}^{-1} = \sigma ^{-2} \sum _{i = 1}^{n} \varvec{z}_{i} \varvec{z}_{i}^{\prime }$, and $\varvec{b}_{1} = \sigma ^{-2} \varvec{B}_{1} \sum _{i = 1}^{n} e_{i} \varvec{z}_{i}$ (see Eq. (8) for $e_{i}$).

Step 4. Generate $\sigma ^{2}$ conditional on $\alpha , \beta _{0}, \varvec{\beta }, \{ \lambda _{i}^{A} \}_{i \in {\mathcal {C}}_{0}}$.

Because the full conditional density for $\sigma ^{2}$ is also nonstandard, the MH algorithm is applied as well. The proposal for this generation is a mixture of the inverse gamma distributions $IG (r_{1}/2, S_{1}/2)$ and $IG ((r_{1}+1)/2, S_{1}/2)$ with the respective weights w and $1-w$, where $r_{1} = k + 2 + n + n_{0}$, $S_{1} = \sum _{i = 1}^{n} (d_{i} - \alpha y_{i})^{2}$, and

$$\begin{aligned} w = \frac{ q_{1} }{ q_{1} + \sqrt{ 2 n_{1} \left( 1 - \frac{n_{1}}{n} \right) } }. \end{aligned}$$

The acceptance probability for a candidate ${\tilde{\sigma }}^{2}$ in terms of the previous sample $\sigma ^{2, (-1)}$ is given by

$$\begin{aligned} \min \left[ 1, \frac{ g \left( {\tilde{\sigma }}^{2} \right) }{ g \left( \sigma ^{2, (-1)} \right) } \right] , \quad \text {where } g (\sigma ^{2}) = \frac{ \sqrt{ \left( 1 - w \right) ^{2} + w^{2} \sigma ^{2} } }{ 1 - w + w \sigma }. \end{aligned}$$

Step 5. Generate $\lambda _{i}^{A}$ conditional on $\alpha , \beta _{0}, \varvec{\beta }, \sigma ^{2}$ for $i \in {\mathcal {C}}_{0}$.

The full conditional distribution for $\lambda _{i}^{A}$ is the truncated normal distribution, which is given by

$$\begin{aligned} \lambda _{i}^{A} \mid \alpha , \beta _{0}, \varvec{\beta }, \sigma ^{2} \sim&TN_{(0, \infty )} \left( -p_{i} + \alpha y_{i} + \beta _{0} + \varvec{x}_{i}^{\prime } \varvec{\beta }, \sigma ^{2} \right) . \end{aligned}$$

Step 6. Repeat Step 2 through Step 5.

Appendix D: Chain Paths

To have a closer look at the convergence of the Markov chain, this section provides chain paths of model parameters for the numerical analysis as well as the empirical analysis.

First, Fig. 10 shows chain paths for samples under the Jeffreys’ prior with the original dataset.

Fig. 10

Chain paths with the original dataset under the Jeffreys’ prior

These paths seem to achieve convergence. Because other paths (ones under the proper priors or ones with transformed dataset) are very similar, we suppress them.

Fig. 11

Chain paths for tomato

Next, Fig. 11 shows chain paths for the empirical analysis. Because the number of prefectures of no production amount is small, the MCMC samples for tomato are picked up to draw this figure. From this figure, all chains seem to reach convergence, and our proposed method performs well for this dataset.

Appendix E: Histograms of Climate-Related Variables

Figure 12 shows the histograms of climate-related variables that are used in the empirical analysis.

Fig. 12

Histograms of climate-related variables

The width of intervals is determined by the rule proposed by Freedman and Diaconis (1981), that is, $2 L / n^{1/3}$ where L and n are the IQR length and the sample size, respectively. Each histogram is overlaid with the kernel density estimate that uses the Epanechnikov kernel. From this figure, the variation in the temperature is higher than those of other variables in terms of the unit we use.

Appendix F: Analysis of Other Parameters

The regression coefficients except for the intercept are summarized in Fig. 13.

Fig. 13

Posterior summary I

The broader credible interval for $\gamma$ is attributed to the larger number of prefectures of no production. Marginal posterior distributions for remaining parameters $\delta _{0}$ and $\tau$ are summarized in Fig. 14.

Fig. 14

Posterior summary II