Vegetable market and wholesale market in Japan

The average farm size in Japan is very small relative to many Western countries at 3.1 ha (MAFF 2021a). As a result, Japanese producers rely on cooperatives to collect and ship agricultural products (OECD 2014). In Japan, collection and shipment are coordinated by the prefecture-level producer cooperatives such as the Japan Agricultural Cooperative.Footnote ¹ Rather than the producer negotiating on their own, producer cooperatives collect farm commodities at the prefecture level and ship to wholesalers (Higaki, Gunjal, and Coffin Reference Higaki, Gunjal and Coffin2001), which have the advantage of increasing countervailing market power (Sexton and Xia Reference Sexton and Xia2018).

While average farm sizes remain small, production areas are becoming more concentrated in specific prefectures that enable local cooperatives to exercise market power (Matsuda and Kurokawa Reference Matsuda and Kurokawa1996). Therefore, the high market share of oligopolistic producer cooperatives is a vital policy concern in the vegetable market (MAFF 2021b). Table 1 provides the Hirschman–Herfindahl index (HHI) and the four-firm concentration ratio (CR4).Footnote ² Based on the HHI, the producer side of the market appears to be reasonably competitive. However, both HHI and CR4 have increased over time for most items. Thus, there is a need to examine the possibility of increasing oligopolistic power on the producer side.

Table 1. HHI and CR4 in 1992 and 2014

Source: MAFF (2014a) and Matsuda and Kurokawa (Reference Matsuda and Kurokawa1996).

Notes: CR4 is the sum of market shares for the four largest producers at the prefecture level. CR4 ∈ (0, 100 percent]; HHI is the sum of the squared percentage shares for all 47 prefectures. HHI ∈ (0, 10, 000]. ${\rm HHI} = \mathop \sum \limits_{k = 1}^{47} \left({{{q_k} \over {\mathop \sum \nolimits_{i = 1}^{47} q_l}} \times 100} \right)^2\semicolon \;k, \;\;l\in \{ {1, \;\ldots 47} \}$, where q _l is the production volume from the lth prefecture. The HHI in1992 is the Herfindahl index from Matsuda and Kurokawa (Reference Matsuda and Kurokawa1996) multiplied by 1,000.

On the retail side of the vegetable market, the supermarket is the primary entity that sells to consumers. According to the report by the Organisation for Economic Co-operation and Development (OECD), the 5-firm concentration ratio in terms of sales in the retail food market is about 12 percent at the national level in Japan, which means low concentration at the national level. However, at the regional level, local supermarkets often have significant market shares (50 percent or more, Kojima and Fuchikawa Reference Kojima and Fuchikawa2010; OECD 2014).

The supply chain of vegetables is summarized in Figure 1. While some fresh vegetables are sold directly from producers to retailers or consumers, most are mainly sold through the wholesale market in Japan. In 2016, the latest data we have, the wholesale market accounted for almost 70 percent of fresh vegetables sold. Imported vegetables accounted for 21 percent of Japan's total vegetable supply in 2019. However, the majority (74 percent, MAFF 2019) of imports are processed vegetables rather than fresh vegetables. Therefore, we ignore imported vegetables in this study.

Figure 1. Flow of Vegetables in the Japanese Market.

In the wholesale market in Japan, there are two players: one is the producer (seller), who cosigns their products to the wholesaler. The other is the retailer (buyer), who purchases the products from the wholesaler through the intermediate wholesaler. The Wholesale Market Act regulates vegetable trading in the wholesale market. Generally, the wholesale market in Japan sells vegetables, fruits, and seafood. This wholesale market system was established to increase the efficiency of domestic distribution in vegetables (Higaki, Gunjal, and Coffin Reference Higaki, Gunjal and Coffin2001) by reducing transaction costs (Ichinose Reference Ichinose2018). Vegetables sold at the Ota Market, one of the wholesale markets in Tokyo, creates a reference price for the domestic vegetable market (Byung and Nagaki Reference Byung and Nagaki2006). At the wholesale market, both pair-wise trading and auction trading take place. Wholesalers approved by the local authority sell vegetables consigned by producers primarily through pair-wise trading. By the principle of prohibition of entrustment refusal in the Wholesale Market Act, wholesalers are restricted from manipulating the price through quantity adjustments. To participate as a buyer, one must be registered as an intermediate wholesaler and gain approval from the local authority (Tokyo Central Wholesale Market 2021). The registered intermediate wholesalers are allowed to purchase products from the registered wholesaler in the wholesale market and resell them to buyers (retailers).

Previous market power studies

Many economists have examined issues of concentration, competition, and market power in the food supply chain. The New Industrial Organization (NEIO) framework has been widely used to investigate market power issues in agricultural industries (e.g., Schroeter Reference Schroeter1988; Suzuki and Kaiser Reference Suzuki and Kaiser2006; Lopez, He, and Azzam Reference Lopez, He and Azzam2018). To date, several studies have investigated bilateral imperfect competition (e.g., Azzam Reference Azzam1996; Schroeter, Azzam, and Zhang Reference Schroeter, Azzam and Zhang2000; Raper, Love, and Shumway Reference Raper, Love and Shumway2000; Boetel and Liu Reference Boetel and Liu2010; Chung and Tostão Reference Chung and Tostão2012; Park, Chung, and Raper Reference Park, Chung and Raper2017). Azzam (Reference Azzam1996) presents the bilateral oligopoly model to estimate the degree of vertical power balance at two stages.

Sexton and Xia (Reference Sexton and Xia2018) argue that there are shortcomings in the estimation of market power parameters: (a) the conduct parameters may result in inaccurate market power measurements (Corts Reference Corts1999), (b) the simplistic estimation of the identification methods due to the lack of data (e.g., Perekhozhuk et al. Reference Perekhozhuk, Glauben, Grings and Teuber2017), and (c) a data limitation, especially the lack of retailers’ wholesale cost data. However, despite these criticisms, the original model by Azzam (Reference Azzam1996) is still advantageous because the model is simple and highly extensible, requiring fewer data. Recently, this model has been applied to the US beef processors and retailers (Chung, Eom, and Yang Reference Chung, Eom and Yang2014), the Belgian pork production chain (Maes, Vancauteren, and Van Passel Reference Maes, Vancauteren and Van Passel2019), and international trade of agricultural commodities (Yamaura and Xia Reference Yamaura and Xia2016). In addition, KSK extended this model to simultaneously estimate the vertical and horizontal competition among three stages in the Japanese milk market.

In the vegetable market in Japan, rising concentration needs to be taken into account for the producer side due to the existence of large producer cooperatives. The oligopoly behavior on the producer side has been empirically estimated in the Japanese onion market (Matsuda and Kurokawa Reference Matsuda and Kurokawa1996) and potato market (Higaki, Gunjal, and Coffin Reference Higaki, Gunjal and Coffin2001). Both studies have found that producers do not exert monopolistic power. However, these findings might be biased as the imperfect competition on the retailer side is not considered.

In this study, we develop and extend the bilateral oligopoly approach pioneered by KSK to estimate the market power in the vegetable market in Japan. Specifically, KSK did not take the retailers’ marginal variable costs into account, but we do. This study adopts an economic model of a vertical and horizontal relationship between producers and retailers that considers the retailers’ marginal cost, which enables the derivation of the unique parameter.

Setup

Consider two players in the wholesale market. The sellers are producer cooperatives, while the buyers are retailers (e.g., supermarkets). We assume imperfect competition for both buyers and sellers. In this model, price is determined by the “vertical power balance.” Furthermore, the model assumes that there exists an upper limit and lower limit on the price. The upper limit is the maximum price the producer cooperatives can extract from retailers, while the lower limit is the minimum price the retailers can extract from cooperatives. For instance, if cooperatives have market power over retailers, the price should be closer to the upper-limit level.

Here, we define that PW^U as the maximum price on the producer side and PW^L as the minimum price on the retailer side in terms of the range of price formation. The maximum price and the minimum price denote the limit of the possible value range of the wholesale price. With the introduction of vertical power parameters relative to producers, we can express the actual wholesale price as follows:

(1)$$\matrix{ {{\rm PW} = {\rm \omega P}{\rm W}^{\rm U} + ( {1-{\rm \omega }} ) {\rm P}{\rm W}^{\rm L}.} \cr } $$

Consequently, the producers’ degree of vertical power parameters relative to the retailers’ ω is 1 when the producers possess a completely dominant vertical market power. The formation of the maximum price PW^U also considers a horizontal market power parameter, θ, that indicates the producers’ degree-of-horizontal-competition parameter, which ranges from zero to one, described in equation (5). The formation of the minimum price PW^L takes into account a horizontal market power parameter, λ, that indicates the retailers’ degree-of-horizontal-competition parameter, which ranges from zero to one, described in equation (9). Following Azzam (Reference Azzam1996), our study defines the maximum price as the seller monopoly price and the minimum price as the buyer monopsony price.

Here, we make the following assumptions:

• Assumption 1: This study models vegetable producers at the prefecture level following Higaki, Gunjal, and Coffin (Reference Higaki, Gunjal and Coffin2001). As mentioned in the section “Vegetable market and wholesale market in Japan”, the producer cooperatives collect and ship produce consigned by producers at the prefecture level. Following Matsuda and Kurokawa (Reference Matsuda and Kurokawa1996), the degree of horizontal power parameter measures the competition/coordination relationship mainly among prefectures as production units on vegetable pricing. Therefore, this article considers the market power of producers as the producers’ collective power, not the market power of individual producers.

• Assumption 2: Because of the Wholesale Market Act and the regulation in each market, wholesalers and intermediate wholesalers cannot manipulate the price and the quantity.

• Assumption 3: We assume that all vegetables are homogeneous products among the producers at the prefecture level, to focus on not product differentiation but competition in the vegetable market in Japan. Because of the limitation of sellers’ and buyers’ transaction data, our model estimates the market power parameters using aggregated market data. This differs from the concept of bargaining power, which considers product differentiation (e.g., Bonnet and Bouamra-Mechemache Reference Bonnet and Bouamra-Mechemache2016).

• Assumption 4: We assume that the degree of oligopolistic power measures the deviation from purely monopolistic and competitive behavior modes (Appelbaum Reference Appelbaum1982).

Theoretical model

The theoretical model extended from KSK begins with the formulation of price-setting behavior at the producer cooperative level. When producer cooperatives sell vegetables to retailers at the wholesale market, profit for the producer side is given by

(2)$$\mathop {\max }\limits_Q {\rm \pi }_{i, t} = {\rm P}{\rm W}_{i, t}Q_{i, t}-{\rm TC}_{i, t}^{\rm p} , \;$$

where PW_i,t = PW(Q _i,t) is the wholesale price of the product i at the period t, Q _i,t is the quantity of the product i to sell, and ${\rm TC}_{i, t}^{\rm p}$ is the total cost of producers.

The first-order condition for profit maximization with the producers who have market power can be rewritten as follows:

(3)$$\matrix{ {{\rm P}{\rm W}_{i, t}\left({1-\displaystyle{1 \over {{\rm \eta }_i^{\rm p} }}} \right) = {\rm MC}_{i, t}^{\rm p} , \;} \cr } $$

where ${\rm MC}_{i, t}^{\rm p}$ is the producers’ marginal cost, and ${\rm \eta }_i^{\rm p}$ is the absolute value of the product i's price elasticity of demand faced with producers. This study follows the framework of KSK, Azzam (Reference Azzam1996), and Matsuda and Kurokawa (Reference Matsuda and Kurokawa1996); hence, the price elasticity of demand is assumed to be constant.

By introducing the degree-of-competition parameter θ_i,t with 0 ≤ θ_i,t ≤ 1, equation (3) can be generalized as follows:

(4)$$\matrix{ {{\rm P}{\rm W}_{i, t}\left({1-\displaystyle{{{\rm \theta }_{i, t}} \over {{\rm \eta }_i^{\rm p} }}} \right) = {\rm MC}_{i, t}^{\rm p} , \;} \cr } $$

where θ_i,t denotes the producers’ degree-of-horizontal-competition parameter following KSK. Equation (4) expresses the oligopoly market between perfect competition and monopoly. If θ_i,t is equal to 1, the producer side acts as a monopolist; if θ_i,t is equal to zero, the market is characterized by perfect competition.

In equation (2), we assume that the vertical power balance between producer cooperatives and retailers is 1:0. Hence, PW_i,t, wholesaler price in equation (4), denotes the upper price at which producer cooperatives can sell to retailers. PW^U denote the upper price in the wholesale market:

(5)$$\matrix{ {{\rm PW}_{i, t}^{\rm U} = \displaystyle{{{\rm MC}_{i, t}^{\rm p} } \over {\left({1-\displaystyle{{{\rm \theta }_{i, t}} \over {{\rm \eta }_i^{\rm p} }}} \right)}}.} \cr } $$

Next, consider the retail level. When retailers buy vegetables from producer cooperatives at the wholesale market, the profit of the retailer side is given by

(6)$$\mathop {{\rm max}}\limits_Q {\rm \pi }_{i, t} = ( {\rm P}{\rm R}_{i, t}-{\rm P}{\rm W}_{i, t}) Q_{i, t}-{\rm TC}_{i, t}^{\rm r} , \;$$

where PW_i,t = PW(Q _i,t) is the purchase price of the product i at the period t, PR_i,t = PR(Q _i,t) is the retail price, Q _i,t is the quantity of the product i to buy, and ${\rm TC}_{i, t}^{\rm r}$ is the total cost of retailers.

The first-order condition for profit maximization with the retailers who have monopsony power can be rewritten as follows:

(7)$$\matrix{ {{\rm P}{\rm R}_{i, t}\left({1-\displaystyle{1 \over {{\rm \eta }_i^{\rm r} }}} \right) = {\rm P}{\rm W}_{i, t}\left({1 + \displaystyle{1 \over {{\rm \varepsilon }_i}}} \right) + {\rm MC}_{i, t}^{\rm r} , \;} \cr } $$

where ${\rm MC}_{i, t}^{\rm r}$ is the retailers’ marginal cost for selling and ${\rm \eta }_i^{\rm r}$ is the absolute value of the price elasticity of demand faced with retailers. Subsequently, ɛ_i is the absolute value of the price elasticity of supply faced with retailers. While KSK estimated the range of power parameters, our approach identifies the parameter uniquely by considering both V ^r and PR in the theoretical model.

Following the framework of KSK, Azzam (Reference Azzam1996), and Matsuda and Kurokawa (Reference Matsuda and Kurokawa1996), the price elasticity of supply is also assumed to be constant.

By introducing the degree-of-competition parameter λ_i,t with 0 ≤ λ_i,t ≤ 1, equation (7) can be generalized as follows:

(8)$$\matrix{ {{\rm P}{\rm R}_{i, t}\left({1-\displaystyle{{{\rm \lambda }_{i, t}} \over {{\rm \eta }_i^{\rm r} }}} \right) = {\rm P}{\rm W}_{i, t}\left({1 + \displaystyle{{{\rm \lambda }_{i, t}} \over {{\rm \varepsilon }_i}}} \right) + {\rm MC}_{i, t}^{\rm r} , \;} \cr } $$

where λ_i,t is the retailers’ horizontal-competition parameter and 0 ≤ λ_i,t ≤ 1. Equation (8) expresses the oligopoly market between perfect competition and monopoly. In this study, we assume that the competition intensity in a wholesale market is equal to that in a retail market based on assumption 2.Footnote ³

In equation (6), we assume that the vertical power balance between retailers and producer cooperatives is 1:0. Hence, PW_i,t, purchase price in equation (8), denotes the lower price at which retailers can buy from producers. PW^L denotes the lower price in the wholesale market:

(9)$$\matrix{ {{\rm PW}_{i, t}^{\rm L} = \displaystyle{{{\rm P}{\rm R}_{i, t}\left({1-\displaystyle{{{\rm \lambda }_{i, t}} \over {{\rm \eta }_i^{\rm r} }}} \right)-{\rm MC}_{i, t}^{\rm r} } \over {\left({1 + \displaystyle{{{\rm \lambda }_{i, t}} \over {{\rm \varepsilon }_i}}} \right)}}.} \cr } $$

Let the actual price at the wholesale market in Japan be set between the upper price and the lower price by the market power of producers and retailers. Taking into account the above relations of (5) and (9) enables us to rewrite equation (1) into the practical trading price equation at the wholesale market as follows:

(10)$$\matrix{ {{\rm P}{\rm W}_{i, t} = {\rm \omega }_{i, t}{\rm PW}_{i, t}^{\rm U} + ( {1-{\rm \omega }_{i, t}} ) {\rm PW}_{i, t}^{\rm L} , \;} \cr } $$

where ω_i,t is the product i's vertical power parameter between producers and retailers at the period t (0 ≤ ω_i,t ≤ 1). Equation (10) is consistent with the theoretical consequences of the generalized Nash bargaining model of Prasertsri and Kilmer (Reference Prasertsri and Kilmer2008).

Therefore,

(11)$$\matrix{ {{\rm P}{\rm W}_{i, t} = {\rm \omega }_{i, t}\displaystyle{{{\rm MC}_{i, t}^{\rm p} } \over {\left({1-\displaystyle{{{\rm \theta }_{i, t}} \over {{\rm \eta }_i^w }}} \right)}} + ( {1-{\rm \omega }_{i, t}} ) \displaystyle{{{\rm P}{\rm R}_{i, t}\left({1-\displaystyle{{{\rm \lambda }_{i, t}} \over {{\rm \eta }_i^r }}} \right)-{\rm MC}_{i, t}^{\rm r} } \over {\left({1 + \displaystyle{{{\rm \lambda }_{i, t}} \over {{\rm \varepsilon }_i}}} \right)}}.} \cr } $$

Following KSK, we assume that TC  =  VQ +  F, where TC is the total cost, V is the average variable cost, F is the fixed cost, and Q is the quantity of the product. In this case, the producers’ marginal cost ${\rm MC}_{i, t}^{\rm p}$ is equal to V ^p. For the retailers’ marginal cost, KSK assumed that the retailers’ marginal cost MC^r is equal to zero because most milk retail costs can be considered as fixed costs. In this study, however, MC^r is equal to V ^r due to vegetables’ packaging and shipment fees, assuming no economies of scale. Like KSK, we also assume that the demand elasticity in the wholesale markets and the demand elasticity in the retail markets are the same; ${\rm \eta }_i^{\rm p} = {\rm \eta }_i^{\rm r} = {\rm \eta }_i$. It is also assumed that the elasticities of price η_i and ɛ_i are constant to simplify the estimation. For instance, if this assumption is relaxed, η_i increases or ɛ_i increases, ω_i,t decreases (see Appendix B for more details).

Rewriting equation (11) yields

(12)$$\matrix{ {{\rm P}{\rm W}_{i, t} = {\rm \omega }_{i, t}\displaystyle{{V_{i, t}^{\rm p} } \over {\left({1-\displaystyle{{{\rm \theta }_{i, t}} \over {{\rm \eta }_i}}} \right)}} + ( {1-{\rm \omega }_{i, t}} ) \displaystyle{{{\rm P}{\rm R}_{i, t}\left({1-\displaystyle{{{\rm \lambda }_{i, t}} \over {{\rm \eta }_i}}} \right)-V_{i, t}^{\rm r} } \over {\left({1 + \displaystyle{{{\rm \lambda }_{i, t}} \over {{\rm \varepsilon }_i}}} \right)}}, \;} \cr } $$

where

(13)$$\matrix{ {\;0 \le {\rm \omega }_{i, t} \le 1, \;\;0 \le {\rm \theta }_{i, t} \le 1, \;\;0 \le {\rm \lambda }_{i, t} \le 1.} \cr } $$

In equation (12), while KSK assumes that $V_{i, t}^{\rm r} = 0$, we relax this assumption and extend the model to include both V ^r and PR. This allows us to uniquely obtain market power parameters instead of obtaining a range of parameters as in KSK. However, equation (12) has two theoretically similar parameters, which can lead to multicollinearity problems (see Appendix C for more details). Thus, we use a nonlinear programming method, as described below, instead of regression methods in this study.

With constraints (13), and ${\rm PW}_{i, t}^{\rm U}$ from equation (5) and ${\rm PW}_{i, t}^{\rm L}$ from equation (9) should be positive and ${\rm PW}_{i, t}^{\rm U}$ should be larger than ${\rm PW}_{i, t}^{\rm L}$, a range of values can be examined by adding the following constraints:

(14)$$\matrix{ {\;0 \le {\rm \omega }_{i, t} \le 1\;, \;\;0 \le {\rm \theta }_{i, t} \le {\rm \eta }_i, \;\;0 \le {\rm \lambda }_{i, t} \le {\rm \eta }_i, \;} \cr {\displaystyle{{{\rm P}{\rm R}_{i, t}\left({1-\displaystyle{{{\rm \lambda }_{i, t}} \over {{\rm \eta }_i}}} \right)-V_{i, t}^{\rm r} } \over {\left({1 + \displaystyle{{{\rm \lambda }_{i, t}} \over {{\rm \varepsilon }_i}}} \right)}} \le \displaystyle{{V_{i, t}^{\rm p} } \over {\left({1-\displaystyle{{{\rm \theta }_{i, t}} \over {{\rm \eta }_i}}} \right)}}.} \cr } $$

Accordingly, this model estimates the degree of vertical power balance between producers and retailers based on the degree of horizontal competition in each stage of the market. The degree of power balance can be obtained as a mean value in the designed period through this method.

Data

We use annual data for the period 2007–2014 for 14 vegetables: radish, carrot, Chinese cabbage, cabbage, spinach, leek, eggplant, tomato, cucumber, paprika, taro, onion, lettuce, and potato, shown in Table 2. These 14 vegetables are chosen because the producers’ and retailers’ cost data are available for the 14 vegetables from the MAFF's survey. Annual data are used since seasonal fluctuations are stronger at the daily, weekly, monthly, or even quarterly levels. Further, producers’ and retailers’ cost data are available only at the annual level. Data on wholesale prices, retail prices, and retail variable costs are aggregated from Tokyo, representing the consumption area. The producer variable costs are at the prefecture-level data.

Table 2. Descriptive statistics for prices and variable costs by items

Note: The unit of dependent variables is JPY per kilogram. 100 yen = 0.92 US dollars in 2007.

The wholesale price PW is based on the average wholesale price from the “Market Statistics” (Tokyo Metropolitan Government 2021) collected by the Metropolitan Central Wholesale Market of the Tokyo Metropolitan Government. We use the average wholesale price (yen/kg) each year for the 14 items.

The retail price PR is obtained from the “Retail Price Survey” (MIC 2014) from the MIC.

Producer variable costs V ^p are based on the “Management Statistics Survey by Item” from the “Agricultural Management Statistics Survey” (MAFF 2013) obtained by the MAFF. Producer variable costs are the average value for several production areas based on this surveyFootnote ⁴ and are calculated by dividing the agricultural fluid cost in each fiscal year by the sales volume (per household). To transform into real values, we use the material price index from the “Statistical Survey on Prices in Agriculture” (MAFF 2014b) in 2007 as the base year.

Retail variable costs V ^r are taken from the “Price Formation Survey by Food Distribution Stage” (MAFF 2017) from MAFF each year. Retail variable costs are measured as the packaging material costs and payment fare for each vegetable. The MAFF conducts this survey on distributive costs from three market levels: producers, wholesalers, and retailers. To obtain the price per unit of weight, we multiplied the total retail costs by the ratio of retail variable cost per retailer. The total retail cost is based on the retail cost of the “Price Formation Survey by Food Distribution Stage.” In the survey, the MAFF conducts preliminary calculations of the distribution expense at three stages on 14 vegetables per 100 kg. The ratio of retail variable cost per retailer is calculated by dividing the sum of packaging material costs and payment fare by retail costs from each year of the “Price Formation Survey by Food Distribution Stage.”

The price elasticities of demand and supply come from Nagakubo et al. (Reference Nagakubo, Hui, Sato and Suzuki2018) and are listed in Table 3.

Table 3. Price elasticity of demand and supply

Source: Nagakubo et al. (Reference Nagakubo, Hui, Sato and Suzuki2018)

Note: η is the absolute value.

The sales ratio of products via the wholesale markets is calculated by dividing the wholesale quantity shipped to the wholesale market by the gross domestic product (Table 4). When the value is greater than 1, it is set to 1. The wholesale quantity is based on annual statics of the wholesale quantity of vegetables from the “Fruit and Vegetable Wholesale Market Research” (MAFF 2014c). We use a total quantity for 14 items shipped to all wholesale markets in Japan. We use the gross domestic product data for 14 items from the “Food Balance Sheet” (MAFF 2014d).

Table 4. Descriptive statistics for the sales ratio of products traded via the wholesale markets and concentration ratio of production

The concentration ratio of production is based on the “Crop Survey” (MAFF 2014e). Following Matsuda and Kurokawa (Reference Matsuda and Kurokawa1996), we use the market share of the top four prefectures as the concentration ratio.

Model estimation

Market power in the vegetable market

Although KSK estimates power parameters ω_i,t, θ_i,t, and λ_i,t by regressions, we use a nonlinear programming model to reflect parameter constraints explicitly. That is, there should be a constraint on the upper and lower limits of parameters, as shown in Table 5. Another advantage of nonlinear programming is that the model is applicable even when some variables are highly correlated that could cause multicollinearity problems using the conventional regression approach (Judd, Maliar, and Maliar Reference Judd, Maliar and Maliar2011).

Table 5. Lower and upper limits of the parameters

The estimated value of PW can be calculated from Equation (12) as follows:

$$\matrix{ {\widehat{{{\rm P}{\rm W}_{i, t}}} = {\rm \omega }_{i, t}\displaystyle{{V_{i, t}^{\rm p} } \over {\left({1-\displaystyle{{{\rm \theta }_{i, t}} \over {{\rm \eta }_i}}} \right)}} + ( {1-{\rm \omega }_{i, t}} ) \displaystyle{{{\rm P}{\rm R}_{i, t}\left({1-\displaystyle{{{\rm \lambda }_{i, t}} \over {{\rm \eta }_i}}} \right)-V_{i, t}^{\rm r} } \over {\left({1 + \displaystyle{{{\rm \lambda }_{i, t}} \over {{\rm \varepsilon }_i}}} \right)}}.} \cr } \eqno ( 12^{\prime}) $$

Similarly, the estimated values of PR, V ^p, and V ^r are calculated from equation (12) as follows:

$$\matrix{ {\widehat{{{\rm P}{\rm R}_{i, t}}} = \displaystyle{1 \over {\left({1-\displaystyle{{{\rm \lambda }_{i, t}} \over {{\rm \eta }_i}}} \right)}}\left\{{\displaystyle{1 \over {( {1-{\rm \omega }_{i, t}} ) }}\left({1 + \displaystyle{{{\rm \lambda }_{i, t}} \over {{\rm \varepsilon }_i}}} \right)\left[{{\rm P}{\rm W}_{i, t}-{\rm \omega }_{i, t}\displaystyle{{V_{i, t}^{\rm p} } \over {\left({1-\displaystyle{{{\rm \theta }_{i, t}} \over {{\rm \eta }_i}}} \right)}}} \right] + V_{i, t}^{\rm r} } \right\}, \;} \cr } $$

$$\matrix{ {\widehat{{V_{i, t}^{\rm p} }} = \displaystyle{{\left({1-\displaystyle{{{\rm \theta }_{i, t}} \over {{\rm \eta }_i}}} \right)} \over {{\rm \omega }_{i, t}}}\left[{{\rm P}{\rm W}_{i, t}-( {1-{\rm \omega }_{i, t}} ) \displaystyle{{{\rm P}{\rm R}_{i, t}\left({1-\displaystyle{{{\rm \lambda }_{i, t}} \over {{\rm \eta }_i}}} \right)-V_{i, t}^{\rm r} } \over {\left({1 + \displaystyle{{{\rm \lambda }_{i, t}} \over {{\rm \varepsilon }_i}}} \right)}}} \right], \;} \cr } \eqno ( 12 ^{\prime\prime} ) $$

$$\matrix{ {\widehat{{V_{i, t}^{\rm r} }} = {-}\left\{{\displaystyle{1 \over {( {1-{\rm \omega }_{i, t}} ) }}\left({1 + \displaystyle{{{\rm \lambda }_{i, t}} \over {{\rm \varepsilon }_i}}} \right)\left[{{\rm P}{\rm W}_{i, t}-{\rm \omega }_{i, t}\displaystyle{{V_{i, t}^{\rm p} } \over {\left({1-\displaystyle{{{\rm \theta }_{i, t}} \over {{\rm \eta }_i}}} \right)}}{\rm \;}} \right]-{\rm P}{\rm R}_{i, t}\left({1-\displaystyle{{{\rm \lambda }_{i, t}} \over {{\rm \eta }_i}}} \right)} \right\}\;.} \cr } $$

For 14 vegetables within each period, the estimated value of the price and cost was simultaneously calculated from the sets of parameters. As the optimum value for 14 vegetables, respectively, we selected the combination of parameters that had the smallest sum of the difference between the actual price and cost and the estimated value of the price and cost given as

(15)$$ \eqalign{ \matrix{ {{\rm sum\;of\;difference} = {\left({\displaystyle{{\widehat{{{\rm P}{\rm W}_{i, t}}}-{\rm P}{\rm W}_{i, t}} \over {{\rm P}{\rm W}_{i, t}}}} \right)}^2 + {\left({\displaystyle{{\widehat{{V_{i, t}^{\rm p} }}-V_{i, t}^{\rm p} } \over {V_{i, t}^{\rm p} }}} \right)}^2 } \cr \qquad \qquad \qquad\qquad { + {\left({\displaystyle{{\widehat{{{\rm P}{\rm R}_{i, t}}}-{\rm P}{\rm R}_{i, t}} \over {{\rm P}{\rm R}_{i, t}}}} \right)}^2 + {\left({\displaystyle{{\widehat{{V_{i, t}^{\rm r} }}-V_{i, t}^{\rm r} } \over {V_{i, t}^{\rm r} }}} \right)}^2.} }} $$

The nonlinear programming solves the following minimization problem for each period t and the product i:

$$ \eqalign{ {\rm min}\,{\rm sum\;of\;difference} = \left({\displaystyle{{\widehat{{{\rm P}{\rm W}_{i, t}}}-{\rm P}{\rm W}_{i, t}} \over {{\rm P}{\rm W}_{i, t}}}} \right)^2 + \left({\displaystyle{{\widehat{{V_{i, t}^{\rm p} }}-V_{i, t}^{\rm p} } \over {V_{i, t}^{\rm p} }}} \right)^2 \cr + \left({\displaystyle{{\widehat{{{\rm P}{\rm R}_{i, t}}}-{\rm P}{\rm R}_{i, t}} \over {{\rm P}{\rm R}_{i, t}}}} \right)^2 + \left({\displaystyle{{\widehat{{V_{i, t}^{\rm r} }}-V_{i, t}^{\rm r} } \over {V_{i, t}^{\rm r} }}} \right)^2, \;} $$

(16)$$\matrix{ {{\rm s}{\rm .t}{\rm .\;}0 \le {\rm \theta }_{i, t} \le {\rm \eta }_i, \;\;0 \le {\rm \lambda }_{i, t} \le {\rm \eta }_i, \;\;0 \le {\rm \omega }_{i, t} \le 1, \;} \cr {\displaystyle{{{\rm P}{\rm R}_{i, t}\left({1-\displaystyle{{{\rm \lambda }_{i, t}} \over {{\rm \eta }_i}}} \right)-V_{i, t}^{\rm r} } \over {\left({1 + \displaystyle{{{\rm \lambda }_{i, t}} \over {{\rm \varepsilon }_i}}} \right)}} \le \displaystyle{{V_{i, t}^{\rm p} } \over {\left({1-\displaystyle{{{\rm \theta }_{i, t}} \over {{\rm \eta }_i}}} \right)}}, \;} \cr } $$

which is derived from equations (14) and (15). The initial values for the power parameters are taken from the range in steps of 0.1. By solving this model, we can derive the power parameters ω_i,t, θ_i,t, and λ_i,t.

Vertical power parameters

After deriving the vertical power parameter ω through solving the nonlinear programming model, we investigate the determinants of ω using Ordinary Least Squares (OLS). Specifically, we estimate the following equation using OLS:

(17)$$ \eqalign{ \matrix{ {{\rm \omega }_{i, t} = {\rm \beta }_0 + {\rm \beta }_1{\rm via\;the\;wholesale\;market}{\rm s}_{i, t} + {\rm \beta }_2{\rm top}4_{i, t} + {\rm \beta }_3d_{2011} + {\rm \beta }_4d_{2012} }\cr + {\rm \beta }_5{\rm ite}{\rm m}_i + u.} \cr } $$

For equation (17), we identify the following independent variables that are hypothesized to impact the vertical market power parameter for each product i in time period t: via the wholesale markets is the sales ratio of products traded in the wholesale markets relative to all markets, top4 is the market concentration ratio of the top four producers, d ₂₀₁₁ and d ₂₀₁₂ are time dummy variables for the time period of the Tohoku earthquake in 2011 and an emergency demand and supply adjustment performed by the MAFF in 2012 due to the price decline (MAFF 2012), and item_i is the dummy variable for the product i.