2. Theoretical Framework

According to Muthoo (Reference Muthoo1999, p. 2), “bargaining is any process through which the players on their own try to reach an agreement.” Bargaining can be bilateral or multilateral depending on whether two or more agents are involved in the process. Previous studies investigate either the general economic theory of bargaining or how it is modeled in the context of a specific industry. Furthermore, bargaining problems can be analyzed theoretically using axiomatic or strategic approaches. The axiomatic approach considers a unique solution, referred to as symmetric Nash bargaining, to a set of axiomatic assumptions (Nash, Reference Nash1950). In other words, when a two-person bargaining problem satisfies the axioms of scale invariance, symmetry, efficiency, and independence of irrelevant alternatives, the Nash bargaining solution is the unique solution.

Roth (Reference Roth1979) extends the axiomatic approach to solve for asymmetric Nash bargaining solutions. However, this approach does not provide the details of the bargaining process such as the bargaining periods and the number of the suggested strategies. Rubinstein (Reference Rubinstein1982) introduces the strategic approach, which addresses more details of the bargaining process, and Roth (Reference Roth1985) adds a risk component to this approach. Binmore, Rubinstein, and Wolinsky (Reference Binmore, Rubinstein and Wolinsky1986) advance the relationship between axiomatic and strategic approaches. Oczkowski (Reference Oczkowski1991) proposes a method to describe agents’ behavior under disequilibrium situations. Although this method is originally applied to disequilibrium in markets with quantity controls, the application can be broadened to nonquota equilibrium markets.

We extend the framework proposed in Oczkowski (Reference Oczkowski1991) to model the bargaining environment and determination of milk prices in Iran. To the best of our knowledge, there is no previous research that has modeled price determination in Iran's agro-food sector in general and the milk market specifically, using a bargaining framework.

In bilateral bargaining, two players (agents) have incentives to increase their joint payoffs while competing for higher gains. In this study, bilateral bargaining refers to the negotiations between (representatives of) the farmers and the processors to determine the national price of raw milk. Because there is no friction in the offer process and the settled price does not depend on the duration of negotiations, the generalized axiomatic Nash approach is used to characterize the bargaining process.

The generalized axiomatic Nash approach is the asymmetric form of a Nash solution. It is proper to use this model when the symmetry assumption does not hold for the game. The following function N represents the generalized Nash bargaining model for price negotiation between two players:

(1) $$\begin{equation} N = \max {\left( {{\pi _1} - {d_1}} \right)^\tau }{\left( {{\pi _2} - {d_2}} \right)^{1 - \tau }}, \end{equation}$$

where π₁ and π₂ are the payoffs for the farmers and the processors upon reaching an agreement. In the case where the bargaining process fails, the outcomes for the farmers and the processors would be d₁ and d₂, respectively. Similar to Bonnet and Bouamra-Mechemache (Reference Bonnet and Bouamra-Mechemache2015), the d_i’s are the payoffs that could be realized outside the negotiations. In that case, farmers could potentially seek other buyers, and processors could look for other suppliers. However, the negotiations are held at the national level, and this option is unattainable. Therefore, in practice d₁ = d₂ = 0. The relative bargaining power of farmers is denoted by τ and that of processors by 1 − τ in this model. If τ = 0.5, both players have equal bargaining power, and the asymmetric Nash bargaining solution is equivalent to the symmetric solution.

The agreement payoff for each party is a function of negotiated price P and the reservation price, which will be defined as

(2) $$\begin{equation} \begin{array}{@{}l@{}} {{\rm{\pi }}_{\rm{1}}}\; = \;P - {P^s} \qquad {\rm{farmers}}\\ {{\rm{\pi }}_2} = {P^d} - P \qquad {\rm{ processors,}} \end{array} \end{equation}$$

where P^s denotes the minimum price at which farmers are willing to sell the raw milk (i.e., supply reservation price). On the other hand, P^d is the maximum price at which processors would buy the raw milk (i.e., demand reservation price).

Therefore, farmers prefer any negotiated price above P^s (P > P^s), and processors prefer any price below P^d (P < P^d). If the suggested price in the negotiations stands below P^s or exceeds P^d, the bargaining process fails. Hence, both parties continue to negotiate until they each gain payoffs greater than zero. Because d₁ = d₂ = 0, equation (1) can be written as follows:

(3) $$\begin{equation} N = \max {\left( {P - {P^s}} \right)^\tau }{\left( {{P^d} - P} \right)^{1 - \tau }}. \end{equation}$$

To maximize the generalized Nash bargaining model, we differentiate equation (3) with respect to the negotiated price, P, and set the first-order condition equal to zero. Therefore,

(4) $$\begin{equation} P = \tau {P^d} + \left( {1 - \tau } \right){P^s}, \end{equation}$$

which implies that the negotiated price is a function of reservation prices and agents’ bargaining power. However, we cannot directly estimate this equation.

In order to estimate the generalized Nash bargaining solution to the bargaining model, following Oczkowski (Reference Oczkowski1991), we define the determinants of demand reservation price, P^d, and supply reservation price, P^s, and form a system that lays out the bargaining model as follows:

(5) $$\begin{equation} \begin{array}{@{}l@{}} P_t^s = f\left( {X_t^s} \right) + u_t^s\\ P_t^d = f\left( {X_t^d} \right) + u_t^d\\ {P_t} = \tau P_t^d + (1 - \tau )P_t^s + u_t^p, \end{array} \end{equation}$$

in which X^d_t and X^s_t are nonstochastic independent variables, and u^s_t, u^d_t, and u^p_t represent error terms with zero mean and constant variance of σ²_s, σ²_d, and σ²_p, respectively.

The supply reservation price of milk, P^s_t, is the minimum price that is acceptable by farmers and depends on cost factors and the quantity produced. Feed and labor are the major components of the total cost of milk production, about 70% and 12%, respectively. We specify the supply of raw milk as

(6) $$\begin{equation} P_t^s = {\beta _0} + {\beta _1}F{C_t} + {\beta _2}{L_t} + {\beta _3}Q_t^{pe} + u_t^s, \end{equation}$$

where FC_t and L_t denote the adjusted price of concentrated feedFootnote ⁷ and the wage rate, respectively. When feed and labor prices rise, we expect farmers to demand a higher price for raw milk in the negotiations. So, feed and labor prices have positive impacts on P^s_t. Hence, higher production cost results in upward pressure on the supply reservation price of milk.

Because equation (6) is specifying the supply side of the raw milk market, we expect a positive relationship between quantity of supply, Q^pe_t, and the price. The “actual” quantity of supply is unknown when negotiations are conducted at the beginning of each calendar year. So, we have to use the “estimated” total quantity of raw milk production in time t, which is unobserved at the time of decision making. To approximate this unobservable variable, following Nerlove and Bessler (Reference Nerlove, Bessler, Gardner and Rausser2001), we use data on the “observed” supply of raw milk (i.e., supply in previous time periods) and estimate Q_t = b ₁Q _{t − 1} + b ₂Q _{t − 2} + . . . + b_pQ _{t − p} + μ_t + c ₁μ_{t − 1} + . . . + c_qμ_{t − q} using an autoregressive integrated moving average (ARIMA) process of order (p, d, q).Footnote ⁸ Then we replace ${\hat{Q}_t}$ for Q^pe_t in equation (6).

Dairy processors utilize raw milk as the main input to produce dairy products. The maximum price that processors are willing to offer for the raw milk is the demand reservation price, P^d_t, which depends on the income of processors, the price of substitute inputs, and the quantity of demand at time t. Therefore, we specify the demand side of the raw milk market as

(7) $$\begin{equation} P_t^d = {\alpha _0} + {\alpha _1}W{P_t} + {\alpha _2}D{M_t} + {\alpha _3}Q_t^{de} + u_t^d, \end{equation}$$

where the unit price of processed milk, WP_t, represents income from selling dairy products, and DM_t and Q^de_t denote the price of dry milk as the substitute input and the quantity of demand for raw milk, respectively. An increase in the price of dairy products encourages processors to demand more raw milk for processing and thus offer a higher price for raw milk in the negotiations. Hence, we expect any increase in WP_t to positively affect P^d_t. Moreover, when the price of dry milk rises, demand for raw milk and, consequently, P^d_t would increase. Higher quantity of expected demand, Q^de_t, will negatively affect P^d_t. Because the negotiations are focused only on the price of raw milk, we assume that there is no excess demand or supply in the market (i.e., Q^de_t = Q_t ^pe). We can use ${\hat{Q}_t}$ to also represent Q^de_t when estimating equation (7).

Two econometric issues arise when estimating the bilateral bargaining model specified in the system of equations in equation (5). The first issue is the correlation between dairy price index (WP_t) and the negotiated price of raw milk (P_t). Dairy price index is the weighted average of prices for pasteurized milk, yogurt, and cheese where quantity of demand for each dairy product serves as the respective weight for that product. To address the correlation issue, we use a two-stage estimation method. In the first stage, we regressed WP_t on all of the exogenous variables in the system as follows:

(8) $$\begin{equation} W{P_t} = {\gamma _0} + {\gamma _1}{Q_t} + {\gamma _2}D{M_t} + {\gamma _3}F{C_t} + {\gamma _4}{L_t} + {\gamma _5}{K_t} + {\varepsilon _t}. \end{equation}$$

The notation is similar to before except that here Q_t is the “observed” quantity of milk that is processed into dairy products, and we equate it to the firms’ demand for raw milk. The price of capital inputFootnote ⁹ in the dairy industry is denoted by K_t. Machinery, buildings, land, vehicles, and durable goods are considered as capital assets.

According to Griliches and Jorgenson (Reference Griliches and Jorgenson1966), if we assume that there is no direct taxation, then the price of the jth capital service for ith dairy firm in each period satisfies the following relationship:

(9) $$\begin{equation} {P_{ij}} = {q_{ij}}\left( {r + {\delta _j} - \frac{{{{\dot{q}}_{ij}}}}{{{q_{ij}}}}} \right){\rm{for}}\;j = {\rm{ }}1,{\rm{ }}2,{\rm{ }} \ldots ,{\rm{ }}5 \ {\rm{ and}}\;i = {\rm{ }}1,{\rm{ }} \ldots ,{\rm{ }}n, \end{equation}$$

where q_ij is the price of the jth investment good for the firm i, r is the rate of return on all capital, δ_j is the rate of replacement of the jth investment good, and $\frac{{{{\dot{q}}_{ij}}}}{{{q_{ij}}}}\ $is the firm's rate of capital gain on each investment good. Therefore, ${\bar{P}_i} = \frac{{\mathop \sum \nolimits_{j = 1}^5 {\omega _{ij}}{P_{ij}}}}{{{Q_i}}}$ is the price (or cost) of capital input for firm i to process one unit of milk, ω_ij is the relative share of the capital service j in the value of all capital services for firm i, and Q_i is the demand for milk by firm i. To calculate the capital input price (K_t) for the dairy industry in time t, we aggregate the capital price (${\bar{P}_i}$) for all firms in each period using the arithmetic mean. In each period, we use the 5-year investment deposit rate and linear depreciation rate for r and δ_j, respectively. Using all the information, we estimate equation (8) and replace ${\widehat {WP}_t}$ in equation (7) to address the simultaneity problem.

The second problem in estimating the bilateral bargaining model is that both P^s_t and P^d_t are latent variables in the system. Dempster, Laird, and Rubin (Reference Dempster, Laird and Rubin1977) proposed an iterative algorithm that computes maximum likelihood estimates when the observations are incomplete or there are latent variables in the model. Similar to Dempster, Laird, and Rubin (Reference Dempster, Laird and Rubin1977), we use the Monte Carlo expectation maximization (MCEM) algorithm, which has two iterative steps. The first step is called the expectation step (E step), which uses the current estimates of the parameters to create an expected log-likelihood function. Then, the second step, or the maximization step (M step), computes parameters that maximize the previously created log-likelihood function. The new estimated maximizing parameters are then used to determine the distribution of the latent variables, and the first step is repeated. The alternation between the E and M steps is continued until the sequence of likelihood values converges.

To implement the MCEM algorithm, we follow Wei and Tanner (Reference Wei and Tanner1990). Let the complete data set be defined by ${X_t} = {[ {\begin{array}{*{20}{c}} {P_t^s}&{P_t^d}&{{P_t}} \end{array}} ]^T}$. Then, the log-likelihood function with complete information, l(θ|X_t), is

(10) $$\begin{equation} l\left( {\theta |{X_t}} \right) = \log L(\theta |{X_t}) = \mathop \sum \limits_{i = 1}^n {\rm{log}}\left[ {{f_{\theta ,t}}\left( {{X_t}} \right)} \right], \end{equation}$$

where θ is the vector of parameters of interest, and f _{θ, t}(X_t) is the probability density function (pdf) of the random vector X_t. In the MCEM algorithm, the E step uses θ to form l(θ|X_t), and the M step finds the θ^(k) that maximizes l(θ|X_t) where k simply denotes the number of iterations. Thus, at the (k + 1)th iteration, the E step computes Q (θ;θ^(k)), where

(11) $$\begin{equation} \mathcal{Q}\left( {\theta ,{\theta ^{\left( k \right)}}} \right) = \mathop \sum \limits_{t = 1}^n {E_{{\theta ^{\left( k \right)}}}}\left\{ {{\rm{log}}\left[ {{f_\theta }\left( {{X_t}} \right)} \right]} \right\} \end{equation}$$

(12) $$\begin{equation} {E_{{{\bm{\theta }}^{\left( k \right)}}}}\left\{ {{\rm{log}}\left[ {{f_{\bm{\theta }}}\left( {{X_t}} \right)} \right]} \right\} = \mathop \smallint \limits_{{P_t}}^{ + \infty } \mathop \smallint \limits_{ - \infty }^{{P_t}} {\rm{log}}\left[ {{f_{\bm{\theta }}}\left( {{X_t}} \right)} \right]f_{{{\bm{\theta }}^{\left( k \right)}}}^*\left( {p_t^s,p_t^d} \right)dp_t^sdp_t^d, \end{equation}$$

in which f*_θ^(k)(p_t ^s, p^d_t) denotes the joint pdf of latent variables, which in our case are P^s_t and P^d_t. The right-hand side in equation (12) is hard to evaluate, leaving the E step untraceable. To resolve the problem, we modify the E step in equation (11) using a Monte Carlo approach as follows:

(13) $$\begin{equation} \mathcal{Q}\left( {{\bm{\theta }},{{\bm{\theta }}^{\left( k \right)}}} \right) \cong \frac{1}{m}\mathop \sum \limits_{t = 1}^n \mathop \sum \limits_{i = 1}^m {\rm{log}}\left[ {{f_{\bm{\theta }}}\left( {P_{i,t}^d,P_{i,t}^s,{P_t}} \right)} \right]. \end{equation}$$

The M step then chooses θ^{(k + 1)} to be any value of θ ∈ Ω that maximizes$\ \mathcal{Q}( {\theta ,{\theta ^{( k )}}} )$ such that

(14) $$\begin{equation} {\theta ^{\left( {k + 1} \right)}} = arg\mathop {\max }\nolimits_\theta \mathcal{Q}\left( {\theta ,{\theta ^{\left( k \right)}}} \right). \end{equation}$$

The key point in the MCEM algorithm approach is the pdf of the random vector X_t, which consists of price variables. Prices are assumed to be normally distributed in economics; however, we assume that negotiated price is bound between reservation prices (i.e., P^s_t < P_t < P^d_t) in our bargaining model. Thus, we use a truncated multivariate normal distribution with the density function of

(15) $$\begin{equation} {f_{{\bm{\theta }},t}}\left( {{X_t}} \right) = {g_t}{\left( {\bm{\theta }} \right)^{ - 1}}{\left( {2\pi } \right)^{\frac{3}{2}}}{\left| {{\bf \Sigma }} \right|^{ - \frac{1}{2}}}{\rm{exp}}\left[ { - \frac{1}{2}{{\left( {{X_t} - {{\bm{\mu }}_t}} \right)}^{\prime}}{{{\bf \Sigma }}^{ - 1}}\left( {{X_t} - {{\bm{\mu }}_t}} \right)} \right], \end{equation}$$

$$\begin{equation*} {g_t}\left( {\bm{\theta }} \right) = \mathop \smallint \limits_{ - \infty }^{ + \infty } \mathop \smallint \limits_{p_t^s}^{ + \infty } \mathop \smallint \limits_{p_t^s}^{p_t^d} {\left( {2\pi } \right)^{\frac{3}{2}}}{\left| {{\bf \Sigma }} \right|^{ - \frac{1}{2}}}{\rm{exp}}\left[ { - \frac{1}{2}{{\left( {{X_t} - {{\bm{\mu }}_t}} \right)}^{\prime}}{{{\bf \Sigma }}^{ - 1}}\left( {{X_t} - {{\bm{\mu }}_t}} \right)} \right]d{p_t}dp_t^ddp_t^s. \end{equation*}$$

To test whether the estimated parameters are statistically significant, we use the bootstrap methodFootnote ¹⁰ to calculate the standard error of the parameters (θ*) as follows:

(16) $$\begin{equation} SE\left( {\hat{\theta }} \right) = {\left[ {{\raise0.7ex\hbox{$1$} \! / \!\lower0.7ex\hbox{$m$}}\mathop \sum \limits_{i = 1}^{m} {{\left( {\theta _i^{*} - \hat{\theta }} \right)}^{2}}} \right]^{{\raise0.7ex\hbox{$1$} \! / \!\lower0.7ex\hbox{$2$}}}}, \end{equation}$$

where $\hat{\theta }$ is the estimated parameter using the total number of observations. After calculating the standard error of the parameters, we form the 95% confidence interval as $\bar{\theta } \pm 1.96*SE( {\hat{\theta }} )$, in which $\bar{\theta }$ is the mean of bootstrap estimated parameters.

3. Data

Time series data for the dairy industry in Iran for the period of 1990–2013 published by the Statistical Center of Iran in statistical yearbooks are used for empirical estimation. Because the quarterly data are not available for some variables, such as milk production, we employ annual data. We estimate the expected quantity of milk, Q_t, using the annual information on milk production from 1965 to 2013. Then we include ${\hat{Q}_t}$ (t = 1990–2013) in the bargaining model in equation (5).

We use an index price for dairy products as a proxy for the unit price of processed milk, WP_t. Therefore, WP_t is calculated as the weighted average of prices for pasteurized milk, yogurt, and cheese, where the quantity demanded for each dairy product in time t serves as the respective weight for the product.