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\begin{document}

\appendix

\section{Model}

All output and input prices along the supply chain are determined
endogenously through market-clearing conditions. To focus on the vertical
integration of the model, we only include inputs that are part of
the supply chain, while other inputs such as labor, capital, and land
are excluded from the profit function. Therefore, labor, capital,
and land are fixed factors of production and the value of these input
captures the returns-to-scale of a segment.

\subsection{Corn/Soymeal Segment}

The corn and soymeal segment of the supply chain is independent from
the integrator. To focuses on the broiler industry, we model the residual
corn and soymeal supply to the broiler industry. Therefore, residual
supply of corn, $S^{C}$, and soymeal, $S^{SM}$, to broiler feed
mills are represented as reduced-form supply functions:
\begin{eqnarray}
S^{C} & = & \psi^{C}\left(p^{C}\right)^{\epsilon^{C}},\label{eq:SCorn}\\
S^{SM} & = & \psi^{SM}\left(p^{SM}\right)^{\epsilon^{SM}},\label{eq:SSoymeal}
\end{eqnarray}
where $p^{C}$ and $p^{SM}$ are the prices of corn and soymeal, respectively,
and $\psi$s and $\epsilon$s are residual supply scale parameters
and elasticities.

\subsection{Feed Mills}

The feed mill segment is controlled by the integrator because feed
formulas are tightly controlled by integrators. Thus, feed is not
purchased from large feed companies. Feed mills buy corn and soymeal
from the independent farmers or grain elevators to produce broiler
feed. The profit function, $\Pi^{F}$, for feed mills is
\begin{eqnarray}
\Pi^{F} & = & \rho^{F}Q^{F}-C^{F}\left(p^{C},p^{SM},Q^{F}\right),\label{eq: Profit Feed}
\end{eqnarray}
where $\rho^{F}$ is the transfer price of feed and $Q^{F}$ is quantity
of feed. The cost function, $C^{F}\left(\cdot\right)$, for feed is
\begin{equation}
C^{F}\left(\cdot\right)=\left(\frac{Q^{F}}{A^{F}}\right)^{\frac{1}{\nu^{F}}}\left[\left(\alpha_{C}^{F}\right)^{\frac{1}{1+\eta^{F}}}\left(p^{C}\right)^{\frac{\eta^{F}}{1+\eta^{F}}}+\left(\alpha_{SM}^{F}\right)^{\frac{1}{1+\eta^{F}}}\left(p^{SM}\right)^{\frac{\eta^{F}}{1+\eta^{F}}}\right]^{\frac{1+\eta^{F}}{\eta^{F}}},\label{eq:CFeed}
\end{equation}
where $p^{C}$ is cost price of corn, $p^{SM}$ is cost price of soymeal,
$A^{F}$ is productivity, $\nu^{F}$ is returns to scale, $\alpha_{C}^{F}$
and $\alpha_{SM}^{F}$ are share parameters, and $\eta^{F}$ is the
CES parameter. Define $\chi^{F}\equiv\left(\alpha_{C}^{F}\right)^{\frac{1}{1+\eta^{F}}}\left(p^{C}\right)^{\frac{\eta^{F}}{1+\eta^{F}}}+\left(\alpha_{SM}^{F}\right)^{\frac{1}{1+\eta^{F}}}\left(p^{SM}\right)^{\frac{\eta^{F}}{1+\eta^{F}}}$.
Then, taking the partial derivative of profits with respect to $Q^{F}$
and setting it equal to zero gives the first-order condition:
\begin{eqnarray}
\frac{\partial\Pi^{F}}{\partial Q^{F}} & = & \rho^{F}-\left(\frac{Q^{F}}{A^{F}}\right)^{\frac{1}{\nu^{F}}-1}\frac{\left[\chi^{F}\right]^{\frac{1+\eta^{F}}{\eta^{F}}}}{\nu^{F}A^{F}}=0.\label{eq:FOCFeed}
\end{eqnarray}
Solving for $Q^{F}$ yields supply as a function of the output price
and input prices:
\begin{equation}
Q^{F}=\left(\frac{1}{\rho^{F}\nu^{F}}\left(A^{F}\right)^{\frac{-1}{\nu^{F}}}\left[\chi^{F}\right]^{\frac{1+\eta^{F}}{\eta^{F}}}\right)^{\frac{\nu^{F}}{\nu^{F}-1}}.\label{eq:SFeed}
\end{equation}
Using Shepard's lemme, take partial derivatives of the cost function
with respect to each input price ($p^{C}\text{ and }p^{SM}$) to obtain
input demand functions for corn ($D_{C}^{F}$) and soymeal ($D_{SM}^{F}$),
respectively:
\begin{eqnarray}
D_{C}^{F}=\frac{\partial C^{F}\left(\cdot\right)}{\partial p^{C}} & = & \left(\frac{Q^{F}}{A^{F}}\right)^{\frac{1}{\nu^{F}}}\left(\chi^{F}\right)^{\frac{1}{\eta^{F}}}\left(\alpha_{1}^{F}\right)^{\frac{1}{1+\eta^{F}}}\left(p^{C}\right)^{\frac{-1}{1+\eta^{F}}},\label{eq:DFC}\\
D_{SM}^{F}=\frac{\partial C^{F}\left(\cdot\right)}{\partial p^{SM}} & = & \left(\frac{Q^{F}}{A^{F}}\right)^{\frac{1}{\nu^{F}}}\left(\chi^{F}\right)^{\frac{1}{\eta^{F}}}\left(\alpha_{2}^{F}\right)^{\frac{1}{1+\eta^{F}}}\left(p^{SM}\right)^{\frac{-1}{1+\eta^{F}}}.\label{eq:DFSM}
\end{eqnarray}

The feed mill segment is downstream in the supply chain from the corn
and soymeal segment because corn and soymeal are sold as inputs to
the feed mill segments. Mathematically, in the corn and soymeal segments,
the corn and soymeal prices are \emph{output} prices, while in the
feed mill segment these are \emph{input} prices to feed production.

\subsection{Breeders}

The breeders segment of the supply chain is independent from the integrators.
Breeder companies produce new lines of breeder stock to meet specific
market requirements. After producing a new type of breeder stocks,
they sell eggs to pullet farmers (contracted), who raise the chicken
until (about 1 year) it can start producing eggs for the hatchery
segment. \emph{Pullets} are not explicitly modeled here; one can think
of pullets as part of the breeders' costs in which feed is the main
requirement for pullets to produce eggs. The profit function, $\Pi^{B}$,
for breeders is
\begin{eqnarray}
\Pi^{B} & = & p^{B}Q^{B}-C^{B}\left(\rho^{F},Q^{B}\right),\label{eq:ProfitBreeder}
\end{eqnarray}
where $p^{B}$ is the price of fertilized eggs and $Q^{B}$ is the
quantity of fertilized eggs. The cost function, $C^{B}\left(\cdot\right)$,
for producing fertilized eggs is
\begin{equation}
C^{B}\left(\cdot\right)=\left(\frac{Q^{B}}{A^{B}}\right)^{\frac{1}{\nu^{B}}}\rho^{F},\label{eq:CBreeders}
\end{equation}
where $\rho^{F}$ is\textbf{ }transfer cost price for feed, $A^{B}$
is productivity, and $\nu^{B}$ is returns to scale. Taking the partial
derivative of profit with respect to $Q^{B}$ and setting it equal
to zero gives the first-order condition:
\begin{eqnarray}
\frac{\partial\Pi^{B}}{\partial Q^{B}} & = & p^{B}-\left(\frac{Q^{B}}{A^{B}}\right)^{\frac{1}{\nu^{B}}-1}\frac{\rho^{F}}{\nu^{B}A^{B}}=0.\label{eq:FOCBreeder}
\end{eqnarray}
Solving for $Q^{B}$ yields supply as a function of the output price
and input prices:
\begin{equation}
Q^{B}=\left(\frac{1}{p^{B}\nu^{B}}\left(A^{B}\right)^{\frac{-1}{\nu^{B}}}\rho^{F}\right)^{\frac{\nu^{B}}{\nu^{B}-1}}.\label{eq:SBreeder}
\end{equation}
The partial derivatives of the cost function with respect to $\rho^{F}$
yield input demand functions for feed ($D_{F}^{B}$):
\begin{eqnarray}
D_{F}^{B}=\frac{\partial C^{B}\left(\cdot\right)}{\partial\rho^{F}} & = & \left(\frac{Q^{B}}{A^{B}}\right)^{\frac{1}{\nu^{B}}}.\label{eq:DFF}
\end{eqnarray}

The breeder segment is downstream from the feed mills segment because
the broiler feed is sold as an input to the breeders. Mathematically,
in the feed mills segment, the feed transfer price is the output price,
while in the breeders segment, the feed transfer price is the input
price to fertilized eggs production.

\subsection{Hatcheries}

The hatcheries are controlled by the integrator. In the hatching segment,
fertilized eggs are purchased from the breeder to produce day-old
chickens (DOC). The profit function for hatchery $\Pi^{H}$ is 
\begin{eqnarray}
\Pi^{H} & = & \rho^{H}Q^{H}-C^{H}\left(p^{B},Q^{H}\right),\label{eq:ProfitHatchery}
\end{eqnarray}
where $\rho^{H}$ is the transfer price of DOC and $Q^{H}$ is the
number of DOCs. The cost function, $C^{H}\left(\cdot\right)$, for
producing DOC:
\begin{equation}
C^{H}\left(\cdot\right)=\left(\frac{Q^{H}}{A^{H}}\right)^{\frac{1}{\nu^{H}}}p^{B},\label{eq:Chatcheries}
\end{equation}
where $A^{H}$ is productivity and $\nu^{H}$ is returns to scale.
Taking the partial derivative of profits with respect to $Q^{H}$
and setting it equal to zero gives the first-order condition:
\begin{eqnarray}
\frac{\partial\Pi^{H}}{\partial Q^{H}} & = & \rho^{H}-\left(\frac{Q^{H}}{A^{H}}\right)^{\frac{1}{\nu^{H}}-1}\frac{p^{B}}{\nu^{H}A^{H}}=0.\label{eq:FOCHatchery}
\end{eqnarray}
Solving for $Q^{H}$ yields supply as a function of the output price
and input prices:
\begin{equation}
Q^{H}=\left(\frac{1}{\rho^{H}\nu^{H}}\left(A^{H}\right)^{\frac{-1}{\nu^{H}}}p^{B}\right)^{\frac{\nu^{H}}{\nu^{H}-1}}.\label{eq:SHatchery}
\end{equation}
The partial derivatives of the cost function with each input price
($p^{B}$) yields input demand functions for fertilized eggs ($D_{B}^{H}$):
\begin{eqnarray}
D_{B}^{H}=\frac{\partial C^{H}\left(\cdot\right)}{\partial p^{B}} & = & \left(\frac{Q^{H}}{A^{H}}\right)^{\frac{1}{\nu^{H}}}.\label{eq:DFB}
\end{eqnarray}

The hatchery segment is downstream from the breeders segment because
the fertilized eggs are sold as an input to the hatcheries. Mathematically,
in the breeders segment, the price of fertilized eggs is the output
price, while in the hatchery segment, the price of fertilized eggs
is the input price to produce DOC.

\subsection{Grow-Out Farms}

The grow-out farms are independent from the integrator. More than
90\% of chicken production in the United States is raised by contract
farmers. The farmers enter into a contract with the integrators. Although
there are different types of agreements between integrators and farmers
based on different integrators, the main structure of agreement is
similar, in which integrators provide chicks, feed, technical service,
managerial expertise, and a guaranteed market. The farmers are responsible
for the land, labor, utilities, and other operating expenses, which
we model as fixed factors of production reflected in the returns-to-scale
parameter. The farmers are supplied feed and DOC from integrators
based on their contracts and are the primary costs for grow-out farmer.
Then, they sell their broiler chickens when they reach desired weight.
The profit function for grow-out farmer, $\Pi^{G}$, is
\begin{eqnarray}
\Pi^{G} & = & p^{G}Q^{G}-C^{G}\left(\rho^{H},\rho^{F},Q^{G}\right),\label{eq:ProfitGrow-out}
\end{eqnarray}
where $p^{G}$ is the price of full grown live chickens and $Q^{G}$
is the number of full grown live chickens. The cost function, $C^{G}\left(\cdot\right)$,
for full grown live chickens:
\begin{equation}
C^{G}\left(\cdot\right)=\left(\frac{Q^{G}}{A^{G}}\right)^{\frac{1}{\nu^{G}}}\left[\left(\alpha_{1}^{G}\right)^{\frac{1}{1+\eta^{G}}}\left(\rho^{H}\right)^{\frac{\eta^{G}}{1+\eta^{G}}}+\left(\alpha_{2}^{G}\right)^{\frac{1}{1+\eta^{G}}}\left(\rho^{F}\right)^{\frac{\eta^{G}}{1+\eta^{G}}}\right]^{\frac{1+\eta^{G}}{\eta^{G}}},\label{eq:Cgrow-out}
\end{equation}
where $A^{G}$ is productivity, $\nu^{G}$ is returns to scale, $\alpha_{1}^{G}$
and $\alpha_{2}^{G}$ are share parameters, and $\eta^{G}$ is the
CES parameter. Define $\chi^{G}\equiv\left(\alpha_{1}^{G}\right)^{\frac{1}{1+\eta^{G}}}\left(\rho^{H}\right)^{\frac{\eta^{G}}{1+\eta^{G}}}+\left(\alpha_{2}^{G}\right)^{\frac{1}{1+\eta^{G}}}\left(\rho^{F}\right)^{\frac{\eta^{G}}{1+\eta^{G}}}$.
Then, taking the partial derivative of profit with respect to $Q^{G}$
and setting it equal to zero gives the first-order condition:
\begin{eqnarray}
\frac{\partial\Pi^{G}}{\partial Q^{G}} & = & p^{G}-\left(\frac{Q^{G}}{A^{G}}\right)^{\frac{1}{\nu^{G}}-1}\frac{\left[\chi^{G}\right]^{\frac{1+\eta^{G}}{\eta^{G}}}}{\nu^{G}A^{G}}=0.\label{eq:FOCGrow-Out}
\end{eqnarray}
Solving for $Q^{G}$ yields supply as a function of the output price
and input prices:
\begin{equation}
Q^{G}=\left(\frac{1}{p^{G}\nu^{G}}\left(A^{G}\right)^{\frac{-1}{\nu^{G}}}\left[\chi^{G}\right]^{\frac{1+\eta^{G}}{\eta^{G}}}\right)^{\frac{\nu^{G}}{\nu^{G}-1}}.\label{eq:SGrow-out}
\end{equation}
The partial derivatives of the cost function with respect to each
transfer price ($\rho^{H}\text{ and }\rho^{F}$) yield input demand
functions for hatching eggs ($D_{H}^{G}$) and feed ($D_{F}^{G}$),
respectively:
\begin{eqnarray}
D_{H}^{G}=\frac{\partial C^{G}\left(\cdot\right)}{\partial\rho^{H}} & = & \left(\frac{Q^{G}}{A^{G}}\right)^{\frac{1}{\nu^{G}}}\left(\chi^{G}\right)^{\frac{1}{\eta^{G}}}\left(\alpha_{1}^{G}\right)^{\frac{1}{1+\eta^{G}}}\left(\rho^{H}\right)^{\frac{-1}{1+\eta^{G}}},\label{eq:DFH}\\
D_{F}^{G}=\frac{\partial C^{G}\left(\cdot\right)}{\partial\rho^{F}} & = & \left(\frac{Q^{G}}{A^{G}}\right)^{\frac{1}{\nu^{G}}}\left(\chi^{G}\right)^{\frac{1}{\eta^{G}}}\left(\alpha_{2}^{G}\right)^{\frac{1}{1+\eta^{G}}}\left(\rho^{F}\right)^{\frac{-1}{1+\eta^{G}}}.\label{eq:DFF-G}
\end{eqnarray}

The grow-out farms are downstream from the feed mills and the hatchery
segments because feed and DOC are sold as inputs to the grow-out farmer.
Mathematically, the prices of feed and DOC are output prices for feed
mill and hatcheries, respectively, while these prices are input prices
for grow-out farmer to produce broiler chickens. Horizontal linkages
arise between breeders and hatcheries because both segments are downstream
from feed mills.

\subsection{Processing Plants}

The processing plants are controlled by the integrator. A processing
plant is the main segment in which broiler chickens are prepared for
selling in the market as a whole chicken and chicken parts by slaughtering\textbf{
}and packaging. Also, in this section, live broiler chickens are prepared
for value-added processing segments. Because chicken parts and preparing
chicken value-added products require more processing than whole birds,
we model whole chickens separately from chicken parts and processing
for value-added products.

\subsubsection{Whole Chickens}

For whole chicken, the main input is broiler chicken transferred from
grow-out farms based on the contract agreements. The profit function
for a whole chicken $\Pi^{W}$ is
\begin{eqnarray}
\Pi^{W} & = & p^{W}Q^{W}-C^{W}\left(p^{G},Q^{W}\right)\label{ProfitWhole}
\end{eqnarray}
where $p^{W}$ is price per pound of whole chickens and $Q^{W}$ is
number of pounds of whole chickens produced. The cost function, $C^{W}\left(\cdot\right)$,
of whole chickens is
\begin{equation}
C^{W}\left(\cdot\right)=\left(\frac{Q^{W}}{A^{W}}\right)^{\frac{1}{\nu^{W}}}p^{G},\label{eq:Cwholechicken}
\end{equation}
where $A^{W}$ is productivity and $\nu^{W}$ is returns to scale.
Taking the partial derivative of profits with respect to $Q^{W}$
and setting it equal to zero gives the first-order condition:
\begin{equation}
\frac{\partial\Pi^{W}}{\partial Q^{W}}=p^{W}-\left(\frac{Q^{W}}{A^{W}}\right)^{\frac{1}{\nu^{W}}-1}\frac{p^{G}}{\nu^{W}A^{W}}=0.\label{eq:FOCWhole}
\end{equation}
Solving for $Q^{W}$ yields supply as a function of the output price
and input prices:
\begin{equation}
Q^{W}=\left(\frac{1}{p^{W}\nu^{W}}\left(A^{W}\right)^{\frac{-1}{\nu^{W}}}p^{G}\right)^{\frac{\nu^{W}}{\nu^{W}-1}}.\label{eq:SWhole}
\end{equation}
Using Shepard's lemme, taking the partial derivatives of the cost
function with respect to $p^{G}$ to obtain the input demand function
for whole chickens ($D_{G}^{W}$):
\begin{equation}
D_{G}^{W}=\frac{\partial C^{W}\left(\cdot\right)}{\partial p^{G}}=\left(\frac{Q^{W}}{A^{W}}\right)^{\frac{1}{\nu^{W}}}.\label{eq:DFG}
\end{equation}


\subsubsection{Chicken Parts}

Next, we define the profit function for chicken parts. One chicken
yields $\omega$ pounds of chicken breasts, $Q^{Br}$, $\gamma$ pounds
of chicken wings, $Q^{Wi}$, and $\eta$ pounds of dark meat, $Q^{D}$,
which implies the following relationships
\begin{equation}
Q^{Br}=\omega Q^{P},\,Q^{Wi}=\gamma Q^{P},\text{ and }Q^{D}=\eta Q^{P}.\label{eq:QB_to_QWD}
\end{equation}
The total pounds of chicken parts is
\begin{equation}
Q^{P}=Q^{Br}+Q^{Wi}+Q^{D}.\label{eq:QParts}
\end{equation}
Total revenue of chicken parts is obtained by multiplying quantity
by price:
\begin{equation}
p^{P}Q^{P}=p^{Br}Q^{Br}+p^{Wi}Q^{Wi}+p^{D}Q^{D}\label{eq:RevParts}
\end{equation}
where $p^{P}$ is a price index for parts, $p^{Br}$ is price per
pound of chicken breast, $p^{Wi}$ is price of per pound of chicken
wings, and $p^{D}$ is price per pound of chicken dark meat. We can
define the price index for parts by substituting (\ref{eq:QB_to_QWD})
into (\ref{eq:RevParts}) which yields: $p^{P}Q^{P}=\left(\omega p^{Br}+\gamma p^{Wi}+\eta p^{D}\right)Q^{P}$.
The price index for chicken parts $p^{P}$ is
\[
p^{P}=\omega p^{Br}+\gamma p^{Wi}+\eta p^{D}.
\]

The profit function for chicken parts $\left(\Pi^{P}\right)$ is
\begin{eqnarray}
\Pi^{P} & = & p^{P}Q^{P}-C^{P}\left(p^{G},p^{PI},Q^{P}\right)\label{eq:ProfitParts}
\end{eqnarray}
where $Q^{P}$ is quantity of chicken parts produce. The cost function,
$C^{P}\left(\cdot\right)$, of chicken parts given by
\begin{equation}
C^{P}\left(\cdot\right)=\left(\frac{Q^{P}}{A^{P}}\right)^{\frac{1}{\nu^{P}}}p^{G},\label{eq:CchickenParts}
\end{equation}
where $A^{P}$ is productivity and $\nu^{P}$ is returns to scale.
Taking the partial derivative of profits with respect to $Q^{P}$
and setting it equal to zero gives the first-order condition:
\begin{equation}
\frac{\partial\Pi^{P}}{\partial Q^{P}}=p^{P}-\frac{1}{\nu^{P}}\left(\frac{Q^{P}}{A^{P}}\right)^{\frac{1}{\nu^{P}}-1}\frac{p^{G}}{A^{P}}=0.\label{eq:FOCParts}
\end{equation}
Solving for $Q^{P}$ yields supply as a function of the output price
and input prices:
\begin{equation}
Q^{P}=\left(\frac{1}{p^{P}\nu^{P}}\left(A^{P}\right)^{\frac{-1}{\nu^{P}}}\left[p^{G}\right]\right)^{\frac{\nu^{P}}{\nu^{P}-1}}.\label{eq:SParts}
\end{equation}
The partial derivatives of the cost function with respect to $p^{G}$
yields the input demand function for grown whole chicken ($D_{G}^{P}$):
\begin{equation}
D_{G}^{P}=\frac{\partial C^{P}\left(\cdot\right)}{\partial p^{G}}=\left(\frac{Q^{P}}{A^{P}}\right)^{\frac{1}{\nu^{P}}}.\label{eq:DFG-P}
\end{equation}


\subsubsection{Chicken Processing for the Value-added Segment}

For chickens processed for value-added products, the main input is
broiler chicken bought from the grow-out farms. The profit function
$\Pi^{PV}$ is
\begin{eqnarray}
\Pi^{PV} & = & \rho^{PV}Q^{PV}-C^{PV}\left(p^{G},Q^{PV}\right),\label{ProfitProcessing V-A}
\end{eqnarray}
where $\rho^{PV}$ is the transfer price per pound of whole chickens
and $Q^{PV}$ is number of pounds for the value-added segment. The
cost function, $C^{PV}\left(\cdot\right)$, of whole chickens is
\begin{equation}
C^{PV}\left(\cdot\right)=\left(\frac{Q^{PV}}{A^{PV}}\right)^{\frac{1}{\nu^{PV}}}p^{G},\label{eq:CprocessingforV_A}
\end{equation}
where $A^{PV}$ is productivity and $\nu^{PV}$ is returns to scale.
Taking the partial derivative of profits with respect to $Q^{PV}$
and setting it equal to zero gives the first-order condition:
\begin{equation}
\frac{\partial\Pi^{PV}}{\partial Q^{PV}}=\rho^{PV}-\left(\frac{Q^{PV}}{A^{PV}}\right)^{\frac{1}{\nu^{PV}}-1}\frac{p^{G}}{\nu^{PV}A^{PV}}=0.\label{eq:FOCprocessingV-A}
\end{equation}
Solving for $Q^{PV}$ yields supply as a function of the output price
and input prices:
\begin{equation}
Q^{PV}=\left(\frac{1}{\rho^{PV}\nu^{PV}}\left(A^{PV}\right)^{\frac{-1}{\nu^{PV}}}p^{G}\right)^{\frac{\nu^{PV}}{\nu^{PV}-1}}.\label{eq:SProcessingV-A}
\end{equation}
The partial derivatives of the cost function with respect to $p^{G}$
yields the input demand function for demand for grow-out chicken ($D_{G}^{PV}$):
\begin{equation}
D_{G}^{PV}=\frac{\partial C^{PV}\left(\cdot\right)}{\partial p^{G}}=\left(\frac{Q^{PV}}{A^{PV}}\right)^{\frac{1}{\nu^{PV}}}.\label{eq:DFG-VA}
\end{equation}

The processing segment of whole chickens, chicken parts, and processing
for value-added products is downstream in the supply chain from the
grow-out farms because broiler chickens are sold as inputs to the
processor. Mathematically, the price of broiler chicken is the output
price for grow-out farms, while the price of broiler chicken is the
input price for whole chickens, chicken parts, and processing for
value-added. In addition, there are horizontal linkages in the processing
segments of the broiler supply chain between whole chickens, chicken
parts, and processing for value-added. The horizontal linkage arises
because whole chickens, chicken parts, and processing for value-added
segments all demand broiler chickens, implying that they compete with
each other for the available supply.

\subsection{Value-Added Plants}

The value-added plants are controlled by the integrator. Broilers
slaughtered for value-added processing are transformed into ready-to-cook
products (e.g., chicken nuggets, chicken burgers, chicken noodle soup,
etc.). Profit function for value-added plants $\Pi^{VA}$ is
\begin{eqnarray}
\Pi^{VA} & = & p^{VA}Q^{VA}-C^{VA}\left(\rho^{PV},Q^{VA}\right),\label{eq:ProfitFurther}
\end{eqnarray}
where $p^{VA}$ is price of value-added chicken and $Q^{VA}$ is the
quantity value-added products. The cost function, $C^{VA}\left(\cdot\right)$,
for value-added chicken:
\begin{equation}
C^{VA}\left(\cdot\right)=\left(\frac{Q^{VA}}{A^{VA}}\right)^{\frac{1}{\nu^{VA}}}\rho^{PV},\label{eq:CRCchicken}
\end{equation}
where $A^{VA}$ is productivity and $\nu^{VA}$ is returns to scale.
Taking the partial derivative of profits with respect to $Q^{VA}$
and setting it equal to zero gives the first-order condition:
\begin{equation}
\frac{\partial\Pi^{VA}}{\partial Q^{VA}}=p^{VA}-\frac{1}{\nu^{VA}}\left(\frac{Q^{VA}}{A^{VA}}\right)^{\frac{1}{\nu^{VA}}-1}\frac{\rho^{PV}}{A^{VA}}=0.\label{eq:FOCFurther}
\end{equation}
Solving for $Q^{VA}$ yields supply as a function of the output price
and input prices:
\begin{equation}
Q^{VA}=\left(\frac{\left(A^{VA}\right)^{\frac{-1}{\nu^{VA}}}}{p^{VA}\nu^{VA}}\rho^{PV}\right)^{\frac{\nu^{VA}}{\nu^{VA}-1}}.\label{eq:SFurther}
\end{equation}
The partial derivatives of the cost function with respect to $\rho^{PV}$
yields the input demand function for processed chickens ($D_{PV}^{VA}$):
\begin{equation}
D_{PV}^{VA}=\frac{\partial C^{VA}\left(\cdot\right)}{\partial\rho^{PV}}=\left(\frac{Q^{VA}}{A^{VA}}\right)^{\frac{1}{\nu^{VA}}},\label{eq:DFG-RC}
\end{equation}

The value-added segment is downstream from the processing segment
because the processed broilers are sold as an input to the value-added
segment. Mathematically, in the processing segment, the price of processed
broilers are output prices, while in the value-added segment, the
price of processed broilers are input prices.

\subsection{Market-Clearing Conditions}

This section provides market-clearing conditions, which determine
prices endogenously, for each segment of the broiler supply chain.
For the corn and soybean segment, the market-clearing conditions are\footnote{To simplify the model, we exclude beef and pork cross prices and adjust
the elasticity of demand to be less responsive to price changes.}
\begin{equation}
S^{C}\left(p_{C}\right)=D_{C}^{F}\left(p_{C}\right),\label{eq:CornMarket}
\end{equation}
\begin{equation}
S^{SM}\left(p_{SM}\right)=D_{SM}^{F}\left(p_{SM}\right).\label{eq:SoymealMarket}
\end{equation}
The first equation implies total supply of corn to the broiler industry
is equal to the total demand of corn by feed mills. The second equation
equates total supply of soymeal to the broiler industry to the total
demand of soymeal by the feed mills. The price of feed is determined
by the market-clearing condition,
\begin{equation}
Q^{F}=D_{F}^{G}+D_{F}^{B},\label{eq:FeedMarket}
\end{equation}
where total quantity of feed is equal to feed demand by breeders and
grow-out farms, which establishes the horizontal linkage between breeders
and grow-out farms. The price of fertilized eggs in the breeders is
determined by 
\begin{equation}
Q^{B}=D_{B}^{H},\label{eq:BreederMarket}
\end{equation}
where total number of fertilized eggs equals the total demand by hatcheries.
The price of DOCs are determined by
\begin{equation}
Q^{H}=D_{H}^{G},\label{eq:HatcheryMarket}
\end{equation}
where the number of DOCs equal the total demand of the DOC by grow-out
farms. The price of fully grown (grow-out) chickens is determined
by
\begin{equation}
Q^{G}=D_{G}^{PV}+D_{G}^{P}+D_{G}^{W},\label{eq:Grow-outMarket}
\end{equation}
where the total production of broiler chickens is equal to the demand
by whole chicken, chicken parts, and chickens for further processing,
which establishes the horizontal linkage between whole chicken, chicken
parts, and chickens for further processing.

The prices of whole chickens and chicken parts (chicken breasts, wings,
and dark meat) are determined individually by their market-clearing
conditions. For whole chickens $\left(W\right)$, chicken breast $\left(Br\right)$,
chicken wings $\left(Wi\right)$, dark meat $\left(D\right)$, and
value-added chicken $\left(VA\right)$, the market-clearing condition
is
\begin{equation}
Q^{k}=D_{k}^{USC}\left(p^{k}\right)+D_{k}^{MexC}\left(p^{k}\right)+D_{k}^{CanC}\left(p^{k}\right)+D_{k}^{ROWC}\left(p^{k}\right),\label{eq:BreastMarket}
\end{equation}
where $k=W,Br,Wi,D,VA$, $D_{k}^{USC}\left(p^{k}\right)=\delta_{k}^{USC}\left(p^{k}\right)^{\varepsilon_{k}^{USC}}$
is the reduced-form demand function of US consumers, $D_{k}^{MexC}\left(p^{k}\right)=\delta_{k}^{MexC}\left(p^{k}\right)^{\varepsilon_{k}^{MexC}}$
is the reduced-form demand function of Mexican consumers, $D_{k}^{CanC}\left(p^{k}\right)=\delta_{k}^{CanC}\left(p^{k}\right)^{\varepsilon_{k}^{CanC}}$
is the reduced-form demand function of Canadian consumers, $D_{k}^{ROWC}\left(p^{k}\right)=\delta_{k}^{ROWC}\left(p^{k}\right)^{\varepsilon_{k}^{ROWC}}$
is the reduced-form demand function of ROW consumers for chicken products
produced in the United States, $\delta$s are scale parameters, and
$\varepsilon$s are elasticities of demand. This market-clearing condition
indicates that quantity of chicken-product $k$ produced in the United
States equals the quantity demanded by US, Mexican, Canadian, and
ROW consumers. Finally, the price of processing for value-added products
is determined by
\begin{equation}
Q^{PV}=D_{VA}^{PV},\label{eq:PVMarket}
\end{equation}
where total supply of slaughtered chicken equals demand for slaughtered
chicken for value-added products.

\section{Calibration}

Next, we discuss the calibration of the parameters in the model.

\subsection{Corn/Soymeal Segment}

According to \citet{roberts2013identifying}, the elasticities of
supply for corn and soymeal are relatively inelastic at $\epsilon^{C}=0.207$
and $\epsilon^{SM}=0.705$. Given these elasticities and data on the
corn and soymeal prices and quantities, we calibrate the corn and
soymeal supply parameters as
\begin{eqnarray*}
\psi^{C} & = & \frac{S^{C}}{\left(p^{C}\right)^{\epsilon^{C}}},\\
\psi^{SM} & = & \frac{S^{SM}}{\left(p^{SM}\right)^{\epsilon^{SM}}}.
\end{eqnarray*}


\subsection{Feed mills}

To impose a low degree of substitutability between corn and soymeal,
we assume the CES parameter $\eta^{F}$ to positive at $4$, implying
an elasticity of substitution of $\sigma^{F}=\frac{1}{1+\eta^{F}}=0.2$.
Given $\sigma^{F}$ and data on the corn price, corn use, soymeal
price, and soymeal usse, we calibrate the corn and soymeal CES share
parameters as
\[
\alpha_{C}^{F}=\frac{p^{C}D_{C}^{F}{}^{\frac{1}{\sigma^{F}}}}{p^{C}D_{C}^{F}{}^{\frac{1}{\sigma^{F}}}+p^{SM}D_{C}^{SM}{}^{\frac{1}{\sigma^{F}}}},
\]
\[
\alpha_{SM}^{F}=\frac{p^{SM}D_{SM}^{F}{}^{\frac{1}{\sigma^{F}}}}{p^{C}D_{C}^{F}{}^{\frac{1}{\sigma^{F}}}+p^{SM}D_{SM}^{F}{}^{\frac{1}{\sigma^{F}}}}.
\]
The returns to scale parameter is given by input expenditures on corn
and soymeal ($\tilde{C}^{F}=p^{C}D_{C}^{F}+p^{SM}D_{SM}^{F}$) divided
by total revenue
\[
v^{F}=\frac{\tilde{C}^{F}}{\rho^{F}Q^{F}}.
\]
Given the above parameters and data total production, input expendutures,
and input prices, we calibrate the productivity parameter. To do this,
note that the total cost function equals input expenditures, which
implies that $\tilde{C}^{F}=C^{F}\left(\cdot\right)$. Then, using
this relationship and equation \ref{eq:CFeed}, the productivity parameter
is calibrated as the residual:
\[
A^{F}=Q^{F}\left(\frac{\tilde{C}^{F}}{\left[\left(\alpha_{C}^{F}\right)^{\frac{1}{1+\eta^{F}}}\left(p^{C}\right)^{\frac{\eta^{F}}{1+\eta^{F}}}+\left(\alpha_{SM}^{F}\right)^{\frac{1}{1+\eta^{F}}}\left(p^{SM}\right)^{\frac{\eta^{F}}{1+\eta^{F}}}\right]^{\frac{1+\eta^{F}}{\eta^{F}}}}\right)^{-\nu^{F}}.
\]


\subsection{Breeders}

For breeders, the returns to scale parameter is calibrated as input
expenditure on feed ($\tilde{C}^{B}=\rho^{F}D_{F}^{B}$) divided by
total revenue
\[
v^{B}=\frac{\tilde{C}^{B}}{p^{B}Q^{B}}.
\]
Using the data on total number of fertilized eggs, price of feed,
and input expenditure, we calibrate the productivity parameter. As
with the feed segment, total cost function equals input expenditure,
which implies that $\tilde{C}^{B}=C^{B}\left(\cdot\right)$. Then,
using this relationship and equation \ref{eq:CBreeders}, the productivity
parameter is calibrated as the residual:
\[
A^{B}=Q^{B}\left(\frac{\tilde{C}^{B}}{\rho^{F}}\right)^{-\nu^{B}}.
\]


\subsection{Hatcheries}

Similarly, for hatcheries, we calibrate the returns to scale parameter
as input expenditure on fertilized eggs ($\tilde{C}^{H}=p^{B}D_{B}^{H}$)
divided by total revenue
\[
v^{H}=\frac{\tilde{C}^{H}}{\rho^{H}Q^{H}}.
\]
Based on the data on total number of DOC, price of fertilized eggs,
input expenditure, and equation (\ref{eq:Chatcheries}), the productivity
parameter is calibrated as the residual:
\[
A^{H}=Q^{H}\left(\frac{\tilde{C}^{H}}{p^{B}}\right)^{-\nu^{H}}.
\]


\subsection{Grow-out Farms}

Similar to the feed mills, we assume the CES parameter $\eta^{G}$
to be $4$, implying an elasticity of substitution of $\sigma^{G}=\frac{1}{1-\eta^{G}}=0.2$,
which allows for a low degree of substitutability between DOC and
feed in producing broiler chickens. Given $\sigma^{G}$ and data on
DOC and feed prices and use, we calibrate the share parameters as
\[
\alpha_{H}^{G}=\frac{\rho^{H}D_{H}^{G}{}^{\frac{1}{\sigma^{G}}}}{\rho^{H}D_{H}^{G}{}^{\frac{1}{\sigma^{G}}}+\rho^{F}D_{F}^{G}{}^{\frac{1}{\sigma^{G}}}},
\]
\[
\alpha_{F}^{G}=\frac{\rho^{F}D_{F}^{G}{}^{\frac{1}{\sigma^{G}}}}{\rho^{H}D_{H}^{G}{}^{\frac{1}{\sigma^{G}}}+\rho^{F}D_{F}^{G}{}^{\frac{1}{\sigma^{G}}}}.
\]
The returns to scale parameter is calibrated as input expenditures
on corn and soymeal ($\tilde{C}^{G}=\rho^{H}D_{H}^{G}+\rho^{F}D_{F}^{G}$)
divided by total revenue
\[
v^{G}=\frac{\tilde{C}^{G}}{p^{G}Q^{G}}.
\]
With the above parameters and data on input expenditures and broiler
production and price, we calibrate the productivity parameter, using
equation \ref{eq:Cgrow-out}, as
\[
A^{G}=Q^{G}\left(\frac{\tilde{C}^{G}}{\left[\left(\alpha_{H}^{G}\right)^{\frac{1}{1+\eta^{G}}}\left(\rho^{H}\right)^{\frac{\eta^{G}}{1+\eta^{G}}}+\left(\alpha_{F}^{G}\right)^{\frac{1}{1+\eta^{G}}}\left(\rho^{F}\right)^{\frac{\eta^{G}}{1+\eta^{G}}}\right]^{\frac{1+\eta^{G}}{\eta^{G}}}}\right)^{-\nu^{G}}.
\]


\subsection{Processing Plants}

For whole chickens, chicken parts, and value-added chicken, the returns
to scale parameters are calibrated as input expenditure ($\tilde{C}^{k}=p^{G}D_{G}^{k}$,
where $k=W,P,VA$) divided by total revenue
\[
v^{k}=\frac{\tilde{C}^{k}}{p^{k}Q^{k}}
\]
Using data on the total production and output price, we calibrate
the productivity parameters, using equations (\ref{eq:Cwholechicken}),
(\ref{eq:CchickenParts}), (\ref{eq:CprocessingforV_A}), as the residual:
\[
A^{W}=Q^{W}\left(\frac{\tilde{C}^{W}}{p^{G}}\right)^{-\nu^{W}}.
\]

Based on data from \citet{USDA2018Quick}, one chicken yields $\omega=1.81$
pounds of chicken breasts,$Q^{Br}$, $\gamma=0.45$ pounds of chicken
wings, $Q^{Wi}$, and $\eta=2.12$ pounds of dark meat, $Q^{D}$.
Once the quantity of processed chicken is determened, the quantites
of chicken breasts, chicken wings, and dark meat can be determined
using equation (\ref{eq:QB_to_QWD}).

\subsection{Value Added}

The returns to scale parameter for value-added production is calibrated
as input expenditure on broiler chicken ($\tilde{C}^{VA}=p^{G}D_{G}^{VA}$)
divided by total revenue
\[
v^{VA}=\frac{\tilde{C}^{VA}}{p^{VA}Q^{VA}}.
\]
Using data on the total quantity and price of value added chicken
and input expenditure, using equation \ref{eq:CprocessingforV_A},
the productivity parameter is calibrated as 
\[
A^{VA}=Q^{VA}\left(\frac{\tilde{C}^{VA}}{p^{G}}\right)^{-\nu^{VA}}.
\]


\subsection{Reduced-Form Demand Functions}

For US demand functions, whole chicken, chicken breast, wings, dark
meat, and value-added products, based on estimates from \citep{lusk2016meat},
we average elasticities for low, middle, and high income consumers
$\varepsilon^{USC}$ for breast meat (1.504) and wing meat (0.93).
Since the literature does not provid demand elasticity estimates for
whole chicken and dark meat, we assumed an elasticity that is $50\%$
higher than that of chicken breast and wings. Given consumption and
price data and demand elasticities, the scale parameter is calibrated
as
\[
\delta_{j}^{USC}=\frac{D_{j}^{USC}}{\left(p^{j}\right)^{\varepsilon_{j}^{USC}}},
\]
where $j=W,Br,Wi,D,VA$. For Mexico, according to \citet{salazar20052025},
chicken demand elasticity is on average $-0.862$ across income groups.
Also, \citet{williams1996liberalized} found that Mexican chicken
meat demand elasticity is inelastic at $-0.33$ and $-0.37$ based
on Marshallian and Hicksian demand functions, respectively. However,
these studies do not provide elasticities based on chicken parts.
Thus, we assumed based on the literature that elasticities of chicken
part $\varepsilon_{j}^{i},\,i=MexC$ and $j=W,Br,Wi,D,VA$ are between
$-0.25$ and $-0.9$. Since Mexican consumers prefer to dark meat
instead of chicken breast, we assume dark meat is more inelastic $\left(-0.25\right)$
compared to chicken breast $\left(-0.9\right)$. Given consumption
and price data and demand elasticities, the scale parameter is calibrated
as
\[
\delta_{j}^{MexC}=\frac{D_{j}^{MexC}}{\left(p^{j}\right)^{\varepsilon_{j}^{MexC}}}.
\]
For Canada and ROW, the literature does not provide demand elasticities.
Therefore, we assume that Canadian elasticities are similar to those
of the United States, and ROW elasticities are similar to those of
Mexico. Given consumption and price data and demand elasticities,
the scale parameter is calibrated as
\[
\delta_{j}^{CanC}=\frac{D_{j}^{CanC}}{\left(p^{j}\right)^{\varepsilon_{j}^{CanC}}}.
\]

\renewcommand{\thetable}{A\arabic{table}}

\begin{table}
\caption{Calibrated and Assumed Parameters\label{tab:Calibrated-Parameters}}

\centering{}%
\begin{tabular}{ccc}
\hline 
Parameters & Description & Value\tabularnewline
\hline 
\hline 
$\psi^{C}$ & Supply Scala parameter for corn production. & $0.0123$\tabularnewline
\hline 
$\psi^{SM}$ & Supply Scala parameter for soybean production. & $0.0002$\tabularnewline
\hline 
$\epsilon^{C}$ & Elasticity for corn  & $0.207$\tabularnewline
\hline 
$\epsilon^{SM}$ & Elasticity for soybean production. & $0.705$\tabularnewline
\hline 
$A^{F}$ & Productivity for Feed Mills & $0.62$\tabularnewline
\hline 
$\nu^{F}$ & Return to scale for Feed Mills & $0.55$\tabularnewline
\hline 
$\alpha_{1}^{F}$ & Share parameter for Feed Mills & $0.93$\tabularnewline
\hline 
$\alpha_{2}^{F}$ & Share parameter for Feed Mills & $0.06$\tabularnewline
\hline 
$\alpha_{3}^{F}$ & Share parameter for Feed Mills & $0$\tabularnewline
\hline 
$\eta^{F}$  & Constant elasticity of substitution parameters & $4$\tabularnewline
\hline 
$A^{B}$ & Productivity for Breeders & $4.57$\tabularnewline
\hline 
$\nu^{B}$ & Return to scale for Breeders & $0.25$\tabularnewline
\hline 
$\alpha_{1}^{B}$ & Share parameter for Breeders & $1$\tabularnewline
\hline 
$\alpha_{2}^{B}$ & Share parameter for Breeders & $0$\tabularnewline
\hline 
$\eta^{B}$  & Constant elasticity of substitution parameters & $4$\tabularnewline
\hline 
$A^{H}$ & Productivity for Hatcheries & $0.78$\tabularnewline
\hline 
$\nu^{H}$ & Return to scale for Hatcheries & $0.99$\tabularnewline
\hline 
$\alpha_{1}^{H}$ & Share parameter for Hatcheries & $1$\tabularnewline
\hline 
$\alpha_{2}^{H}$ & Share parameter for Hatcheries & $0$\tabularnewline
\hline 
$\eta^{H}$  & Constant elasticity of substitution parameters & $4$\tabularnewline
\hline 
$A^{G}$ & Productivity for Grow-out Farms & $117.17$\tabularnewline
\hline 
$\nu^{G}$ & Return to scale for Grow-out Farms & $0.91$\tabularnewline
\hline 
$\alpha_{1}^{G}$ & Share parameter for Grow-out Farms & $0.998$\tabularnewline
\hline 
$\alpha_{2}^{G}$ & Share parameter for Grow-out Farms & $0.001$\tabularnewline
\hline 
$\alpha_{3}^{G}$ & Share parameter for Grow-out Farms & $0$\tabularnewline
\hline 
$\eta^{G}$  & Constant elasticity of substitution parameters & $4$\tabularnewline
\hline 
$A^{W}$ & Productivity for Whole Chicken & $1.63$\tabularnewline
\hline 
$\nu^{W}$ & Return to scale for Whole Chicken & $0.52$\tabularnewline
\hline 
$\alpha_{1}^{W}$ & Share parameter for Whole Chicken & $1$\tabularnewline
\hline 
$\alpha_{2}^{W}$ & Share parameter for Whole Chicken & $0$\tabularnewline
\hline 
$\eta^{W}$  & Constant elasticity of substitution parameters & $4$\tabularnewline
\hline 
\end{tabular}
\end{table}

\begin{table}
\caption{Calibrated and Assumed Parameters Continued\label{tab:Calibrated-Parameters-2}}

\begin{longtable}{ccc}
\hline 
Parameters & Description & Value\tabularnewline
\hline 
\hline 
$A^{P}$ & Productivity for Chicken Parts & $5.34$\tabularnewline
\hline 
$\nu^{P}$ & Return to scale for Chicken Parts & $0.33$\tabularnewline
\hline 
$\alpha_{1}^{P}$ & Share parameter for Chicken Parts & $1$\tabularnewline
\hline 
$\alpha_{2}^{P}$ & Share parameter for Chicken Parts & $0$\tabularnewline
\hline 
$\eta^{P}$  & Constant elasticity of substitution parameters & $4$\tabularnewline
\hline 
$A^{PV}$ & Productivity for Processing Value-added  & $1.46$\tabularnewline
\hline 
$\nu^{PV}$ & Return to scale for Processing Value-added  & $0.77$\tabularnewline
\hline 
$\alpha_{1}^{PV}$ & Share parameter for Processing Value-added  & $1$\tabularnewline
\hline 
$\alpha_{2}^{PV}$ & Share parameter for Processing Value-added  & $0$\tabularnewline
\hline 
$\eta^{PV}$  & Constant elasticity of substitution parameters & $4$\tabularnewline
\hline 
$A^{VA}$ & Productivity for Value added & $1.3$\tabularnewline
\hline 
$\nu^{VA}$ & Return to scale for Value added & $0.90$\tabularnewline
\hline 
$\alpha_{1}^{VA}$ & Share parameter for Value added & $1$\tabularnewline
\hline 
$\alpha_{2}^{VA}$ & Share parameter for Value added & $0$\tabularnewline
\hline 
$\eta^{VA}$  & Constant elasticity of substitution parameters & $4$\tabularnewline
\hline 
\end{longtable}
\end{table}

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\end{document}
