\documentclass[12pt]{article}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage[dcucite]{harvard}
\usepackage{amsmath}
\usepackage{color}
\usepackage{setspace}
\usepackage{booktabs}
\usepackage{hyperref}
\usepackage[T1]{fontenc}
\usepackage{threeparttable}
\usepackage{array}
\usepackage{longtable}
\usepackage{subcaption}
\usepackage{comment}
\usepackage{chngcntr}
\counterwithin*{equation}{section}
\setcounter{MaxMatrixCols}{10}
\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\setlength{\topmargin}{0.1in}
\setlength{\headheight}{0in}
\setlength{\headsep}{0in}
\setlength{\topskip}{0in}
\setlength{\textheight}{8.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\evensidemargin}{0in}
\setlength{\textwidth}{6.5in}
\makeatletter
\setlength{\@fptop}{0pt}
\makeatother
\begin{document}
\title{A Simple Supply Chain Model with Bertrand Competition\vspace{0.5in}%
}
\author{Ross Hallren\thanks{U.S. International Trade Commission.\newline Contact emails: Ross.Hallren@usitc.gov}}
\date{\vspace{1.5in}%
\today}
\thispagestyle{empty}
{ % set font to helvetica (arial) to make it 508-compliant
\fontfamily{phv}\selectfont
\begin{center}
{\Large ALONG CAME A MONEMPORIST: \vspace{0.125in}} \\
{\Large A MODEL OF BERTRAND-MONOPSONY COMPETITION \vspace{0.25in}} \\
\vspace{1.5in}
{\large Ross J. Hallren \\}
\vspace{1.5in}
{\large ECONOMICS WORKING PAPER SERIES}\\
Working Paper 2019--12--A \\ \vspace{0.5in}
U.S. INTERNATIONAL TRADE COMMISSION \\
500 E Street SW \\
Washington, DC 20436 \\
\vspace{0.25in}
December 2019
\end{center}
\vfill
\noindent The author is grateful to David Riker and Tilsa Ore-Monago for helpful comments and suggestions.
\vspace{0.25in}
\noindent Office of Economics working papers are the result of ongoing professional research of USITC Staff and are solely meant to represent the opinions and professional research of individual authors. These papers are not meant to represent in any way the views of the U.S. International Trade Commission or any of its individual Commissioners. Working papers are circulated to promote the active exchange of ideas between USITC Staff and recognized experts outside the USITC and to promote professional development of Office Staff by encouraging outside professional critique of staff research.
\newpage
\thispagestyle{empty} % remove headers, footers, and page numbers from cover page
\begin{flushleft}
Along Came a Monemporist: A Model of Bertrand-Monopsony Competition\\
Ross J. Hallren\\
Office of Economics Working Paper 2019--12--A\\
December 2019\\~\\
\end{flushleft}
\vfill
\begin{abstract}
\noindent
In this paper, we derive an industry specific, supply chain model with Bertrand competition in the final demand goods sector and monopsony demand in the value added inputs markets. The literature broadly defines this type of model, a monempory model. This model allows us to explore the within supply-chain feedback effects of trade shocks in industries with oligopolistic, price competition in the final demand goods market. We conduct a series of experiments to test the performance of the model against a similarly designed Armington supply chain model and a simple, non-supply chain, Bertrand model. When tariffs are removed on imports of final demand goods, the simple Bertrand and supply chain Bertrand models generate similar estimates of the effects of tariff removal on prices, quantities demanded, and profits. However, the results diverge as we increase the market share of subject variety imports and the Armington elasticity in the final demand market. When we eliminate a tariff on imports of intermediate goods, the supply chain Bertrand and supply chain Armington models predict similar changes in final demand and intermediate goods prices.
\end{abstract}
\vfill
\begin{flushleft}
Ross J. Hallren, PhD, Office of Economics\\
\href{mailto:ross.hallren@usitc.gov}{ross.hallren@usitc.gov}\\
\vspace{0.5in}
\end{flushleft}
} % end of helvetica (arial) font
\clearpage
\newpage
\doublespacing
\setcounter{page}{1}
\section{Introduction}
We derive a simple supply chain model with Bertrand (price) competition in the final demand market and monospony competition in the value added markets. The literature broadly defines models with seller's power in the downstream market and buyer's power in the upstream market as "monempory" models. (\citeasnoun{Nichol1943}) To do this we combine a supply chain model similar to those derived in \citeasnoun{HosoeETAL2015}, \citeasnoun{HallrenRiker2018}, and \citeasnoun{DesaiETAL2019} with a modified version of the Bertrand competition model from \citeasnoun{Riker2019}. We explicitly model the domestic firm's supply chain and allow for monopsony demand in the labor and capital goods markets. By including these features in a Bertrand competition model, we are able to capture an important feature of Bertrand oligopoly firms, namely that they often exhibit market power in both the final demand and domestic factor markets. Moreover, we are able to simulate how tariff shocks ripple through the supply chain and determine how different the predictions are from those of the simple Bertrand model in \citeasnoun{Riker2019}.
In the textbook Bertrand case, with one firm in each national market, firms operate as monopsonists in their respective domestic factor markets, conditional on factors being mostly untraded across other domestic industries or across countries. (\citeasnoun{Robinson1932}) A market with Bertrand competition that utilizes specialized labor with country specific licensing could easily generate this outcome. (\citeasnoun{BoalRansom1997}) The implication of the Bertrand firm operating as a monopsonists in the factor market is that if it faces an upward sloping market supply curve, then the firm will pay a lower price and utilize less of the factor than in the perfect competition case. (\citeasnoun{Robinson1932})
Additionally, this simple supply chain Bertrand model allows us to capture the vertical and international linkages throughout the production chain. In doing so we are able to estimate not only the direct final demand effects of a policy shock but also the within supply chain feedback effects in the intermediate inputs, labor, and capital markets. Allowing for Bertrand competition creates a model that is more appropriate for industries, such as autos; wide-body jumbo jets; travel services; etc., than a similarly constructed model with Armington CES demand in the final demand sector.
In this paper we run a series of experiments to demonstrate the performance of the model. Additionally, we run the same experiments on the Bertrand model from \citeasnoun{Riker2019} and an Armington CES supply chain model that is similar to \citeasnoun{DesaiETAL2019}. We then compare the predictions of the three models to demonstrate the sensitivity of estimates to relaxing the perfect competition assumption in the final demand portion of the model, explicitly including the supply chain of domestic producers, and allowing for the domestic firm to exhibit market power in the domestic labor market.
The paper proceeds as follows. In section 2, we derive a simple supply chain Bertrand partial equilibrium model. In section 3, we conduct a series of experiments to demonstrate the performance of the model. Section 4 summarizes the results and section 5 concludes with a discussion of potential applications.
\section{A Bertrand CES Supply Chain Model \label{sec: section2}}
Figure 1 illustrates the conceptual structure of our simple Bertrand supply chain model. In this model, the firms engage in Bertrand competition in the final demand sector. The supply chain for the domestic firm is explicitly included. The domestic firm produces output by combining labor (L), capital (K), and an aggregate intermediate input (INT) via Cobb-Douglas production technology.
We assume that the domestic firm operates as a monopsonist in the value added factor markets. Further, we assume that the labor and capital markets exhibit upward sloping market supply equations. We assume that the industry supply of capital is highly inelastic such that demand shocks will primarily translate into changes in price.
In the intermediate goods portion of the model, products are differentiated by country of origin, and there are three sources for intermediate inputs, the domestic market (D), foreign countries subject to a policy shock (S), and foreign countries not subject to a policy shock (N). In the intermediate goods market, we assume that these inputs are sufficiently traded across industries and across countries such that the perfect competition assumption in the aggregate intermediate goods market is reasonable. The domestic firm combines these country specific intermediate input varieties into an aggregate intermediate input via CES technology.
The final demand portion of the model is as in \citeasnoun{Riker2019}. Therefore, we omit the derivation.\footnote{A detailed derivation is included in the appendix.} Instead, we discuss how we integrate the upstream supply portion of the Armington CES supply chain model from \citeasnoun{DesaiETAL2019} with the Bertrand model from \citeasnoun{Riker2019}.
In this model, three firms supply the domestic final demand sector, a domestic firm (D), a foreign firm subject to a policy change (S), and a foreign firm not subject to a policy change (N). Firms are profit maximizers, produce imperfect substitutes, and engage in price (Bertrand) competition. Each firm faces constant marginal cost of production and a fixed cost for entering the market.
Buyers substitute between each firm variety at a constant rate of substitution (CES). Across sectors, we assume that preferences are Cobb-Douglas. Given these assumptions, the system of equations is as in \citeasnoun{Riker2019}.
The final demand CES price index is
\begin{equation}\label{eq:1}
P_*=\left ( p_D^{1-\sigma}+b_S(p_St_S)^{1-\sigma}+b_N(p_Nt_N)^{1-\sigma}) \right )^{\frac{1}{1-\sigma}}
\end{equation}
Demand for each firm's variety is
\begin{equation}\label{eq:2}
q_D=k(P_*)^{1-\sigma}(p_D)^{-\sigma}
\end{equation}
\begin{equation}\label{eq:3}
q_S=kb_S(P_*)^{1-\sigma}(p_St_S)^{-\sigma}
\end{equation}
\begin{equation}\label{eq:4}
q_N=kb_N(P_*)^{1-\sigma}(p_N)^{-\sigma}
\end{equation}
The variable $t_S$ is the power of the import tariff on variety S. The power of the tariff is equal to one plus the ad-valorem tariff rate. $b_S$ and $b_N$ are model parameters that are calibrated to initial equilibrium conditions to capture differences in preferences and product quality across varieties. The parameter k is calibrated to the initial size of the market. The firm price is ($p_j$) for each variety j.
Through some algebraic manipulation, we can calibrate the demand parameter using initial expenditure data, tariff rates, and industry prices as follows.
\begin{equation}\label{eq:5}
b_h=\left (\frac{V_{h0}}{V_{D0}} \right )\left ( \frac{p_{h0}t_{h0}}{p_{D0}} \right )^{\sigma-1} \ for\ h \in S,N
\end{equation}
Given the CES preferences, the market share updating equation is equal to equation (6) for the domestic variety and equation (7) for imported varieties. The initial market shares are calibrated via the initial equilibrium expenditure data.\footnote{See \citeasnoun{Armington1969} and \citeasnoun{Riker2019}}
\begin{equation}\label{eq:6}
m_D=\frac{(p_D)^{1-\sigma}}{p_D+\sum b_h(p_ht_h)^{1-\sigma}} \ for\ h \in S,N
\end{equation}
\begin{equation}\label{eq:7}
m_h=\frac{b_h(p_ht_h)^{1-\sigma}}{p_D+\sum b_h(p_ht_h)^{1-\sigma}} \ for\ h \in S,N
\end{equation}
As in \citeasnoun{Riker2019}, firms produce differentiated products and engage in profit maximizing Bertrand competition. Firms face constant marginal costs and a fixed cost to entering the market. We assume perfect competition in the input markets; and therefore, marginal costs are equal to the price index from the supply chain. Therefore, the firms' equations are as follows:
\begin{equation}\label{eq:8}
\pi_j=(p_j-c_j)q_j-f_j \ for\ j \in \{D,S,N\}
\end{equation}
\begin{equation}\label{eq:9}
c_D=\prod p_l^{\alpha_l} \ for\ l \in \{L, K, INT\}
\end{equation}
$\alpha_l$ is the cost share of each composite input, labor (L), capital (K), and intermediate input (INT); and each price ($p_l$) is the market price for the corresponding composite input. In this paper, equation (9) describes the marginal cost for domestic firms. In the calibration phase, the marginal cost for the domestic firm is set equal to the initial equilibrium price index from the supply chain. The initial marginal cost for all other varieties (S,N) are set to 1, and prices are then adjusted until the model matches initial market conditions.\footnote{This deviates from \citeasnoun{Riker2019} where initial prices are set to 1 and marginal costs are adjusted until the model matches the initial equilibrium data.}
The supply chain portion of a vertical supply chain model is thoroughly presented in \citeasnoun{Hallren2019a} and \citeasnoun{DesaiETAL2019}.\footnote{In these papers, the domestic producers combines labor (L) and capital (K) into a composite valued added (VA) input and then combines the composite VA input with a composite intermediate goods input (INT) to produce output. Notably, the substitution elasticities at the VA-INT and L-K nodes are the same. Because of this, we can simplify the model by replacing the VA-INT node with a L-K-INT node. This simplifies the algebra when we introduce monopsonistic demand in the labor and capital markets. See \citeasnoun{HallrenRiker2018}} Therefore, we state the key equations and skip the derivation, except where the assumption of monopsony demand in the labor and capital markets requires adjustments to the baseline model.
Domestic firms produce output by combining a labor (L), capital (K), and a composite intermediate input (INT) through Cobb-Douglas technology. Therefore, demand for each factor is:
\begin{equation}\label{eq:10}
Q_{l} = \beta_{l}\frac{p_{D}}{P_l}q_{D} \ for\ l \in \{L, K, INT\}
\end{equation}
We assume constant returns to scale such that $\sum{\beta_{l}}$ $= 1$.
We assume monopsonistic demand in the value added factor markets, L and K. To allow for this type of competition, we must explicitly specify supply equations for each value added input. We consider a simple upward sloping supply relationships for labor and capital.\footnote{Assuming an upward sloping supply curve is important because when the supply of labor is perfectly elastic, (i.e. Firms can hire as many workers as required at the predominant wage (w).) the monopsony demand outcome, in-terms of quantity of labor demanded and wage paid, is identical to the perfectly competitive market outcome.}
\begin{equation}\label{eq:11}
Q_{f}=k_{f}(P_{f})^{1/\epsilon_{f}} \ for\ f \in \{L, K\}
\end{equation}
In this equation, we restrict $\epsilon_{f}$ to be greater than zero so that the inverse supply function is increasing in quantity. When an input's (e.g. labor) inverse supply function is increasing in quantity, then the monosponist will utilize less labor than in the competitive equilibrium and use its market power to pay a wage lower than the competitive equilibrium wage. (\citeasnoun{Robinson1932} and \citeasnoun{BoalRansom1997}) In our application, we assume that the supply of capital is highly price inelastic, and the supply of labor is nearly unitary elastic in price.
In equilibrium, the quantity of each value added input utilized is determined by the intersection of the marginal revenue product (MRP) and marginal cost (MC) curves, with respect to each input. However, as the only purchaser, the monopsony firm's buying price is determined by the inverse supply curve for each factor. Additionally, as a monemporist, the firm has to take into account how input procurement decisions affect the price of the final demand good. Therefore, to derive the monopsony equilibrium factor prices and quantities, we write the firm's profit function in terms of labor; capital; the composite intermediate input; the price of labor; the price of capital; the price of intermediate goods; the domestic final demand good production function; and the inverse demand function for domestic output, in lieu of the price of domestic output. (\citeasnoun{MonagoTavera2018})
\begin{equation}\label{eq:12}
\max \pi = \left({\frac{k{P}_{*}^{\theta+\sigma}}{q_D}}\right)^{1/\sigma} \left( Q_{L}^{\beta_{L}}Q_{K}^{\beta_{K}}Q_{INT}^{\beta_{INT}} \right) - Q_{L}P_{L} - Q_{K}P_{K} - Q_{INT}P_{INT}
\end{equation}
We then take the partial derivative with respect to each factor and solve the first order conditions. As illustrated in figure 2, the MRP, MC, and inverse supply equations allow us to determine the monempory equilibrium factor prices and quantities $(p_{f}^M,q_{f}^M)$. (\citeasnoun{BoalRansom1997})\footnote{For a concise example of this type of derivation see \citeasnoun{Shelburne2004}.}
\begin{equation}\label{eq:13}
MRP_{f}=\beta_{f}\left ( \frac{\sigma - 1}{\sigma} \right )\left ( \frac{q_Dp_D}{Q_{f}^M} \right ) \ for\ f \in \{L, K\}
\end{equation}
\begin{equation}\label{eq:14}
MC_{f}=P_{f}^M+\epsilon_{f}P_{f}^M \ for\ f \in \{L, K\}
\end{equation}
\begin{equation}\label{eq:15}
P_{f}^M=\left ( \frac{Q_{f}^M}{k_{f}} \right )^{\epsilon_{f}} \ for\ f \in \{L, K\}
\end{equation}
By contrast, in the Armington supply model we assume perfect competition throughout the supply chain and that the domestic industry utilizes a small portion of the labor force. Therefore, domestic producers can hire as many workers as necessary at the exogenous market wage $(p_L^{PC})$. In the capital goods market, producers face the same highly price inelastic inverse supply equation as specified above, and the competitive capital rent $(p_{K}^{PC})$ is determined by supply and demand. Given these conditions, the following will be true: $(p_{f}^M