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% Changed: 
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% (5) Empirical writeup


\begin{document}
\title{Collusion Along the Learning Curve: \\Theory and Evidence from the Semiconductor Industry}
%\date{March 15, 2015} % Fixed date placeholder here
\author{Danial Asmat\footnote{Economic Analysis Group, Antitrust Division, U.S. Department of Justice (\texttt{danial.asmat@usdoj.gov}). I am grateful to my advisors Maggie Levenstein, Valerie Suslow, Dan Ackerberg, Francine LaFontaine, and Brian Wu as well as Joe Harrington, Kai-Uwe K{\"u}hn, David Miller, Jagadeesh Sivadasan, Greg Werden, Nathan Wilson, and participants in several seminars, IIOC 2016, CRESSE 2016, and EARIE 2017 for comments. Data for this study was purchased with financial support from the Michigan Ross Doctoral Studies Office. I also thank Steve King of Gartner Inc., Jim Handy of Objective Analysis, and Sarim Shah of GlobalFoundries for numerous discussions about data and industry details. The empirical methodology for this paper was constructed with the aid of publicly available media and academic sources while the author was a Ph.D. student at the University of Michigan, prior to any employment with the U.S. Department of Justice. The views expressed are not purported to reflect those of the U.S. Department of Justice.}}

\maketitle
\thispagestyle{empty}

\begin{abstract} 
\noindent
\onehalfspacing
This paper studies the effectiveness of collusion in the DRAM cartel. Like other high technology products, DRAM is characterized by learning-by-doing and multiproduct competition. I hypothesize that collusion is more difficult to sustain on a new generation, where learning is high, than an old generation, where learning is low. A higher learning rate makes defection from a collusive equilibrium more attractive by reducing future cost. Empirical analysis exploits variation between cartelization and competition to estimate the change in firms' output decisions on each generation. Consistent with the hypothesis, cartel participants are estimated to cut output more on the oldest generation than newer generations. Output decisions on the newest generation also show evidence consistent with defection from collusive equilibria. Lastly, the paper presents a theoretical framework to analyze collusive equilibria with learning-by-doing and multiproduct competition. The model motivates various pieces of evidence that competition authorities can compile to guide antitrust investigations in high technology markets. 
~\\
~\\
\noindent
%Latest Version: \href{https://536621ad-a-ab15346e-s-sites.googlegroups.com/a/umich.edu/danial\_asmat/cabinet/DA\_JMP\_website.pdf?attachauth=ANoY7cp7mx2QEv-njTrZlKMCcwrNjlZrJpMvHCayb66wl5yffzQZkLOfUbIAtP42425HD3TmZi\_Jb2yqpwRL0-X9C4Dlvv4F6nPwQmTQjChVKZNmBO7TfTO5vzL4TUGXyJ282fA-TcOVgf\_Wa9SMrajNr-ef2rY31vDM9K4wuFFr-0KBIaUjns9Jpaqw4i8UdZrEbvlA-F04FP2HysATNl9Az887fwEtvBpkGosZcgERwIE-O\_qZysY\%3D\&attredirects=0} {umich.edu/danial\_asmat/JMP} \\
JEL Classification: D43, L13,  L41, L63

\end{abstract}

\clearpage

%%%%%%%%%%%%%%%%
%%%% INTRO SECTION %%%%
%%%%%%%%%%%%%%%%



\section{Introduction} \label{Intro}

%\renewcommand{\thepage}{\arabic{page}}
\pagenumbering{arabic}
Innovation is the foundational element of competition for a wide variety of high-technology products. It is particularly noteworthy that recent years have seen a steady discovery of price fixing cartels in dynamic, innovation-intensive industries: electrical auto parts, capacitors, LCD panels, cathode ray tubes, disk drives, and three different types of memory chips.\footnote{Fines from these cases are among the largest in U.S. antitrust history. AU Optronics, LG Display, and Samsung Electronics have each received penalties of \$300 million or more since 2006: \url{http://www.justice.gov/atr/public/criminal/sherman10.html}.} Some of these markets feature characteristics long known to enable collusion, including high barriers to entry and cooperative research and development. Yet they also share two additional characteristics less familiar in the academic literature on collusion. First, manufacturing displays learning-by-doing: a firm's cumulative output reduces the marginal cost of its future output. Second, firms steadily release new product generations based on technological advancements, such as those predicted by Moore's Law. Product generations with varying qualities therefore overlap on the market.%\footnote{In 2016, for example, buyers could choose between 4Gb, 8Gb, and 16Gb chips. See \url{http://www.techspot.com/article/1043-8gb-vs-16gb-ram/}.}


To study how firms collude in a high-technology industry with these two features, this article examines firm behavior during an illegal price fixing cartel in the Dynamic Random Access Memory (DRAM) market. Specifically, I hypothesize that collusion is more effective on older product generations than newer ones. To explore this hypothesis, consider that producing a unit of output on one product reduces its future cost. But the \textit{rate} of cost decrease depends on whether the product is old or new. In an older product, because engineers have spent years engaging in trial-and-error, the learning rate is low. But in a newer product the learning rate is high. The faster is the learning rate, the more firms gain by defecting from collusion, and the more patient they must be to sustain it. These ideas are later formalized in a simple theoretical model. 



%I first develop a parsimonious model of learning with multiproduct competition. The model shows that collusive equilibria are determined by the rate at which firms learn and the extent to which products are vertically differentiated. Next, I use data from an illegal price fixing cartel in the Dynamic Random Access Memory (DRAM) market to find evidence consistent with the model's hypothesis. 

%The model shows conditions under which collusion is more successful for an older product (generation) than a newer product. To see this, consider that producing a unit of output on one product reduces its future cost. But the \textit{rate} of cost decrease depends on whether the product is old or new. In an older product, because engineers have spent years engaging in trial-and-error, the learning rate is low. But in a newer product, in contrast, the learning rate is high. The faster is the learning rate, the more firms gain by defecting from collusion, and the more patient they must be to sustain it. This result holds unless the newer product has a significantly large and persistent quality advantage, and is maintained whether firms use a single- or multiproduct punishment strategy.

Like numerous international cartels, the DRAM case also featured a fringe of smaller firms outside of the cartel agreement. The empirical model utilizes the presence of an exogenously competitive fringe to generate a testable hypothesis. The more effective is collusion on a product, the more the output of the fringe expands relative to the output of members. The key empirical hypothesis of the paper is that during the cartel, the fringe gains more share on the old generation than on newer generations. 

DRAM production is a classic example of learning-by-doing: output allows firms to reduce cost-per-chip by reducing the rate of defective chips in manufacturing. The learning process repeats as chip generations are released every two to three years, with long product life cycles creating overlapping generations. Moreover, the discovery and eventual antitrust litigation on the cartel has created evidence on its duration, participants, and non-participants.

To identify the output decisions of fringe versus cartel members, I construct a reduced form model that predicts firm output. The model exploits two levels of variation: conduct changed across time, and some firms joined the cartel while others did not. This permits a ``difference-in-differences" estimate: the average change in members' output during the cartel period less the average change in non-members' output, separately for each generation. %By comparing the estimates \textit{between} generations, the model ties disparities in cartel effects to disparities in supply side dynamics. %Because each firm eventually produces all major generations, this approach circumvents endogeneity threats that would otherwise inhibit studying the dynamic behavior of a cartel.


%To estimate this difference-in-differences model, it is necessary to impose a limited degree of structure that governs learning-by-doing, cannibalization, and market demand. The most important assumption attributes differences in firm shares to differences in firm learning rates, while restricting those learning rates to behave systematically across time. I test the plausibility of this restriction by permitting learning to vary across a subset of the sample period and showing that the estimates are stable. I also test the model's robustness to other assumptions, including firm exit and alternative effects of cannibalization. 

Descriptive statistics show that the price of the 16Mb generation rose up to 25\% during the cartel, whereas the price fell for the newer 64Mb and 128Mb generations. Consistent with the hypothesis, the DiD model estimates that cartel members reduced output by more on the 16Mb generation than the 64Mb and 128Mb generations, at a significance level of up to 1\%. Although it is more challenging to estimate by how much cartel members reduced output on a single generation, the model's estimates can be used to construct a 95\% confidence interval for total output reduction on the 16Mb generation between 3.1\% and 24.6\%. 


%For example, the model rejects the null hypothesis of equality between member output reduction on the 16Mb generation and the 128Mb generation at the 1\% level.


To better understand the mechanism driving these results, I also formulate a straightforward test for defection from collusive equilibria in learning-by-doing industries. This test relies on the high correlation between each firm's order of entry into a generation and its subsequent market share ranking on that generation. If an unsuccessful attempt to collude stems from realized deviation by one or more firms, then there is basis for the correlation between order of entry and subsequent ranking to fall. This test shows that the correlation of 0.43 was significantly lower for the 128Mb DRAM generation than any other generation in the sample. The finding is consistent with the underlying hypothesis that collusion is most difficult to sustain on the newest generations during a cartel.  

%\clearpage

\textit{Literature Discussion} This study's primary contribution is to generate empirical evidence from a cartel in a highly innovative market. It focuses on the classic problem of how cartels meet incentive compatibility constraints (\citealp{porter83}; \citealp{ellison94}; \citealp{lev97}; \citealp{morton97}; \citealp{genesove01}; \citealp{roller_steen06}; \citealp{walsh13}; \citealp{clark_houde13, clark_houde14}, and \citealp{isugaya}).\footnote{See \cite{levsus06} for a broader review of empirical literature examining incentive compatibility in cartels.} Many of these studies test canonical models of repeated games by exploiting the historical legality of cartels. Consequently, they are geared toward industries with straightforward supply functions, such as those in railroads, sugar, and cement. 

%\footnote{DRAM, and semiconductors at large, have been well studied in the economics literature owing to their strategic importance for international trade, industrial policy, and antitrust. Empirical studies include \cite{irwinkl94}, who find evidence of significant learning, and \cite{dick91}, who evaluates the plausibility of antidumping laws in a setting with evolving marginal cost.}

In contrast, the present study examines a modern, high-technology product market in DRAM. It shows that a supply side dynamic can be the primary driver of incentive compatibility. \cite{roos04} uses a dynamic oligopoly model to explain the lysine cartel through the lens of entry deterrance and post-entry renegotiation. Similarly, this article shows that the dynamic incentives stemming from learning-by-doing should be considered in characterizing the DRAM cartel's behavior.



%Secondarily, it develops a parsimonious theoretical model that carries insights for antitrust policy in high technology markets. 



%Empirically, it joins literature that studies a ``hard core'' price fixing cartel to assess how firms reach incentive compatibility in collusive equilibria, which is otherwise difficult to assess. Finally, it extends literature on the learning-intensive semiconductor industry as it enters an era of increased concentration, intellectual property disputes and antitrust scrutiny. 

%


 %The model reveals how learning-by-doing can induce firms to collude in one product market while competing in another product market. and \citealp{fershtman00} for R\&D; and \citealp{sully16} for product differentiation).


%``semi-collusion," in which firms collude along one dimension but compete along another dimension

%It differs from existing models of learning-by-doing in focusing on collusive equilibria rather than long-run industry dynamics such as increasing dominance, natural monopoly and predatory pricing.

%(\citealp{paha} later)



%Finally, it is worth noting that the present study is the first to 

%the empirical analysis shows that learning-by-doing shapes collusion in the semiconductor industry, where previous evidence is restricted to competition. \cite{irwinkl94} specify a quantity-setting model to estimate that a firm's marginal cost of chip production is reduced by an average of 20\% following a 100\% increase in output, but that there is no significant learning effect between new product generations. \cite{zul03} and \cite{siebertz13} examine the rate of learning and competitiveness in DRAM using data from the 1970s to the mid-1990s, before firms began explicitly colluding.\footnote{Several additional studies have found evidence consistent with significant learning effects in DRAM and other memory chips. See \cite{dick91}, \cite{nye96}, \cite{gruber98}, and \cite{cabralleib01}.} \cite{gar16} uses more recent data to estimate a structural model of imperfect information and finds that DRAM manufacturers share demand information in equilibrium.\footnote{The role of memory and microprocessor chips, as well as technology standards in memory chips, is explained in \Cref{Ind_DRAM}.}

To assist in drawing more general insights, the paper also constructs a theoretical model of oligopoly centered on a two-period game. Following \cite{sieb10} and \cite{ft83}, firms choose quantities in a learning period knowing that they gain a cost reduction as a function of output in a post-learning period.\footnote{\cite{gar16} also models firms as choosing quantities in the DRAM market. Other quantity-setting models of learning include \cite{spence81}, \cite{ghem85}, \cite{dasgupta88} and \cite{cabralrio97}.} This structure is then augmented to include vertically differentiated products and a collusive strategy. The model motivates various pieces of evidence that competition authorities can compile to guide horizontal investigations in high-technology markets.\footnote{\cite{mookh91} also propose a model of collusion that includes scale economies and learning-by-doing, but only a single product. The present model emphasizes the interaction between learning and multi-product competition.} 

%The benefit of the two-period premise is to model dynamic strategic incentives within a collusive regime, without relying on the numerical computation of Markov perfect equilibria from a fully specified dynamic stochastic game (as in \citealp{fershtman00}). \textbf{Concluding sentence about policy implications and/or ivaldi}

The remainder of the paper is outlined as follows. \Cref{Ind_DRAM} and \Cref{Data_DRAM} describe the features of the industry, the cartel, and the data. \Cref{RF} estimates the empirical model and tests for alternative hypotheses. \Cref{Theory_DRAM} presents a model of collusion over the product life cycle, and summarizes its implications for antitrust policy. I conclude in \Cref{Conc_DRAM} by discussing the limitations and future work. 



%%%%%%%%%%%%%%%%%%%%%%
%%%% INDUSTRY OVERVIEW SECTION %%%%
%%%%%%%%%%%%%%%%%%%%%%

\section{Cartel Operations} \label{Ind_DRAM}


Most cartel cases end before jury trial, limiting the amount of public information about their evolution. The DRAM cartel is noteworthy for leading to a federal criminal trial as well as civil class action trials. This section summarizes what the trial record implies about the cartel's timeline and its potential effectiveness, which are important to understand before proceeding to empirical analysis.  


%summarizes the competitive forces for DRAM identified in the trial records, before describing the detailed evidence of collusion. The legal sources are available in \href{https://sites.google.com/site/danialasmat/config/pagetemplates/attachments/DRAM_cartel.zip}{Online Appendix D}.

%\footnote{The criminal trial ended with a hung jury and the defendant acquitted. Over a dozen other individual defendants pled guilty to the conspiracy. The civil trials ended in settlement before a final decision was made.} 



%Other than density, the secondary innovation channel for DRAM is through the \textit{technology} that underlies the chips. Unlike the density innovation process, technology standards are jointly developed by industry participants. For semiconductors, the JEDEC SSO openly standardizes next generation technologies many years in advance of a rollout.\footnote{See \url{http://searchsolidstatestorage.techtarget.com/definition/JEDEC}} Different generations of DRAM chips can be layered on top of any type of underlying technology, although newer generations are more likely to use newer technologies. \textbf{Their role is discussed further in empirical robustness analysis.} 

\begin{figure}[h]
	\centering
  	\includegraphics[height=3.0in, keepaspectratio]{./pics1/timeline_scrnsht_VS_comments_v3}
	\caption{Events During Cartel Period}
  	\label{fig:timeline}
Sources: Noll (2014) and DOJ Press Releases
\end{figure}

\Cref{fig:timeline} sequentially depicts the main events in the DRAM cartel. First, firm communication began in response to excess capacity. Next, cartel members cooperated to raise price from at least the second quarter of 1999 through the first quarter of 2001. From April through November 2001, however, they switched to active price competition in a bid to eliminate South Korean firm Hynix from the market. After state-owned banks bailed out the company, members reverted to the price fixing strategy until the DOJ issued a subpoena in June 2002 (\citealp{noll14}).\footnote{Communications between cartel members remained cooperative throughout these events, including the ``Kill Hynix'' period. For example, on September 20, 2001 executives from two other members discuss Hynix' position: `On the assumption that Hynix gets new money, they would consider taking supply out of the market if others do the same.' Third Amended Class Action Complaint, MDL No. 1486, $\P$ 214.}


%the DOJ dates the cartel from April 1, 1999 through June 15, 2002. From 


%It is well known that negotiated dates of collusion may not correspond to actual dates of collusion (\citealp{levsus11}). In this case, detailed evidence of inter-firm communications indicates that cartel members were cooperating to raise price from at least the second quarter of 1999 until the first quarter of 2001. 

%\footnote{``DRAM Rivals Question Korean Government's Role in Hynix Bailout,'' EE Times, 11/5/2001 \url{http://www.eetimes.com/document.asp?doc_id=1131477}}

%Both the DOJ and Noll cite June 2002 as the end of the cartel, after Michael Dell, CEO of Dell Inc., publicly alleged collusive behavior and the DOJ issued supoenas to major manufacturers. Finally, \cite{noll14} also cites documentary evidence of external communication between firms in 1998.

These events illustrate three distinct phases of firm conduct: the cartel's first phase; the ``Kill Hynix'' phase; and the cartel's third phase. Empirical analysis will consider the dates corresponding to each phase as fixed. The ``Kill Hynix'' phase is treated as competitive, and the other two as collusive.%\footnote{The DOJ officially dates the cartel from April 1, 1999 through June 15, 2002, inclusive of the ``Kill Hynix'' phase. It is well known that negotiated dates of collusion may not correspond to actual dates of collusion (\citealp{levsus11}).} 

%Talk about how I will treat these dates as fixed, based on others' exhaustive review of hundreds of pages of written communications. Whereas the alternating behavior of the cartel creates fewer periods to analyze collusion, it provides more periods to analyze change in conduct from competition to collusion or vice versa. 


%List Noll result about overall price increase during this cartel period, using RF model. But no price increase during ``Kill Hynix'' period. He marks Q1-2001 as a ``transition'', but results are robust to excluding it. Finally, discuss exit. 

During its operation, cartel members communicated to raise price in three ways: specific price quotes to individual customers; general pricing levels across customers; and production volumes and inventories. Mid-level managers directed their sales representatives to communicate with rival counterparts in quoting prices to major OEMs. There is substantial email evidence documenting these price quotes, including a reference to a shared spreadsheet that tracked different firms' price targets. The evidence does not detail any specific monitoring or punishment strategy, or identify instances of cheating. 

The scope to make strategic output decisions, including those factoring in future cost reductions, was likely limited to high-level executives. Senior executives held in-person meetings to discuss production figures and supply reduction. These meetings appear to have been organized between executives of two or more firms at a time rather than through a trade association or industry group. The evidence does not detail the content of the meetings, except where it was later memorialized through email, as in this example:

 %in-person meetings among high-level executives. 

%In the DRAM cartel, 

%Like other international cartels, 

\begin{quote} 
\onehalfspacing
\textit{Mr. Chang confirmed that [firms] `had reached an agreement ... to push up DRAM prices to US \$3 a chip by stopping dumping.' }
  --- Third Amended Class Action Complaint, $\P$ 203-214
\end{quote} 

Official plea agreements state that the effectiveness of collusion varied by product:

\begin{quotation}
\onehalfspacing
\noindent ...\textit{at certain times during the Relevant Period, DRAM prices decreased significantly. Nevertheless, the Defendant, or its Corporate Founders, and their coconspirators reached agreements to limit the rate of price declines, which were achieved with varying levels of effectiveness.}\\\ 
\; \; \; --- Plea agreement with [firm], March 22, 2006
\end{quotation}

The litigation record does not appear to link such statements to specific products or generations. The most suggestive remark comes from defendants' economic expert. Using price data from the market research firm IDC, the expert testified that prices during the third phase of the cartel rose significantly more for the 16Mb generation than newer ones. Testimony does not appear to analyze \textit{why} such disparities may have arisen across generations, which motivates the present study.\footnote{The transcript of cross examination on this point is included in \cref{fig:swanson}.}

%This article attempts to study the effectiveness of collusion across generations with greater detail. 

%Referring to the third phase of the cartel, defendants' economic expert testified to the following description of price trends:
%\begin{quotation}
%\onehalfspacing
%\noindent \textit{The price for the 64 and 128 size went up during [Q4-2001 and Q1-2002], but not as much [as the 16 size].}\footnote{Transcript of Proceedings as to Criminal Trial of Gary Swanson, Case No. 06-0692, U.S. District Court N.D. Cal., 12/17/2007, pg. 97.}
%\end{quotation}

 %To do so, it primarily focuses on comparing the output levels of cartel members to non-members during the cartel period. The data, identification strategy, and limitations of this approach are discussed in the two sections to follow.


%\subsection{Industry} \label{industry}
% 1 Microchips are a bedrock of the electronics revolution: they provide the processing, memory, speed, and performance ability of computers and many electronic devices. DRAM, one type of chip, allows a microprocessor, a second type of chip, to function effectively by refreshing the data contained within it. A critical feature of DRAM is its \textit{bit density}: the number of transistors per square inch. Increases in density reduce the effective cost of a microchip, or, equivalently, increase its speed. A new generation of DRAM is equivalent to doubling its density. 

% Capacity has risen sharply since the industry's inception in the 1970s, famously corresponding to ``Moore's Law'': the maximum capacity of an individual chip doubles every 18-24 months.

% 2 The primary route to innovation occurs through density \textit{photolithography}.\footnote{The cost per chip can also be reduced by increasing the size of the wafer itself, which allows more chips per lithographic batch of DRAM. Such changes occurred about half as frequently as density advancements during the 1980s and 1990s; see \cite{kang2010} Table 3-2.} Each new generation initially produces a yield of zero chips because of hundereds of nanoscale level-steps requiring precise light, air, and dust conditions. \cite{mowery98} find that the most important adjustments engineers make to increase yield are in \textit{parametric processing}, determining the minute range of chemicals to be applied at each step, followed by adjustments in air particle contamination. Problem diagnoses in these two areas gradually increase the number of usable chips, and increasing yield drives the cost of each chip down as a function of output.  




%In the early 2000s, two different types of new technologies were competing to become the next industry standard. Double Data Rate Synchronous DRAM (``DDR''), preferred by the industry SSO, competed with Rambus DRAM (``RDRAM''), designed and patented by the Silicon Valley firm of the same name. \Cref{dram_tech} discusses competition between these technologies and shows that neither technology began to proliferate until after the cartel period, which precludes the possibility of competition over technologies confounding any of the empirical results to follow.


%\subsection{Market} \label{mkt}


% 3 DRAM demand was overwhelmingly driven by PCs for the data period under study. PCs constituted the only significant end use throughout the 1980s and early 1990s \citep{flamm_reiss93}; they comprised about 90\% of the market in the late 1990s and early 2000s (Third Amended Class Action Complaint, MDL No. 1486, para 76 \nocite{CA_MDL}); and 80\% of the market in 2006 \citep{kang2010}.\footnote{In recent years, the market for DRAM has begun to shift from traditional PCs toward mobile smart phones and computer tablets, which use lower-power memory but sell many times the units as PCs. See ``Why Growth in Mobile Devices Will Fuel Micron's DRAM Shipments,'' Forbes, 1/15/2013 \url{http://www.forbes.com/sites/greatspeculations/2013/01/15/why-growth-in-mobile-devices-will-fuel-microns-dram-shipments/}.}

%In the late 1990s, the industry faced overcapacity despite steady PC growth.\footnote{``But too many new plants were built several years ago in a rush to benefit from a boom many thought would never end...The glut led to a free fall in prices. In-Stat expects a 21 percent decline in revenue this year, to \$15.6 billion.'' (\textit{Dallas Morning News}, June 1998 \url{http://articles.chicagotribune.com/1998-06-29/business/9807100053_1_memory-in-personal-computers-dram-memory-chip-business}).}

% 4 Investment is lumpy: each new fabrication facility costs billions of dollars.\footnote{By the end of the data period, fab cost had reached at least \$3 billion (``Semi Industry Fab Costs Limit Industry Growth'', \textit{EE Times}, October 2012  \url{http://www.eetimes.com/document.asp?doc_id=1264577}).} In the late 1990s, demand growth was insufficient to keep pace with capacity increases. 


%%%%%%%%%%%%%%%
%%%% DATA SECTION %%%%
%%%%%%%%%%%%%%%

\section{Data and Descriptive Evidence} \label{Data_DRAM}

I use a proprietary dataset from market research firm Gartner Research that lists quarterly shipments by firm and density generation, and quarterly market price by density generation, for all firms in the DRAM market from 1974-2011. Firm names allow accurate tracking of mergers, entry, and exit throughout the data sample. Separately, Gartner research reports list the quarterly worldwide PC shipments from 1997 onward. I consider the overlap of these two datasets, which spans from 1997-2011.

%\textbf{\cite{noll14} highlights the fundamental role of innovation to DRAM competition. The initial microchips of a new generation are defective: each firm must increase its yield of usable chips by repeating hundreds of nanoscale-level steps requiring precise light, air, and dust conditions. This process restarts for each generation, so most learning-by-doing takes place \textit{within} a DRAM generation.\footnote{Consistent with the DRAM engineering process, \cite{irwinkl94} find significant learning-by-doing within rather than across generations.} The prices of each generation are available worldwide and the chips are easy to transport.}


\begin{figure}[h]
\centering
\subfloat[]{
        \label{subfig:prof_1}
        \includegraphics[height=2in, keepaspectratio]{./pics1/price_log_ext2} } 
\subfloat[]{
        \label{subfig:prof_2}
        \includegraphics[height=2in, keepaspectratio]{./pics1/price_index_ext2} }
\caption{DRAM Price Trends: Cartel Active Periods Shaded}
\label{fig:price_index}
\end{figure} 

%\begin{figure}
%	\centering
%   \includegraphics[height=3.75in]{./pics1/price_index.png}
%	\caption{DRAM Price Trends by Generation}
%  	\label{fig:price_index}
%\end{figure}

%DRAM price is tracked at the generation-time level.

To gauge preliminary evidence, it is useful to consider the DRAM price during and after the cartel period. \Cref{fig:price_index} depicts this trend in two different ways. Panel (a) is the log of the module price, separately for each generation. Panel (b) standardizes the price level of each generation by its value in the the first quarter of 1999, prior to the official start of the cartel. During the first phase of the cartel, the price of the 16Mb chip rose up to 25\% above its starting value. After the cartel ended, the 16Mb price declined, whereas the 64Mb and 128Mb price increased. 

This pattern is consistent with a hypothesis in which the 16Mb price transitioned from a collusive to a competitive value after the cartel. However, price data suffers from small sample size because prices are available only at the generation-quarter level. There are 10 data points for cartel prices in each generation, and no firm-level variation. Consequently, I use firm-generation-quarter level shipments going forward.  


\begin{figure}[h]
\centering
\subfloat[Total Output, by Generation]{
        \label{subfig:volume}
        \includegraphics[height=1.75in, keepaspectratio]{./pics1/qty_by_gen} } \\
\subfloat[Market Share, Select Firms]{
        \label{subfig:share_evo}
        \includegraphics[height=1.75in, keepaspectratio]{./pics1/share_evo} }            
\subfloat[Number Firms Active, by Generation]{
        \label{subfig:gen_N}
        \includegraphics[height=1.75in, keepaspectratio]{./pics1/gen_N} }  
\caption{}
\label{fig:agg}
\end{figure} 

\Cref{fig:agg} represents several long-term trends in the DRAM market. Panel (a) depicts the total output (across all firms) by generation. Output takes an inverse U-shape for each generation and the total output of DRAM increased over time, consistent with increasing global demand for computers and electronics. Panels (b) and (c) highlight that the industry became steadily more concentrated over time. Panel (b) shows that the five largest firms accounted for an increasing share of industry revenues; (c) shows that with each new generation, some firms failed to advance (the three cartel generations are shaded in black).  In the analysis that follows, I consider all active firms in each generation.\footnote{\Cref{extra} also displays the market share evolution of the largest firms separately for each of the three cartel generations.}  

%To maintain a consistent set of firms in the sample, I narrow to those that remained active through the 128Mb generation. Of the seven firms that produced 16Mb or 64Mb, but not 128Mb, five exited before the start of the cartel. Firm exit is formally considered in the robustness analysis following empirical results.

% \begin{figure}[h]
% 	\centering
%   \includegraphics[height=2.75in]{./pics1/qty_by_gen}
%   	\caption{Total Output by Generation: 256Kb - 4Gb}
%  	\label{fig:unlog_qty}
%  \end{figure}

%Moving to the firm level, FIGURE Y shows the industry-wide market shares of the four largest DRAM manufacturers over time. During the cartel period, the C4 concentration ratio was (x) and the HHI was (y); it rose to (w) and (z) by 2011. 


The empirical analysis focuses on firm output during the cartel period. Cartel members are identified by combining reports from the two largest antitrust authorities, the EC and DOJ, with additional firms named in the private class action complaint. This yields 11 cartel members, and the remaining firms are taken to comprise a competitive fringe. \Cref{DD_stats} summarizes the differences between cartel members and non-members from 1998 through the end of the cartel period, 2002.


\begin{table}[h]\centering
\begin{threeparttable}
\captionsetup{justification=centering}
 \caption{Firm Averages by Cartel Membership, 1998-2002}
 \label{DD_stats}
\def\onepc{$^{\ast\ast\ast}$} \def\fivepc{$^{\ast\ast}$}
\def\tenpc{$^{\ast}$}
\begin{tabular}{lcccccccc} \toprule \toprule
 {\extracolsep{0.2in}}
%p{3.2cm}

    & \multicolumn{2}{c}{N Firms$^{*}$} & \multicolumn{2}{c}{Total Share$^\dagger$} & \multicolumn{2}{c}{Age (qtr)$^\ddagger$}  & \multicolumn{2}{c}{Log Cum Output$^\ddagger$}  \\
    \cmidrule(lr){2-3} \cmidrule(lr){4-5} \cmidrule(lr){6-7} \cmidrule(lr){8-9}  \\
    &1  &0 &1  &0 &1  &0 &1  &0   \\
    \midrule

16  &11  &11 &0.76 &0.24  &26.2 &20.2  &12.5 &10.6 \\[7.5pt]
64  &10 &9  &0.90 &0.10  &14.0 &8.9  &11.4 &8.9 \\[7.5pt]
128 &10 &6 &0.97 &0.03  &7.7 &4.8   &9.7 &6.7 \\[7.5pt]
    \bottomrule
\end{tabular}

\begin{tablenotes}
$^*$ Sum across firms in group.
$^\dagger$ Mean across time, all firms in group $^\ddagger$ Mean across time and firms \\
1 = cartel member; 0 = non-member. %\textit{0: Etron; Fujitsu; Powerchip; Vanguard; IBM (16, 64, 128); Rohm (16, 64)}
\end{tablenotes}
\end{threeparttable}

\end{table}

Members have higher average shares, older ages, and more cumulative output than non-members. For this reason, the fringe is stronger the older is the generation. More generally, during the cartel period the 16Mb, 64Mb, and 128Mb generations accounted for 93\% of total DRAM share by revenue. %Nine different product generations, from 1Mb to 2Gb, appeared after 1997.

\begin{figure}[h]
	\centering
   \includegraphics[height=2.0in]{./pics1/share_1.png}
	\caption{DRAM Cartel Member Share by Generation}
  	\label{fig:share_1}
\end{figure}

%As reflected in the descriptive table, members have the largest share in the 128Mb generation, followed by the 64Mb and 16Mb generations.

\Cref{fig:share_1} plots the time series of total member share relative to all firms in the generation, before and after the cartel period. As reflected in the descriptive table, members have the largest share in the 128Mb generation, followed by 64Mb and 16Mb. In each generation, the cartel member share tends to decline over time, as smaller, later-arriving firms increase their production whereas larger firms eventually decrease it. This graph highlights the need to account for each firm's stage in the product life cycle of its generation. %For the 16Mb and 64Mb chips, this pattern is reversed in the ``Kill Hynix'' period, when member share rises. The same effect does not appear present for the 128Mb chip. 

%The figure reveals a mixed pattern: during the cartel's first phase, member share of the 16Mb and 64Mb generations \textbf{tended to be lower} than during the ``Kill Hynix'' phase. On the other hand, the 16Mb member share stayed constant during the cartel's third phase, whereas 64Mb share decreased. \textbf{closing sentence}

%By inspection, the 16Mb trend exhibits a pattern consistent with collusion. Members decrease share during the first phase of the cartel, and increase share during the ``Kill Hynix'' phase. The 64Mb generation shows a similar, though higher variance, pattern. In contrast, the member share in the 128Mb generation stays relatively constant, and increases during the cartel's third phase. 



%The price trends are materially identical to those constructed from the alternative data provider IDC and presented at trial.

%Finally, FIG Y disaggregates cartel member shares separately for the 16Mb and 64Mb generations. For each generation, it conditions on the time period of entry, (describe). Conditioning on the period of entry shows a different picture. For both chips, the cartel member share among the later group of entrants reaches its lowest point during the cartel's first phase. The member share rises in the ``Kill Hynix'' period, consistent with 

\begin{figure}[h]
\centering
\subfloat[16Mb]{
        \label{subfig:split_share_3}
        \includegraphics[height=1.75in, keepaspectratio]{./pics1/split_share_3} }            
\subfloat[64Mb]{
        \label{subfig:split_share_4}
        \includegraphics[height=1.75in, keepaspectratio]{./pics1/split_share_4} }  
\caption{Cartel Member Shares, by Generation and Entry Group}
\footnotesize{Share = $x_{et} (1 - \alpha_{et}) / \left[ x_{et} (1 - \alpha_{et}) + y_{et} \alpha_{et} \right]$ , where $e$=entry group, $t$=quarter \\ $x$=total member output, $y$=total non-member output, $\alpha$= $n_{members}$ / $n_{all}$}
\label{fig:splits}
\end{figure} 

%\\ \footnotesize{Share = $x_{et} (1 - \alpha_{et}) / \left[ x_{et} (1 - \alpha_{et}) + y_{et} \alpha_{et} \right]$ where $e$=entry group, $t$=quarter \\ $x$=total member output, $y$=total non-member output, $\alpha$=number of members divided by all firms}

To get a clearer picture, \cref{fig:splits} plots the cartel member share separately based on entry period for (respectively) the 16Mb and 64Mb generations. Each graph groups firms into two categories: early and later entrants. The cartel member share for each group is computed as the ratio of cartel member output to the sum of member and non-member output within the entry group.\footnote{The caption also conveys that the output for non-members is weighted by the ratio of the number of firms inside to outside the cartel within the entry group at any time. This weighting standardizes to a per-firm level if the number of cartel members differs from the number of non-members. It also controls for the effect of any firm exiting the generation, which can occur in the final quarters of an older chip like 16Mb. Results are similar without weighting.} Conditioning on entry period shows that, among the newer group of firms, member share reached its lowest point from 1998-2004 during the first phase of the cartel, and rebounded afterward. This effect is present in both generations. It is not driven by entry of new firms after the cartel started, as it excludes entrants to each generation on or after 1999-Q2. 

The pattern in \cref{fig:splits} suggests that at least during the cartel's first phase, certain cartel members reduced output relative to outsiders on these two generations. However, the graph does not condition on cumulative output. It also ignores the role of firm decisions across product generations and of demand. To account for the effect of these variables, \cref{RF} formulates a regression model that compares the output of firms in the cartel to those outside. %\Cref{ID} describes the identification assumptions sufficient to estimate the model. 

%Although these patterns are suggestive of collusion being most effective in the 16Mb generation, they face natural limitations. The price trends cannot capture underlying costs, and the cartel member shares do not account for the selection effects indicated by \cref{DD_stats}. Hence \cref{ID} describes the identification assumptions sufficient to estimate a difference-in-differences model of the cartel's effect on its members' output.   

%To gain a more precise understanding of the difference in behavior on each generation across the two groups of firms, I estimate a difference-in-differences model of firm output. 


%Restricting the sample to firms producing output during the first 10 years of a product life cycle, there are 26 different firms in the dataset, with a maximum of 19 firms in 1993. 

%%%%%%%%%%%%%%%%%%%%
%%%% Reduced FORM SECTION %%%%%
%%%%%%%%%%%%%%%%%%%%

\section{Empirical Results} \label{RF}

\subsection{Empirical Model} \label{ID}

For each generation of interest, the research design estimates the cartel's effect on member output. This challenge is pursued by exploiting two sources of variation. Firm conduct is hypothesized to change across time: from competition to collusion, and vice versa. Firms are also hypothesized to differ in conduct: cartel participants and the competitive fringe. 

%Notably, the empirical strategy ties all variation in conduct across generations to the dynamic effect of learning-by-doing. It cannot distinguish between alternative dynamic explanations that may explain some of the evidence.

%\textbf{The regression model is descriptive in nature: it intends to examine whether controlling for the key determinants of firm output yields a similar or different cartel effect across generations.} 


% if pushed, can argue that these explanations, if fully fleshed out, would not necessarily lead to predictions that I estimate in the data. If members are trying to drive weak firms out, then this should be observed on all generations, not just new ones. 

%For example, incumbents on newer generations enjoy greater profits than those on older generations. An alternative explanation is renegotiation (de Roos), although both 16Mb and 64Mb were active before cartel began, suggesting that this is not only explanation. Also discuss exit, and viability of fringe as a control: assumes that cartel member and non-member identities are fixed across time

 %\textbf{Natural variation across firms, generations, and time is paramount to the research design at large. With a single product generation, for example, estimating the effect of a cartel would require imposing a fully specified dynamic structural model of oligopoly with Markov perfect equilibrium. The well-known computational challenges of that approach may account for the paucity of such studies.}

%Formally, consider the following semiparametric panel regression:
%\begin{align}
%	y_{ikt}= w_{kt}' \gamma + f( x_{ikt}) + \sum_{i} g_{i}( z_{kt}) +  \alpha_{i} \times \eta_{k} + \nu_{t} + \epsilon_{ikt} \label{eqref:id_eq} 
%\end{align}
%Let $\gamma_{6 \times 1}$ consist of two types of indicator variables, for each of the three cartel generations. The first indicates a cartel period, and the second interacts cartel period with cartel membership.\footnote{Primary effects on cartel membership are absorbed in firm fixed effects $\alpha_{i}$.} Each interacted parameter of $\gamma$ estimates the average output effect on cartel \textit{members}, relative to the average effect on \textit{non-members}. 

% It also includes fixed effects at the firm-generation and quarter-of-year levels.

%\begin{align}
%	y_{ikt}= m\left( x_{ikt}' \right) + h\left( z_{kt}' \right) + w_{kt}' \gamma + \alpha_{i} (+ \phi_{k} ) + \nu_{t} + \epsilon_{ikt} 
%\end{align}

%\subsection{Identification Assumptions} \label{id_s}

Index firms by $i$, generations by $k$, and time by $t$. The panel data specification is given by a DiD model that accounts for the effect of accumulated output, cannibalization, and demand:
\begin{align}
\begin{split}
ln(q_{ikt}) =& \, \beta_{0} + ln ( x_{ikt}) \beta_{1} + \sum_{i=2}^{I} \Big[  ln(x_{ikt}) \times \alpha_{i} \Big] \beta_{i}  \\ 
&+ ln \left(q_{i r t} \right) \omega_{1} + ln(PC_{t}) \eta_{1} + \sum_{k=2}^{K} \Big[ ln(PC_{t}) \times GenRank_{kt} \Big] \eta_{k} \\ 
&+ \sum_{k=1}^{k=3} \bigg\{ \mathbbm{1}(Coll_{kt}) \gamma_{k} + \Big[ \mathbbm{1}(Coll_{kt}) \times \mathbbm{1}(Memb_{i}) \Big] \gamma_{k+3}  \bigg\}  + \nu_{t} + \epsilon_{ikt} \label{eqref:id_eq} 
\end{split}
\end{align}
The first line models the effects of accumulated output, which leads the firm to increase current output. Denote $ln(x_{ikt})$ = $ ln \left( \sum_{\tau=1}^{t-1} q_{ik \tau} \right) $ the log of cumulative output accrued by firm $i$ on $k$ at $t-1$, and $\alpha_{i}$ firm fixed effects. The last term in the line interacts log cumulative output with firm fixed effects. 

This specification restricts the effect of accumulated output to be equal across generations for each firm. This restriction is violated to the extent that firm learning changes over time, or firms experience significant changes to capacity levels. Its plausibility is examined by testing for pre-cartel parallel trends, and by a robustness test which considers a differential effect of cumulative output across the sample period. 

The second line models the effect of cannibalization, which eventually leads the firm to reduce output; and demand, which can increase or decrease its output. Considering cannibalization, $ln(q_{i r t})$ denotes industry output on the succeeding generation $r=k+1$. The estimation of cannibalization is discussed further below.


Considering demand, define $ln(PC_{t})$ as the log of worldwide PC shipments at $t$. The last term interacts this measure with $GenRank_{kt}$: fixed effects for the ranking of each generation from newest to oldest at $t$. PC shipments constituted the primary application for DRAM during the sample, and are taken as exogeneous to the supply of the component as in \cite{igami17}.\footnote{Highlighting the importance of PC shipments, \cref{fig:output_PC} shows that there is little correlation between the release dates of major Windows operating systems and DRAM output, except through PC growth.} Because the marginal value of a frontier generation could be greater for new than old PCs, this specification allows an exogenous shift in demand to vary by generation ranking.


The regressor $ln(q_{irt})$ is expected to be endogenous because of unobserved factors in firm output that influence both $ln(q_{irt})$ and the dependent variable $ln(q_{ikt})$. These include temporary changes to a firm's costs such as plant outages or financial shocks in the home country. For example, a power outage at a Japanese plant forced Toshiba to cut its flash memory production by up to 20\% in Q1-2010.\footnote{See https://ca.reuters.com/article/technologyNews/idCATRE6B865620101209.}

%Products are assumed to be homogeneous within a generation, but imperfect substitutes (based on preferences for speed) across generations. Assumption~\ref{ida3} models this incentive by assuming that the succeeding generation $r=k+1$ cannibalizes sales of the current generation $k$. It focuses on the demand-side effect of cannibalization by modeling the sum of industry output on $r$, $\sum_{i}^{I} q_{i r t}$. It also restricts the underlying own- and cross-price inverse demand elasticities to remain constant across time. These assumptions are later releaxed by distinguishing between industry output on $r$ and firm output on $r$.



To account for endogeneity, $ln(q_{irt})$ is instrumented by the elapsed quarters of shipment of firm $i$ on $r$, denoted $FirmAge_{irt}$. $FirmAge_{irt}$ is positively correlated with $ln(q_{irt})$ over the relevant portion of $r$'s life cycle. The condition for instrument exogeneity assumes that the initial release (the first quarter of positive output) on $r$ is not impacted by cost shocks. This condition is more likely to hold in the semiconductor industry than others, because the predictable nature of Moore's Law permits multi-year forecasts, and each firm begins a new generation with \textit{de minimus} production. %Moreover, because the initial quarter's output generates \textit{de minimus} revenue for the firm, it is incentivized to begin production as soon as the generation 


Finally, the last line estimates the effect of the cartel on output. $\mathbbm{1}(Coll_{kt})$ equals one for each cartel generation during collusive periods, and $\mathbbm{1}(Coll_{kt}) \times \mathbbm{1}(Memb_{i})$ equals one for cartel member firms in those periods. The three parameters $\{ \gamma_{4}, \gamma_{5}, \gamma_{6}\}$ estimate the effect of the cartel on $k$, and $\nu_{t}$ denotes quarter-of-year fixed effects.


%The appropriate specification will be governed by empirical diagnostic tests. It rests on the degree of information conferred by other firms or generations: if such information is statistically low-value, then (iii) is preferred; if it is informative, then (i) or (ii) is preferred.




%But it does not allow the effect of a given level of PC volume to vary between generations of the same age. 

%Assumption~\ref{ida1} also carries two restrictions. First, it restricts the effect of a change in demand to be equal across firms in a generation. This corresponds to the structure of the theoretical model, in which DRAM firms sell homegeneous products.

%The practical intepretation of this assumption is that a positive (negative) demand realization for a firm-generation results in an output increase (decrease) in its opening quarter, whereafter it is constant.} 

%\textbf{Moreover, the dynamic nature of the output cycle renders the restriction implied by Assumptions~\ref{ida1}-\ref{ida1b} amenable to empirical testing. To the extent that firms receive persistently positive (negative) demand shocks in a generation, errors from the regression model should follow an AR($t$) process, $t > 1$.} 


%Finally, note that whereas price-adjusted quality increases in Intel CPUs help drive demand for PCs, and 32-bit Intel CPUs prevailed throughout the first 10 quarters of the cartel.

%The set of assumptions outlined above draws heavily upon the institutional elements that comprise the theoretical model: DRAM is a homogeneous product; learning occurs within firms and generations; and demand is driven by common trends that affect all firms in the industry. 


\subsection{Primary Results} \label{qty}

To begin working with the empirical model outlined above, I estimate a ``pre-treatment'' version of the linear regression model in equation \ref{eqref:id_eq}. This specification treats $\gamma$ as the vector of dummy variables estimating the effect of future cartel membership \textit{prior to} the start of cartel. If the data meets the parallel trends condition of the DiD model, there should be no effect of future membership on firm output. This test can be conducted for two generations, 16Mb and 64Mb, which were active prior to the start of the cartel in 1999. 
 

\begin{table}[htbp]\centering
\begin{threeparttable}
\captionsetup{justification=centering}
 \caption{Estimates of Future Cartel Membership Status on Log Firm Output:\\During Pre-Cartel Years}
 \label{placebo}
\def\onepc{$^{\ast\ast\ast}$} \def\fivepc{$^{\ast\ast}$}
\def\tenpc{$^{\ast}$}
\begin{tabular}{lcccccc} \toprule \toprule
 {\extracolsep{0.15in}}
%p{3.2cm}

    & \multicolumn{4}{c}{OLS} & \multicolumn{2}{c}{2SLS}   \\
    \cmidrule(lr){2-5} \cmidrule(lr){6-7}  \\
    & \multicolumn{2}{c}{(1)}  & \multicolumn{2}{c}{(2)}  & \multicolumn{2}{c}{(3)}     \\
    \midrule

Constant & -9.407 & (2.584) & -5.035 & (2.482) & -3.735 & (2.643) \\[5pt]
Log Cum Output$^\dagger$ & 0.420 & (0.041) & 0.366 & (0.040) & 0.436 & (0.056)  \\[5pt]
Log Output$_{k+1}$$^{\ddagger}$ & 0.029 & (0.021) & 0.002 & (0.019) & -0.096 & (0.041)  \\[5pt]
Log PC Ship & 0.816 & (0.153) & 0.586 & (0.147) & 0.481 & (0.161) \\[5pt]
Log PC Ship $\times$ Gen Rank$_{2}$ & 0.030 & (0.010) & 0.043 & (0.011) & 0.051 & (0.012)  \\[5pt]
Log PC Ship $\times$ Gen Rank$_{3}$ & -0.001 & (0.013) & 0.030 & (0.014) & 0.055 & (0.017) \\[5pt]
Log PC Ship $\times$ Gen Rank$_{4}$ & -0.069 & (0.013) & -0.035 & (0.013) & -0.001 & (0.019) \\[5pt]
Log PC Ship $\times$ Gen Rank$_{5}$ & -0.100 & (0.014) & -0.072 & (0.015) & -0.038 & (0.018) \\[7.5pt]
I(1997)$_{16}$ & 0.168 & (0.193) & 0.532 & (0.311) & 0.519 & (0.324) \\[5pt]
I(1997)$_{16}$ $\times$ I(Memb)$_{16}$ & 0.271 & (0.176) & 0.055 & (0.350) & 0.102 & (0.371) \\[5pt]
I(1998)$_{16}$  & -0.054 & (0.206) & 0.174 & (0.216) & 0.222 & (0.225)\\[5pt]
I(1998)$_{16}$ $\times$ I(Memb)$_{16}$ & 0.413 & (0.257) & 0.254 & (0.244) & 0.289 & (0.249) \\[5pt]
\; \; \; Wald \{ 97 $\times$ I(Memb) + 98 $\times$ I(Memb) = 0 \} & &(0.060)   & &(0.570) &  & (0.501)  \\[5pt]
I(1997)$_{64}$  & -0.520 & (0.182) & -0.094 & (0.213) & -0.010 & (0.238)  \\[5pt]
I(1997)$_{64}$ $\times$ I(Memb)$_{64}$ & 0.559 & (0.241) & 0.127 & (0.263) & -0.069 & (0.261)  \\[5pt]
I(1998)$_{64}$ & 0.203 & (0.256) & 0.549 & (0.206) & 0.480 & (0.262) \\[5pt]
I(1998)$_{64}$ $\times$ I(Memb)$_{64}$ & 0.512 & (0.247) & 0.126 & (0.222) & -0.044 & (0.272) \\[5pt]
\; \; \; Wald \{ 97 $\times$ I(Memb) + 98 $\times$ I(Memb) = 0 \} & &(0.003)	  & &(0.526) &  & (0.802) \\[5pt]
\; \; \; rk Wald F Stat$^\ast$ & &  & & &139.76 &  \\
    \hline
~\\
	Firm FE $\times$ Log Cum Output & \multicolumn{2}{c}{N} & \multicolumn{2}{c}{Y} & \multicolumn{2}{c}{Y} \\	
	Quarter FE & \multicolumn{2}{c}{Y} & \multicolumn{2}{c}{Y} & \multicolumn{2}{c}{Y}  \\[2.5pt]
	\hline 
~\\
R-squared & \multicolumn{2}{c}{0.592} & \multicolumn{2}{c}{0.686} & \multicolumn{2}{c}{0.660}  \\
N &\multicolumn{2}{c}{2,387} &\multicolumn{2}{c}{2,387} &\multicolumn{2}{c}{2,387} \\
Clusters &\multicolumn{2}{c}{136} &\multicolumn{2}{c}{136} &\multicolumn{2}{c}{136}   \\
    \bottomrule
\end{tabular}

\begin{tablenotes}
Coefficients and standard errors displayed. Standard errors clustered at firm-generation level. \\
$^\dagger$ Effect of log cumulative output is given for average firm. \\
$^\ddagger$ (2) instruments Log Output$_{k+1}$ with elapsed quarters of firm $i$'s output on $k+1$ \\
$^\ast$ F statistic valid for clustering at firm-generation level, based on \cite{af}. \\
The dependent variable is logged firm-level output of DRAM units. I(Year)$_{k}$ $\times$ I(Memb)$_{k}$ estimates the yearly difference in output for future cartel members relative to non-members in generation $k$. Sample for each regression is restricted to firm shipments within first 12 years of generation activity and at least 1000 units of quarterly shipments. 
\end{tablenotes}
\end{threeparttable}

\end{table}

%It contains 2 nested specifications to characterize learning-by-doing. Specification (1) imposes equal learning across all firms and generations; (2) imposes equality across firms but not generations.

\Cref{placebo} displays the results of the regression model estimated by OLS and 2SLS. The OLS specification (1) does not include interactions between firm fixed effects and log cumulative output. Both OLS specifications estimate a negligible effect of $Log Output_{k+1}$ on $Log Output_{k}$. In contrast, 2SLS specification (3) instruments firm $i$'s output on $k+1$ with elapsed quarters of $i$'s output on $k+1$. Consistent with cannibalization, the estimate in (3) implies that if $i$ doubles its output on $k+1$, it reduces its output on $k$ by an average of about 10\%. The instrument is strongly correlated with the endogenous variable (\cref{first_stage} summarizes the full first stage estimates).

%pieces include firm-specific learning curves on the supply side, functional form and shape of current and (expected) future demand across products, as well as parameters for the discount factor, market size,

By accounting for cannibalization, specification (3) also estimates a higher log cumulative output effect than specification (2). A heuristic exercise can help assess its plausibility: assuming single-product static quantity-setting, symmetric firms, linear demand, and an approximately unit market elasticity, a cost reduction of 15-25\% corresponds to an output increase of 30-50\% with 18 firms.\footnote{There were an average of 18.25 firms active across the sample period. \cite{gar16} and \cite{sieb10} both estimate market-wide DRAM elasticities near -1, and \cite{irwinkl94} estimate DRAM learning at 20\% across generations, though it varies from 10-27\%. \label{lbd_DRAM}} The 44\% coefficient estimate falls within this range. Similarly, observe that the effect of PC shipments varies significantly by generation rank as hypothesized. A 1\% increase in PC shipments leads to a 0.48\% increase in firm output on the newest generation, when production is ramping up. The magnitude of this effect rises for the second- and third-ranked generations, which are typically in the peak of their product life cycles.  

To account for the possibility of serially correlated errors, standard errors are clustered at the firm-generation level. 2SLS estimation fails to reject the null hypothesis of no pre-cartel difference in trend between members and non-members. As another specification test, I relaxed the model's restriction that the effect of cumulative output is equal across all generations for each firm. \Cref{sub} displays the results of a subsample test with the added interaction terms Log Cum Output $\times$ Firm $\times$ $\mathbbm{1}(Post)$, where $\mathbbm{1}(Post) = 1$ if the generation is among the last four. For most firms in the dataset, \cref{sub} shows small changes in the effect of cumulative output in the last four generations. This suggests that the model is appropriately specified. %The p-values of Wald tests for equality with the 16Mb member output are (0.12) for 64Mb and (0.01) for 128Mb. % For example, \cref{sub} shows that one firm, \#14, was estimated to significantly decrease learning in later generations. This firm exited the industry after producing the 512Gb generation, which could explain its low coefficient.

%Finally, it is worth discussing variance estimation of the model's parameters. Firm output increases on each generation until cannibalization induces it to decrease. This leads to the possibility of serial correlation at the firm-generation level.\footnote{A regression of current residuals on lagged residuals, with firm $\times$ generation fixed effects, reveals first-order autocorrelation: $\rho_{t-1} = 0.81$; $\rho_{t-2} = -0.01$; $\rho_{t-3} = 0$.} Errors are therefore clustered at the firm-generation level, with over 100 distinct clusters. 


%On the 16Mb generation, firms are estimated to have produced higher than expected output in 1996 and 1997, with a return to baseline in 1998. This is consistent with the presence of excess capacity in the mid-1990s, when the 16Mb generation accounted for the majority of sales. 

%implies that any unobserved heterogeneity, e.g. in firm-generation level learning rates or demand, does not appear to result in clear disparities across the two comparison groups.} 
%\footnote{Each specification also controls for the cartel effect on each generation, but those coefficients are omitted from display.}


%\footnote{The cubic spline is a baseline form of the partially linear model. For each nonparametric covariate, 5 knots are chosen to fall at equidistant percentiles of its marginal distribution. The estimates do not respond sharply to changes in the location or number of knots.}



%Even after weighting, the inverse U-shaped pattern of output implies that unexplained variation is likely to lead to autocorrelation. \Cref{fig:resid} plots the residuals from 2SLS model (3) in \cref{placebo}, for each of the three cartel generations. 

% if standard errors are deemed too high, then subsequent models can use FGLS. This allows the explicit specification of an AR(1) process in the variance covariance matrix, and represents an efficiency gain relative to fixed effects estimation (Cameron and Miller, 2015; Hansen, 2007). 


\begin{table}[htbp]\centering
\begin{threeparttable}
\captionsetup{justification=centering}
 \caption{Estimates of Cartel Membership Status on Log Firm Output:\\ During Cartel Years}
 \label{firm_all}
\def\onepc{$^{\ast\ast\ast}$} \def\fivepc{$^{\ast\ast}$}
\def\tenpc{$^{\ast}$}
\begin{tabular}{lcccc} \toprule \toprule
 {\extracolsep{0.2in}}
%p{3.2cm}

    & \multicolumn{2}{c}{OLS} & \multicolumn{2}{c}{2SLS}   \\
    \cmidrule(lr){2-3} \cmidrule(lr){4-5}  \\
     & \multicolumn{2}{c}{(1)} & \multicolumn{2}{c}{(2)}   \\
    \midrule

Constant  & -3.061 & (2.207) & -2.181 & (2.336) \\[5pt]
Log Cum Output$^\dagger$  & 0.378 & (0.040) & 0.442 & (0.057)  \\[5pt]
Log Output$_{k+1}$$^{\ddagger}$  & 0.002 & (0.019) & -0.084 & (0.040)  \\[5pt]
Log PC Ship & 0.472 & (0.132) & 0.392 & (0.145) \\[5pt]
Log PC Ship $\times$ Gen Rank$_{2}$ & 0.044 & (0.010) & 0.050 & (0.010) \\[5pt]
Log PC Ship $\times$ Gen Rank$_{3}$ & 0.021 & (0.013) & 0.043 & (0.015) \\[5pt]
Log PC Ship $\times$ Gen Rank$_{4}$  & -0.046 & (0.012) & -0.014 & (0.017) \\[5pt]
Log PC Ship $\times$ Gen Rank$_{5}$  & -0.079 & (0.014) & -0.048 & (0.017) \\[7.5pt]

I(Collusion)$_{16}$ & 0.804 & (0.125) & 0.679 & (0.182) \\[5pt]
I(Collusion)$_{16}$ $\times$ I(Memb)  & -0.712 & (0.225) & -0.611 & (0.251)  \\[5pt]
I(Collusion)$_{64}$ & 0.441 & (0.214) & 0.293 & (0.211) \\[5pt]
I(Collusion)$_{64}$ $\times$ I(Memb) & -0.213 & (0.236) & -0.021 & (0.223)   \\[5pt]
I(Collusion)$_{128}$ & -0.231 & (0.157) & -0.117 & (0.213)   \\[5pt]
I(Collusion)$_{128}$ $\times$ I(Memb)  & 0.264 & (0.196) & 0.323 & (0.233)  \\[5pt]
\; \; \; Wald \{ 16 $\times$ I(Memb) - 64 $\times$ I(Memb) = 0 \}  & &(0.099) &  & (0.077)  \\[5pt]
\; \; \; Wald \{ 16 $\times$ I(Memb) - 128 $\times$ I(Memb) = 0 \}  & &(0.001) &  & (0.005)  \\[5pt]
    \hline
~\\
	Firm FE $\times$ Log Cum Output & \multicolumn{2}{c}{Y} & \multicolumn{2}{c}{Y} \\	
	Quarter FE & \multicolumn{2}{c}{Y} & \multicolumn{2}{c}{Y}  \\[2.5pt]
	\hline 
~\\
R-squared & \multicolumn{2}{c}{0.685} & \multicolumn{2}{c}{0.665}  \\
N &\multicolumn{2}{c}{2,387} &\multicolumn{2}{c}{2,387}  \\
Clusters &\multicolumn{2}{c}{136} &\multicolumn{2}{c}{136}  \\
    \bottomrule
\end{tabular}

\begin{tablenotes}
Coefficients and standard errors displayed, with clustering at firm-generation level. \\
$^\dagger$ Effect of log cumulative output is given for average firm. \\
$^\ddagger$ (2) instruments Log Output$_{k+1}$ with elapsed quarters of firm $i$'s output on $k+1$. \\
The dependent variable is logged firm-level output of DRAM units. I(Collusion)$_{k}$ $\times$ I(Memb) estimates the difference in output for cartel members relative to non-members during collusion in generation $k$. Sample for each regression is restricted to firm shipments within first 12 years of generation activity and at least 1000 units of quarterly shipments. 
\end{tablenotes}
\end{threeparttable}

\end{table}


\Cref{firm_all} presents the results of the model that replaces pre-cartel trends with collusion indicators. In interpreting the results, it is instructive to focus on the difference between the coefficients of interest across generations. Specification (2) estimates that member output in the 16Mb generation declined significantly relative to non-members. In contrast, member outputs for the 64Mb and 128Mb generations cannot be rejected as different from non-members. Two-sided Wald tests for equality between the 16Mb member coefficient and the 64Mb and 128Mb member coefficients are rejected at 10\% and 1\% levels, respectively. %These results suggest that members reduced output more on the older chip than newer chips. 

For several reasons, it is more difficult to estimate how effective collusion was on a single generation than across generations. The coefficients of interest have relatively large standard errors. One cause is that for each firm-generation pair, there were only 10 quarters in which the cartel was active. A second is that the magnitude of output change during the cartel is expected to vary across its members when they differ in costs that are correlated over time (e.g., \citealp{atheyb08}). This type of variation is consistent with \cref{fig:splits}, which shows greater relative output reductions for one group of firms than another. 

If firms choose output levels (as in the theoretical model below), strategic substitution implies that a decrease in output by cartel members is met with an increase by non-members. Consequently, the coefficient estimates should be interpreted such that cartel members reduced output on the 16Mb chip by some amount between $(e^{-0.61} - 1) = -46\%$ and 0\%. The precise value depends on the slope of best response functions between firms, which are not recovered by the model. 

By assuming that the estimated member coefficient of 46\% on 16Mb consists equally of a 23\% output decrease by member firms and 23\% output increase by fringe firms, it is possible to construct a 95\% confidence interval for net output change on the generation during the cartel period. Using the delta method to approximate standard errors, total 16Mb output decreased by between 69,043 to 712,584 units, or 3.1\% to 24.6\%, during the cartel. Allocating a larger share of the 46\% estimate to members' output reduction shifts this interval higher in absolute value.\footnote{Using a monthly dataset with proprietary manufacturer costs, direct purchasers' economic expert estimated the average output reduction at the industry level during the collusive periods. The model yielded a 95\% confidence interval of -0.8\% to -13.6\%. This appears to be the only estimate of the cartel's effect on output conducted by either side during civil litigation; the case was ultimately settled without verdict.}


%Second, the sampling distribution of the 64Mb and 128Mb member estimates contain mass in the region greater than zero. A positive effect of the cartel on output could result if firms successfully collude to reduce output on the 16Mb generation, inducing customers to shift demand to newer chips. Formally testing that possibility is beyond the scope of this article. 





\subsection{Other Empirical Explanations} \label{robust}

%Most generally, the reduced form approach does not explicitly model supply side dynamics. Initial marginal costs, R\&D spending, and the extent of cannibalization may vary non-systematically across firms and time.

%Under identification assumptions 1-3, the difference-in-differences model accounts for any unobserved variation in output across time that impacts \textit{each firm} within a generation equally. Hence, it remains valid if PC shipments do not fully capture changes to demand, as mentioned above. It also remains valid in the presence of unobserved capacity that varies at the generation-level. Anecdotal evidence suggests that, due to lumpy plant investment, capacity does tend to rise and fall symmetrically across firms.\footnote{``During the profitable years...they invest. But they all have their good years at the same time and they all invest in new factories at the same time. Two years later, the factories are all up to speed and you have a glut.'' (Source: ``Memory Industry Shakeup: Q \& A with Jim Handy,'' \textit{CBS News}, April 2009 \url{http://www.cbsnews.com/news/memory-industry-shakeup-q38a-with-jim-handy/}) \label{handy_cbs}} Moreover, capital depreciation accounts for about 80\% of DRAM manufacturing cost, so labor is not a significant component.\footnote{Source: \Cref{handy_cbs}} 

%There are alternative reasons for cartel members to reduce output more on trailing than leading generations. 

There are alternative explanations for the above patterns that do not rely upon incentive compatibility constraints. For example, cartel members in newer generations may have sought to lower costs rapidly to prevent fringe firms from establishing share. Similarly, although there is documented communication between sales representatives on 128Mb pricing, it is possible that higher level executives---who would take learning-by-doing incentives into account---did not have enough volume data to negotiate broader terms. Consistent with these scenarios, \cite{roos06} suggests that a price war in the 1990s lysine cartel could have been caused by either attempted predation or building up market share prior to a future agreement.

%\cite{fershtman00} models a cartel in which leading firms have a dynamic incentive to drive out weaker rivals. 

%in collusion, and 


%Similarly, \cite{fershtman00} model a dynamic incentive of stronger firms to drive out weaker rivals, and \cite{roos06} finds evidence that this type of effect may have caused a price war in the lysine cartel. If these predatory incentives were also at play in the DRAM cartel, then they may be able to explain some variation in conduct across generations. Because of the salience of learning-by-doing to the semiconductor industry, the approach below relies on the position that variation in conduct across generations during the cartel was tied primarily to learning rather than other dynamic incentives. 

To address such possibilities, I construct a test for defection from collusive equilibria by one or more firms in learning-by-doing industries. Assume that firms select quantities with subsequently perfect monitoring. If an unsuccessful attempt to collude stems from IC constraints, then the episode must have featured either the \textit{threat} of deviation or \textit{realized} deviation. If realized, a deviating firm would expand its output relative to rivals and, if detection is not sufficiently fast, overtake rivals who had initial leads. This type of ``rank reversal" is uncommon during the early years of a DRAM generation, so an increase in its level may be observed for the 64Mb and 128Mb generations. 


\begin{figure}[h]
\centering
\subfloat[$r$=0.78]{
        \label{subfig:test2}
        \includegraphics[height=1.5in, keepaspectratio]{./pics1/test2} } 
\subfloat[$r$=0.65]{
        \label{subfig:test3}
        \includegraphics[height=1.5in, keepaspectratio]{./pics1/test3} }           
\subfloat[$r$=0.74]{
        \label{subfig:test4}
        \includegraphics[height=1.5in, keepaspectratio]{./pics1/test4} }  \\
\subfloat[$r$=0.43]{
        \label{subfig:test5}
        \includegraphics[height=1.5in, keepaspectratio]{./pics1/test5} } 
\subfloat[$r$=0.70]{
        \label{subfig:test6}
        \includegraphics[height=1.5in, keepaspectratio]{./pics1/test6} }  
\subfloat[$r$=0.83]{
        \label{subfig:test7}
        \includegraphics[height=1.5in, keepaspectratio]{./pics1/test7} } 

\caption{Firm Order of Entry against Subsequent Ranking, by Generation}
\footnotesize{$r$ = sample correlation coefficient}
\label{fig:rank_switch}
\end{figure} 

\Cref{fig:rank_switch} plots each firm's order of entry into a generation against its subsequent output ranking during the first four years of that generation's life cycle.\footnote{For the 64Mb and 128Mb generations, the set of firms is limited to cartel members. The results are similar if all generations are limited to cartel members.} The scatterplots are shaded to indicate the density of x-y pairs across quarters. The region of highest density tends to track a 45 degree line, reflecting a positive correlational coefficient between the two variables. This pattern is weakest for the 128Mb generation. This strengthens the hypothesis that at least for the 128Mb generation, the regression results above stem from IC constraints rather than alternative explanations.

%\footnote{Cartel member output on the 256Mb generation has not been tested in the DiD models above because firms began production only during the cartel period, yielding low volumes and no viable control group of non-members.}


%To address such possibilities, we construct a test for output ``rank reversal". For any pair of firms $i$ and $j$, denote $RR=1$ if $q_{ikt} > q_{jkt}$ but $q_{ik,t+1} < q_{jk,t+1}$. If firms are competing without punishments, then the rate of rank reversal should be the same during the cartel as it is in adjacent periods. On the other hand, if firms are alternating between collusive outputs and punishments from defection, there is basis for rank reversal to become more frequent.\footnote{For example, suppose the pair of firms agree to restrict output to the level $\bar{q}_{kt}$, but $i$ defects while $j$ adheres. Then $q_{ikt} > q_{jkt}$, but this may reverse in future periods if $j$ has a higher rate of learning.} Formally, we compare the mean rank reversal probability between cartel and non-cartel periods:
%\begin{align*}
%RR_{pkt} = \beta_{0} + \rho_{p} \times \kappa_{k} +  \mathbbm{1}(Coll_{64,t}) +   \mathbbm{1}(Coll_{128,t}) + \epsilon_{ikt}
%\end{align*}


%The dependent variable equals one when any pair of firms $p$ in generation $k$ at quarter $t$ reverses rank, and zero otherwise. The model includes pair-generation fixed effects. The effect of the cartel is estimated relative to adjacent periods separately for the 64 and 128Mb generations.


%Under the null ypothesis, changes in output are not driven by changes in incentive compatibiity constraints. Under the alternative hypothesis, firms defect from agreed upon collusive equilibria. Consequently, firms are more likely to reverse rankings. 

One way to formalize this test is to run the following linear regression model:
\begin{align}
OutputRank_{ikt} =& \, \beta_{0} + EntryOrder_{ik} \beta_{1} + \kappa_{k} \times \tau_{t} +  \sum_{k=2}^{K} \Big[  EntryOrder_{ik} \times \kappa_{k} \Big] \beta_{k}  +  \epsilon_{ikt}
\end{align} 
The output rank and entry order are defined as above. The first interacted term denotes generation $\times$ time fixed effects, which reflect the unit of observation being studied. The coefficient on the second interacted term estimates the average correlation between entry order and output rank separately on any generations of interest $k$. 

\Cref{RR} below presents the results of the regression using two different samples: specifications (1) and (2) include all generations with at least 100 observations, dating to 4Kb. Specification (3) uses only those generations adjacent to the cartel: 4Mb - 512Mb. Specification (1) estimates the mean correlation for all generations. Specifications (2) and (3) estimate the mean correlation across all non-cartel generations, less the mean correlation of each of the two generations of interest.


\begin{table}[h]\centering
\begin{threeparttable}
\captionsetup{justification=centering}
 \caption{Output Rank Regressed on Entry Order, by Generation}
 \label{RR}
\def\onepc{$^{\ast\ast\ast}$} \def\fivepc{$^{\ast\ast}$}
\def\tenpc{$^{\ast}$}
\begin{tabular}{l c c c c c c } \toprule \toprule
& \multicolumn{4}{c}{All Generations} & \multicolumn{2}{c}{Adjacent Generations}  \\
    \cmidrule(lr){2-5} \cmidrule(lr){6-7}   \\
    & \multicolumn{2}{c}{(1)} & \multicolumn{2}{c}{(2)}   & \multicolumn{2}{c}{(3)}  \\
    \cmidrule(lr){2-3} \cmidrule(lr){4-5} \cmidrule(lr){6-7}  \\[2.5pt]	
    %\midrule 
Constant &1.402 &(0.114) 	& 1.369	&(0.113)	&2.045 &(0.188)   \\[2.5pt] 
Entry Order & & 	&0.802	&(0.017)	&0.680	&(0.032)  \\[2.5pt] 
Entry Order $\times$ I(4\textbf{ Kb}) &0.918 &(0.064) 	  \\[2.5pt]
\; \; \; \; \; \; \; \, \, \, $\times$ I(16) &0.910 &(0.039) 		  \\[2.5pt]
\; \; \; \; \; \; \; \, \, \, $\times$ I(64) &0.798 &(0.060) 		  \\[2.5pt]
\; \; \; \; \; \; \; \, \, \, $\times$ I(256) &0.932 &(0.047) 	  \\[2.5pt]
\; \; \; \; \; \; \; \, \, \, $\times$ I(1\textbf{ Mb}) &0.815 &(0.049)   \\[2.5pt]
\; \; \; \; \; \; \; \, \, \, $\times$ I(4) 	& 0.716 & (0.040)   \\[2.5pt]
\; \; \; \; \; \; \; \, \, \, $\times$ I(16) &0.596 &(0.053) 	 \\[2.5pt]
\; \; \; \; \; \; \; \, \, \, $\times$ I(64) &0.721 &(0.082) 	&-0.081 &(0.085)	&0.030 &(0.100) \\[2.5pt]
\; \; \; \; \; \; \; \, \, \, $\times$ I(128) &0.364 &(0.073) 	&-0.438 &(0.076)	&-0.321 &(0.091) \\[2.5pt]
\; \; \; \; \; \; \; \, \, \, $\times$ I(256) &0.661 &(0.068) 	 \\[2.5pt]
\; \; \; \; \; \; \; \, \, \, $\times$ I(512) &0.802 &(0.117) 		 \\[2.5pt]
\; \; \; \; \; \; \; \, \, \, $\times$ I(1\textbf{ Gb}) &0.715 &(0.188)  \\[2.5pt]
    \hline
~\\
	Gen FE $\times$ Time FE & \multicolumn{2}{c}{Y} & \multicolumn{2}{c}{Y}  & \multicolumn{2}{c}{Y} \\[2.5pt]	
	\hline 
~\\
R-squared & \multicolumn{2}{c}{0.625} & \multicolumn{2}{c}{0.613}  & \multicolumn{2}{c}{0.428}   \\
N &\multicolumn{2}{c}{1,590} &\multicolumn{2}{c}{1,590} &\multicolumn{2}{c}{781}  \\
    \bottomrule
\end{tabular}
\begin{tablenotes}
Coefficients and standard errors displayed. Specifications (1) and (2) include all generations with at least 100 firm-time observations. (3) includes only 4Mb-512Mb generations. In all specifications, 64Mb and 128Mb generations limited to cartel members, and all generations limited to first 4 years of observations. Ties in order of entry broken based on initial output level.
\end{tablenotes}
\end{threeparttable}
\end{table}

%Spec (x) estimates using default standard errors, which may be biased downward based on correlation across time within generations. Spec (y) corrects using FGLS. This is chosen because there are not enough generations to cluster on. 

In both (2) and (3), the coefficient of the 128Mb interaction is negative and significant at the 1\% level. To determine whether the larger or smaller sample is more appropriate, I conduct an F-test for the joint equality of coefficients on all non-cartel generations. The null hypothesis of equality is rejected if all generations are included (as in (1)), but fails to be rejected if only 4Mb-512Mb generations are included (p-value=0.38). This suggests that the regression sample should be limited to generations adjacent to the cartel, at the cost of a smaller sample size. More distant generations are more likely to differ in the number and type of firms, which likely impacts the relevant correlation.   

It is also important to test the model's sensitivity to the chosen time period for the expansion phase of a generation's life cycle. Because the test assumes that each firm is increasing its output, a longer time period increases noise if some firms begin to cut output on $k$ to mitigate cannibalization on $k+1$. The time period above is chosen as four years, and the results are not sensitive to alternative definitions of three to five years. 

%There are several factors that limit the interpretation of this test: the appropriate time period should approximate each generation's initial expansion phase; this phase may differ across generations as demand varies.

This test may be revisited in future empirical studies of collusion in high technology markets. In addition to the considerations above, the test's explanatory power increases with the pace of firm learning and the frequency of data collection relative to the detection lag. It is also more informative if the identities of cheating firms are known. These firms may be named directly in the litigation record, or inferred by the researcher who matches participants' private price or share agreements to public data.\footnote{One potential candidate for this type of study is the mid-2000s cartel in LCD screens. If cartel members cheated from agreements, then they may be identifiable through meetings logs which contain the cartel's target prices. These records from the so-called ``crystal meetings" are included in public expert reports.} 

%these findings may suggest that cheating did not occur in equilibrium; that cheating did occur but punishments were made at monthly or weekly levels; or that firms were not attempting to collude on these generations at all. 










% -O- Finally, other obvious explanations appear unlikely to explain the output shift during the cartel. The model precludes cartel firms from communicating to share learning gains. \cite{mowery98} and \cite{mowery03} find that organizational management, data analysis, and equipment technology impede the pace of learning. Firms are thus limited in the learning they can successfully translate between plants; it is unlikely that different firms could coordinate the transition to a new generation without immense and costly effort, planning, team-oriented communication and problem solving. There is no evidence to suggest any attempt was made in the DRAM cartel.\footnote{Moreover, it is unlikely that any increase in profits from collusion allowed firms to notably expand 128Mb output. Annual SEC filings show that U.S. firm Micron incurred a net income loss in both 1998 and 1999, in the leadup to 128Mb introduction. Net income reached about \$1.5 billion in 2000 before again falling negative in 2001.}


%%%%%%%%%%%%%%%%%%%%
%%%% THEORY MODEL SECTION %%%%
%%%%%%%%%%%%%%%%%%%%


\section{Theoretical Model and Policy Implications} \label{Theory_DRAM}

\subsection{Preliminaries} \label{prelims}

%There are two key features of the model. First, firms on one product engage in learning between the two periods, whereas 

The preceding empirical sections suggest that owing to incentive compatibility constraints, the DRAM cartel restricted output more significantly in the older 16Mb generation than newer generations. To expand upon these findings, this section formalizes the conditions under which it is strictly more difficult to collude on newer generations. It then discusses several implications for antitrust policy in high technology markets. 

I present a two-period model of learning based on \cite{sieb10} and \cite{ft83}. The most important feature of this model is that learning occurs only once: between a product's first and second phase. This framework permits a simple representation of dynamic effects: several years of firm learning are compressed into just one period within the model.  %This framework simplifies the dynamic game into a tractable model while preserving the key feature of learning.


%To overcome the computational challenges inherent in computing supra-competitive equilibria within a dynamic game, the model imposes supply side structure on the evolution of costs, the timing of entry, and the type of collusive strategies played. It , and several additional years of slower learning into a second period. 
 

%\footnote{\cite{kreps83} show that static equilibria from a quantity game with one period are equivalent to those from a capacity-pricing game with two periods, under plausible technical conditions.}

% (isomorphic part of R\&S or H\&H)

Consider a multi-period duopoly between two firms $i$ and $j$. Each firm produces two products \textit{old} and \textit{new}, indexed by $k \in \{o,n\}$. Within each period, $i$ and $j$ set their output simultaneously a la Cournot. In the following period, each firm perfectly observes its rivals' previous output decisions. The timing of the game, which is symmetric between $i$ and $j$, is depicted visually in \cref{fig:timing_game}:



%The fringe setup in the present model is motivated by these and other complementary results.


%\footnote{A model of quantity-setting leadership with a price-taking fringe was posited by \cite{econo86}. \textbf{MORE ABOUT COLLUSION.}}


\begin{figure}[h]
	\centering
   \includegraphics[height=2.125in]{./pics1/timing_game_v7b.png}
	\caption{Timing of Product Life Cycles and Learning}
  	\label{fig:timing_game}
\end{figure}

There is only one supply side difference between the two products: firms begin producing $n$ one period after $o$. Consequently, when firms on $n$ engage in learning, firms on $o$ have already finished learning. Across periods, firm learning lowers firm $i$ or $j$'s marginal costs from the starting point $\bar{c}_{k}$ to $ \bar{c}_{k} \cdot \phi \left( q_{i/j,k,t-1} \right)$, where $\phi \left( 0 \right) = 1$ and $\phi \left( \cdot \right)$ is decreasing. To maintain an infinite horizon, the model adds a third phase of repeated static competition. Between phases two and three, any further learning effects on $k$ diffuse to both firms.\footnote{\Cref{tech_c} describes further conditions governing how cost enters the model.} 


%[perfect info]\footnote{This assumption is invoked in \cref{payoffs} below to model collusive punishments within a subgame perfect equilibrium framework. Widespread knowledge of fabrication capacity levels as well as market research reports detailing quarterly firm-level shipments allow firms to monitor output.}

Next, consider the demand for each product to be given by the function $P_{k}(Q_{ot}, Q_{nt})$, where $Q_{kt} = \sum_{i,j} q_{i/j,kt}$. Demand is fixed across time and there is only one demand side difference between the two products: $n$ is higher quality than $o$. A fraction of consumers derive greater utility from higher quality, whereas the rest are indifferent. This relation is mapped onto the inverse demand curves by assuming that there is a price intercept; that $n$ has a higher price intercept than $o$; and that they have identical slopes.\footnote{For example, a linear demand curve of this type has the form $P_{kt} = a_{k} - \eta Q_{kt} - \gamma Q_{-kt}$, where the intercept varies by product but the slope parameters do not.} Define the difference in price intercepts as $\mu(a) = P_{n}(0,0) - \underbrace{P_{o}(0,0)}_{a} > 0$. When $\mu(a)=0$, all consumers are indifferent; as $\mu(a)$ increases, an increasing share prefer $n$. The overall size of the market increases with $a$.

%To model this relation, it is useful to impose structure on the shape of the demand curves. Assume the existence of a price intercept for each product, $P_{k}(0,0)$. Let the own- and cross-price inverse demand elasticity parameters be negative and symmetric on both products, such that their slopes are equal. 


%Let $\bar{\mu}$ be sufficiently large such that when the entire market prefers higher quality, 

%$\mu$ takes the domain $[0 ,\bar{P}_{n}]$, with higher values indicating strictly higher preference toward $n$. Hence $\mu=0$ is indifferent, whereas $\mu=\bar{P}_n$ prefers $n$ to $o$ at any price. The \textit{extent} of vertical differentiation is indexed by the intercept of the inverse demand function. Higher intercepts are equivalent to 

%provided by the cumulative distribution function $G(\mu)$, strictly increasing and continuously differentiable. The greater is the mass of this distribution toward the high end of the support, the greater is the share of the market that purchases $n$. 


%Thus higher quality products are purchased by higher types, and lower quality products by lower types.

%For any pair of $o$ and $n$ outputs $(x,y) \in \Re^{2}_{+}$, $P_{n}(x,x) - P_{o}(x,x) > 0$ and $P_{n}(y,y) - P_{o}(y,y) > 0$. At the same output, demand is greater for the new product, leading to a higher price. 




%This does not imply, however, that $P_{n}(\cdot) > P_{o}(\cdot)$ in general. If $x<y$, then it is possible that $P_{n}(\cdot) < P_{o}(\cdot)$ by downward-sloping demand. Let the parameter $\mu(x,y)$ denote the extent of vertical differentiation at any point along the demand curve. 

%For any pair of equal outputs across generations $(x,x) \in \Re^{2}_{+}$, let $P_{n}(x,x) - P_{o}(x,x) = \mu(x) > 0$. 


Given a common discount factor $\delta$, the supply and demand side of the game give rise to the following profit function for firm $i$ on product $k$ at time $t$:
\begin{align}
	\operatorname*{\textit{max }}_{q_{ikt}} \Pi_{i} = \sum\limits_{k = o}^{n} \sum\limits_{t = 1}^{\infty} \delta^{t - 1} &\Big[ P\left(Q_{kt}; Q_{-kt}\right)q_{ikt} - C \left(q_{ikt}, q_{i k t-1} \right) \Big] 
	 \label{eqref:obj}	
\end{align}
Firms can play a variety of repeated game strategies, and the relevant equilibrium concept is subgame perfection: strategies must constitute a Nash equilibrium at every period of the game. \Cref{tech_c} gives the technical conditions on supply and demand sufficient for the existence of a subgame perfect equilibrium. 


\subsection{Collusive Equilibrium} \label{delta}

Next, I assess the incentives of firms $i$ and $j$ to enforce a collusive equilibrium in the presence of learning-by-doing and multiproduct competition. Assume that firms can agree to produce the joint profit maximizing output, and to punish deviation from this output with a ``grim trigger.'' The joint profit maximizing output is considered as a benchmark: if firms cannot reach it on a given generation, then they must resort to a sub-optimal collusive strategy or no collusive strategy at all.\footnote{Other types of trigger and stick-and-carrot strategies share the property that the extent of punishment is based on marginal cost. By focusing on the relationship between learning and marginal cost, the model below highlights the intuition for a variety of collusive strategies.} 


Formally, denote the following strategy $\sigma_{ik}'$: firms $i$ and $j$ maximize the discounted sum of joint profits on $k$ beginning from period 1, and punish deviation on product $k$ with Cournot reversion on $k$ forever after. Equilibrium requires incentive compatibility:
\begin{mydef}
Define $^{*}$ and $^{**}$ respectively as noncooperative and cooperative actions of the repeated game. To constitute a subgame perfect equilibrium, $\sigma_{ik}'$ must satisfy:
\begin{align}
	\operatorname*{\textit{max    }}_{q_{ikt} , q_{jkt}} \sum\limits_{t = 1}^{\infty} \delta^{t-1} \Big[ \Pi_{ikt} + \Pi_{jkt} \Big] \; \; \; \; \; \text{subject to} \; \; \; \; \; \sum\limits_{t = 1}^{\infty} \delta^{t-1} \Big[ \Pi_{ikt}^{**} - \Pi_{ikt}^{*} \Big] \geq 0 
%	\operatorname*{\textit{max }}_{q_{ikt} , q_{jkt}} \Pi_{i} + \Pi_{j}  = \sum\limits_{i}^{j} \sum\limits_{k = o}^{n} &\sum\limits_{t = 1}^{\infty} \delta^{t - 1} \Big[ P\left(Q_{kt}; Q_{-kt}\right) \cdot q_{ikt}  - C \left(q_{ikt} ; q_{i k t-1} \right) \Big] \\[5pt]
%	\textit{subject to   } &	\sum\limits_{t = 2}^{\infty} \delta^{t - 1} \Big[ \Pi_{ikt}^{**}  - \Pi_{ikt}^{*}  \Big] \geq \Pi_{i1}^{*} ( q_{j1}^{**}) - \Pi_{i1}^{**} 
 \label{eqref:icc}
\end{align}
\end{mydef}

%Note that $i$'s cost reduction for generation $k$ is still based on its own previous-period output $q_{ik t-1}$, not total previous-period output $Q_{kt-1}$. Firms can collude in the product market but they cannot collude in the learning process to share gains from output across the two firms.

%\footnote{Consistent with the framework, \cite{mowery98} and \cite{mowery03} find that learning is facilitated primarily by organizational knowledge rather than individual engineers.}



Before assessing the viability of this SPE, it is useful to define the statistic $\tilde{\mu}_{t}(a)$. This statistic marks a convenient guide post for the extent of vertical differentiation toward $n$.

\begin{mydef} \label{propos0}
Given $(c_{iot}, c_{int}), \; \exists$ unique $\mu(a)$ such that $ \Pi^{*}_{int} = \Pi^{*}_{iot}$. 
Call this $\tilde{\mu}_{t}(a)$.
\end{mydef}
\begin{proof}
If $\mu(a) = 0$, $\Pi^{*}_{iot} >  \Pi^{*}_{int}$. As $\mu \rightarrow \infty$, $\Pi^{*}_{int} > \Pi^{*}_{iot} \rightarrow 0$. By continuity of profits, there exists an intersection point. By decreasing differences, $\Pi^{*}_{int} - \Pi^{*}_{iot}$ is increasing in $\mu$, so the intersection is unique.
\end{proof}
%\indent (i) \; $\Pi^{*}_{int}(\cdot) - \Pi^{*}_{iot}(\cdot)$ is strictly increasing in $\mu(x)$ \\[5pt]
%\indent (ii) \; $\exists \; \mu(x) \equiv \tilde{\mu}(q_{t}^{*})$ such that  $\Pi^{*}_{int}(\cdot) = \Pi^{*}_{iot}(\cdot)$

Definition~\ref{propos0} shows that under competitive equilibrium, in each period there is a unique level of vertical differentiation such that the profits on each generation are equal. This occurs because firms produce $n$ at a cost disadvantage relative to $o$ in each period. Given this definition, we are ready to prove the main result:


% produces more than firm $j$ in the subsequent period because it has lower costs: $f \left( q_{in \tau}^{*} \left( \cdot \right) \right) < f \left( \bar{q}_{jn \tau}^{**} \left( \cdot \right) \right) $

%Deviation from $q_{i n 1}^{**}$ increases $\Pi_{i 2}^{*}$ relative to playing a static symmetric Cournot game. This is because :  $i$'s costs are lower than $j$'s costs based on $i$'s increased learning in the first period.

\begin{prop} \label{propos1}
Define $\hat{\delta}_{k}$ as the minimum discount factor that satisfies the ICC for $\sigma'_{ik}$. \\
 If $\mu(a)  \leq \tilde{\mu}_{1}(a)$ , then $\hat{\delta}_{o} < \hat{\delta}_{n} $.
%(ii) Further, if $ \hat{\delta}_{o} < \delta < \hat{\delta}_{n} $, then $\Delta \Delta_{ot} > \Delta \Delta_{nt}
\end{prop}
\begin{proof} See \Cref{proofof1}.
\end{proof}

Proposition~\ref{propos1} formalizes the intuition that it is strictly more difficult to collude on a new generation than an older generation if the learning rate decreases monotonically over time. The sufficient condition is that vertical differentiation is small enough that profits on $n$ are no greater than profits on $o$ at the start of collusion. This condition assumes symmetric firms and is sufficient but not necessary.\footnote{At the start of the DRAM cartel in Q2-1999, the sufficient condition was partially satisfied: $\Pi_{128} < \Pi_{16} < \Pi_{64}$.}

%Given the ex ante symmetry of firms in this model, the condition carries a convenient empirical counterpart: it holds as long as industry profits on $n$ are no greater than those on $o$. At the start of the DRAM cartel in Q2-1999, 

%When the condition binds, the foresaken profit to defecting from the collusive path is equally large in the $n$ scenario as it is in the $o$ scenario, absent any learning effects. When learning is added, foresaken profits are strictly lower in the $n$ scenario. To acheive incentive compatibility, the discount factor $\delta$ must therefore be greater on $n$ than $o$. 

%\footnote{When $\mu_{t} > (c_{int} - c_{iot}) $, foresaken profits are lower in the $n$ scenario than in the $o$ scenario even without the learning advantage.}

\begin{figure}
	\centering
  	\includegraphics[height=2.5in, keepaspectratio]{./pics1/IC_delta_v1}
	\caption{Minimum Discount Factor, \textit{n} and \textit{o}}
  	\label{fig:MDF}
\end{figure}

\Cref{fig:MDF} depicts the result graphically with a set of parameters that satisfy the condition. As $\delta$ increases, so does the profit on the incentive constraint, eventually making $\sigma_{ik}'$ sustainable. Because slack on $o$ lies strictly above $n$, $\hat{\delta}_{o} < \hat{\delta}_{n}$. When $\hat{\delta}_{o} < \delta <  \hat{\delta}_{n}$, $\sigma_{ik}'$ is sustainable for $o$ but not $n$. %The size of this interval increases with the speed of the learning curve, $\phi(\cdot)$, and decreases with the degree of vertical differentiation, $\mu(a)$.

Proposition~\ref{propos1} also assumes that firms formulate strategies that treat different products independently. \Cref{MMC} permits firms to punish deviation on $n$ with conduct on $o$, and shows that joint collusion on $n$ and $o$ remains strictly more difficult than collusion on $o$. 

%The second point is how large the area of the distribution $\hat{delta}_{o} < \delta <  \hat{delta}_{n}$ is.... 

%Can have surf plot of (delta area) wrt (b-a) and (speed of learning). 

%The size of this area increases with the speed of learning, and decreases . To the latter point, The RIGHT PANEL shows that $\hat{delta}_{o} <  \hat{delta}_{n}$ for values of VD very high. Intuitively, the advantage accrued from learning dominates the incrementally higher profits in the statically repeated phase of the game. 

%A resulting question is how large is the area $\hat{\delta}_{o} < \delta <  \hat{\delta}_{n}$, relative to the distribution of $\delta$.  In taking this model to data, it is ultimately an empirical question whether firm conduct is consistent with the collusive equilibrium on $o$ but the competitive equilibrium on $n$. 

%This depends on the degree of vertical differentiation toward $n$, the cost disadvantage against $n$, the shape of demand, and the rate of learning. 


%\textbf{Also cover convexities in demand?}.

\subsection{Discussion}

The modeling framework can be used to highlight wider issues relevant to antitrust policy. In a report to the European Commission, \cite{ivo_03} present a model of collusion in innovative markets. Their model shows that as the exogenous probability that any firm achieves a future innovation increases, collusion between duopolists is more difficult to sustain. 

%consider a price setting duopoly in which there is an exogenous probability that any firm achieves an innovation in each future period. As the probability of future innovation increases, the incentive to deviate also increases, rendering collusion more difficult. 

Many high technology markets are characterized by innovation through learning-by-doing as well as overlapping product generations.\footnote{Prominent examples include microprocessors, servers, LCD panels, hard drives, and numerous other computer hardware and electronics products.}  Innovation through learning-by-doing takes place endogenously through a firm's output. The effect of innovation on collusion can therefore be separated from the effect of future demand on collusion; competition agencies may find both forces important in assessing coordinated effects in high technology mergers. %Both forces may be important for an antitrust agency to evaluate.  In such markets, the present model of collusion can add context to this type of policy recommendation.

\begin{figure}[h]
\centering
\subfloat[$  P_{ot} = a - 2 Q_{ot} -  Q_{nt} $ ; $  P_{nt} = \bm{b} - 2 Q_{nt} -  Q_{ot} $   
$ a = 476$ ; $\bm{b} \in {[476, 521 ]}$ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \;
$ \phi(q) = (\bm{l}q)^{-1} + \underline{c}_n / \bar{c}_{n}$; $\bm{l} \in {[1, 10 ]} $ ]{
        \label{subfig:shift_1}
        \includegraphics[height=2.5in, keepaspectratio]{./pics1/wire_prod_o_v2} } 
\subfloat[]{
        \label{subfig:shift_2}
        \includegraphics[height=2.25in, keepaspectratio]{./pics1/LBD_256} }\\ 
\caption{}
\label{fig:sim_lrn}
\end{figure}

To illustrate, \cref{fig:sim_lrn} plots a theoretical numerical simulation (panel a) alongside empirical data from the 256Mb generation (panel b). Panel (a) plots the minimum discount factor necessary to sustain collusion on generation $o$ during its learning phase. By inspection, the critical value increases with the pace of learning and the degree of vertical differentiation toward $n$. Moreover, the effect of increased learning on $\hat{\delta}_{o}$ intensifies as vertical differentiation is higher. The reason is that if $n$ is significantly more attractive than $o$, a reduction in $n$'s cost speeds up the the obsolescence of $o$. 

Panel (b) plots a simplified representation of learning-by-doing against the total revenue share of the 256Mb generation, the first after the discovery of the cartel. Learning is modeled as $C_{kt} = a X_{kt}^{-b}$, where $X_{kt} = \sum_{i} x_{ikt}$ is the cumulative output accrued by the industry on generation $k$ at time $(t-1)$; $b = -0.2$ indicates a 20\% learning rate; and $a=1$ is the initial cost.\footnote{As \cref{lbd_DRAM} notes, \cite{irwinkl94} estimate DRAM learning to be 20\% on average across generations.} The graph shows that the revenue share of the 256Mb generation is highest when most learning has likely been completed, and remains significant for several more quarters. This suggests that vertical differentiation is sufficiently low that DRAM generations are long-lived, and consequently, that collusion may pose a threat after several years of production. Agencies can construct this type of measure using commonly available data: learning rates may appear in firms' business documents, proprietary engineering estimates, or the academic literature. Similarly, the speed of obsolescence can be gauged by calculating revenues or conducting customer surveys.

%This paper highlights the importance of empirical analysis into cartels that operate in modern, innovative environments.

%As technology becomes increasingly expensive to downsize at the nanoscale level, it is well understood that semiconductors of all types are reaching a limit to cost reduction

The effects of learning-by-doing on collusion should also be considered separately from other types of innovation. It is well understood that the limits of nanoscale technology are beginning to inhibit semiconductors from achieving the rapid cost gains of the preceding decades. On the other hand, the same phenomenon is spurring investment in R\&D geared toward new chip technologies.\footnote{For example, ``Computer Chips Evolve to Keep Up with Deep Learning,'' \textit{Wall Street Journal} 1/11/2017.} The former trend should increase the viability of collusion whereas the latter should decrease it (as in \citealp{ivo_03}). In evaluating coordinated effects, the competition agency can investigate managers' projections of future learning rates, the most likely future innovators, and the stage of new product development. It is possible that this evaluation would suggest a larger threat of coordination in the short- versus long-term, which the agency should weigh in light of the appropriate competition law environment. 

%The extent to which the agency considers a coordinated threat may depend on how strongly it weighs the short- versus long-term innovation trends. 

%To the extent that new technologies are unlikely to reach market until the long-term, the threat of coordination from reduced learning may be higher in the short-term. 

%This information can help guide the agency toward how much to weigh a coordinated threat in the short- or long-term. 



%The probability of achieving a major new innovation may differ substantially across product markets, so it is unlikely that a ``blanket'' policy would generate optimal enforcement decisions. Instead, the competition agency may benefit from investigating  Each of these variables could be important in evaluating the viability of collusion in the short- or long-term. 

%Focusing narrowly on learning, this phenomenon implies an increase in the likelihood and effectiveness of collusion.


%this means that overall learning rates may be slowing down while other structural features that facilitate coordination---product homogeneity, multimarket contact, R\&D cooperation---remain in place. Competition authorities should be aware that the conditions favoring shakeout may also raise the likelihood of firms reaching collusive equilibria. 



%To see this, observe that standard intuition implies that with overlapping technology products, collusion is harder to sustain on the product with exogenously lower demand


%captures two distinct possibilities: if $\mu(a) \leq \hat{\mu}(a)$, then collusion is less difficult to sustain on an older, less innovative product; but otherwise, collusion is \textit{more difficult} on the less innovative product. The latter result captures the same intuition from \cite{ivo_03}: if the less innovative product 



%In addition, it is important to point out that the possibility of increased output of the new generation at $\tau$ carries significant further dynamic effects. Because of learning-by-doing, $c_{in \tau+1} = \bar{c}_{n} \cdot f \left( q_{in \tau} \right) < c_{i n \tau}$: future costs decline as a function of current output. When current output increases, firms learn a greater amount than they otherwise would, which reduces cost in the future. Reduced cost imposes downward pressure on prices, although it is ambiguous whether resulting price equilibrium is greater than or less than Cournot-Nash. Decreased prices of the newer generation are consistent with increased accrued learning through the demand shift described above, and they will also be tested for in the empirical analysis to follow. 

%%%%%%%%%%%%%%%%%%%%%%%%
% Result II: Add Fringe and Testable Prediction %
%%%%%%%%%%%%%%%%%%%%%%%%


\section{Conclusion} \label{Conc_DRAM}

If firms in innovative industries successfully refrain from competition, the cost to society could be especially large, because firms would also delay the rate at which they reduce long-term costs. The fundamental insight of this paper is that the effectiveness of collusion in such markets is determined by the rate of learning. In particular, when firms simultaneously produce multiple products, it can be more difficult to sustain collusion on the newer product(s). Empirical analysis of the DRAM cartel is consistent with this scenario. 

More generally, this paper illustrates that learning-by-doing is one characteristic through which cartels in a dynamic setting may face different challenges than those in static ones. This possibility depends on the institutional elements of each market, including firm patience, the pace of learning, the number of products each firm sells, and the extent of vertical differentiation between them. Nonetheless, several other known collusive episodes in high-technology markets raise fruitful opportunities for further research. Careful study of those cases may reveal more about the challenges and strategies taken by modern cartels. 

%Empirical analysis of the DRAM cartel shows evidence consistent with the model. Firms are estimated to produce up to 30\% less output for the 16Mb generation during cartelization than competition. On the other hand, firms produced substantially more output for the frontier 64Mb and 128Mb generations. The results imply that firms colluded successfully on the 16Mb generations, which shifted demand toward the 64Mb, 128Mb, and 256Mb generations. They further imply that firms learned more during 10 quarters of cartelization than they would have during 10 quarters of Cournot competition.  



%It remains vital to understand the viability of collusion using data from modern cartels like DRAM, which operate in vastly different legal and technological settings from those studied in the past. 

\newpage


\bibliographystyle{aea}
\bibliography{JMP_refs}

%\processdelayedfloats

%%%%%%%%%%%%%%%%%%%%%%%

\clearpage 

\appendix

\renewcommand{\thesection}{\Alph{subsection}}

\section{Appendices}

\renewcommand{\thesubsection}{\Alph{subsection}}

%%%%%%%%%%%%%%%%%%%%%%%%
% Result IA: Adding Multiproduct Punishments %
%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Collusion with Multiproduct Punishments} \label{MMC}

   \setcounter{prop}{0}
    \renewcommand{\theprop}{\Alph{section}\arabic{prop}}
    

%This section allows firms to coordinate collusive strategies across separate product generations. It shows that relative to single-product punishment, multiproduct punishment may increase the scope for collusion on product $n$. This possibility depends on the level of the firm-invariant discount factor $\delta$. %but that collusion is never more sustainable on both generations under a multiproduct strategy as it is for generation $o$ under a single-product strategy. 

The discussion in the body permitted only one type of collusive strategy: $\sigma_{ik}^{'}$. Now, permit a second collusive strategy $\sigma_{i}^{''}$. This strategy also requires each firm to maximize the discounted sum of joint profits, but punishes deviation on generation $k$ with Cournot reversion on $k$ \textit{and} $-k$. When punishments occur across products, firm $i$ will consider either adhering to the collusive path on both generations or deviating from the collusive path on both generations.\footnote{Deviation on $k$ and adherence on $-k$ is strictly dominated by deviation on both.} 

%Assume that $\delta$ is within the range of interest: $\hat{\delta}_{x \tau} \leq \delta < \hat{\delta}_{y \tau}$.

For subgame perfection, firms must satisfy the following condition: 
\begin{align}
	\operatorname*{\textit{max    }}_{q_{ikt} , q_{jkt}} \Pi_{i} + \Pi_{j} \; \; \; \; \; \textit{subject to} \; \; \; \; \; \sum\limits_{k=o}^{n}  \sum\limits_{t = 1}^{\infty} \delta^{t-1} \Big[ \Pi_{ikt}^{**} - \Pi_{ikt}^{*} \Big] \geq 0
\label{eqref:icc2}
\end{align}
\Cref{eqref:icc2} differs from the single-product case, \cref{eqref:icc}, in only one way: firms can pool IC constraints between products (\citealp{bern90}). Define the IC constraints corresponding to $o$ and $n$ as $g_{o}(\delta)$ and $g_{n}(\delta)$, respectively. The constraint in \cref{eqref:icc2} can be decomposed as:
\begin{align}
g_{o}(\delta) + g_{n}(\delta) &\geq 0 \label{eqref:icc2_mmc} \\[5pt]
g_{o}(\delta) &\geq - g_{n}(\delta) \label{eqref:icc3_mmc}
\end{align} 

%It follows that $g_{x}(\delta)) > 0$ and $g_{y}(\delta)< 0$.
\Cref{eqref:icc2_mmc} shows that $\sigma_{i}^{''}$ is sustainable only if the \textit{sum} of the individual IC constraints is greater than or equal to zero. If $\delta > \hat{\delta}_{o}$, then $g_{o}(\delta) > 0$: there is ``complementary slack'' on $o$. \Cref{eqref:icc3_mmc} shows that this slack can be redistributed: if $g_{o}(\delta)$ is sufficiently large relative to (negative) slack $-g_{n}(\delta)$, then collusion can be sustained on both generations. This logic leads to the following result:
\begin{prop} \label{proposa1}
Define $\hat{\hat{\delta}}$ as the minimum discount factor that satisfies the ICC for $\sigma''_{i}$.\\[5pt]
If  $\mu(a)  \leq \tilde{\mu}_{1}(a)$, then $\hat{\delta}_{o}  <  \hat{\hat{\delta}}  <  \hat{\delta}_{n}$.
\end{prop}
%\in ( \hat{\delta}_{o}, \hat{\delta}_{n})
\begin{proof}
See \Cref{proofofa1}.
\end{proof}

Proposition~\ref{proposa1} imposes the same vertical differentiation condition as Proposition~\ref{propos1}, and carries two implications. First, collusion on both products under $\sigma_{i}^{''}$ is \textit{more} difficult to sustain than collusion on product $o$ alone. The reason is simple: to generate any improvement relative to the single-product case, the multiproduct case requires at least some slack profit on $o$, which in turn requires firms to be more patient. Second, and for the same reason, collusion on both products is \textit{more} sustainable with multiproduct than single-product punishment. 

In sum, the effect of Proposition~\ref{proposa1} is that it ``narrows the gap'' between minimum discount thresholds demonstrated in Proposition~\ref{propos1}. However, whereas the gap is reduced, it cannot be completely eliminated. For $\delta \in (\hat{\delta}_{o}, \hat{\hat{\delta}})$, $\sigma''_{i}$ and $\sigma'_{in}$ are unsustainable, but $\sigma'_{io}$ remains sustainable.

\clearpage




\makeatletter
%\efloat@restorefloats
\makeatother




\subsection{Proofs of Results} \label{proofs}

%%%%%%%%%%%
% Technical Conditions
%%%%%%%%%%%

\subsubsection{Technical Conditions Guaranteeing Equilibrium} \label{tech_c}

On the supply side, each firm's marginal costs evolve based on $c_k \in  [\underline{c}_k ,  \bar{c}_k]$. At phase one, firms have equal marginal cost $\bar{c}_k$ within a generation, but asymmetric costs across generations: $\bar{c}_o < \bar{c}_n$. Between phases one and two, marginal cost declines from $\bar{c}_{k}$ to $ \bar{c}_{k} \cdot \phi \left( q_{ikt-1} \right)$. There are diminishing returns to learning, such that $\phi \left( 0 \right) = 1$, $ \dydx{\phi}{q} < 0$, $ \dtydxt{\phi}{q} > 0$. Marginal cost is bounded from below such that $\lim_{q \to \infty} \phi(q) = \underline{c}_k / \bar{c}_{k}$. At phase three, marginal cost reaches its minimum $\underline{c}_k$, $\underline{c}_o < \underline{c}_n$. 

On the demand side, define $\textbf{P}(Q_{ot}, Q_{nt})$, $\textbf{P} : \Re^{2}_{+} \mapsto \Re^{2}_{+}$, where $Q_{kt} = \sum_{i,j} q_{i/j,kt}$. Assume that $\textbf{P}$ is invertible and twice continuously differentiable. Each element $P_{k} \in [0, \bar{P}_{k} ]$ with $\dydx{P_{k}}{Q_{k}}<0$ and $\dydx{P_{k}}{Q_{-k}}<0$ $\forall \; Q_{k}>0$: the two products are substitutes. Let the inverse demand slopes be symmetric across products and fixed across time. 

With constant marginal costs, restrictions on the demand curve are sufficient to guarantee equilibrium. Assume the multiproduct version of the downward-sloping marginal revenue assumption. That is, define $\text{MR}_{ik} = P_{k} + \dydx{P_{k }}{Q_{k}} q_{ik} + \dydx{P_{-k}}{Q_{k}} q_{i-k}$ and assume:
\begin{align}
\dydx{\text{MR}_{ik}}{q_{ik}} \leq 0 \; \; \text{and} \; \; \dydx{\text{MR}_{ik}}{q_{i-k}} \leq 0 \label{eqref:MR_2ds}   
\end{align}
Marginal revenue on $k$ is weakly decreasing in $k$ as well as $-k$. The first inequality implies that that profits display decreasing differences on $k$ across firms, and the second that profits display decreasing differences across products $k$ and $-k$ (within or across firms). With two firms, condition \ref{eqref:MR_2ds} guarantees the existence and uniqueness of a Cournot-Nash equilibrium at each period (\citealp{JM06}). A sufficient condition for \ref{eqref:MR_2ds} to hold is log-concave inverse demand, e.g. with the exponential form.

%\footnote{Convex demand functions, e.g. log-linear, will also satisfy the marginal revenue condition, if they are sufficiently``non-convex" such that best response outputs are decreasing everywhere in the support. See proof in \cref{proofof2}.}

%\footnote{Note that the condition is weaker than assuming quasi-concavity of profits: \ref{eqref:MR_2ds} $\notimplies H(\Pi_{i})$ negative semi-definite.} 

%If $P_{n}=P_{o}$, let the indifferent types choose $n$ to $o$. It follows that $P_{n}^{*}>P_{o}^{*}$ when $Q_{k}^{*} > 0$. 

\clearpage

%%%%%%%%%%%
% Proof of Proposition 1
%%%%%%%%%%%

\subsubsection{Proof of Proposition 1} \label{proofof1}

Separate the ICC for collusion on generations $k \in \{o, n \}$ into two parts, as follows: 
\begin{align*}
	 \sum\limits_{t = 2}^{\infty} \delta^{t - 1} \left(\Pi_{ikt}^{**} - \Pi_{ikt}^{*}  \right)   \geq   \Pi_{ik 1}^{*}  - \Pi_{ik 1}^{**} 
\end{align*}

By construction, the left hand side is increasing in $\delta$, and it follows that there is a critical discount factor $\hat{\delta}_{k}$ such that $\sigma_{ik}'$ is sustainable for $\delta \geq \hat{\delta}_{k}$ and unsustainable for  $\delta < \hat{\delta}_{k}$. %Analysis proceeds by comparing $\hat{\delta}_{o \tau}$ to $\hat{\delta}_{n \tau}$. %For notational clarity, expressions analyzing the former scenario will condition on $o$, and the latter on $n$.

The sufficient condition is that $\mu(a) \leq \tilde{\mu}_{1}(a)$. Assume first that this constraint is binding. Further, assume the absence of any learning effects on $n$. Both of these assumptions will be relaxed in turn. 

%By symmetry of own- and cross-price elasticities, 
In assessing the incentive to deviate, we must consider two different scenarios: deviation on $n$, and deviation on $o$. Through symmetry of demand, the respective on- and off-equilibrium path profits under $\sigma_{ik}'$ are equal in both scenarios, which implies equality of deviation profits:
\begin{align}
\begin{split}
	 \Pi_{iot}^{**} &=  \Pi_{int}^{**} \\[5pt]
	 \Pi_{iot}^{*}  &= \Pi_{int}^{*}  \label{eqref:eqq1} 
\end{split}
\end{align} 
\begin{align}
	 \, \sum\limits_{t = 2}^{\infty} \delta^{t - 1} \left[ \Pi_{int}^{**}  - \Pi_{int}^{*}    \right]  \,    &=  \,  \sum\limits_{t = 2 }^{\infty} \delta^{t - 1} \left[ \Pi_{iot}^{**}  - \Pi_{iot}^{*}   \right]  \label{eqref:p3_proof}
\end{align} 

%There are two cases permitted within the sufficient condition: , and $\mu(\cdot) < \hat{\mu}(\cdot) $ for at least one $ t \in [\tau, \infty]$. 

%\begin{align}
%\left[ c_{iot} \, | \, k = o \right] \; < \; \left[ c_{int} \, | \, k = n \right] \; = \label{eqref:MCs}
%\end{align}

%In Part III, both generations are in the post-learning phase of their product life cycles, and cost has reached its minimum $\underbar{c}_{k}$. Consider the difference in payoff, $\Delta \pi$, between competition and collusion in Phase III for $o$. The analysis is symmetric for the same decision on $n$. Marginal revenue with respect to output is:

%\[
%\begin{bmatrix}
% P_{o t} + \dydx{P_{o t}}{q_{io t}}q_{io t} + \dydx{P_{n t}}{q_{io t}} Q_{n t}  \\
% P_{n t} + \dydx{P_{n t}}{q_{i n t}}Q_{n t} + \dydx{P_{o t}}{q_{in t}} q_{i o t} 
%\end{bmatrix}
%\]

Next, consider any period $t$ in which the sufficient condition strictly holds: $\mu(a) < \tilde{\mu}_{t}(a)$. By definition, $\Pi_{iot}^{*} > \Pi_{int}^{*}$. Then the marginal profitability of reducing output is greater on $o$ than $n$:
\begin{align}
 \dydx{\pi_{io t}}{q_{io t}}  &> \dydx{\pi_{in t}}{q_{in t}}
\end{align}
It follows that total profitability in reducing output from the Cournot-Nash equilibrium to the collusive equilibrium is greater on $o$ than $n$:
\begin{align}
  \int_{q_{io t}^{**}}^{q_{io t}^{*}} \; \dydx{\pi_{io t}}{q_{io t}} \, dq    &>   \int_{q_{in t}^{**}}^{q_{in t}^{*}} \; \dydx{\pi_{in t}}{q_{in t}} \, dq   \\[5pt]
 \Pi_{iot}^{**} - \Pi_{iot}^{*}   &>\Pi_{int}^{**} - \Pi_{int}^{*}   \label{eqref:p1_proof}
\end{align}

Finally, reintroduce the learning effect for product $n$. Consider period 2 and assume for the sake of contradiction that:
\begin{align}
 \left( \Pi_{io 2}^{**} - \Pi_{io 2}^{*}  \right)  &\leq  \left( \Pi_{in 2}^{**} - \Pi_{in 2}^{*}  \right)  \label{eqref:p2_s1}
\end{align}

Inequality \ref{eqref:p2_s1} implies:
\begin{align}
\begin{split}
& \left( P_{o 2}^{**} ( \cdot ) - \underbar{c}_o \right) \cdot q_{io 2}^{**}  -  \left( P_{o 2}^{*} ( \cdot ) - \underbar{c}_o \right) \cdot q_{io 2}^{*}    \; \; \; \; \; \leq  \\
&  \left( P_{n 2}^{**} ( \cdot ) - c_{in 2}\left( q_{in 1}^{**} \right)  \right) \cdot q_{in 2}^{**}  -  \left( P_{n 2}^{*} ( \cdot ) - c_{in 2} \left( q_{in 1}^{*} \right)   \right) \cdot q_{in 2}^{*}   \label{eqref:p2_s2}
\end{split}
\end{align}

Inequality \ref{eqref:p2_s2} implies that $ c_{in 2}\left( q_{in 1}^{**} \right) \leq c_{in 2}\left( q_{in 1}^{*} \right) $. However, $ q_{in 1}^{**} < q_{in 1}^{*} $ implies:
\begin{align}
\begin{split}
c_{1} \cdot \phi \left(  q_{in 1}^{**}  \right)   &> 	c_{1} \cdot \phi \left(  q_{in 1}^{*}  \right)   \\[5pt]
c_{in 2}\left( q_{in 1}^{**}  \right) &> c_{in 2}\left( q_{in 1}^{*}  \right) 	\contradiction
\end{split}
\end{align}
It follows that slack at period 2 is greater on $o$ than $n$:
\begin{align}
\Pi_{io 2}^{**}  - \Pi_{io 2}^{*}  > \Pi_{in 2}^{**} - \Pi_{in 2}^{*}   \label{eqref:p2_proof}
\end{align}

Combining \ref{eqref:p3_proof}, \ref{eqref:p1_proof}, and \ref{eqref:p2_proof}, it follows that the minimum discount factor necessary to sustain $\sigma_{io}'$ is less than that to sustain $\sigma_{in}'$ :
\begin{align}
\hat{\delta}_{o} < \hat{\delta}_{n } \label{eqref:delta_show} \; \; \; \blacksquare
\end{align}

\clearpage

%Define Total Revenue for generation $k$ at time t $TR_{k t}$. Let Part I be $\Delta TR_{k \tau}$: the within-period $\tau$ benefit to deviating from a collusive agreement on generation $k$, holding fixed collusion on generation $-k$.  
%Because marginal cost is constant within each period, change in profit is $P \times Q $


%%%%%%%%%%%%
% Proof of Proposition A1
%%%%%%%%%%%%


\subsubsection{Proof of Proposition A.1} \label{proofofa1}

\Cref{proofof1} proves that $\hat{\delta}_{n } > \hat{\delta}_{o }$. Also recall from \Cref{MMC} that the pooled ICC corresponding to $\sigma_{i}^{''}$ is:
\begin{align}
	g_{o}(\delta) \geq -g_{n}(\delta) \label{eqref:icc_mmc_rep}
\end{align}
The proof is divided into two parts to correspond to two distinct cases. First, I show that $\hat{\hat{\delta}}$ is strictly greater than $\hat{\delta}_{o }$. Consider increasing $\delta$ from the left. As $\delta \rightarrow \hat{\delta}_{o }$ from the left, $g_{o}( \delta ) \rightarrow 0$ from the left and \ref{eqref:icc_mmc_rep} is unsatisfied:
\begin{align}
\lim_{\delta \to \hat{\delta}_{o }} &g_{o}(\delta) = 0 \\[5pt]
\lim_{\delta \to \hat{\delta}_{o }} &g_{o}(\delta) +  g_{n}(\delta) < 0 \\[5pt]
\lim_{\delta \to \hat{\delta}_{o }} &g_{o}(\delta)  <  -  g_{n}(\delta) 
\end{align}
And it follows that $\hat{\hat{\delta}} > \hat{\delta}_{o }$.

Second, I show that $\hat{\hat{\delta}}$ is strictly less than $\hat{\delta}_{n }$. Define $g_{o}(\hat{\delta}_{n }) - g_{o}(\hat{\delta}_{o }) = \gamma > 0$. Consider the range $ \hat{\delta}_{o } < \delta < \hat{\delta}_{n } $. As $\delta \rightarrow \hat{\delta}_{n }$ from the left, $g_{n} ( \delta ) \rightarrow 0$ from the left and \ref{eqref:icc_mmc_rep} is satisfied:
\begin{align}
\lim_{\delta \to \hat{\delta}_{n }} &g_{n}(\delta) = 0 \\[5pt]
\lim_{\delta \to \hat{\delta}_{n }} g_{o}(\delta) + &g_{n}(\delta) = \gamma + 0 \\[5pt]
\lim_{\delta \to \hat{\delta}_{n }} g_{o}(\delta) - \gamma = -&g_{n}(\delta) 
%g_{x}(\delta) > &-g_{y}(\delta) \label{eqref:x_equal}
\end{align}
Define $\epsilon > 0$ as an arbitrarily small constant. By the continuity of $g_{o}\left( \delta \right)$, there exists an $\epsilon$ such that: 
\begin{align}
g_{o}(\hat{\delta}_{n }) - g_{o}(\hat{\delta}_{n } - \epsilon) &\leq  \gamma  \\[5pt]
g_{o}(\hat{\delta}_{n } - \epsilon) &\geq  g_{o}(\hat{\delta}_{n }) - \gamma =  \underbrace{-g_{n}(\hat{\delta}_{n })}_{=0} > -g_{n}(\hat{\delta}_{n } - \epsilon)  \\[5pt]
g_{o}(\hat{\delta}_{n } - \epsilon) &>  -g_{n}(\hat{\delta}_{n } - \epsilon)
\end{align}
And it follows that $\hat{\hat{\delta}} < \hat{\delta}_{n }$. $\blacksquare$

\clearpage



%\begin{table}[h]\centering
%\begin{threeparttable}
%\captionsetup{justification=centering}
%\caption{DRAM Firms by Cartel Status\\1992-2011}\centering
%\label{firm_active}
%
%\bgroup
%\def\arraystretch{2}%  1 is the default, change whatever you need
%
%\tabcolsep=0.55cm
%
%\begin{tabular}{| l | c | c  || l | c | c  |} \hline
%{\textbf{Firm ID$^*$}} & {\textbf{Years Active$^\dagger$}} & {\textbf{Cartel}} & {\textbf{Firm ID$^*$}} & {\textbf{Years Active$^\dagger$}} & {\textbf{Cartel}} \\[5pt] \hline 
%1  & 2001- &Y & 21 &1992-2000  &Y  \\[2.5pt]
%2  & 1992-2005  &   & 22 & 1992-1998 & \\[2.5pt]
%3  &  1992-  &Y   & 23 & 1992-1993 &  \\[2.5pt]
%4  & 2002-2004   &   & 24 & 1992-1998 & \\[2.5pt]
%5  &  1992-2009  &Y   & 25 &1992-1997  &  \\[2.5pt]
%6  &  1992-1999 &   &  &  & \\[2.5pt]
%7  & 1992-  &Y   & &  &  \\[2.5pt]
%8  &  1992-2003 &Y   &   &  & \\[2.5pt]
%9  & 2001-  &Y   &   &  & \\[2.5pt]
%10  & 1999-  &   &   &   & \\[2.5pt]
%11  & 1992- &Y    &  &  & \\[2.5pt]
%12  & 1992-2003  &   &  &    &   \\[2.5pt]
%13  &  1992-  &Y   &   &   &  \\[2.5pt]
%14  & 1992-   &Y  &  &    &  \\[2.5pt]
%15  &  2001-2004  &   &  &    &  \\[2.5pt]
%16  &  1999-2008 &Y   &  &   &  \\[2.5pt]
%17  & 1992-2000  &Y   &   & &   \\[2.5pt]
%18  &  1993-2000 &   &   &  &   \\[2.5pt]
%19  & 1992-1998  &Y  &   &  &   \\[2.5pt]
%20  & 1992-1997  &   &   &  &  \\ \hline
%\end{tabular}
%
%\egroup
%
%\begin{tablenotes}
%	\item [*] Firm names are masked, per DOJ clearance policies, in order to avoid conveying the impression of private information about cartel participants. \\
%	\item [$\dagger$] Database treats all exit or merger as occurring at the start of a calendar year. \\
% \item Restricted to 1992-; entry within first 4 years of cycle; total product cyle 10 years
%\end{tablenotes}
%\end{threeparttable}
%\end{table}

%\clearpage

\subsection{Additional Figures and Tables}\label{extra}


\setcounter{table}{0}
\setcounter{figure}{0}
\renewcommand{\thetable}{C\arabic{table}}
\renewcommand{\thefigure}{C\arabic{figure}}


\begin{figure}[h]
\centering
\subfloat[]{
        \label{subfig:prof_1}
        \includegraphics[height=1.5in, keepaspectratio]{./pics1/firm_share_evo_1} } 
\subfloat[]{
        \label{subfig:prof_2}
        \includegraphics[height=1.5in, keepaspectratio]{./pics1/firm_share_evo_2} } \\
\subfloat[]{
        \label{subfig:prof_3}
        \includegraphics[height=1.5in, keepaspectratio]{./pics1/firm_share_evo_3} } 
\caption{Market Share by Generation, Largest Firms}
\footnotesize{Firm names are masked to avoid conveying the impression of private information about cartel participants.}
\label{fig:firm_gen_evo}
\end{figure} 

\begin{figure}[h]
	\centering
 \includegraphics[height=2in]{./pics1/output_PC_yoy.png}
	\caption{Industry DRAM Output against PC Shipment Change}
  	\label{fig:output_PC}
\end{figure}


\begin{table}[htbp]\centering
\begin{threeparttable}
\captionsetup{justification=centering}
 \caption{First Stage Estimate of Log Output$_{k+1}$ on Exogenous Variables}
 \label{first_stage}
\def\onepc{$^{\ast\ast\ast}$} \def\fivepc{$^{\ast\ast}$}
\def\tenpc{$^{\ast}$}
\begin{tabular}{l p{1.5cm} c p{1.5cm} c} \toprule \toprule
 {\extracolsep{5.5in}}
%p{3.2cm}

    & \multicolumn{2}{c}{~~~~~OLS~~~~~}  \\
    \midrule

Constant &17.349 &(5.793)	  \\[5pt]
Firm Age$_{k+1}$$^{\ddagger}$  &0.194 &(0.016)  \\[5pt]
Log Cum Output$^\dagger$ &0.510 &(0.075) \\[5pt]
Log PC Ship &-1.207 &(0.329) \\[5pt]
Log PC Ship $\times$ Gen Rank$_{2}$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ &0.069 &(0.020)  \\[5pt]
Log PC Ship $\times$ Gen Rank$_{3}$ &0.190 &(0.029)    \\[5pt]
Log PC Ship $\times$ Gen Rank$_{4}$ &0.213 &(0.031)    \\[5pt]
Log PC Ship $\times$ Gen Rank$_{5}$ &0.121 &(0.031)    \\[7.5pt]
I(1997)$_{16}$  &0.041 &(0.692)   \\[5pt]
I(1997)$_{16}$ $\times$ I(Memb)$_{16}$ &0.417 &(0.903)	  \\[5pt]
I(1998)$_{16}$  &0.867 &(1.064)	 \\[5pt]
I(1998)$_{16}$ $\times$ I(Memb)$_{16}$  &0.270 &1.265	\\[5pt]
I(1997)$_{64}$  &0.553 &(0.571)	    \\[5pt]
I(1997)$_{64}$ $\times$ I(Memb)$_{64}$  &-1.621 &(0.611) \\[5pt]
I(1998)$_{64}$ & -0.654 &(0.573)	  \\[5pt]
I(1998)$_{64}$ $\times$ I(Memb)$_{64}$ &-1.429 &(0.785)	   \\[5pt]
    \hline
~\\
	Firm FE $\times$ Log Cum Output & \multicolumn{2}{c}{Y}  \\	
	Quarter FE & \multicolumn{2}{c}{Y}  \\[2.5pt]
	\hline 
~\\
R-squared & \multicolumn{2}{c}{0.754}   \\
N &\multicolumn{2}{c}{2,387} \\
Clusters &\multicolumn{2}{c}{136}  \\
    \bottomrule
\end{tabular}

\begin{tablenotes}
Coefficients and standard errors displayed. Standard errors clustered at firm-generation level. \\
$^\ddagger$ Instrumental variable equals elapsed quarters of firm $i$ producing $k+1$ \\
$^\dagger$ Effect of log cumulative output is given for average firm. \\
The dependent variable is logged firm-level output of units on generation $k+1$. Sample for each regression is restricted to firm shipments within first 12 years of generation activity and at least 1000 units of quarterly shipments. 
\end{tablenotes}
\end{threeparttable}

\end{table}


\begin{table}[h]\centering
\begin{threeparttable}
\captionsetup{justification=centering}
 \caption{Firm Log Cumulative Output on Log Current Output:\\ Subsample Test - 2SLS Model, 1997-2011}
 \label{sub}
\def\onepc{$^{\ast\ast\ast}$} \def\fivepc{$^{\ast\ast}$}
\def\tenpc{$^{\ast}$}
\begin{tabular}{lcccc} \toprule \toprule
 {\extracolsep{0.2in}}
%p{3.2cm}

   Firm & \multicolumn{2}{c}{$\times$ Log Cum Output$^\dagger$} & \multicolumn{2}{c}{$\times$ Log Cum Output $\times$ I(Post)$^\ddagger$} \\
    \midrule

2 & -0.028 & (0.033)  & 0.021 & (0.033)  \\[7.5pt]
3 & 0.044 & (0.035)  & -0.029 & (0.013)  \\[7.5pt]
4 & 0.038 & (0.031)  & 0.012 & (0.014)  \\[7.5pt]
5 & -0.026 & (0.032)  & 0.035 & (0.031)  \\[7.5pt]
6 & 0.032 & (0.031)  & -0.021 & (0.015) \\[7.5pt]
7 & -0.067 & (0.031)  & -0.041 & (0.021) \\[7.5pt]
8 & -0.010 & (0.031)  & 0.008 & (0.018)  \\[7.5pt]
9 & -0.084 & (0.034)  & 0.012 & (0.022)  \\[7.5pt]
10 & -0.040 & (0.030)  & 0.034 & (0.020)  \\[7.5pt]
11 & -0.017 & (0.035)  & 0.026 & (0.025)  \\[7.5pt]
12 & -0.002 & (0.032)  & 0.043 & (0.018)  \\[7.5pt]
13 & -0.070 & (0.033)  & -0.013 & (0.023)  \\[7.5pt]
14 & -0.002 & (0.033)  & -0.248 & (0.035)  \\[7.5pt]
15 & -0.016 & (0.033)  & -0.024 & (0.019)  \\[7.5pt]
    \hline
~\\
	Log Cum Output  & \multicolumn{4}{c}{0.454   (0.059)}  \\
	Log PC Ship  & \multicolumn{4}{c}{Y}    \\
	Log PC Ship $\times$ Gen Rank$_{k}$ & \multicolumn{4}{c}{Y} \\ 	
	Log Output$_{k+1}$  & \multicolumn{4}{c}{Y} \\
	Quarter FE & \multicolumn{4}{c}{Y}  \\[2.5pt]
	\hline 
~\\
R-squared & \multicolumn{4}{c}{0.644}  \\
N &\multicolumn{4}{c}{2,376}  \\
Clusters &\multicolumn{4}{c}{131}  \\
    \bottomrule
\end{tabular}

\begin{tablenotes}
Coefficients and standard errors (clustered at firm-generation level) displayed. \\
$^\dagger$ estimates firm $i$ learning effect for 1Mb - 128Mb \\
$^\ddagger$ estimates incremental firm $i$ learning learning effect for 256Mb - 2Gb \\
The dependent variable is logged firm-level output of DRAM units. Regression estimated via 2SLS by instrumenting Log Output$_{k+1}$ by Firm Age$_{k+1}$. Excludes firm-generation pairs with less than or equal to 4 quarters of data. Firm names are masked to avoid conveying the impression of private information about cartel participants.
\end{tablenotes}
\end{threeparttable}

\end{table}








%\begin{figure}[h]
%	\centering
% \includegraphics[height=3.5in]{./pics1/cartel_MS_v2}
%	\caption{Cartel Member Market Share: 16Mb - 128Mb}
 % 	\label{fig:cart_share}
%\end{figure}


\clearpage

\subsection{Trial Transcripts}


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\begin{figure}[h]
	\centering
 \includegraphics[height=4.0in]{./pics1/swanson}
	\caption{Excerpt from Transcript of Proceedings as to Criminal Trial of Gary Swanson, Case No. 06-0692, U.S. District Court N.D. Cal., 12/17/2007, pg. 97.}
  	\label{fig:swanson}
\end{figure}


\end{document}








