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\title{\bigskip \bigskip \bigskip \bigskip \textsc{Technical Appendix to}\\
\textsc{"Multiple-Product Firms and Product Switching"}\bigskip \bigskip
\bigskip \bigskip }
\date{\bigskip \bigskip \textbf{January 2009}}
\author{Andrew Bernard, Stephen J. Redding, and Peter K. Schott\thanks{%
Bernard:\ Tuck School at Dartmouth \&\ NBER, 100 Tuck Hall, Hanover, NH
03755, andrew.b.bernard@dartmouth.edu. Redding:\ Yale School of Management
\&\ CEPR, 135 Prospect St., New Haven, CT 06520, stephen.redding@yale.edu.
Schott: Yale School of Management \ \&\ NBER, 135 Prospect St., New Haven,
CT 06520, peter.schott@yale.edu. Bernard and Schott (SES-0241474) and Schott
(SES-0550190) thank the National Science Foundation, and Redding thanks the
Centre for Economic Performance, the Leverhulme Trust and Princeton
University for research support. We thank Evan Gill and Justin Pierce for
excellent research assistance, and Jim Davis for timely disclosure. We thank
the editor, three anonymous referees and Esteban Rossi-Hansberg for insight
and suggestions. We are also grateful to Marc Melitz, Sharon Oster, Peter
Neary, John Sutton, John Van Reenen, Stephen Yeaple and seminar participants
at AEA, CEPR, Dartmouth, LSE, Princeton, Stanford and Yale for helpful
comments. Any opinions, findings, and conclusions or recommendations
expressed in this material are those of the authors and do not necessarily
reflect the views of the NSF, NBER, CEPR, or the U.S. Census Bureau. Results
have been screened to insure that no confidential data are revealed.}}
\maketitle
\title{Technical Appendix to Multi-Product Firms and Product Switching}

%TCIMACRO{\TeXButton{appendix}{\appendix}}%
%BeginExpansion
\appendix%
%EndExpansion

\section{Introduction}

This technical appendix contains a more detailed exposition of the model and
additional supplementary material for the paper. The remainder of the
technical appendix is structured as follows. Section B develops the model.
Section C derives the features of product switching at the firm, product and
firm-product level that we examine empirically in the paper. For further
economic intuition, Section D considers a particularly tractable special
case of the model that abstracts from serial correlation in consumer tastes
and productivity. Though this setup does not give rise to scale and age
dependence in product dropping, it nevertheless provides insight into many
of the other features of product switching found in the more general model
and observed in our census data. Section E characterizes the model's
implications for revenue and quantity-based measures of productivity.
Section F reports additional empirical results. An appendix at the end of
this document contains technical derivations and the proofs of propositions.

\section{Dynamic Firm-Product Selection Model}

In this section, we develop in further detail the model of multi-product
firms and product switching discussed in the paper. The model builds closely
on existing theories of industry dynamics, in which firms produce a single
product and profitability varies stochastically across firms over time (see
for example Jovanovic 1982, Hopenhayn 1992, Ericson and Pakes 1995, and
Melitz 2003). In these existing models, firms that are heterogeneous in
productivity are assumed to produce a single product, with the result that
firm and product-market entry and exit are equivalent. Here, we develop a
natural extension of such models in which firms choose to produce an
endogenous range of products in response to evolving firm and product
characteristics. The model highlights a number of features of product
switching at the firm, product and firm-product level that we use to guide
our empirical research and that are derived below.

\subsection{Preferences and Endowments}

The representative consumer's utility is defined over the consumption of a
fixed continuum of products, which we normalize to the interval $[0,1]$.
Utility is assumed to be a constant elasticity of substitution function of
the consumption of each of the continuum of products:%
\begin{equation}
U=\left[ \int_{0}^{1}\left( a_{i}C_{i}\right) ^{\nu }di\right] ^{\frac{1}{%
\nu }},\qquad 0<\nu <1.  \label{utility}
\end{equation}%
where $i$ indexes products, $a_{i}>0$ is a demand parameter that allows the
relative importance of products in utility to vary, and $\kappa =\frac{1}{%
1-\nu }>1$ is the elasticity of substitution between products.

As discussed further below, firms decide whether or not to participate in
each product market, and if they choose to participate, to supply a
horizontally differentiated variety of a product. Therefore $C_{i}$ is a
consumption index for product $i$, which is defined over the horizontally
differentiated varieties of firms, and is also assumed to take the constant
elasticity of substitution form:%
\begin{equation}
C_{i}=\left[ \int_{\omega \in \Omega _{i}}\left( \lambda _{i}\left( \omega
\right) c_{i}\left( \omega \right) \right) ^{\rho }d\omega \right] ^{\frac{1%
}{\rho }},\qquad 0<\rho <1.  \label{cindex}
\end{equation}%
where $\omega $ indexes firm varieties within product $i$. The demand
parameter $\lambda _{i}\left( \omega \right) \geq 0$ captures the
representative consumer's tastes for a firm's variety within product $i$ and
allows demand to vary across the varieties of a given product supplied by
different firms.\footnote{%
One interpretation of the parameter $\lambda _{i}\left( \omega \right) $ is
product quality, though it also captures other more subjective
characteristics of a firm's variety that influence the representative
consumer's demand for that variety.} Although not central to our results, we
make the natural assumption that the elasticity of substitution across
varieties within products is greater than the elasticity of substitution
across products, $\sigma =\frac{1}{1-\rho }>\kappa =\frac{1}{1-\nu }>1$.
Similarly, we also assume for simplicity that the elasticity of substitution
across varieties within products, $\sigma =\frac{1}{1-\rho }$, is the same
for all products. The price index dual to (\ref{cindex}) is:%
\begin{equation}
P_{i}=\left[ \int_{\omega \in \Omega _{i}}\left( \frac{p_{i}\left( \omega
\right) }{\lambda _{i}\left( \omega \right) }\right) ^{1-\sigma }d\omega %
\right] ^{\frac{1}{1-\sigma }}.  \label{pindex}
\end{equation}

Consumer expenditure minimization implies that revenue for a variety of a
product $r_{i}\left( \omega \right) $ depends upon the own variety price $%
p_{i}\left( \omega \right) $, consumer tastes $\lambda _{i}\left( \omega
\right) $, the product price index $P_{i}$ and aggregate product revenue $%
R_{i}$, while aggregate product revenue $R_{i}$ depends on the product price
indices $P_{i}$ and aggregate revenue for the economy as whole $R$:%
\begin{equation}
r_{i}\left( \omega \right) =p_{i}\left( \omega \right) ^{1-\sigma }\lambda
_{i}\left( \omega \right) ^{\sigma -1}P_{i}^{\sigma -1}R_{i},\qquad R_{i}= 
\left[ \frac{\left( P_{i}/a_{i}\right) ^{1-\kappa }}{\int_{0}^{1}\left(
P_{i}/a_{i}\right) ^{1-\kappa }di}\right] R  \label{consrev}
\end{equation}

Labor is the sole factor of production and is assumed to be in inelastic
supply $L$ (which also indexes the size of the economy).

\subsection{Production Technology}

A firm is defined by its production technology (productivity $\varphi $) and
its product attributes that influence demand (consumer tastes $\lambda _{i}$%
).\footnote{%
We follow existing models of industry dynamics in taking a technological
approach to the boundaries of the firm. As such we refrain from endogenizing
the ownership of the knowledge assets that embody technology, as in the
incomplete contracts literature following Grossman and Hart (1986).
Implicitly, knowledge assets are assumed to be intangible so that they
cannot be transferred beyond the boundaries of the firm using arms-length
transactions.} Selection across firms is driven by heterogeneity in
productivity $\varphi $, while selection within firms arises as a result of
heterogeneity in consumer tastes across a firm's products $\lambda _{i}$.
Productivity and consumer tastes evolve stochastically over time, and firms
make endogenous decisions whether to enter and exit and whether to
participate in individual product markets as productivity and consumer
tastes evolve.

We assume that there is a competitive fringe of potential firms who are
identical prior to entry. In order to enter, firms must incur a sunk entry
cost of $f_{e}>0$ units of labor, which can be interpreted as an upfront
investment in research and development (R\&D). Incurring the sunk entry cost
creates a firm brand and a blueprint for one horizontally differentiated
variety of each product that can be produced using this brand. The firm's
productivity and the demand for its variety of each product are uncertain
prior to entry. Once the sunk cost has been incurred, the firm observes its
productivity $\varphi $ and consumer tastes for its products $\lambda _{i}$,
and decides whether to enter or exit. If the firm decides to exit, the
knowledge embodied in its productivity and product blueprints is lost, and
the sunk cost must be incurred again in order for the firm to re-enter. If
the firm decides to enter, productivity and consumer tastes evolve over time
according to stochastic processes whose properties are known prior to entry.

We choose a specification for the evolution of productivity and consumer
tastes that is both tractable and sufficiently general to match key features
of the firm-product data. After the sunk entry cost is paid, firm
productivity, $\varphi \in \lbrack \underline{\varphi },\bar{\varphi}%
]\subset \lbrack 0,\infty )$, is drawn from a continuous distribution $%
g_{e}\left( \varphi \right) $, and consumer tastes for the firm's variety, $%
\lambda _{i}\in \lbrack \underline{\lambda },\bar{\lambda}]\subset \lbrack
0,\infty )$, are drawn from a continuous distribution $z_{ei}\left( \lambda
_{i}\right) $ for each product $i$. The subscript $e$ is mnemonic for entry
and the corresponding cumulative distributions are denoted by $G_{e}\left(
\varphi \right) $ and $Z_{ei}\left( \lambda _{i}\right) $ respectively. To
make the firm's problem an interesting one, we assume that the intervals for
productivity and consumer tastes are sufficiently wide that we have an
interior equilibrium, in which a firm's entry decision depends on its
productivity and the decision to participate in individual product markets
depends on consumer tastes. To make use of law of large numbers results, the
productivity and consumer taste distributions are assumed to be independent
across firms, which enables idiosyncratic risk to be perfectly diversified
across firms. Similarly, we assume that the consumer taste distributions are
independent across products and the consumer taste and firm productivity
distributions are uncorrelated with one another, which simplifies the
characterization of the equilibrium range of products produced by a firm.%
\footnote{%
While the consumer taste and firm productivity distributions are independent
of one another, there is interdependence in a firm's profitability across
products, because firm productivity is common across products.}

Following entry, a firm faces a constant Poisson probability $\varepsilon
_{i}>0$ of a shock to consumer tastes $\lambda _{i}$ for its variety of
product $i$, in which case a new value for consumer tastes $\lambda
_{i}^{\prime }$ is drawn from a continuous conditional distribution $%
z_{ci}\left( \lambda _{i}^{\prime }|\lambda _{i}\right) $. Both the
probability of a consumer taste shock ($\varepsilon _{i}$) and the
conditional distribution for consumer tastes ($z_{ci}\left( \lambda
_{i}^{\prime }|\lambda _{i}\right) $) are allowed to vary across products.
Following entry, the firm also faces a constant Poisson probability $\theta
>0$ of a shock to productivity $\varphi $, in which case a new value for
productivity $\varphi ^{\prime }$ is drawn from a continuous conditional
distribution $g_{c}\left( \varphi ^{\prime }|\varphi \right) $. The
subscript $c$ is mnemonic for conditional, and we denote the corresponding
cumulative distributions by $Z_{ci}\left( \lambda _{i}^{\prime }|\lambda
_{i}\right) $ and $G_{c}\left( \varphi ^{\prime }|\varphi \right) $
respectively. While the probabilities $\varepsilon _{i}$ and $\theta $ are
independent across time, our formulation allows for serial correlation in
consumer tastes and firm productivity, because the new draws for these
variables following a stochastic shock depend on their existing values
through the conditional distributions $z_{ci}\left( \lambda _{i}^{\prime
}|\lambda _{i}\right) $ and $g_{c}\left( \varphi ^{\prime }|\varphi \right) $%
. We make the natural assumption of positive serial correlation, $\partial
Z_{ci}\left( \lambda _{i}^{\prime }|\lambda _{i}\right) /\partial \lambda
_{i}<0$ and $\partial G_{c}\left( \varphi ^{\prime }|\varphi \right)
/\partial \varphi <0$ for $\lambda _{i}^{\prime }\in \left[ \underline{%
\lambda },\bar{\lambda}\right] $ and $\varphi ^{\prime }\in \left[ 
\underline{\varphi },\bar{\varphi}\right] $, so that the probability of
drawing a new value for consumer tastes (productivity) below $\lambda
_{i}^{\prime }$ ($\varphi ^{\prime }$) is decreasing in the existing value
of consumer tastes (productivity). In addition to stochastic shocks to
productivity and consumer tastes, the firm faces a constant Poisson
probability of death $\delta >0$, which captures \textit{force majeure}
events unrelated to the profitability of the firm.\footnote{%
The model can be extended to allow firms to make endogenous investments in
improving productivity and enhancing consumer tastes, but these extensions
are not central to the model's key predictions, which are driven by
selection. Our focus on selection within firms mirrors the emphasis on
selection across firms in existing industry dynamics models of firm creation
and destruction, such as Hopenhayn (1992), Ericson and Pakes (1995) and
Melitz (2003).}

Once the sunk cost has been incurred and productivity and consumer tastes
are observed, the firm decides whether to enter and which products to
produce. We assume that there is a fixed corporate headquarters cost of $%
f_{h}>0$ units of labor, which the firm must incur irrespective of the
number of products that it chooses to produce, and a fixed production cost
of $f_{pi}>0$ units of labor for each product $i$ that is produced. There is
a constant marginal cost of production for each product, which depends upon
the firm's productivity. Total labor employed by a firm with productivity $%
\varphi $\ is therefore:%
\begin{equation}
l\left( \varphi \right) =f_{h}+\int_{0}^{1}I_{i}\left[ f_{pi}+\frac{%
q_{i}\left( \varphi ,\lambda _{i}\right) }{\varphi }\right] di,
\label{prodtech}
\end{equation}%
where $I_{i}$ is an indicator variable which equals one if a firm produces
product $i$ and zero otherwise, and $q\left( \varphi ,\lambda _{i}\right) $
denotes output of product $i$ by a firm with productivity $\varphi $ and
demand parameter $\lambda _{i}$.

\subsection{Firm-Product Profitability}

Demand for a firm's variety of a product depends upon the own variety price,
the price index for the product and the price indices for all other
products. If a firm produces a product, it supplies only one of a continuum
of varieties, and so is unable to influence the price index for the product.
Additionally, the price of a firm's variety of one product only influences
the demand for its varieties of other products through the price indices.
Therefore, the firm's inability to influence the price indices implies that
its profit maximization problem reduces to choosing the price of each
product variety separately to maximize the profits derived from that product
variety.\footnote{%
The structure of our model eliminates strategic interaction within or
between firms. This choice of model structure enables us to isolate the role
of selection within and across firms from considerations of strategic
interaction.} This optimization problem yields the standard result that the
equilibrium price of a product variety is a constant mark-up over marginal
cost:

\begin{equation}
p_{i}\left( \varphi ,\lambda _{i}\right) =\frac{1}{\rho }\frac{w}{\varphi },
\label{varprice}
\end{equation}%
where we choose the wage for the numeraire and so $w=1$.

From consumer expenditure minimization, substituting for the pricing rule
and using the choice of numeraire, equilibrium revenue and profits from a
product variety are:%
\begin{equation}
r_{i}\left( \varphi ,\lambda _{i}\right) =R_{i}\left( \rho P_{i}\varphi
\lambda _{i}\right) ^{\sigma -1},\qquad \pi _{i}\left( \varphi ,\lambda
_{i}\right) =\frac{r_{i}\left( \varphi ,\lambda _{i}\right) }{\sigma }%
-f_{pi},  \label{varrev}
\end{equation}%
where $R_{i}$ denotes aggregate expenditure on product $i$.

From equation (\ref{varrev}), differences in firm productivity have exactly
the same effects on equilibrium revenue and profits from a product as
differences in consumer tastes for the firm's variety, because prices are a
constant mark-up over marginal costs and demand exhibits a constant
elasticity of substitution.\footnote{%
Therefore $\lambda _{i}$ has an equivalent interpretation as a component of
a firm's productivity that is specific to individual products. Under this
alternative interpretation, which leaves the determination of general
equilibrium unchanged, stochastic shocks to $\lambda _{i}$ capture changes
in firm productivity that are specific to individual products.} From the
expression for equilibrium product revenue, the ratio of revenue for two
varieties of the same product depends solely on the relative productivities
of the firms producing those varieties and the relative strength of consumer
demand for those varieties:%
\begin{equation}
r_{i}\left( \varphi ^{\prime \prime },\lambda _{i}^{\prime \prime }\right)
=\left( \varphi ^{\prime \prime }/\varphi ^{\prime }\right) ^{\sigma
-1}\left( \lambda _{i}^{\prime \prime }/\lambda _{i}^{\prime }\right)
^{\sigma -1}r_{i}\left( \varphi ^{\prime },\lambda _{i}^{\prime }\right) .
\label{relvarrev}
\end{equation}

The key economic decisions of a firm in the model are whether to enter or
exit and in which product markets to participate. Under our assumptions, the
firm's decision whether or not to produce a product takes a very tractable
form. Fixed and marginal production costs for individual product varieties
have no sunk component and consumer tastes for a firm's variety of a product
evolve independently of whether the firm's variety is actually produced. The
firm's decision whether or not to produce its variety of a product thus
reduces to a period-by-period comparison of contemporaneous revenue and
production costs.\footnote{%
In contrast, the costs of firm entry are sunk, which introduces an option
value to firm entry, as discussed further below. While our analysis can be
extended to also allow for sunk costs of entering each product, the
resulting option value to producing each product complicates the analysis
without substantively changing the features of the model that we examine in
our empirical work.} The existence of product fixed production costs implies
that there is a \textbf{zero-profit consumer taste cutoff }$\lambda
_{i}^{\ast }\left( \varphi \right) $ for a firm with productivity $\varphi $%
, such that the firm will produce the product if consumer tastes $\lambda
_{i}$ are equal to or greater than $\lambda _{i}^{\ast }\left( \varphi
\right) $. From the expression for equilibrium profits above, the value of $%
\lambda _{i}^{\ast }\left( \varphi \right) $ is defined by the following
zero-profit condition:%
\begin{equation}
r_{i}\left( \varphi ,\lambda _{i}^{\ast }\left( \varphi \right) \right)
=R_{i}\left( \rho P_{i}\varphi \lambda _{i}^{\ast }\left( \varphi \right)
\right) ^{\sigma -1}=\sigma f_{pi}.  \label{zppp}
\end{equation}

Using the expression for relative product variety revenue in equation (\ref%
{relvarrev}) above, equilibrium revenue and profits from a product can be
expressed relative to the revenue of a variety with the zero-profit consumer
taste:%
\begin{eqnarray}
r_{i}\left( \varphi ,\lambda _{i}\left( \varphi \right) \right) &=&\left\{ 
\begin{array}{l}
\left( \frac{\lambda _{i}}{\lambda _{i}^{\ast }\left( \varphi \right) }%
\right) ^{\sigma -1}\sigma f_{pi} \\ 
0%
\end{array}%
\right. \qquad \left. 
\begin{array}{l}
\text{for}\qquad \lambda _{i}\geq \lambda _{i}^{\ast }\left( \varphi \right)
\smallskip \\ 
\text{otherwise}%
\end{array}%
\right. ,  \label{eqvarrev} \\
\pi _{i}\left( \varphi ,\lambda _{i}\left( \varphi \right) \right)
&=&\left\{ 
\begin{array}{l}
\left[ \left( \frac{\lambda _{i}}{\lambda _{i}^{\ast }\left( \varphi \right) 
}\right) ^{\sigma -1}-1\right] f_{pi} \\ 
0%
\end{array}%
\right. \left. 
\begin{array}{l}
\text{for}\qquad \lambda _{i}\geq \lambda _{i}^{\ast }\left( \varphi \right)
\smallskip \\ 
\text{otherwise}%
\end{array}%
\right. .  \notag
\end{eqnarray}

As consumer tastes for a firm's variety of each product change over time,
previously profitable products become unprofitable and are dropped when $%
\lambda _{i}$ falls below the zero-profit cutoff value $\lambda _{i}^{\ast
}\left( \varphi \right) $. Similarly, previously unprofitable products
become viable and are added when $\lambda _{i}$ rises above $\lambda
_{i}^{\ast }\left( \varphi \right) $.

The equilibrium of the model features an endogenous stationary distribution
for consumer tastes $\lambda _{i}$ across firms within each product $i$,
which we denote by $\gamma _{zi}\left( \lambda _{i}\right) $, and which is
influenced by both the entry and conditional distributions for consumer
tastes and the probability of a consumer taste shock. The stationary
distribution for consumer tastes is independent of firm productivity, since
the entry and conditional distributions for consumer tastes are by
assumption independent of the corresponding distributions for firm
productivity. The stationary distribution of consumer tastes within each
product is characterized by the requirement that the inflow of firms to
consumer taste $\lambda _{i}$ equals the corresponding outflow from consumer
taste $\lambda _{i}$ for all $\lambda _{i}\in \lbrack \underline{\lambda },%
\bar{\lambda}]$:%
\begin{equation}
z_{ei}\left( \lambda _{i}\right) \left[ 1-G_{e}\left( \varphi ^{\ast
}\right) \right] M_{e}+\left[ \int_{\underline{\lambda }}^{\bar{\lambda}%
}\varepsilon _{i}z_{ic}\left( \lambda _{i}|\lambda _{i}^{\prime }\right)
\gamma _{zi}\left( \lambda _{i}^{\prime }\right) d\lambda _{i}^{\prime }%
\right] M=\left( \delta +\chi +\varepsilon _{i}\right) \gamma _{zi}\left(
\lambda _{i}\right) M,  \label{invarianttaste}
\end{equation}%
where $M_{e}$ denotes the mass of entrants each period; $M$ denotes the mass
of firms producing; $\varphi ^{\ast }$ is the zero-value cutoff productivity
below which a firm exits as determined below; $\chi \equiv \theta
\int_{\varphi _{\ast }}^{\bar{\varphi}}G_{c}\left( \varphi ^{\ast }|\varphi
\right) \gamma _{g}\left( \varphi \right) d\varphi $ is the aggregate
probability of firm exit due to stochastic productivity shocks, which
depends on the stationary distribution of firm productivity $\gamma
_{g}\left( \varphi \right) $ that is characterized below.

The equality of inflows and outflows in equation (\ref{invarianttaste}) has
an intuitive interpretation. The first and second terms on the left-hand
side are the inflow of firms to consumer taste $\lambda _{i}$ as a result of
firm entry and consumer taste shocks, respectively. The first, second and
third terms on the right-hand side are respectively the outflow of firms
from consumer taste $\lambda _{i}$ as a result of firm death, productivity
shocks that induce firm exit and consumer taste shocks. The stationary
distribution for consumer tastes determined by the equality of inflows and
outflows is defined over the interval $[\underline{\lambda },\bar{\lambda}]$%
, such that $\int_{\underline{\lambda }}^{\bar{\lambda}}\gamma _{zi}\left(
\lambda _{i}\right) d\lambda _{i}=1$, and we denote the corresponding
cumulative distribution by $\Gamma _{zi}\left( \lambda _{i}\right) $. There
is also a corresponding stationary distribution for consumer tastes
conditional on a product being produced, which is a truncation of the
distribution $\gamma _{zi}\left( \lambda _{i}\right) $ at the zero-profit
cutoff $\lambda _{i}^{\ast }\left( \varphi \right) $.

\subsection{Firm Profitability}

Having characterized a firm's decision whether to produce a product, we now
examine firm profitability across the continuum of products as a whole. The
total revenue and profits of a firm with productivity $\varphi $ are as
follows:%
\begin{equation}
r\left( \varphi \right) =\int_{0}^{1}I_{i}r_{i}\left( \varphi ,\lambda
_{i}\right) di,\qquad \pi \left( \varphi \right) =\int_{0}^{1}I_{i}\pi
_{i}\left( \varphi ,\lambda _{i}\right) di-f_{h},  \label{firmrevprof}
\end{equation}%
where $I_{i}$ is again the indicator variable which equals one if a firm
produces product $i$ and zero otherwise.

With a continuum of products and independent distributions for consumer
tastes, the law of large numbers implies that a firm's expected revenue
across the continuum of products equals the sum of its expected revenues
from each product. Total firm revenue and profits across the continuum of
products are thus:

\begin{eqnarray}
r\left( \varphi \right) &=&\int_{0}^{1}\left[ \left[ 1-\Gamma _{zi}\left(
\lambda _{i}^{\ast }\left( \varphi \right) \right) \right] \int_{\lambda
_{i}^{\ast }\left( \varphi \right) }^{\bar{\lambda}}r_{i}\left( \varphi
,\lambda _{i}\right) \left( \frac{\gamma _{zi}\left( \lambda _{i}\right) }{%
1-\Gamma _{zi}\left( \lambda _{i}^{\ast }\left( \varphi \right) \right) }%
\right) d\lambda _{i}\right] di,  \label{expfirmrevprof} \\
\pi \left( \varphi \right) &=&\int_{0}^{1}\left[ \left[ 1-\Gamma _{zi}\left(
\lambda _{i}^{\ast }\left( \varphi \right) \right) \right] \int_{\lambda
_{i}^{\ast }\left( \varphi \right) }^{\bar{\lambda}}\pi _{i}\left( \varphi
,\lambda _{i}\right) \left( \frac{\gamma _{zi}\left( \lambda _{i}\right) }{%
1-\Gamma _{zi}\left( \lambda _{i}^{\ast }\left( \varphi \right) \right) }%
\right) d\lambda _{i}\right] di-f_{h}.  \notag
\end{eqnarray}

While consumer tastes for a firm's variety of a product are stochastic, the
law of large numbers implies that all firms with the same productivity
experience the same flow of total profits across the continuum of products.
Stochastic shocks to consumer tastes generate fluctuations in the
profitability of individual products, which lead them to be added and
dropped over time. However, these fluctuations in the profitability of
individual products average out at the level of the firm, so that the
evolution of total firm profits over time is determined solely by stochastic
shocks to firm productivity.

Once a firm observes its productivity and hence total profits across the
continuum of products, it decides whether or not to enter. The lower a
firm's productivity $\varphi $, the higher the zero-profit cutoff for
product expertise $\lambda _{i}^{\ast }\left( \varphi \right) $, and so the
lower the probability of having a value for consumer taste sufficiently high
to profitably produce a product $\left( 1-\Gamma _{zi}\left( \lambda
_{i}^{\ast }\left( \varphi \right) \right) \right) $. With a continuum of
products and independent distributions for consumer tastes, the fraction of
products produced by a firm with productivity $\varphi $ equals the sum of
the probabilities of producing each product:%
\begin{equation}
\Lambda \left( \varphi \right) =\int_{0}^{1}\left( 1-\Gamma _{zi}\left(
\lambda _{i}^{\ast }\left( \varphi \right) \right) \right) di
\label{productscope}
\end{equation}

As a firm with lower productivity has a lower probability of producing each
product, it produces a smaller range of products than another firm with
higher productivity. Therefore, the range of products produced by a firm is
monotonically increasing in its productivity. Additionally, for sufficiently
low values of a firm's productivity, the excess of revenue over fixed
production costs in the small range of profitable products falls short of
the fixed headquarters cost, and the instantaneous flow of total firm
profits across the continuum of products is negative.

The existence of a sunk entry cost combined with stochastic shocks to
productivity generates an option value to firm entry, so that a firm's entry
decision does not simply depend on a comparison of total firm profits and
the fixed headquarters cost. If a firm chooses to exit, it forgoes both the
net present value of its instantaneous flow of profits and also the option
of experiencing stochastic productivity shocks. Therefore, with this option
value to firm entry, the threshold productivity for firm entry and exit will
in general lie below the productivity at which the instantaneous flow of
total firm profits is equal to zero. The value of a firm with productivity $%
\varphi $ is determined according to the following Bellman equation:

\begin{equation}
v\left( \varphi \right) =\left\{ 
\begin{array}{l}
\frac{\pi \left( \varphi \right) +\theta \left[ \int_{\varphi ^{\ast }}^{%
\bar{\varphi}}\left[ v\left( \varphi ^{\prime }\right) -v\left( \varphi
\right) \right] g_{c}\left( \varphi ^{\prime }|\varphi \right) d\varphi
^{\prime }\right] }{\delta +\theta G_{c}\left( \varphi ^{\ast }|\varphi
\right) }\qquad \\ 
\\ 
0%
\end{array}%
\right. \left. 
\begin{array}{l}
\text{for }\varphi \geq \varphi ^{\ast } \\ 
\bigskip \\ 
\text{otherwise}%
\end{array}%
\right. ,  \label{bellvalfirm}
\end{equation}%
where $\varphi ^{\ast }$ is the zero-value cutoff productivity below which a
firm exits; $\delta $ is the exogenous probability of firm death; $\theta
G_{c}\left( \varphi ^{\ast }|\varphi \right) $ is the endogenous probability
of firm exit, which depends on productivity $\varphi $, and equals the
probability of a stochastic shock to productivity $\theta $ times the
probability of drawing a new value for productivity below the zero-value
cutoff $G\left( \varphi ^{\ast }|\varphi \right) $.

The Bellman equation (\ref{bellvalfirm}) has an intuitive interpretation.
The value of a firm with productivity $\varphi $ is the flow of current
profits plus the expected value of capital gains or losses as a result of a
stochastic productivity shock, discounted by the probability of firm exit.
To determine the value of a firm with productivity $\varphi $, it proves
convenient to re-write the Bellman equation in the following way:%
\begin{equation}
v\left( \varphi \right) =\left\{ 
\begin{array}{l}
\frac{\pi \left( \varphi \right) }{\delta +\theta }+\frac{\theta }{\delta
+\theta }\int_{\varphi ^{\ast }}^{\bar{\varphi}}v\left( \varphi ^{\prime
}\right) g_{c}\left( \varphi ^{\prime }|\varphi \right) d\varphi ^{\prime
}\qquad \\ 
\\ 
0%
\end{array}%
\right. \left. 
\begin{array}{l}
\text{for }\varphi \geq \varphi ^{\ast } \\ 
\bigskip \\ 
\text{otherwise}%
\end{array}%
\right. ,  \label{bellvalfirmalt}
\end{equation}%
which, as shown in the appendix at the end of this document, has the
following solution:%
\begin{equation}
v\left( \varphi \right) =\left\{ 
\begin{array}{l}
\frac{\pi \left( \varphi \right) }{\delta +\theta }+\frac{\theta }{\delta
+\theta }\int_{\varphi ^{\ast }}^{\bar{\varphi}}\eta \left( \varphi ^{\prime
}\right) \pi \left( \varphi ^{\prime }\right) g_{c}\left( \varphi ^{\prime
}|\varphi \right) d\varphi ^{\prime }\qquad \\ 
\\ 
0%
\end{array}%
\right. \left. 
\begin{array}{l}
\text{for }\varphi \geq \varphi ^{\ast } \\ 
\bigskip \\ 
\text{otherwise}%
\end{array}%
\right. ,  \label{bellvalfirmsol}
\end{equation}%
where $\eta \left( \varphi \right) $ is determined in the appendix.

The zero-value cutoff productivity $\varphi ^{\ast }$ below which exit
occurs is defined by $v\left( \varphi ^{\ast }\right) =0$, which implies:%
\begin{equation}
-\pi \left( \varphi ^{\ast }\right) =\theta \int_{\varphi ^{\ast }}^{\bar{%
\varphi}}\eta \left( \varphi ^{\prime }\right) \pi \left( \varphi ^{\prime
}\right) g_{c}\left( \varphi ^{\prime }|\varphi ^{\ast }\right) d\varphi
^{\prime },  \label{zerovalcut}
\end{equation}%
so that at productivity $\varphi ^{\ast }$\ the flow of current losses
exactly equals the probability of a stochastic shock to productivity times
expected profits following a productivity shock.

As well as an endogenous stationary distribution for consumer tastes within
each product, the equilibrium of the model also features an endogenous
stationary distribution for firm productivity $\varphi $, which we denote by 
$\gamma _{g}\left( \varphi \right) $. This stationary distribution for firm
productivity is determined by the requirement that the inflow of firms to
each productivity equals the corresponding outflow from each productivity
for all $\varphi \geq \varphi ^{\ast }$:

\begin{equation}
g\left( \varphi \right) M_{e}+\int_{\varphi ^{\ast }}^{\bar{\varphi}}\theta
g_{c}\left( \varphi |\varphi ^{\prime }\right) \gamma _{g}\left( \varphi
^{\prime }\right) d\varphi ^{\prime }M=\left( \delta +\theta \right) \gamma
_{g}\left( \varphi \right) M\quad  \label{invariantprod}
\end{equation}

The equality of inflows and outflows from each productivity in equation (\ref%
{invariantprod}) also has an intuitive interpretation. The first and second
terms on the left-hand side are the inflow of firms to productivity $\varphi 
$ from entry and productivity shocks, respectively. The first term on the
right-hand side is the outflow of firms from productivity $\varphi $ due to
exogenous firm death. The second term on the right-hand side is the outflow
of firms from productivity $\varphi $\ to other values of productivity as a
result of productivity shocks, including productivity shocks that induce
endogenous firm exit. We define the stationary distribution for firm
productivity $\gamma _{g}\left( \varphi \right) $ as conditional upon entry,
so that $\int_{\varphi ^{\ast }}^{\bar{\varphi}}\gamma _{g}\left( \varphi
\right) =1$, and we denote the corresponding cumulative distribution by $%
\Gamma _{g}\left( \varphi \right) $.

\subsection{Firm Entry}

The equilibrium zero-value cutoff productivity $\varphi ^{\ast }$ is
determined by the free entry condition that requires the expected value of
entry to equal the sunk entry cost. The expected value of entry is equal to
the probability of successful entry times the expected value of the firm
conditional on successful entry:%
\begin{equation}
V=\left[ 1-G_{e}\left( \varphi ^{\ast }\right) \right] \bar{v}=f_{e},\qquad 
\bar{v}\equiv \int_{\varphi ^{\ast }}^{\bar{\varphi}}v\left( \varphi \right)
\left( \frac{g_{e}\left( \varphi \right) }{1-G_{e}\left( \varphi ^{\ast
}\right) }\right) d\varphi ,  \label{freeentry}
\end{equation}%
where $\left[ 1-G_{e}\left( \varphi ^{\ast }\right) \right] $ is the \textit{%
ex ante} probability of drawing a productivity above the zero-value cutoff $%
\varphi ^{\ast }$, and $\bar{v}$ is the expected value of the firm
conditional on entry, which depends on the equilibrium value of the firm for
each productivity from (\ref{bellvalfirmsol}) and the \textit{ex ante}
productivity distribution $g_{e}\left( \varphi \right) $.

Firms are assumed to finance the sunk entry cost by issuing equity to the
representative consumer. As there is a continuum of firms, and the
productivity and consumer tastes distributions are independent across firms,
consumers can perfectly diversify the idiosyncratic risk of stochastic
shocks to a firm's productivity and consumer tastes. Therefore the
equilibrium rate of return received by the representative consumer on firm
equity is equal to the aggregate risk of firm death and exit due to
productivity shocks, which is equal to $\delta +\chi \equiv \delta +\theta
\int_{\varphi _{\ast }}^{\bar{\varphi}}G_{c}\left( \varphi ^{\ast }|\varphi
\right) \gamma _{g}\left( \varphi \right) d\varphi $.

\subsection{Goods and Labor Markets}

While the analysis so far has characterized firm-level decisions about entry
and production, we now turn to the determination of aggregate variables,
such as the price indices and product revenue, which depend on the mass of
firms. The stationary equilibrium of the model is characterized by a
constant mass of firms entering each period, $M_{e}$, and a constant mass of
firms producing, $M$. Each of the firms producing is active in a subset of
product markets, with the steady-state mass of firms producing a given
product, $M_{pi}$, equal to a constant fraction of the mass of firms
producing.

To determine the mass of firms producing a given product, we note that of
the mass of firms with productivity $\varphi $, a fraction $\left[ 1-\Gamma
_{zi}\left( \lambda _{i}^{\ast }\left( \varphi \right) \right) \right] $
produce product $i$. Therefore the total mass of firms producing product $i$
can be determined from this fraction by integrating over values for
productivity using the stationary productivity distribution:%
\begin{equation}
M_{pi}=\left[ \int_{\varphi ^{\ast }}^{\bar{\varphi}}\left[ 1-\Gamma
_{zi}\left( \lambda _{i}^{\ast }\left( \varphi \right) \right) \right]
\gamma _{g}\left( \varphi \right) d\varphi \right] M,  \label{massproduct}
\end{equation}

The mass of firms producing, $M$, can be in turn determined from aggregate
revenue, $R$, and the expected revenue of a firm conditional on production, $%
\bar{r}$:%
\begin{equation}
M=\frac{R}{\bar{r}},\qquad \bar{r}=\int_{\varphi ^{\ast }}^{\bar{\varphi}%
}r\left( \varphi \right) \gamma _{g}\left( \varphi \right) d\varphi
\label{massfirms}
\end{equation}%
where revenue across the continuum of products for a firm with productivity $%
\varphi $, $r\left( \varphi \right) $, is given by (\ref{expfirmrevprof}).

The mass of firms entering each period, $M_{e}$, can be determined from the
mass of firms producing, $M$. With a constant mass of firms producing in the
stationary equilibrium, the mass of firms that enter and draw a productivity
above the zero-value cutoff must equal the mass of firms that exit, which
equals the mass of firms that die or draw a new productivity below the
zero-value cutoff:%
\begin{equation}
\left[ 1-G_{e}\left( \varphi ^{\ast }\right) \right] M_{e}=\left( \delta
+\chi \right) M.  \label{sss}
\end{equation}%
where $\chi \equiv \theta \int_{\varphi _{\ast }}^{\bar{\varphi}}G_{c}\left(
\varphi ^{\ast }|\varphi \right) \gamma _{g}\left( \varphi \right) d\varphi $%
.

The price indices for each product, $P_{i}$, can be determined from the mass
of firms producing each product, $M_{pi}$. From the equilibrium pricing rule
(\ref{varprice}), the price index for a product can be written as a function
of the mass of firms producing the product and the price of a variety with a
weighted average of firm productivity and consumer tastes:%
\begin{equation}
P_{i}=M_{pi}^{\frac{1}{1-\sigma }}\frac{1}{\rho \tilde{\varphi}_{i}},
\label{eqpindex}
\end{equation}%
where the weighted average $\tilde{\varphi}_{i}$ depends on the stationary
distributions of firm productivity and consumer tastes and is defined in the
appendix at the end of this document.

Aggregate revenue for each product, $R_{i}$, can be in turn determined from
the price indices, $P_{i}$, and aggregate revenue, $R$, using equation (\ref%
{consrev}). Finally, labor market clearing requires that the demand for
labor in production and entry equals the economy's supply of labor:%
\begin{equation}
L_{q}+L_{e}=\bar{L},  \label{labmktclear}
\end{equation}%
where the subscripts $q$ and $e$ denote labor used in production and entry
respectively.

\subsection{General Equilibrium}

General equilibrium is referenced by the quadruple \{$\varphi ^{\ast }$, $%
\lambda _{i}^{\ast }\left( \varphi ^{\ast }\right) $, $P_{i}$, $R_{i}$\},
together with a stationary distribution for firm productivity $\gamma
_{g}\left( \varphi \right) $ for $\varphi \geq \varphi ^{\ast }$, and a
stationary distribution for consumer tastes $\gamma _{zi}\left( \lambda
_{i}\right) $ for $\lambda _{i}\in \left[ \underline{\lambda },\bar{\lambda}%
\right] $. We begin by characterizing the stationary distributions of firm
productivity and consumer tastes, before turning to consider the elements of
the quadruple \{$\varphi ^{\ast }$, $\lambda _{i}^{\ast }\left( \varphi
^{\ast }\right) $, $P_{i}$, $R_{i}$\}.

\subsubsection{Stationary Distributions for Consumer Tastes and Productivity 
\label{geexistence}}

The stationary distributions for firm productivity and consumer tastes are
determined by the requirements of an equality of inflows to and outflows
from each value of productivity and consumer tastes in equations (\ref%
{invariantprod}) and (\ref{invarianttaste}). These equations imply that the
stationary distributions of firm productivity and consumer tastes depend on
the distributions from which these variables are drawn upon entry and the
conditional distributions from which they are drawn following a stochastic
shock. Since the entry and conditional distributions are continuous and have
bounded support, there exist unique stationary distributions for firm
productivity and consumer tastes, as long as the probabilities of a
stochastic shock to consumer tastes and productivity do not exceed the
thresholds specified in the proofs below.

\begin{proposition}
\label{PropInvProd}For $\theta <\bar{\theta}$, there exists a unique
stationary distribution of firm productivity conditional upon entry, $\gamma
_{g}\left( \varphi \right) $, for each value of the aggregate variables $%
\left\{ M_{e}\text{, }M\text{, }\varphi ^{\ast }\right\} $.
\end{proposition}

\begin{proof}
See the appendix below.
\end{proof}

\begin{proposition}
\label{PropInvTaste}For $\varepsilon _{i}<\bar{\varepsilon}$, there exists a
unique stationary distribution of consumer tastes, $\gamma _{zi}\left(
\lambda _{i}\right) $, for each product $i$ for each value of the aggregate
variables $\left\{ M_{e}\text{, }M\text{, }\varphi ^{\ast }\right\} $.
\end{proposition}

\begin{proof}
See the appendix below.\bigskip
\end{proof}

The stationary distributions of firm productivity and consumer tastes
characterized in the proofs of the propositions are weighted averages of the
entry and conditional distributions, with the weights depending on the
equilibrium values of the mass of entrants $M_{e}$, the mass of firms $M$,
the zero-value cutoff productivity below which firms exit $\varphi ^{\ast }$%
, and the probabilities $\theta $ and $\varepsilon _{i}$ of a stochastic
shock to firm productivity and consumer tastes respectively.

\subsubsection{Zero-profit Consumer Taste Cutoffs\label{gemp}}

Having characterized the stationary distribution of firm productivity and
consumer tastes, we now turn to consider the equilibrium value of the
zero-profit consumer taste cutoff for each product, $\lambda _{i}^{\ast
}\left( \varphi \right) $, which determines the equilibrium range of
products produced by firms. As discussed above, the zero-profit consumer
taste cutoff $\lambda _{i}^{\ast }\left( \varphi \right) $ for each product
varies across firms depending on their productivity $\varphi $. Using the
expression for the relative revenues of product varieties in equation (\ref%
{relvarrev}), the zero-profit consumer taste cutoff for each firm
productivity can be expressed relative to that of a firm with the zero-value
cutoff productivity:%
\begin{equation}
\lambda _{i}^{\ast }\left( \varphi \right) =\left( \frac{\varphi ^{\ast }}{%
\varphi }\right) \lambda _{i}^{\ast }\left( \varphi ^{\ast }\right) .
\label{rel_lambdas}
\end{equation}

The above relationship has an intuitive interpretation. The zero-profit
consumer taste cutoff $\lambda _{i}^{\ast }\left( \varphi \right) $ is
decreasing in firm productivity $\varphi $ because the revenue derived from
a product variety is increasing in firm productivity. Therefore, as a firm's
productivity increases, it generates sufficient revenue to cover product
fixed production costs at a lower value of consumer tastes. In contrast, the
zero-profit consumer taste cutoff $\lambda _{i}^{\ast }\left( \varphi
\right) $ is increasing in the zero-value cutoff productivity $\varphi
^{\ast }$. A higher value of $\varphi ^{\ast }$ increases the average
productivity of rival firms, which intensifies product market competition
and reduces product variety revenue, so that a higher value of consumer
tastes is required to generate sufficient revenue to cover product fixed
production costs. Similarly, an increase in the zero-profit consumer taste
cutoff for the lowest productivity firm $\lambda _{i}^{\ast }\left( \varphi
^{\ast }\right) $ increases $\lambda _{i}^{\ast }\left( \varphi \right) $
for all firm productivities, because it raises the average consumer tastes
of rival firms, which intensifies product market competition and reduces
product variety revenue.

The zero-profit consumer taste cutoff $\lambda _{i}^{\ast }\left( \varphi
^{\ast }\right) $ for a firm with the zero-value cutoff productivity $%
\varphi ^{\ast }$\ on the right-hand side of (\ref{rel_lambdas}) can be
determined from the zero-profit consumer taste cutoff condition (\ref{zppp})
for productivity $\varphi ^{\ast }$:%
\begin{equation}
R_{i}\left( \rho P_{i}\varphi ^{\ast }\lambda _{i}^{\ast }\left( \varphi
^{\ast }\right) \right) ^{\sigma -1}=\sigma f_{pi}.  \label{eqprodscope}
\end{equation}

Together equations (\ref{rel_lambdas}) and (\ref{eqprodscope}) determine the
zero-profit consumer taste cutoff $\lambda _{i}^{\ast }\left( \varphi
\right) $\ for firms of all productivities as a function of \{$\varphi
^{\ast }$, $P_{i}$, $R_{i}$\}. Since the zero-profit consumer taste cutoffs $%
\lambda _{i}^{\ast }\left( \varphi \right) $ determine the probability that
each product is produced $\left[ 1-\Gamma _{zi}\left( \lambda _{i}^{\ast
}\left( \varphi \right) \right) \right] $, these equations also determine
the equilibrium range of products produced by firms of all productivities.

\subsubsection{Zero-value Productivity Cutoff\label{gefe}}

To determine the zero-value cutoff productivity below which firms exit, $%
\varphi ^{\ast }$, we use the free entry condition (\ref{freeentry}) that
requires the expected value of entry to equal the sunk entry cost.
Substituting for the value of a firm from (\ref{bellvalfirmsol}), the free
entry condition can be re-written as follows:

\begin{equation}
V=\left( \frac{1}{\delta +\theta }\right) \int_{\varphi ^{\ast }}^{\bar{%
\varphi}}\left[ \pi \left( \varphi \right) +\theta \int_{\varphi ^{\ast }}^{%
\bar{\varphi}}\eta \left( \varphi ^{\prime }\right) \pi \left( \varphi
^{\prime }\right) g_{c}\left( \varphi ^{\prime }|\varphi \right) d\varphi
^{\prime }\right] g_{e}\left( \varphi \right) d\varphi =f_{e},
\label{eqfreeentry}
\end{equation}%
where the expected value of entry depends on the entry and conditional
distributions for productivity and firm profits as a function of
productivity.

\subsubsection{Product Price Indices and Revenues\label{gegl}}

The two remaining elements of the quadruple \{$\varphi ^{\ast }$, $\lambda
_{i}^{\ast }\left( \varphi ^{\ast }\right) $, $P_{i}$, $R_{i}$\} are the
price indices, $P_{i}$, and aggregate revenues, $R_{i}$, for each product.
To characterize the equilibrium values of these variables for each product,
we combine consumer and producer optimization with the free entry,
steady-state stability and labor market clearing conditions.

We begin with the product price indices in equation (\ref{eqpindex}), which
depend on a weighted average of firm productivity and consumer tastes, $%
\tilde{\varphi}_{i}$, and the mass of firms that produce a product, $M_{pi}$%
. The weighted average, $\tilde{\varphi}_{i}$, can be determined from the
zero-profit consumer taste cutoff, $\lambda _{i}^{\ast }\left( \varphi
\right) $, the zero-value productivity cutoff, $\varphi ^{\ast }$, and the
stationary distributions for consumer tastes, $\gamma _{zi}\left( \lambda
_{i}\right) $, and productivity, $\gamma _{g}\left( \varphi \right) $, as
shown in the appendix. The mass of firms that produce a product, $M_{pi}$,
is a constant fraction of the mass of active firms, $M$, as shown in
equation (\ref{massproduct}). The mass of active firms, $M$, can be
determined from aggregate economy-wide revenue, $R$, and average firm
revenue conditional upon entry, $\bar{r}$, as shown in equation (\ref%
{massfirms}).

Average firm revenue conditional upon entry, $\bar{r}$, is solely a function
of the zero-value cutoff productivity, $\varphi ^{\ast }$, the zero-profit
consumer taste cutoff for a firm with this productivity, $\lambda _{i}^{\ast
}\left( \varphi ^{\ast }\right) $, and the stationary distributions for firm
productivity and consumer tastes, $\gamma _{zi}\left( \lambda _{i}\right) $
and $\gamma _{g}\left( \varphi \right) $, as can be seen from equations (\ref%
{eqvarrev}), (\ref{expfirmrevprof}) and (\ref{rel_lambdas}). Aggregate
economy-wide revenue, $R$, can be determined by combining the free entry,
steady-state stability and labor market clearing conditions. To do so, we
solve for payments to labor used in entry and payments to labor used in
production, and show that these sum to aggregate economy-wide revenue. Using
the steady-state stability condition (\ref{sss}) to substitute for the
probability of successful entry $\left[ 1-G_{e}\left( \varphi ^{\ast
}\right) \right] $ in the free entry condition (\ref{freeentry}), we obtain
the following relationship between payments to labor used in entry and the
expected value of the firm conditional on successful entry:%
\begin{equation}
L_{e}=M_{e}f_{e}=\left( \delta +\chi \right) M\bar{v}  \label{labentry}
\end{equation}%
where $\chi \equiv \theta \int_{\varphi _{\ast }}^{\bar{\varphi}}G_{c}\left(
\varphi ^{\ast }|\varphi \right) \gamma _{g}\left( \varphi \right) d\varphi $
is the aggregate probability of firm exit due to productivity shocks.

Equation (\ref{labentry}) has an intuitive interpretation. Payments to labor
used in entry on the left-hand side equal the flow rate of return received
by the representative consumer on the average value of the firm conditional
on entry on the right-hand side. Only when this condition is satisfied will
the representative consumer break even on the equity issued by firms to
finance the sunk costs of entry.

Payments to labor used in production equal aggregate economy-wide revenue
minus the flow rate of return received by the representative consumer on the
average equity value of the firm conditional on entry:%
\begin{equation}
L_{p}=R-\left( \delta +\chi \right) M\bar{v}  \label{labprod}
\end{equation}

Substituting the expressions for labor used in entry and production in
equations (\ref{labentry}) and (\ref{labprod}) into the labor market
clearing condition (\ref{labmktclear}), aggregate economy-wide revenue
equals the economy's labor endowment, $R=L$, and the labor market clears.
Therefore, as we have characterized $R,$ $\bar{r}$, $M$, $M_{pi}$ and $%
\tilde{\varphi}_{i}$, we have characterized the equilibrium price indices $%
P_{i}$ in equation (\ref{eqpindex}).

The final element of the quadruple \{$\varphi ^{\ast }$, $\lambda _{i}^{\ast
}\left( \varphi ^{\ast }\right) $, $P_{i}$, $R_{i}$\} is aggregate product
revenue $R_{i}$, which can be determined from consumer expenditure
minimization, using the equilibrium price indices $P_{i}$ and aggregate
economy-wide revenue $R$:%
\begin{equation}
R_{i}=\frac{R\left( P_{i}/a_{i}\right) ^{1-\kappa }}{\int_{0}^{1}\left(
P_{i}/a_{i}\right) ^{1-\kappa }di}  \label{prodaggrev}
\end{equation}

\subsubsection{Existence and Uniqueness}

The equilibrium quadruple \{$\varphi ^{\ast }$, $\lambda _{i}^{\ast }\left(
\varphi ^{\ast }\right) $, $P_{i}$, $R_{i}$\} and the stationary
distributions for firm productivity $\gamma _{g}\left( \varphi \right) $\
and consumer tastes $\gamma _{zi}\left( \lambda _{i}\right) $ are determined
by the following six equations: the equality of inflows and outflows for
productivity and consumer tastes ((\ref{invariantprod}) and (\ref%
{invarianttaste})), the free entry condition (\ref{eqfreeentry}), the
zero-profit consumer taste condition for the lowest productivity firm (\ref%
{eqprodscope}), the equilibrium price index (\ref{eqpindex}), and
equilibrium product revenue (\ref{prodaggrev}).

\begin{proposition}
\label{Prop1}For $\theta <\bar{\theta}$ and $\varepsilon _{i}<\bar{%
\varepsilon}$, there exists a unique equilibrium referenced by the quadruple
\{$\varphi ^{\ast }$, $\lambda _{i}^{\ast }\left( \varphi ^{\ast }\right) $, 
$P_{i}$, $R_{i}$\}, a stationary distribution for productivity $\gamma
_{g}\left( \varphi \right) $ for $\varphi \geq \varphi ^{\ast }$, and a
stationary distribution for consumer tastes $\gamma _{zi}\left( \lambda
_{i}\right) $ for $\lambda _{i}\in \left[ \underline{\lambda },\bar{\lambda}%
\right] $.
\end{proposition}

\begin{proof}
See the appendix below.\bigskip
\end{proof}

The general equilibrium of the model features steady-state product switching
by surviving firms and steady-state firm creation and destruction. Each
period a measure of new firms incur the sunk entry cost. Of these new firms,
those with a productivity draw above the zero-value cutoff enter, while
those with a productivity draw below the zero-value cutoff exit. Among
existing firms, a firm with unchanged productivity produces a constant range
of products, but idiosyncratic shocks to consumer tastes for individual
products induce the firm to drop a measure of the products previously
produced and add an equal measure of the products not previously produced.
As stochastic shocks to an existing firm's productivity occur, the range of
products produced expands with rises in productivity and contracts with
declines in productivity. An existing firm exits endogenously when
productivity falls below the zero-value cutoff or exogenously when death
occurs as a result of force majeure considerations beyond the control of the
firm.

\section{Empirical Patterns of Product Switching}

The model highlights a number of features of product switching at the firm,
product and firm-product level, which we use to guide our empirical
research. In this section, we show how these features of product switching
can be derived from the model.

\subsection{Firm-level Evidence}

The model features steady-state adding and dropping of products within firms
as a result of idiosyncratic shocks to the profitability of a firm's
products. To determine the magnitude of steady-state product adding,
consider the measure of products not currently produced by a firm with
productivity $\varphi $, which depends on the stationary distribution of
consumer tastes $\gamma _{zi}\left( \lambda _{i}\right) $ over the interval $%
\lambda _{i}\in \left[ \underline{\lambda },\lambda _{i}^{\ast }\left(
\varphi \right) \right) $. The probability that a product not currently
produced is added by a firm with productivity $\varphi $ equals the
probability of drawing a new value for consumer tastes above the zero-profit
cutoff $\varepsilon _{i}\left( 1-Z_{ci}\left( \lambda _{i}^{\ast }\left(
\varphi \right) |\lambda _{i}\right) \right) $. Integrating over products
and the interval $\left[ \underline{\lambda },\lambda _{i}^{\ast }\left(
\varphi \right) \right) $, the measure of products added by a firm with
productivity $\varphi $ is:%
\begin{equation}
\Lambda ^{a}\left( \varphi \right) =\int_{0}^{1}\left[ \int_{\underline{%
\lambda }}^{\lambda _{i}^{\ast }\left( \varphi \right) }\varepsilon _{i}%
\left[ 1-Z_{ci}\left( \lambda _{i}^{\ast }\left( \varphi \right) |\lambda
_{i}\right) \right] \gamma _{zi}\left( \lambda _{i}\right) d\lambda _{i}%
\right] di.  \label{mseadd}
\end{equation}%
Similarly, the measure of products dropped by a firm with productivity $%
\varphi $ is:%
\begin{equation}
\Lambda ^{d}\left( \varphi \right) =\int_{0}^{1}\left[ \int_{\lambda
_{i}^{\ast }\left( \varphi \right) }^{\bar{\lambda}}\varepsilon
_{i}Z_{ci}\left( \lambda _{i}^{\ast }\left( \varphi \right) |\lambda
_{i}\right) \gamma _{zi}\left( \lambda _{i}\right) d\lambda _{i}\right] di.
\label{msedrop}
\end{equation}

As SIC\ codes can be interpreted as discrete partitions of the model's
continuum of products, product adding in the data reflects firm production
in a new five-digit SIC category, while product dropping in the data
represents abandonment of production in one of the firms' existing
five-digit SIC categories. Simultaneous adding and dropping of products
occurs when both of these actions take place. The interpretation of industry
and sector adding and dropping is analogous.

\subsection{Product-level Evidence}

One of the key features of product switching at the product level is a
positive correlation between the rate of adding and dropping across
products. To determine the measure of surviving firms that add and drop a
product, we begin by considering the measure of firms with a particular
value of productivity and consumer tastes for a product: $\gamma _{zi}\left(
\lambda _{i}\right) \gamma _{g}\left( \varphi \right) M$. The probability
that each of these firms survives after a unit interval of time depends on
the probability of firm death, $\delta $, and the probability that a firm
draws a new value for productivity below the zero-value cutoff, $\theta
G_{c}\left( \varphi ^{\ast }|\varphi \right) $. The measure of firms with a
particular value of productivity and consumer tastes for a product that
survive after a unit interval of time is therefore: 
\begin{equation}
\xi _{i}\left( \varphi ,\lambda _{i}\right) \equiv \left[ 1-\delta -\theta
G_{c}\left( \varphi ^{\ast }|\varphi \right) \right] \gamma _{zi}\left(
\lambda _{i}\right) \gamma _{g}\left( \varphi \right) M.  \label{survive}
\end{equation}

Among this measure of surviving firms with productivity $\varphi $, those
with consumer tastes in the interval $\left[ \lambda _{i}^{\ast }\left(
\varphi \right) ,\bar{\lambda}\right] $ currently produce the product. The
probability that the product is dropped by each of these firms equals the
probability of drawing a new value for consumer tastes below the zero-profit
cutoff $\varepsilon _{i}Z_{ci}\left( \lambda _{i}^{\ast }\left( \varphi
\right) |\lambda _{i}\right) $. Integrating over the interval of
productivities for which firms enter and the interval of values for consumer
tastes for which the product is produced, the total measure of surviving
firms that drop the product after a unit interval of time is as follows:

\begin{equation}
\xi _{i}^{d}=\int_{\varphi ^{\ast }}^{\bar{\varphi}}\left[ \int_{\lambda
_{i}^{\ast }\left( \varphi \right) }^{\bar{\lambda}}\varepsilon
_{i}Z_{ci}\left( \lambda _{i}^{\ast }\left( \varphi \right) |\lambda
_{i}\right) \xi _{i}\left( \varphi ,\lambda _{i}\right) d\lambda _{i}\right]
d\varphi .  \label{measuredrop}
\end{equation}%
Similarly, the total measure of surviving firms that add the product after a
unit interval of time is as follows:%
\begin{equation}
\xi _{i}^{a}=\int_{\varphi ^{\ast }}^{\bar{\varphi}}\left[ \int_{\underline{%
\lambda }}^{\lambda _{i}^{\ast }\left( \varphi \right) }\varepsilon _{i}%
\left[ 1-Z_{ci}\left( \lambda _{i}^{\ast }\left( \varphi \right) |\lambda
_{i}\right) \xi _{i}\left( \varphi ,\lambda _{i}\right) \right] d\lambda _{i}%
\right] d\varphi .  \label{measureadd}
\end{equation}

Comparing equations (\ref{measuredrop}) and (\ref{measureadd}), it is clear
that \textquotedblleft turbulent\textquotedblright\ products with high
probabilities of idiosyncratic shocks $\varepsilon _{i}$ are added and
dropped more frequently, other things equal, than \textquotedblleft
stable\textquotedblright\ products with less frequent idiosyncratic shocks.
Therefore, differences across products in the degree of turbulence (the
probability of idiosyncratic shocks $\varepsilon _{i}$) induce a positive
correlation between the rate at which a product is added and the rate at
which it is dropped. As these idiosyncratic shocks lead some firms to add a
product while other firms simultaneously drop the same product, the gross
changes in output of a product as a result of product switching are
systematically larger than the net changes in output.

\subsection{Firm-Product Level Evidence}

One of the central features of the model is selection within firms across
products as well as selection across firms. The presence of selection within
firms can be seen from the zero-profit cutoff condition for a product to be
produced, which requires variable profits to cover the fixed production
costs:%
\begin{equation*}
r_{i}\left( \varphi ,\lambda _{i}^{\ast }\left( \varphi \right) \right)
=R_{i}\left( \rho P_{i}\varphi \lambda _{i}^{\ast }\left( \varphi \right)
\right) ^{\sigma -1}=\sigma f_{pi}.
\end{equation*}%
From this zero-profit condition, the decision whether to add or drop a
product depends not only on characteristics of the firm (productivity $%
\varphi $) but also on characteristics that are specific to the firm-product
pair (consumer tastes $\lambda _{i}$). Selection therefore operates within
firms and depends on considerations that are idiosyncratic to individual
firm-product pairs.

The process of selection within firms displays scale and age dependence as a
result of the assumption that firm productivity and consumer tastes are
positively serially correlated. Therefore a high existing value for
productivity or consumer tastes reduces the probability of drawing a low new
value for productivity or consumer tastes and dropping a product. As higher
values of productivity and consumer tastes increase a firm's shipments of a
product (from equation (\ref{consrev})), the model implies that the
probability a firm drops a product is decreasing in the firm's shipments of
the product (scale dependence). Additionally, as productivity and consumer
tastes are serially correlated, the longer the length of time for which a
firm has produced a product, the higher the firm's expected productivity and
consumer tastes, and hence the lower the probability of the firm dropping
the product. Therefore the model also implies that the probability a firm
drops a product is decreasing in the length of time for which the firm has
produced the product (tenure or age dependence).\footnote{%
In the model, consumer tastes follow a first-order Markov process, so that
the probability of drawing a new value for consumer tastes depends only on
the current value of consumer tastes. Furthermore, controlling for firm and
product fixed effects, log firm-product shipments are proportional to the
current value of consumer tastes. Therefore, as in much of the firm entry
and exit literature, age or tenure should become insignificant in a
specification that controls appropriately for scale. One natural explanation
for the significance of firm-product tenure in such a specification is that
consumer tastes follow a higher-order Markov process, and the model could be
extended to allow for this possibility.}

\section{A Simple Special Case}

To provide further economic intuition for the workings of the model, this
section considers a simple special case of the more general framework
developed above. This special case makes a number of simplifying assumptions
in order to achieve greater tractability and derive closed form solutions.
The most important of these simplifications is to abstract from serial
correlation in firm productivity and consumer tastes. While this abstraction
implies that the special case of the model does not capture scale and tenure
dependence in the probability that a product is dropped by a firm, it
remains consistent with the other features of product switching examined in
our empirical work.

The additional simplifying assumptions made in the special case of the model
are as follows: (a) all products $i$ receive the same weight in consumer
utility ($a_{i}=1$ for all $i$), though consumer tastes for a firm's variety 
$\omega $ of each product still vary (in general $\lambda _{i}\left( \omega
^{\prime \prime }\right) \neq \lambda _{i}\left( \omega ^{\prime }\right) $
for $\omega ^{\prime \prime }\neq \omega ^{\prime }$); (b) all products have
the same fixed production costs ($f_{pi}=f_{p}$ for all $i$); (c) the
probability of a stochastic shock to consumer tastes is the same across
products ($\varepsilon _{i}=\varepsilon $ for all $i$); (d) the distribution
from which consumer tastes are drawn following a stochastic shock is the
same as upon entry ($z_{ei}\left( \lambda _{i}\right) =z_{ci}\left( \lambda
_{i}|\lambda _{i}^{\prime }\right) =z\left( \lambda _{i}\right) $ for all $i$
and $\lambda _{i}\in \lbrack \underline{\lambda },\bar{\lambda}]$); and (e)
the distribution from which firm productivity is drawn following a
stochastic shock is the same as upon entry ($g_{e}\left( \varphi \right)
=g_{c}\left( \varphi |\varphi ^{\prime }\right) =g\left( \varphi \right) $
for $\varphi \in \lbrack \underline{\varphi },\bar{\varphi}]$).

Since the distributions from which consumer tastes and firm productivity are
drawn following stochastic shocks are the same as the distributions from
which they are drawn upon entry, the stationary distributions for consumer
tastes and productivity take a particularly simple form. The stationary
distribution for consumer tastes is the same as the distribution upon entry:%
\begin{equation}
\gamma _{zi}\left( \lambda _{i}\right) =z\left( \lambda _{i}\right) ,
\label{eg_stattaste}
\end{equation}%
while the stationary distribution for consumer tastes conditional on a
product being produced is a truncation of $z\left( \lambda _{i}\right) $ at
the zero-profit cutoff for consumer tastes $\lambda _{i}^{\ast }\left(
\varphi \right) $.

The stationary distribution for firm productivity conditional upon entry is
a truncation of the distribution $g\left( \varphi \right) $ at the
zero-value cutoff productivity below which firms exit:%
\begin{equation}
\gamma _{g}\left( \varphi \right) =\left\{ 
\begin{array}{l}
\frac{g\left( \varphi \right) }{1-G\left( \varphi ^{\ast }\right) }\qquad
\qquad \text{for}\qquad \varphi \geq \varphi ^{\ast } \\ 
0\hspace{1in}\text{otherwise}%
\end{array}%
\right. .  \label{eg_statprod}
\end{equation}

With symmetric products and independently and identically distributed
consumer tastes, the total profits of a firm with productivity $\varphi $
across the unit continuum of products equal its expected profits from each
product:%
\begin{equation}
\pi \left( \varphi \right) =\int_{\lambda ^{\ast }\left( \varphi \right) }^{%
\bar{\lambda}}\left( \left( \frac{\lambda }{\lambda ^{\ast }\left( \varphi
\right) }\right) ^{\sigma -1}-1\right) f_{p}z\left( \lambda \right) d\lambda
-f_{h}.  \label{eg_firmprof}
\end{equation}%
where we have used $r\left( \varphi ,\lambda ^{\ast }\left( \varphi \right)
\right) =\sigma f_{p}$ and $r\left( \varphi ,\lambda \right) =\left( \lambda
/\lambda ^{\ast }\left( \varphi \right) \right) ^{\sigma -1}r\left( \varphi
,\lambda ^{\ast }\left( \varphi \right) \right) $.

A firm with productivity $\varphi $ draws a consumer taste above the
zero-profit cutoff $\lambda ^{\ast }\left( \varphi \right) $ with
probability $\left[ 1-Z\left( \lambda ^{\ast }\left( \varphi \right) \right) %
\right] $, and so the range of products produced by a firm with productivity 
$\varphi $ equals $\left[ 1-Z\left( \lambda ^{\ast }\left( \varphi \right)
\right) \right] $, which is monotonically increasing in $\varphi $. Firms
with higher productivity generate sufficient revenue to cover the fixed
costs of producing a product at a lower value for consumer tastes, and
therefore manufacture a larger range of products than firms with lower
productivity.

The Bellman equation for the value of a firm with productivity $\varphi $
takes the same form as above: 
\begin{equation}
v\left( \varphi \right) =\frac{\pi \left( \varphi \right) }{\delta +\theta }%
+\left( \frac{\theta }{\delta +\theta }\right) \int_{\varphi ^{\ast }}^{\bar{%
\varphi}}v\left( \varphi ^{\prime }\right) g\left( \varphi ^{\prime }\right)
d\varphi ^{\prime },  \label{eg_bellman}
\end{equation}

Since the distribution from which a new value of productivity is drawn
following a stochastic shock is the same as upon entry and is independent of
a firm's existing value of productivity, the solution to the Bellman
equation takes a particularly simple form in this special case of the model.
Substituting for $v\left( \varphi ^{\prime }\right) $ on the right-hand side
of (\ref{eg_bellman}) using the trial solution $v\left( \varphi \right)
=\alpha \pi \left( \varphi \right) +\beta \int_{\varphi ^{\ast }}^{\bar{%
\varphi}}\pi \left( \varphi ^{\prime }\right) g\left( \varphi ^{\prime
}\right) d\varphi ^{\prime }$, and solving for $\alpha $ and $\beta $,
yields the equilibrium value of a firm:

\begin{equation}
v\left( \varphi \right) =\frac{\pi \left( \varphi \right) }{\delta +\theta }%
+\left( \frac{\theta }{\delta +\theta }\right) \left( \frac{1}{\delta
+\theta G\left( \varphi ^{\ast }\right) }\right) \left[ \int_{\varphi ^{\ast
}}^{\bar{\varphi}}\pi \left( \varphi ^{\prime }\right) g\left( \varphi
^{\prime }\right) d\varphi ^{\prime }\right] ,  \label{eg_solbellman}
\end{equation}

Therefore the equilibrium value of a firm with productivity $\varphi $ is a
weighted average of the current flow of firm profits and the expected flow
of firm profits following a stochastic productivity shock, where the weights
depend on the probability of firm death, the probability of a productivity
shock and the probability that a firm remains active following a
productivity shock. Substituting the expression for the equilibrium value of
a firm into the free entry condition, and rearranging, the free entry
condition can be re-written as follows:%
\begin{equation}
V=\left[ 1-G\left( \varphi ^{\ast }\right) \right] \int_{\varphi ^{\ast }}^{%
\bar{\varphi}}\left[ \frac{\pi \left( \varphi \right) +\left( \frac{\theta }{%
\delta +\theta }\right) \int_{\varphi ^{\ast }}^{\bar{\varphi}}\left[ \pi
\left( \varphi ^{\prime }\right) -\pi \left( \varphi \right) \right] g\left(
\varphi ^{\prime }\right) d\varphi ^{\prime }}{\delta +\theta G\left(
\varphi ^{\ast }\right) }\right] \left( \frac{g\left( \varphi \right) }{%
1-G\left( \varphi ^{\ast }\right) }\right) d\varphi =f_{e},
\label{eg_freeentry}
\end{equation}%
which can be simplified to yield the following expression:%
\begin{equation}
V=\int_{\varphi ^{\ast }}^{\bar{\varphi}}\left( \frac{\pi \left( \varphi
\right) }{\delta +\theta G\left( \varphi ^{\ast }\right) }\right) g\left(
\varphi \right) d\varphi =f_{e},  \label{eg_simpfreeentry}
\end{equation}%
where we have used $\int_{\varphi ^{\ast }}^{\bar{\varphi}}\left[
\int_{\varphi ^{\ast }}^{\bar{\varphi}}\pi \left( \varphi ^{\prime }\right)
g\left( \varphi ^{\prime }\right) d\varphi ^{\prime }\right] g\left( \varphi
\right) d\varphi $ $=$ $\int_{\varphi ^{\ast }}^{\bar{\varphi}}\left[
\int_{\varphi ^{\ast }}^{\bar{\varphi}}\pi \left( \varphi \right) g\left(
\varphi ^{\prime }\right) d\varphi ^{\prime }\right] g\left( \varphi \right)
d\varphi $ \newline
$=$ $\left[ 1-G\left( \varphi ^{\ast }\right) \right] \left[ \int_{\varphi
^{\ast }}^{\bar{\varphi}}\pi \left( \varphi ^{\prime }\right) g\left(
\varphi ^{\prime }\right) d\varphi ^{\prime }\right] $ $=$ $\left[ 1-G\left(
\varphi ^{\ast }\right) \right] \left[ \int_{\varphi ^{\ast }}^{\bar{\varphi}%
}\pi \left( \varphi \right) g\left( \varphi \right) d\varphi \right] $,
which implies that the second term in the numerator of the expression inside
the square parentheses in equation (\ref{eg_freeentry}) is equal to zero.

The expression for the expected value of entry in the free entry condition (%
\ref{eg_simpfreeentry}) has an intuitive interpretation. In the special case
of the model considered here, the distribution from which a new value of
productivity is drawn following a stochastic shock is the same as upon entry
and independent of a firm's existing value of productivity. Therefore the
expected value of entry is equal to expected firm profits discounted by the
sum of the probability of firm death and the probability that a firm remains
active following a productivity shock.

Using the expression for firm profits from equation (\ref{eg_firmprof}), and
noting from the analysis of the general model above that $\lambda ^{\ast
}\left( \varphi \right) =\left( \varphi ^{\ast }/\varphi \right) \lambda
^{\ast }\left( \varphi ^{\ast }\right) $, the free entry condition (\ref%
{eg_simpfreeentry}) can be written solely in terms of parameters and two
unknowns: the zero-value cutoff productivity $\varphi ^{\ast }$ and the
zero-profit consumer taste cutoff for a firm with this productivity $\lambda
^{\ast }\left( \varphi ^{\ast }\right) $,

\begin{equation}
V=\left( \frac{1}{\delta +\theta G_{c}\left( \varphi ^{\ast }\right) }%
\right) \int_{\varphi ^{\ast }}^{\bar{\varphi}}\left[ \int_{\left( \varphi
^{\ast }/\varphi \right) \lambda ^{\ast }\left( \varphi ^{\ast }\right) }^{%
\bar{\lambda}}\left( \left( \frac{\varphi }{\varphi ^{\ast }}\frac{\lambda }{%
\lambda ^{\ast }\left( \varphi ^{\ast }\right) }\right) ^{\sigma
-1}-1\right) f_{p}z\left( \lambda \right) d\lambda -f_{h}\right] g\left(
\varphi \right) d\varphi =f_{e}.\quad  \label{eg_eqfe}
\end{equation}

The free entry condition (\ref{eg_eqfe}) provides one of two equations that
together pin down the zero-value cutoff productivity $\varphi ^{\ast }$ and
the zero-profit consumer taste cutoff for a firm with this productivity $%
\lambda ^{\ast }\left( \varphi ^{\ast }\right) $. The second of the two
equations is obtained from the requirement that the value of a firm with the
zero-value cutoff productivity $\varphi ^{\ast }$ is equal to zero: $v\left(
\varphi ^{\ast }\right) =0$. Using the expression for firm profits from (\ref%
{eg_firmprof}) in the expression for the equilibrium value of a firm in (\ref%
{eg_solbellman}), the requirement that $v\left( \varphi ^{\ast }\right) =0$
can be written as follows:%
\begin{gather}
-\left[ \int_{\lambda ^{\ast }\left( \varphi ^{\ast }\right) }^{\bar{\lambda}%
}\left( \left( \frac{\lambda }{\lambda ^{\ast }\left( \varphi ^{\ast
}\right) }\right) ^{\sigma -1}-1\right) f_{p}z\left( \lambda \right)
d\lambda -f_{h}\right] \hspace{3.6in}  \label{eg_zvcut} \\
=\left( \frac{\theta }{\delta +\theta G\left( \varphi ^{\ast }\right) }%
\right) \int_{\varphi ^{\ast }}^{\bar{\varphi}}\left[ \int_{\left( \varphi
^{\ast }/\varphi \right) \lambda ^{\ast }\left( \varphi ^{\ast }\right) }^{%
\bar{\lambda}}\left( \left( \frac{\varphi }{\varphi ^{\ast }}\frac{\lambda }{%
\lambda ^{\ast }\left( \varphi ^{\ast }\right) }\right) ^{\sigma
-1}-1\right) f_{p}z\left( \lambda \right) d\lambda -f_{h}\right] g\left(
\varphi \right) d\varphi  \notag
\end{gather}

The free entry condition (\ref{eg_eqfe}) is monotonically decreasing in $%
\varphi ^{\ast }$ for a given value of $\lambda ^{\ast }\left( \varphi
^{\ast }\right) $. Similarly, for a given value of $\varphi ^{\ast }$, the
left-hand side of equation (\ref{eg_zvcut}) is monotonically increasing in $%
\lambda ^{\ast }\left( \varphi ^{\ast }\right) $, while the right-hand side
of equation (\ref{eg_zvcut}) is monotonically decreasing in $\lambda ^{\ast
}\left( \varphi ^{\ast }\right) $. Therefore equations (\ref{eg_eqfe}) and (%
\ref{eg_zvcut}) together determine unique equilibrium values of \{$\varphi
^{\ast }$, $\lambda ^{\ast }\left( \varphi ^{\ast }\right) $\}.

In the special case of the model, general equilibrium is again referenced by
the quadruple \{$\varphi ^{\ast }$, $\lambda _{i}^{\ast }\left( \varphi
^{\ast }\right) $, $P_{i}$, $R_{i}$\}, together with a stationary
distribution for firm productivity $\gamma _{g}\left( \varphi \right) $ for $%
\varphi \geq \varphi ^{\ast }$, and a stationary distribution for consumer
tastes $\gamma _{zi}\left( \lambda _{i}\right) $ for $\lambda _{i}\in \left[ 
\underline{\lambda },\bar{\lambda}\right] $. As products are symmetric, we
have $\lambda _{i}^{\ast }\left( \varphi ^{\ast }\right) =\lambda ^{\ast
}\left( \varphi ^{\ast }\right) $, $P_{i}=P$, $R_{i}=R$ and $\gamma
_{zi}\left( \lambda _{i}\right) =\gamma _{z}\left( \lambda \right) $ for all 
$i$. The stationary distributions of consumer tastes and firm productivity \{%
$\gamma _{z}\left( \lambda \right) $, $\gamma _{g}\left( \varphi \right) $\}
were determined above in equations (\ref{eg_stattaste}) and (\ref%
{eg_statprod}). The equilibrium values of \{$\varphi ^{\ast }$, $\lambda
^{\ast }\left( \varphi ^{\ast }\right) $\} can be determined from equations (%
\ref{eg_eqfe}) and (\ref{eg_zvcut}) as discussed above. Finally, the
equilibrium values of the price indices, $P$, and product revenue, $R$, can
be determined in the same way as for the general model in section \ref{gegl}
above. Therefore this completes the characterization of general equilibrium
in the special case of the model.

Despite its simplifying assumptions, the special case of the model still
captures a number of the key features of product switching examined in our
empirical work. Steady-state product switching is driven by idiosyncratic
shocks to consumer tastes, while steady-state firm creation and destruction
is the result of stochastic shocks to firm productivity. The special case of
the model features selection both within and across firms. There is
selection within firms because firms have systematically higher values of
consumer tastes in the products that they choose to produce than in the
products they choose not to produce. There is selection across firms because
firms that draw values of productivity below the zero-value cutoff $\varphi
^{\ast }$ exit.

In the special case of the model, however, a firm's new values for consumer
tastes and productivity following a stochastic shock are by assumption
uncorrelated with its existing values: $g_{e}\left( \varphi \right)
=g_{c}\left( \varphi |\varphi ^{\prime }\right) =g\left( \varphi \right) $\
and $z_{ei}\left( \lambda _{i}\right) =z_{ci}\left( \lambda _{i}|\lambda
_{i}^{\prime }\right) =z\left( \lambda _{i}\right) $. Therefore the special
case of the model does not capture the empirical finding that surviving
firms that drop a product have systematically smaller shipments of the
product than surviving firms that retain the product. To capture this
empirical finding, one requires serial correlation in consumer tastes, as in
the general model, so that $z_{ei}\left( \lambda _{i}\right) \neq
z_{ci}\left( \lambda _{i}|\lambda _{i}^{\prime }\right) $. The special case
of the model also does not capture the empirical finding that exiting firms
have smaller total shipments than surviving firms, because \textit{%
conditional upon entry} the probability that a firm subsequently exits is
independent of total shipments. To capture this second empirical finding,
one requires serial correlation in productivity, as in the general model, so
that $g_{e}\left( \varphi \right) \neq g_{c}\left( \varphi |\varphi ^{\prime
}\right) $.

\section{Measured Quantity and Revenue-based Productivity}

This section examines the relationship between the productivity draw in our
model ($\varphi $)\ and empirical measures of productivity used in the
literature. Following Foster, Haltiwanger and Syverson (2008), we
distinguish between quantity and revenue-based measures of productivity and
show that both are monotonically related to $\varphi $. We start with labor
productivity, move on to index numbers of total factor productivity and
conclude with productivity measures based on production function estimation.
In each case, we first examine measured productivity for a product, before
aggregating across products to derive measured productivity for the firm as
a whole.

\subsection{Measured Quantity-based Labor Productivity}

For simplicity, our model assumes that labor is the sole factor of
production and therefore yields direct implications for measured labor
productivity. A firm's measured \textit{quantity-based} labor productivity
for \textit{product} $i$ ($\theta _{i}^{Q}$) equals its quantity of output
of the product per unit of labor employed:%
\begin{equation}
\theta _{i}^{Q}\equiv \frac{q_{i}\left( \varphi ,\lambda _{i}\right) }{%
l_{i}\left( \varphi ,\lambda _{i}\right) }=\varphi \left( 1-\frac{f_{pi}}{%
l_{i}\left( \varphi ,\lambda _{i}\right) }\right) ,  \label{physprod_p}
\end{equation}%
where we have used $q_{i}\left( \varphi ,\lambda _{i}\right) =\varphi \left(
l_{i}\left( \varphi ,\lambda _{i}\right) -f_{pi}\right) $.

Equilibrium employment for the product ($l_{i}\left( \varphi ,\lambda
_{i}\right) $) is monotonically increasing in the productivity draw ($%
\varphi $) given consumer tastes ($\lambda _{i}$), as can be shown by
combining the CES revenue function from equation (\ref{consrev}) above, the
equilibrium pricing rule from equation (\ref{varprice}) above, and the
production technology for the product:%
\begin{equation}
l_{i}\left( \varphi ,\lambda _{i}\right) =\rho ^{\sigma }\left( \varphi
\lambda _{i}\right) ^{\sigma -1}P_{i}^{\sigma -1}R_{i}+f_{pi},
\label{employ_p}
\end{equation}

Together equations (\ref{physprod_p}) and (\ref{employ_p}) can be used to
characterize the relationship between measured quantity-based labor
productivity for a product ($\theta _{i}^{Q}$) and the firm productivity
draw $\varphi $. As product market conditions ($P_{i}$, $R_{i}$) are the
same for all firms producing the product, measured quantity-based
productivity for a product ($\theta _{i}^{Q}$) only varies across firms
because of variation in the firm productivity draw ($\varphi $) and consumer
tastes ($\lambda _{i}$). From equations (\ref{physprod_p}) and (\ref%
{employ_p}), a firm's measured quantity-based labor productivity for a
product is positively related to its productivity draw $\varphi $ for two
reasons. First, as $\varphi $ rises, a given variable labor input generates
more units of output. Second, as $\varphi $ rises, variable labor input and
output increase, and therefore the fixed labor input is spread over more
units of output.

Following standard empirical methods for productivity aggregation, we define
measured productivity for the firm as a whole as the revenue-share weighted
average of measured productivity for each product. Therefore measured
quantity-based labor productivity for the firm as a whole is:%
\begin{equation}
\Theta ^{Q}\equiv \int_{0}^{1}\theta _{i}^{Q}\frac{r_{i}\left( \varphi
,\lambda \right) }{r\left( \varphi \right) }di,  \label{physprod}
\end{equation}%
where, as $r_{i}\left( \varphi ,\lambda _{i}\right) =R_{i}\left( \rho
P_{i}\varphi \lambda _{i}\right) ^{\sigma -1}$, a change in $\varphi $ has
the same proportionate effect on the revenue of all products, and hence
leaves revenue shares unchanged.

While at the \textit{product} level (from (\ref{physprod_p}) and (\ref%
{employ_p})) measured quantity-based labor productivity depends on both the
firm productivity draw ($\varphi $) and consumer tastes ($\lambda _{i}$),
the independence of the entry and conditional distributions of $\varphi $
and $\lambda _{i}$ implies that the stationary distribution of consumer
tastes ($\gamma _{zi}\left( \lambda _{i}\right) $) is the same for each firm
productivity draw. Therefore, at the \textit{firm} level in (\ref{physprod}%
), measured quantity-based labor productivity depends solely on $\varphi $
and is monotonically increasing in $\varphi $.

\subsection{Measured Revenue-based Labor Productivity}

A firm's measured \textit{revenue-based} labor productivity for \textit{%
product} $i$ ($\theta _{i}^{R}$) equals its revenue from the product per
unit of labor employed:%
\begin{equation}
\theta _{i}^{R}\equiv \frac{p_{i}\left( \varphi ,\lambda _{i}\right)
q_{i}\left( \varphi ,\lambda _{i}\right) }{l_{i}\left( \varphi ,\lambda
_{i}\right) }=\frac{1}{\rho }\left( 1-\frac{f_{pi}}{l_{i}\left( \varphi
,\lambda _{i}\right) }\right) ,  \label{revprod_p}
\end{equation}%
where we have used $q_{i}\left( \varphi ,\lambda _{i}\right) =\varphi \left(
l_{i}\left( \varphi ,\lambda _{i}\right) -f_{pi}\right) $, the equilibrium
pricing rule in equation (\ref{varprice}) above, and our choice of numeraire
($w=1$).

From equations (\ref{employ_p}) and (\ref{revprod_p}), a firm's measured
revenue-based labor productivity for a product is also positively related to 
$\varphi $, but now for only one of the two reasons discussed above. As $%
\varphi $ rises, a given variable labor input generates more units of
output, but the equilibrium pricing rule (\ref{varprice}) implies that
prices are inversely proportional to productivity. Therefore, as $\varphi $
rises, the increase in the number of units of output generated by a given
variable labor input is offset by proportionately lower prices, which leaves
the revenue generated by a given variable labor input unchanged.
Nevertheless, measured revenue-based labor productivity is still increasing
in $\varphi $, because of the presence of fixed costs of production. As $%
\varphi $ rises, variable labor input and revenue increase, and therefore
the fixed labor input is spread over more units of revenue.

Measured productivity for the firm as a whole is again defined as the
revenue-share weighted average of measured productivity for each product:%
\begin{equation}
\Theta ^{R}\equiv \int_{0}^{1}\theta _{i}^{R}\frac{r_{i}\left( \varphi
,\lambda \right) }{r\left( \varphi \right) }di,  \label{revprod}
\end{equation}%
where, as $r_{i}\left( \varphi ,\lambda _{i}\right) =R_{i}\left( \rho
P_{i}\varphi \lambda _{i}\right) ^{\sigma -1}$, a change in $\varphi $ has
the same proportionate effect on the revenue of all products, and hence
leaves revenue shares unchanged.

At the \textit{product} level measured revenue-based labor productivity
depends on both the firm productivity draw ($\varphi $) and consumer tastes (%
$\lambda _{i}$), since from (\ref{employ_p}) and (\ref{revprod_p}) higher
values of $\varphi $ and $\lambda _{i}$ increase variable labor input and
hence the revenue over which the fixed labor input is spread. However, the
independence of the entry and conditional distributions of $\varphi $ and $%
\lambda _{i}$ implies that the stationary distribution of consumer tastes ($%
\gamma _{zi}\left( \lambda _{i}\right) $) is the same for each firm
productivity draw. Therefore, at the \textit{firm} level in (\ref{revprod}),
measured revenue-based labor productivity depends solely on $\varphi $ and
is monotonically increasing in $\varphi $.

\subsection{Relationship Between Quantity and Revenue-based Labor
Productivity}

The model features dispersion in both measured revenue and quantity-based
labor productivity as a result of variation in productivity draws across
firms. The two productivity measures are related at the product level as
follows:%
\begin{equation*}
\theta _{i}^{Q}=\varphi \rho \theta _{i}^{R}.
\end{equation*}%
Since measured revenue-based labor productivity ($\theta _{i}^{R}$) is
positively related to the firm productivity draw ($\varphi $), cov$\left(
\varphi ,\theta _{i}^{R}\right) >0$, which implies the following two
results. First, measured quantity and revenue-based labor productivity are
positively correlated: cov$\left( \theta _{i}^{Q},\theta _{i}^{R}\right) >0$%
. Second, the variance of log measured quantity-based labor productivity is
strictly greater than the variance of log measured revenue-based labor
productivity:%
\begin{equation*}
\text{var}\left( \log \theta _{i}^{Q}\right) =\text{var}\left( \log \varphi
\right) +\text{var}\left( \log \theta _{i}^{R}\right) +2\text{cov}\left(
\varphi ,\theta _{i}^{R}\right) >\text{var}\left( \log \theta
_{i}^{R}\right) >0.
\end{equation*}%
The intuition for this second result is that measured revenue-based labor
productivity depends on both prices and measured quantity-based labor
productivity, and prices are negatively correlated with measured
quantity-based labor productivity, which implies that measured revenue-based
labor productivity has a lower variance than measured quantity-based labor
productivity.

While these results relate to a firm's measured quantity and revenue-based
labor productivity for a product, they are consistent with the empirical
findings for plants in Foster, Haltiwanger and Syverson (2008). Considering
eleven products in the manufacturing sector for which separate data on
plants' revenue and physical quantity of output are available, they find
that measured quantity and revenue-based labor productivity are indeed
positively correlated, and the variance of log measured quantity-based labor
productivity is indeed greater than the variance of log measured
revenue-based labor productivity.

The relationship between measured quantity and revenue-based labor
productivity in our model is the same as that in the benchmark model of firm
heterogeneity (Melitz 2003). As both models have constant elasticity of
substitution preferences, prices are inversely proportional to the
productivity draw $\varphi $, and dispersion in measured revenue-based
productivity occurs because of fixed costs of production. In contrast, in
other demand systems with variable elasticities of substitution, such as the
quasi-linear preferences considered by Melitz and Ottaviano (2008), prices
fall less than proportionately with the productivity draw $\varphi $ because
of endogenous changes in mark-ups. These endogenous changes in mark-ups
therefore provide an additional source of dispersion in measured
revenue-based productivity.

\subsection{Measured Total Factor Productivity}

While our model assumes for simplicity that labor is the sole factor of
production, it can be extended to incorporate additional factors of
production. For example, suppose that both the fixed and variable costs use
two factors of production (labor and capital) according to a Cobb-Douglas
technology.\footnote{%
The production technology here takes the same form as that considered in the
single-product firm model of Bernard, Redding and Schott (2007). While to
simplify the exposition we consider two factors of production, the analysis
generalizes in a straightforward way to more factors of production.}
Therefore the production technology for product $i$ becomes:%
\begin{equation*}
f_{pi}+\frac{q_{i}\left( \varphi ,\lambda _{i}\right) }{\varphi }%
=x_{i}\left( \varphi ,\lambda _{i}\right) ,
\end{equation*}%
where $x_{i}\left( \varphi ,\lambda _{i}\right) $ is a Cobb-Douglas
composite factor input composed of labor and capital:%
\begin{equation*}
x_{i}\left( \varphi ,\lambda _{i}\right) =l_{i}\left( \varphi ,\lambda
_{i}\right) ^{\vartheta }k_{i}\left( \varphi ,\lambda _{i}\right)
^{1-\vartheta },\qquad 0<\vartheta <1.
\end{equation*}%
Given this production technology for a product, standard index number
measures of total factor productivity (TFP) can be constructed. Quantity and
revenue-based measures of TFP can be defined, which are analogous to the
measures of labor productivity in equations (\ref{physprod_p}) and (\ref%
{revprod_p}) above, but are based on the composite factor input ($%
x_{i}\left( \varphi ,\lambda _{i}\right) $) rather than labor input ($%
l_{i}\left( \varphi ,\lambda _{i}\right) $). Since the quantity and
revenue-based measures of TFP take the same form as those for labor
productivity above, the same analysis continues to apply.

\subsection{Productivity Measures from Production Function Estimation}

Quantity and revenue-based measures of productivity can also be constructed
using production function estimation. To simplify the exposition, we return
to the case where labor is the sole factor of production. Assuming that the
physical quantity of output and factor inputs can be observed by firm and
product, and supposing that the functional form of the production technology
is known and appropriate instruments for labor input are available, a firm's
quantity-based productivity for a product can be estimated from the
following regression:%
\begin{equation}
\log q_{i}\left( \varphi ,\lambda _{i}\right) =\log \left( l_{i}\left(
\varphi ,\lambda \right) -f_{pi}\right) +\log \varphi ,  \label{qntyprodfn}
\end{equation}%
which yields the following measure of quantity-based productivity for the
product:%
\begin{equation}
\log \hat{\theta}_{i}^{Q}=\log \varphi .  \label{aphysprod_p}
\end{equation}%
Given data on revenue by firm and product, an analogous measure of
revenue-based productivity can be estimated following the standard approach
for production function estimation using revenue data in differentiated
product markets (see for example Klette and Griliches 1996, Levinsohn and
Melitz 2006, and De Loecker 2008). Deflating a firm's revenue for product $i$
by the aggregate price index for that product, $P_{i}$, yields:%
\begin{equation*}
\tilde{r}_{i}\left( \varphi ,\lambda _{i}\right) \equiv \frac{r_{i}\left(
\varphi ,\lambda _{i}\right) }{P_{i}}=\frac{p_{i}\left( \varphi ,\lambda
_{i}\right) q_{i}\left( \varphi ,\lambda _{i}\right) }{P_{i}},
\end{equation*}%
Using the inverse CES demand curve to substitute for $p_{i}\left( \varphi
,\lambda _{i}\right) $ on the right-hand side in terms of $q_{i}\left(
\varphi ,\lambda _{i}\right) $, we obtain:%
\begin{equation*}
\tilde{r}_{i}\left( \varphi ,\lambda _{i}\right) =\lambda _{i}^{\frac{\sigma
-1}{\sigma }}q_{i}\left( \varphi ,\lambda _{i}\right) ^{\frac{\sigma -1}{%
\sigma }}\left( \frac{R_{i}}{P_{i}}\right) ^{\frac{1}{\sigma }},
\end{equation*}%
Using the production technology ($q_{i}\left( \varphi ,\lambda _{i}\right)
=\varphi \left( l_{i}\left( \varphi ,\lambda _{i}\right) -f_{pi}\right) $)
and taking logarithms, we obtain the following \textquotedblleft revenue
production function\textquotedblright\ for a product:%
\begin{equation}
\log \tilde{r}_{i}\left( \varphi ,\lambda _{i}\right) =\frac{\sigma -1}{%
\sigma }\log \left( l_{i}\left( \varphi ,\lambda \right) -f_{pi}\right) +%
\frac{1}{\sigma }\log \left( \frac{R_{i}}{P_{i}}\right) +\frac{\sigma -1}{%
\sigma }\log \left( \varphi \lambda _{i}\right) ,  \label{revprodfn}
\end{equation}%
where with constant elasticity of substitution demand, aggregate product
market conditions ($R_{i}/P_{i}$) shift the revenue production function for
each firm equi-proportionately.

Assuming that the functional form of the revenue production function is
known, and given appropriate instruments for labor input and controls for
aggregate product market conditions, a firm's revenue-based productivity for
a product can be estimated from the regression (\ref{revprodfn}):%
\begin{equation}
\log \hat{\theta}_{i}^{R}=\left( 1-\frac{1}{\sigma }\right) \log \left(
\varphi \lambda _{i}\right) ,\qquad \sigma >1.  \label{arevprod_p}
\end{equation}%
While quantity-based productivity for a product (\ref{aphysprod_p})
coincides with the firm productivity draw ($\varphi $), revenue-based
productivity for the product (\ref{arevprod_p}) depends on both the firm
productivity draw ($\varphi $) and consumer tastes ($\lambda _{i}$). The
measures of productivity from the production function estimation again
feature dispersion in both revenue and quantity-based productivity. The two
measures of a firm's productivity for a product are related as follows:%
\begin{equation}
\log \hat{\theta}_{i}^{R}=\left( 1-\frac{1}{\sigma }\right) \log \hat{\theta}%
_{i}^{Q}+\left( 1-\frac{1}{\sigma }\right) \log \lambda _{i},
\label{prodfn_revphys}
\end{equation}%
which implies that measured revenue-based productivity is again positively
correlated with measured quantity-based productivity. Furthermore, as the
distributions of firm productivity and consumer tastes are assumed to be
independent of one another, we obtain the following result for the two
measures of a firm's productivity for a product:%
\begin{equation*}
\text{var}\left( \log \hat{\theta}_{i}^{R}\right) =\left( 1-\frac{1}{\sigma }%
\right) ^{2}\text{var}\left( \log \hat{\theta}_{i}^{Q}\right) +\left( 1-%
\frac{1}{\sigma }\right) ^{2}\text{var}\left( \log \lambda _{i}\right) .
\end{equation*}%
Since $0<1-\frac{1}{\sigma }<1$ and var$\left( \log \lambda _{i}\right) >0$,
measures of revenue-based productivity from the production function
estimation can exhibit more or less dispersion at the \textit{product} level
than those of quantity-based productivity, depending on the elasticity of
substitution and the relative variance of consumer tastes and technical
efficiency.\footnote{%
A key reason for this difference between the measures of productivity based
on production function estimation and those considered above is that the
coefficient on labor input is allowed to vary between the regressions with
physical quantity and revenue data in equations (\ref{qntyprodfn}) and (\ref%
{revprodfn}) respectively.}

While revenue-based productivity at the \textit{product} level (\ref%
{arevprod_p}) depends on both the firm productivity draw ($\varphi $) and
consumer tastes ($\lambda _{i}$), revenue-based productivity at the \textit{%
firm} level again depends solely on $\varphi $ and is monotonically
increasing in $\varphi $. Following the same logic as above, the entry and
conditional distributions of consumer tastes and firm productivity are
independently distributed, and hence the stationary distribution of consumer
tastes ($\gamma _{zi}\left( \lambda _{i}\right) $) is the same for each firm
productivity. As a result, integrating over the continuum of products in
equation (\ref{arevprod_p}), the term in consumer tastes becomes a constant.
Therefore the variance of measured revenue-based productivity at the \textit{%
firm} level is strictly less than the variance of measured quantity-based
productivity, since $0<1-\frac{1}{\sigma }<1$.

\section{Additional Empirical Results}

Tables A1-A4 report additional empirical results. To provide a sense of the
relative level of detail between sectors, industries and products, Table A1
lists the thirteen five-digit products in the four-digit industry
\textquotedblleft Nonferrous Wiredrawing and Insulating\textquotedblright\
(SIC 3357). In Table A2, we report the number of five-digit products and
four-digit industries by two-digit sector, as well as the mean and standard
deviation of product capital and skill intensity for each two-digit sector.%
\footnote{%
As discussed in the paper, the CMF\ does not collect information on plant
input usage (or wages) by product, and therefore we measure a product's
capital and skill intensity as the shipment-weighted average of all plants
producing it.} In Table A3, we present decompositions that are analogous to
those reported in Table 6 in the paper, but are for the share of firms
rather the share of output of a product. In Table A4, we summarize several
of our main findings by two-digit sector.\ The table in the paper to which
each set of results relates is noted at the top of each column. Column 1
reports the mean share of multi-product firms across the 1987 to 1997
censuses. Column 2 reports the mean share of firms reporting any product
switching activity across the 1987 to 1997 censuses. Columns 3 and 4 report
mean product add and drop rates. Columns 5 and 7 report the mean share of
products' 1992 output due to firms that recently entered or are about to
exit, respectively. Columns 6 and 8 report the mean share of products' 1992
output accounted for by firms that recently added or are about to drop the
product, respectively. Columns 9 through 12 report decompositions similar to
Columns 6 to 8 but with respect to the average share of firms producing a
product in year $t$. Finally, in Table A5, we compare product and
sector-switching according to whether or not firms concomitantly acquire or
divest a plant.

\section{Appendix to Technical Appendix}

\subsection{Weighted Average of Firm Productivity and Consumer Tastes}

\noindent The aggregate weighted-average of firm productivity and consumer
tastes is:

\begin{equation}
\tilde{\varphi}_{i}\equiv \left[ \int_{\varphi ^{\ast }}^{\overline{\varphi }%
}\tilde{\lambda}_{i}\left( \varphi \right) \gamma _{g}\left( \varphi \right)
d\varphi \right] ^{\frac{1}{\sigma -1}},  \label{phitilde}
\end{equation}%
where the weights are given by the stationary distribution of firm
productivity conditional upon entry $\gamma _{g}\left( \varphi \right) $.
The function $\tilde{\lambda}_{i}\left( \varphi \right) $ is a
weighted-average of consumer tastes for a firm with productivity $\varphi $:%
\begin{equation}
\tilde{\lambda}_{i}\left( \varphi \right) =\frac{1}{1-\Gamma _{zi}\left(
\lambda _{i}^{\ast }\left( \varphi \right) \right) }\int_{\lambda _{i}^{\ast
}\left( \varphi \right) }^{\overline{\lambda }}\left( \varphi \lambda
_{i}\right) ^{\sigma -1}\gamma _{zi}\left( \lambda _{i}\right) d\lambda _{i},
\label{firmphitilde}
\end{equation}%
where the weights given by the stationary distribution of consumer tastes
conditional on a product being produced $\gamma _{zi}\left( \lambda
_{i}\right) /\left[ 1-\Gamma _{zi}\left( \lambda _{i}^{\ast }\left( \varphi
\right) \right) \right] $, and where $\lambda _{i}^{\ast }\left( \varphi
\right) =\left( \varphi ^{\ast }/\varphi \right) \lambda _{i}^{\ast }\left(
\varphi ^{\ast }\right) $.

\subsection{Derivation of the Solution to the Bellman Equation in the
General Model}

Consider the following trial solution to the Bellman equation (\ref%
{bellvalfirmalt}):%
\begin{equation}
v\left( \varphi \right) =\alpha \pi \left( \varphi \right) +\beta
\int_{\varphi ^{\ast }}^{\bar{\varphi}}\eta \left( \varphi ^{\prime }\right)
\pi \left( \varphi ^{\prime }\right) g_{c}\left( \varphi ^{\prime }|\varphi
\right) d\varphi ^{\prime },  \label{appbell1}
\end{equation}%
where $\alpha $, $\beta $ and $\eta \left( \varphi \right) $ are parameters
and functions to be determined below. Using the trial solution to substitute
for $v\left( \varphi ^{\prime }\right) $ on the right-hand side of the
Bellman equation (\ref{bellvalfirmalt}) gives:%
\begin{equation}
v\left( \varphi \right) =\frac{\pi \left( \varphi \right) }{\delta +\theta }%
+\left( \frac{\theta }{\delta +\theta }\right) \int_{\varphi ^{\ast }}^{\bar{%
\varphi}}\left[ \frac{\alpha \pi \left( \varphi ^{\prime }\right) +\beta
\int_{\varphi ^{\ast }}^{\bar{\varphi}}\eta \left( \psi \right) \pi \left(
\psi \right) g_{c}\left( \psi |\varphi ^{\prime }\right) d\psi }{\pi \left(
\varphi ^{\prime }\right) }\right] \pi \left( \varphi ^{\prime }\right)
g_{c}\left( \varphi ^{\prime }|\varphi \right) d\varphi ^{\prime },
\label{appbell2}
\end{equation}%
which yields the following solution:%
\begin{equation}
\alpha =\frac{1}{\delta +\theta },\qquad \beta =\frac{\theta }{\delta
+\theta },\qquad \eta \left( \varphi ^{\prime }\right) =\frac{\pi \left(
\varphi ^{\prime }\right) +\theta \int_{\varphi ^{\ast }}^{\bar{\varphi}%
}\eta \left( \psi \right) \pi \left( \psi \right) g_{c}\left( \psi |\varphi
^{\prime }\right) d\psi }{\left( \delta +\theta \right) \pi \left( \varphi
^{\prime }\right) }.  \label{appbell3}
\end{equation}%
Define the following variables and functions: $x\equiv \varphi ^{\prime }$, $%
y\equiv \psi $, $f\left( x\right) \equiv \eta \left( \varphi ^{\prime
}\right) \pi \left( \varphi ^{\prime }\right) $, $f\left( y\right) \equiv
\eta \left( \psi \right) \pi \left( \psi \right) $, $K\left( x,y\right)
\equiv g_{c}\left( \psi |\varphi ^{\prime }\right) $, $a\left( x\right)
\equiv \pi \left( \varphi ^{\prime }\right) /\left( \delta +\theta \right) $%
, and $\mu \equiv \theta /\left( \delta +\theta \right) $. Using these
definitions, the expression for $\eta \left( \varphi ^{\prime }\right) \pi
\left( \varphi ^{\prime }\right) $ implied by equation (\ref{appbell3}) can
be re-written as follows:%
\begin{equation}
f\left( x\right) =\left( \Omega \left( f\right) \left( x\right) \right)
\equiv a\left( x\right) +\mu \int_{\varphi ^{\ast }}^{\bar{\varphi}}f\left(
y\right) K\left( x,y\right) dy,  \label{appbell4}
\end{equation}%
which takes the form of a \textit{Fredholm }integral equation. The
continuous functions $K\left( x,y\right) $ and $a\left( x\right) $ are
known. For all $x$ and $y$ in the intervals $\varphi ^{\ast }\leq x\leq \bar{%
\varphi}$, $\varphi ^{\ast }\leq y\leq \bar{\varphi}$, we have $\left\vert
K\left( x,y\right) \right\vert \leq \bar{K}<1$, as $K\left( x,y\right) $\ is
a continuous probability density function. Under the assumption $\mu \bar{K}%
\left\vert \bar{\varphi}-\varphi ^{\ast }\right\vert <1$, the mapping $%
\Omega $ is a contraction, since (see for example Marsden and Hoffman 1974):%
\begin{equation}
d\left( \Omega \left( f_{1}\right) ,\Omega \left( f_{2}\right) \right)
=\sup_{x}\left\vert \mu \int_{\varphi ^{\ast }}^{\bar{\varphi}}K\left(
x,y\right) \left[ f_{1}\left( y\right) -f_{2}\left( y\right) \right]
dy\right\vert \leq \mu \bar{K}\left\vert \bar{\varphi}-\varphi ^{\ast
}\right\vert d\left( f_{1},f_{2}\right) ,  \label{appbell5}
\end{equation}%
where $\mu \bar{K}\left\vert \bar{\varphi}-\varphi ^{\ast }\right\vert <1$
is satisfied for $\theta <\bar{\theta}$, where $\bar{\theta}$ is implicitly
defined by $\frac{\bar{\theta}}{\delta +\bar{\theta}}\equiv \frac{1}{\bar{K}%
\left\vert \bar{\varphi}-\varphi ^{\ast }\right\vert }$. In determining the
stationary distribution of firm productivity below, we will assume that the
parameter condition $\theta <\bar{\theta}$ is satisfied. Therefore, the
Fredholm equation (\ref{appbell4}) has a unique solution, which determines
the equilibrium value of $f\left( x\right) \equiv \eta \left( \varphi
^{\prime }\right) \pi \left( \varphi ^{\prime }\right) $. Since $\pi \left(
\varphi ^{\prime }\right) $ is a known function, we have determined the
unique equilibrium value of $\eta \left( \varphi ^{\prime }\right) $.

\subsection{Proof of Proposition \protect\ref{PropInvProd}}

\begin{proof}
The equality of inflows and outflows for each productivity in (\ref%
{invariantprod}) can be re-written as follows:%
\begin{equation}
\gamma _{g}\left( \varphi \right) =\frac{g_{e}\left( \varphi \right) M_{e}}{%
\left( \delta +\theta \right) M}+\left( \frac{\theta }{\delta +\theta }%
\right) \int_{\varphi ^{\ast }}^{\bar{\varphi}}g_{c}\left( \varphi |\varphi
^{\prime }\right) \gamma _{g}\left( \varphi ^{\prime }\right) d\varphi
^{\prime }  \label{prop_invprod1}
\end{equation}%
where \{$M_{e}$, $M$, $\varphi ^{\ast }$\} are aggregate variables.\newline
Define the following variables and functions: $x\equiv \varphi $, $y\equiv
\varphi ^{\prime }$, $f\left( x\right) \equiv \gamma _{g}\left( \varphi
\right) $, $f\left( y\right) \equiv \gamma _{g}\left( \varphi ^{\prime
}\right) $, $K\left( x,y\right) \equiv g_{c}\left( \varphi |\varphi ^{\prime
}\right) $, $a\left( x\right) \equiv g_{e}\left( \varphi \right)
M_{e}/\left( \left( \delta +\theta \right) M\right) $, $\mu \equiv \left(
\theta /\left( \delta +\theta \right) \right) $. Using these definitions,
equation (\ref{prop_invprod1}) can be re-expressed in the following way:%
\begin{equation}
f\left( x\right) =\left( \Omega \left( f\right) \left( x\right) \right)
\equiv a\left( x\right) +\mu \int_{\varphi ^{\ast }}^{\bar{\varphi}}K\left(
x,y\right) f\left( y\right) dy,  \label{prop_invprod2}
\end{equation}%
which takes the form of a \textit{Fredholm }integral equation. The
continuous functions $K\left( x,y\right) $ and $a\left( x\right) $ are
known. For all $x$ and $y$ in the intervals $\varphi ^{\ast }\leq x\leq \bar{%
\varphi}$, $\varphi ^{\ast }\leq y\leq \bar{\varphi}$, we have $\left\vert
K\left( x,y\right) \right\vert \leq \bar{K}<1$, as $K\left( x,y\right) $ is
a continuous probability density function. Under the assumption $\mu \bar{K}%
\left\vert \bar{\varphi}-\varphi ^{\ast }\right\vert <1$, the mapping $%
\Omega $ is a contraction, since (see for example Marsden and Hoffman 1974):%
\begin{equation}
d\left( \Omega \left( f_{1}\right) ,\Omega \left( f_{2}\right) \right)
=\sup_{x}\left\vert \mu \int_{\varphi ^{\ast }}^{\bar{\varphi}}K\left(
x,y\right) \left[ f_{1}\left( y\right) -f_{2}\left( y\right) \right]
dy\right\vert \leq \mu \bar{K}\left\vert \bar{\varphi}-\varphi ^{\ast
}\right\vert d\left( f_{1},f_{2}\right) ,  \label{prop_invprod3}
\end{equation}%
where, from the above, $\mu \bar{K}\left\vert \bar{\varphi}-\varphi ^{\ast
}\right\vert <1$ is satisfied for $\theta <\bar{\theta}$, where $\bar{\theta}
$ is implicitly defined by $\frac{\bar{\theta}}{\delta +\bar{\theta}}\equiv 
\frac{1}{\bar{K}\left\vert \bar{\varphi}-\varphi ^{\ast }\right\vert }$. For
parameters satisfying this condition, the Fredholm equation (\ref%
{prop_invprod2}) has a unique solution, which defines the unique stationary
distribution of productivity conditional upon entry, $f\left( x\right)
\equiv \gamma _{g}\left( \varphi \right) $, for each value of $\left\{ M_{e}%
\text{, }M\text{, }\varphi ^{\ast }\right\} $.
\end{proof}

\subsection{Proof of Proposition \protect\ref{PropInvTaste}}

\begin{proof}
The equality of inflows and outflows for each value of consumer tastes in (%
\ref{invarianttaste}) can be re-written as follows:%
\begin{eqnarray}
\gamma _{zi}\left( \lambda _{i}\right) &=&\frac{z_{ei}\left( \lambda
_{i}\right) \left[ 1-G_{e}\left( \varphi ^{\ast }\right) \right] M_{e}}{%
\left( \delta +\chi +\varepsilon _{i}\right) M}  \label{prop_invtaste1} \\
&&+\left( \frac{\varepsilon _{i}}{\delta +\chi +\varepsilon _{i}}\right) %
\left[ \int_{\underline{\lambda }}^{\bar{\lambda}}z_{ic}\left( \lambda
_{i}|\lambda _{i}^{\prime }\right) \gamma _{zi}\left( \lambda _{i}^{\prime
}\right) d\lambda _{i}^{\prime }\right]  \notag
\end{eqnarray}%
where \{$M_{e}$, $M$, $\varphi ^{\ast }$\} are again aggregate variables and
the functions \{$G_{e}\left( \varphi ^{\ast }\right) $, $z_{ei}\left(
\lambda _{i}\right) $, $z_{ic}\left( \lambda _{i}|\lambda _{i}^{\prime
}\right) $\} are known. The function $\chi \equiv \theta \int_{\varphi
_{\ast }}^{\bar{\varphi}}G_{c}\left( \varphi ^{\ast }|\varphi \right) \gamma
_{g}\left( \varphi \right) d\varphi $ depends upon $G_{c}\left( \varphi
^{\ast }|\varphi \right) $, which is known, and $\gamma _{g}\left( \varphi
\right) $, which is determined in the proof of Proposition \ref{PropInvProd}
above.\newline
Define the following variables and functions: $x\equiv \lambda _{i}$, $%
y\equiv \lambda _{i}^{\prime }$, $f\left( x\right) \equiv \gamma _{zi}\left(
\lambda _{i}\right) $, $f\left( y\right) \equiv \gamma _{zi}\left( \lambda
_{i}^{\prime }\right) $, $K\left( x,y\right) \equiv z_{ic}\left( \lambda
_{i}|\lambda _{i}^{\prime }\right) $, $a\left( x\right) \equiv z_{ei}\left(
\lambda _{i}\right) \left[ 1-G_{e}\left( \varphi ^{\ast }\right) \right]
M_{e}/\left( \left( \delta +\chi +\varepsilon _{i}\right) M\right) $, $\mu
\equiv \varepsilon _{i}/\left( \delta +\chi +\varepsilon _{i}\right) $.
Using these definitions, equation (\ref{prop_invtaste1}) can be re-expressed
as follows:%
\begin{equation}
f\left( x\right) =\left( \Omega \left( f\right) \left( x\right) \right)
\equiv a\left( x\right) +\mu \int_{\underline{\lambda }}^{\bar{\lambda}%
}K\left( x,y\right) f\left( y\right) dy  \label{prop_invtaste2}
\end{equation}%
which again takes the form of a \textit{Fredholm }integral equation. The
continuous functions $K\left( x,y\right) $ and $a\left( x\right) $ are
known. For all $x$ and $y$ in the intervals $\underline{\lambda }\leq x\leq 
\bar{\lambda}$, $\underline{\lambda }\leq y\leq \bar{\lambda}$, we have $%
\left\vert K\left( x,y\right) \right\vert \leq \bar{K}<1$, as $K\left(
x,y\right) $ is a continuous probability density function. Under the
assumption $\mu \bar{K}\left\vert \underline{\lambda }-\bar{\lambda}%
\right\vert <1$, the mapping $\Omega $ is a contraction (see for example
Marsden and Hoffman 1974). The condition $\mu \bar{K}\left\vert \underline{%
\lambda }-\bar{\lambda}\right\vert <1$ is satisfied for $\varepsilon _{i}<%
\bar{\varepsilon}$, where $\bar{\varepsilon}$ is implicitly defined by $%
\frac{\bar{\varepsilon}}{\delta +\chi +\bar{\varepsilon}}\equiv \frac{1}{%
\bar{K}\left\vert \underline{\lambda }-\bar{\lambda}\right\vert }$. For
parameters satisfying this condition, the Fredholm equation (\ref%
{prop_invtaste2}) has a unique solution, which defines a unique stationary
distribution of consumer tastes, $f\left( x\right) \equiv \gamma _{zi}\left(
\lambda _{i}\right) $, for each value of $\left\{ M_{e}\text{, }M\text{, }%
\varphi ^{\ast }\right\} $.
\end{proof}

\subsection{Proof of Proposition \protect\ref{Prop1}}

\begin{proof}
We first determine the equilibrium values of \{$P_{i}$, $R_{i}$\} given \{$%
\varphi ^{\ast }$, $\lambda _{i}^{\ast }\left( \varphi ^{\ast }\right) $\}
and given the stationary distributions \{$\gamma _{g}\left( \varphi \right) $%
, $\gamma _{zi}\left( \lambda _{i}\right) $\}. The product price indices $%
P_{i}$ in equation (\ref{eqpindex}) depend on weighted average productivity $%
\tilde{\varphi}_{i}$ and the mass of firms producing the product $M_{pi}$.
From equations (\ref{rel_lambdas}), (\ref{phitilde}) and (\ref{firmphitilde}%
), $\tilde{\varphi}_{i}$ is uniquely determined by \{$\varphi ^{\ast }$, $%
\lambda _{i}^{\ast }\left( \varphi ^{\ast }\right) $, $\gamma _{g}\left(
\varphi \right) $, $\gamma _{zi}\left( \lambda _{i}\right) $\}. Similarly,
it can be shown that $M_{pi}$ is uniquely determined by \{$\varphi ^{\ast }$%
, $\lambda _{i}^{\ast }\left( \varphi ^{\ast }\right) $, $\gamma _{g}\left(
\varphi \right) $, $\gamma _{zi}\left( \lambda _{i}\right) $\}. To do so,
note that $M_{pi}$ is a constant fraction of the mass of firms producing $M$%
. From equation (\ref{massproduct}), this constant fraction depends only on
\{$\varphi ^{\ast }$, $\lambda _{i}^{\ast }\left( \varphi ^{\ast }\right) $, 
$\gamma _{g}\left( \varphi \right) $, $\gamma _{zi}\left( \lambda
_{i}\right) $\}, and from (\ref{massfirms}) $M=R/\bar{r}$. From equations (%
\ref{eqvarrev}), (\ref{expfirmrevprof}), (\ref{rel_lambdas}) and (\ref%
{massfirms}), $\bar{r}$ is uniquely determined by \{$\varphi ^{\ast }$, $%
\lambda _{i}^{\ast }\left( \varphi ^{\ast }\right) $, $\gamma _{g}\left(
\varphi \right) $, $\gamma _{zi}\left( \lambda _{i}\right) $\}, and we have
shown that $R=L$. Therefore the product price indices $P_{i}$ are uniquely
determined by \{$\varphi ^{\ast }$, $\lambda _{i}^{\ast }\left( \varphi
^{\ast }\right) $, $\gamma _{g}\left( \varphi \right) $, $\gamma _{zi}\left(
\lambda _{i}\right) $\}. With the product price indices determined,
aggregate revenue for each product $R_{i}$ can be solved for from equation (%
\ref{prodaggrev}) using $R=L$.\newline
We next determine the stationary distributions \{$\gamma _{g}\left( \varphi
\right) $, $\gamma _{zi}\left( \lambda _{i}\right) $\}. From the proof of
Propositions \ref{PropInvProd} and \ref{PropInvTaste}, for $\theta <\bar{%
\theta}$ and $\varepsilon _{i}<\bar{\varepsilon}$, there exist unique
stationary distributions for firm productivity, $\gamma _{g}\left( \varphi
\right) $, and consumer tastes, $\gamma _{zi}\left( \lambda _{i}\right) $,
for each value of the aggregate variables $\left\{ M_{e}\text{, }M\text{, }%
\varphi ^{\ast }\right\} $. The equilibrium value of $M=R/\bar{r}$ was
determined above, and the equilibrium value of $M_{e}$ can be determined
from $M$ using equation (\ref{sss}).\newline
Finally, we determine the equilibrium values of \{$\varphi ^{\ast }$, $%
\lambda _{i}^{\ast }\left( \varphi ^{\ast }\right) $\}. The equilibrium
value of $\lambda _{i}^{\ast }\left( \varphi ^{\ast }\right) $ can be
determined given $\varphi ^{\ast }$ from equation (\ref{eqprodscope}), where 
$P_{i}$ and $R_{i}$ were determined above. The equilibrium value of $\varphi
^{\ast }$ can be determined given $\lambda _{i}^{\ast }\left( \varphi ^{\ast
}\right) $ and the stationary distributions \{$\gamma _{g}\left( \varphi
\right) $, $\gamma _{zi}\left( \lambda _{i}\right) $\} from the free entry
condition (\ref{eqfreeentry}). Note that $\lambda _{i}^{\ast }\left( \varphi
\right) =\left( \varphi ^{\ast }/\varphi \right) \lambda _{i}^{\ast }\left(
\varphi ^{\ast }\right) $. Therefore, from equation (\ref{expfirmrevprof}), $%
\pi \left( \varphi \right) \rightarrow \infty $ as $\varphi ^{\ast
}\rightarrow 0$ and $\pi \left( \varphi \right) \rightarrow 0$ as $\varphi
^{\ast }\rightarrow \infty $. Hence, in the free entry condition (\ref%
{eqfreeentry}), $V\rightarrow \infty $ as $\varphi ^{\ast }\rightarrow 0$,
and $V\rightarrow 0$ as $\varphi ^{\ast }\rightarrow \infty $. Since $V$ is
continuous in $\varphi ^{\ast }$, it follows that there exists an
equilibrium value of $\varphi ^{\ast }$ for which the expected value of
entry $V$ equals the sunk entry cost $f_{e}$. Differentiating $V$ with
respect to $\varphi ^{\ast }$ given $\lambda _{i}^{\ast }\left( \varphi
^{\ast }\right) $ and \{$\gamma _{g}\left( \varphi \right) $, $\gamma
_{zi}\left( \lambda _{i}\right) $\} yields:%
\begin{eqnarray*}
\frac{\partial V}{\partial \varphi ^{\ast }} &=&\text{ \ }\underset{\text{%
Term A}}{\underbrace{-\left( \frac{1}{\delta +\theta }\right) \left[ \pi
\left( \varphi ^{\ast }\right) +\theta \int_{\varphi ^{\ast }}^{\bar{\varphi}%
}\eta \left( \varphi ^{\prime }\right) \pi \left( \varphi ^{\prime }\right)
g_{c}\left( \varphi ^{\prime }|\varphi ^{\ast }\right) d\varphi ^{\prime }%
\right] g_{e}\left( \varphi ^{\ast }\right) }} \\
&&+\left( \frac{1}{\delta +\theta }\right) \int_{\varphi ^{\ast }}^{\bar{%
\varphi}}\left[ 
\begin{array}{c}
\underset{\text{Term B}}{\underbrace{\frac{\partial \pi \left( \varphi
\right) }{\partial \varphi ^{\ast }}}}\ \underset{\text{Term C}}{\underbrace{%
+\theta \int_{\varphi ^{\ast }}^{\bar{\varphi}}\frac{\partial \eta \left(
\varphi ^{\prime }\right) }{\partial \varphi ^{\ast }}\pi \left( \varphi
^{\prime }\right) g_{c}\left( \varphi ^{\prime }|\varphi \right) d\varphi
^{\prime }}} \\ 
\underset{\text{Term D}}{\underbrace{+\theta \int_{\varphi ^{\ast }}^{\bar{%
\varphi}}\eta \left( \varphi ^{\prime }\right) \frac{\partial \pi \left(
\varphi ^{\prime }\right) }{\partial \varphi ^{\ast }}g_{c}\left( \varphi
^{\prime }|\varphi \right) d\varphi ^{\prime }}} \\ 
\underset{\text{Term E}}{\underbrace{+\theta \int_{\varphi ^{\ast }}^{\bar{%
\varphi}}\eta \left( \varphi ^{\prime }\right) \pi \left( \varphi ^{\prime
}\right) \frac{\partial g_{c}\left( \varphi ^{\prime }|\varphi \right) }{%
\partial \varphi ^{\ast }}d\varphi ^{\prime }}} \\ 
\underset{\text{Term F}}{\underbrace{-\theta \eta \left( \varphi ^{\ast
}\right) \pi \left( \varphi ^{\ast }\right) g_{c}\left( \varphi ^{\ast
}|\varphi \right) }}%
\end{array}%
\right] g_{e}\left( \varphi \right) d\varphi
\end{eqnarray*}%
where Term A = 0 since $v\left( \varphi ^{\ast }\right) =0$; Term B 
\TEXTsymbol{<} 0 since $\partial \pi \left( \varphi \right) /\partial
\varphi ^{\ast }<0$ from equation (\ref{expfirmrevprof}); and Term F 
\TEXTsymbol{<} 0. Term D \TEXTsymbol{<} 0 since $\partial \pi \left( \varphi
\right) /\partial \varphi ^{\ast }<0$ from equation (\ref{expfirmrevprof}).
In Term E, $\partial g\left( \varphi ^{\prime }|\varphi \right) /\partial
\varphi ^{\ast }=0$ for $\varphi \neq \varphi ^{\ast }$, and so Term E is of
measure zero relative to Term D. Finally, as $\theta \rightarrow 0$, $\eta
\left( \varphi ^{\prime }\right) \rightarrow 1/\delta $, $\partial \eta
\left( \varphi ^{\prime }\right) /\partial \varphi ^{\ast }\rightarrow 0$,
and Term C $\rightarrow $ 0. Therefore, for sufficiently small $\theta <\bar{%
\theta}$, $V$ is monotonically decreasing in $\varphi ^{\ast }$, and there
exists a unique equilibrium value of $\varphi ^{\ast }$.
\end{proof}

\newpage

\section*{References}

\noindent \textbf{Bernard, Andrew B., Stephen J. Redding and Peter K.
Schott. 2007.} \textquotedblleft Comparative Advantage and Heterogeneous
Firms.\textquotedblright\ \textit{Review of Economic Studies}, 74(1): 31-66.

\noindent \textbf{Ericson, Richard and Ariel Pakes. 1995.} \textquotedblleft
Markov-Perfect Industry Dynamics: A\ Framework for Empirical
Work.\textquotedblright\ \textit{Review of Economic Studies}, 62(1): 53-82.

\noindent \textbf{Foster, Lucia, John Haltiwanger and Chad Syverson. 2008}.
\textquotedblleft Reallocation, Firm Turnover and Efficiency: Selection on
Productivity or Profitability.\textquotedblright\ \textit{American Economic
Review}, 98(1): 394-425.

\noindent \textbf{Hopenhayn, Hugo A. 1992.} \textquotedblleft Entry, Exit,
and Firm Dynamics in Long Run Equilibrium.\textquotedblright\ \textit{%
Econometrica}, 60(5): 1127-1150.

\noindent \textbf{Jovanovic, Boyan. 1982.} \textquotedblleft Selection and
the Evolution of Industry.\textquotedblright\ \textit{Econometrica}, 50(3):
649-670.

\noindent \textbf{Klette, Tor J. and Zvi Griliches. 1996.} \textquotedblleft
The Inconsistency of Common Scale Estimators when Output Prices are
Unobserved and Endogenous.\textquotedblright\ \textit{Journal of Applied
Econometrics}, 11(4): 343--361.

\noindent \textbf{De Loecker, Jan. 2008.} \textquotedblleft Product
Differentiation, Multi-Product Firms and Estimating the Impact of Trade
Liberalization on Productivity.\textquotedblright\ Princeton University,
mimeograph.

\noindent \textbf{Levinsohn, James and Marc J. Melitz. 2006.}
\textquotedblleft Productivity in a Differentiated Products Market
Equilibrium.\textquotedblright\ Princeton University, mimeograph.

\noindent \textbf{Marsden, Jerrold E. and Michael J. Hoffman. 1974.} \textit{%
Elementary Classical Analysis.} New York: Freeman, Second Edition.

\noindent \textbf{Melitz, Marc J. 2003.} \textquotedblleft The Impact of
Trade on Intra-Industry Reallocations and Aggregate Industry
Productivity.\textquotedblright\ \textit{Econometrica,} 71(6): 1695-1725.

\noindent \textbf{Melitz, Marc J. and Gianmarco I. P. Ottaviano. 2008.}
\textquotedblleft Market Size, Trade, and Productivity.\textquotedblright\ 
\textit{Review of Economic Studies}, 75(1): 295-316.

\newpage

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