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\parindent 24pt   
         \titlepage   
	     \vspace*{-92pt}
         \begin{center}   
\begin{large}   
         {\bf The Learning Curve in a Competitive Industry     }\\   
 (Forthcoming, {\it RAND Journal of Economics})\\
  \end{large}   
          
May 23, 1996 \\
         \bigskip   
                       Emmanuel Petrakis,
Eric Rasmusen, and
                         Santanu Roy \\   
 \vspace*{24pt} 
        {\it Abstract}   
        \end{center}   
\begin{small} 
      We consider the learning curve in an industry with free entry
and exit, and price-taking firms.  A unique equilibrium exists if the
fixed cost is positive.  Although equilibrium profits are zero, mature
firms earn rents on their learning, and, if costs are convex, no firm
can profitably enter after the date the industry begins. Under some
cost and demand conditions, however, firms may have to exit the
market despite their experience gained earlier.  Furthermore,
identical firms facing the same prices may produce different
quantities.  The market outcome is always socially efficient, even if
it dictates that firms exit after learning. Finally, actual and
optimal industry concentration does not always increase in the
intensity of learning.
     
            \vspace*{ 30pt}
           \noindent 
\hspace*{20pt}	
Petrakis:  Departamento de Economia,
Universidad Carlos III de Madrid,
Calle Madrid 126 
28903 Getafe (Madrid) Spain.  
  Fax:  341-624-9875. Phone: 341-624-9652. 
Internet: Petrakis@eco.uc3m.es. \\
    \hspace*{20pt} Rasmusen: Indiana University School of Business,  1309 East 
  10th Street ,
  Bloomington, Indiana, U.S.A 47405-1701. 
  Phone: (812) 855-9219.  Fax: 812-855-3354. Internet:
Erasmuse@indiana.edu. \\
 Web:  http://ezinfo.ucs.indiana.edu/$\sim$erasmuse. \\ 
 \hspace*{20pt} Roy: Department of Economics, Microeconomics Section, H09-20, 
Erasmus
University, 3000 DR Rotterdam, The Netherlands; Fax: (+31)-10-212-1724,
Tel: (+31)-10-408-1420, Internet: roy@micro.few.eur.nl.\\
 Keywords: Learning curve, industry evolution, perfect competition. 
    
We would like to thank Tai-Yeong Chung  and seminar participants at
Notre Dame, Tilburg University,   the University of Southern California, the 
University of Western
Ontario, the 1992 European Econometric Society Meetings, the Seventh
Annual Congress of the European Economic Association, and the ASSET
Meetings in Toulouse for their comments.  We would especially like to thank the 
referees for their care in  reading this   paper.    The current version has  
benefitted considerably from their insight.   John Spence provided
research assistance.  This work was begun  at the University of
California, Los Angeles.  
 
  \end{small}  
   
\newpage
    
\noindent  
{\bf I.  Introduction}   

  	
Economists have long been aware that a firm's cost curve for
producing a given item may shift down over time as learning occurs.
  The plot of the cost level against cumulative
output is known as the learning curve or experience curve.  
 	Our subject here is learning in an industry of price-taking
firms with free entry and exit-- an arbitrarily large measure of
firms with identical technologies producing   homogeneous good.  Each firm's 
cost curve shifts down with its own accumulated
experience in production, measured by its cumulative output.  

	The assumption of a perfectly competitive market structure
distinguishes our model from much of the existing literature on
learning-by-doing, which has focussed on monopoly and
oligopoly.\footnote{See Spence (1981), Clarke, Darrough \& Heineke
(1982), Fudenberg \& Tirole (1983), Smiley \& Ravid (1983),
Bhattacharya (1984), Dasgupta \& Stiglitz (1988), Jovanovic \& Lach
(1989), Mookherjee \& Ray (1991), and Cabral \& Riordan (1991).} If
the average cost at any point of time is constant in current output,
then learning introduces an intertemporal economy of scale that
creates a natural monopoly. This need not be the case, however, if the
technology displays sufficient decreasing returns. In that case,
learning does not lead to a natural monopoly and is, in fact,
compatible with perfect competition.  Learning-by-doing is distinct
from increasing returns to scale in this sense.\footnote{See Mookherjee \& Ray
(1992) for a discussion of other differences between increasing
returns to scale and learning-by-doing.}
 
  Our model is not part of  that branch of the learning
literature  in  which  an individual firm's
experience spills over to other firms in the industry\footnote{The learning 
spillover literature includes   Arrow
(1962), Ghemawat \& Spence (1985), Romer (1986),  Lucas (1988), and
Stokey (1986, 1988).}  While the market structure in these models
is competitive, the presence of learning spillovers gives rise to
decreasing-cost industries as distinct from decreasing-cost firms.  We
exclude such spillovers, and consider only firm-specific
learning-by-doing.
 
 Our point of departure is  Fudenberg \& Tirole (1983),
which considers learning-by-doing in a competitive industry with
constant instantaneous marginal cost.  In their setting,
learning-by-doing is incompatible with perfect competition, but we
will come to a different conclusion, because we      analyze an industry with 
the usual
textbook assumption of {\it increasing} marginal cost, not constant
marginal cost. We will  show that in a two-period model with a fixed cost,
a unique perfectly competitive equilibrium exists. 
 
 If we make the stronger assumption that costs are convex, then no firm can 
profitably enter after the date at which the industry begins. This reflects the 
entry-deterring aspect of learning-by-doing; however, in our model entry is not 
prevented by any strategic action of incumbent firms, but is a direct 
consequence of dynamic competitive equilibrium with atomistic firms. The unique 
equilibrium takes one of two forms, depending on the demand and cost parameters 
of the economic environment. 

 In the first type of environment, all firms that
enter remain in the industry permanently.  The equilibrium discounted
stream of profits is zero, but mature firms earn quasi-rents on their
learning, compensating for their losses in the first period. 

	In the second type of environment, some firms exit, because
the mature industry cannot sustain the original number of firms with
non-negative profits.    Relatively inelastic demand coupled with a strong 
learning effect
gives rise to this outcome.  Initially identical firms, facing the same
prices, produce different quantities of the homogeneous good in the
first period, and some of them will exit in the second period.
    Firms which exit produce less than the staying firms in the first period.  

This is a new explanation for the endogenous differentiation of ex ante 
identical firms and {\it shakeout} of   small  firms, which does not rely on 
either  aggregate or firm-specific uncertainty.

 There is widespread empirical support for shakeout   (e.g.,  Gort and Klepper 
[1982]) and for   higher exit rates among smaller firms (Dunne, Roberts \& 
Samuelson [1989], Davis \& Haltiwanger [1992]).  The existing literature has 
explained shakeout   as the process  of    ``market selection'' in the presence 
of firm-level uncertainty. In this explanation, firms that are identical ex ante 
become heterogeneous over time through idiosyncratic shocks, uncertain learning, 
innovation and diffusion, and so forth, which lead to exit of the  ex post 
inefficient firms.\footnote{See, in a large literature, Lippman \& Rumelt 
(1982), Jovanovic (1982), Jovanovic \& Lach (1989), Hopenhayn (1992a,b), 1993), 
and Jovanovic \& MacDonald (1994a, b). }  In our model, shakeout is a direct 
result of endogenous technical change at the firm level, through deterministic 
learning and in the presence of stationary market demand.    All firms have 
perfect foresight at their point of entry, and some  enter  fully intending to 
exit later.   
 

The equilibrium outcome  is socially efficient whether it
is characterized by  exit or not.  Even if exit occurs in  equilibrium, a social
planner would choose the same number of firms of each type, the same
quantity produced by each firm in each period, and the same prices as
in the competitive equilibrium.  Thus, the presence of
learning-by-doing implies neither the usefulness of a government
industrial policy to ensure optimal learning, nor the useful effects
of large, innovative monopolies so often attributed to Schumpeter
(1950).  Our model will uncover a pitfall that may exist for
antitrust and regulatory authorities.  Although all firms in our
model are price-takers, one possible feature of equilibrium is that
prices are sometimes below marginal cost, sometimes above marginal
cost, that profits rise over time, small firms drop out of the market
and large firms expand even further, and that the large firms
increase their profits from negative to positive levels without any
new entry occuring.  Government intervention is not only unnecessary, but
possibly harmful.  



 Section II describes the model and discusses its assumptions.
Section III presents theorems on existence and efficiency of the
competitive equilibrium, and discusses the pattern of entry and exit.
  Section IV characterizes the equilibrium under the assumption of
convex costs. Section V contains a numerical example and looks into
special cases where (a) learning reduces only the fixed cost, not the
variable cost of production, and (b) learning reduces just the
marginal cost, not the fixed cost. 
  Section VI   concludes.  



\bigskip \noindent
 {\bf II. The Model}

	An arbitrarily large measure of initially identical firms
compete to enter  a homogeneous-good industry. The measure of firms
actually operating is determined by free entry and exit.  Each firm
is a price taker, since it is infinitesimal compared to the
industry.\footnote{  Other models of learning in which firms are price takers 
include Fudenberg \& Tirole (1983), Boldrin \& Scheinkman (1988), and
Majd \& Pindyck (1989).} Firms are indexed by $i$.  Time is discrete,
and the market lasts for two periods. Firm $i$ produces output
$q_t(i)$ in period $  t = 1,2$. 

Each firm $i$ faces the same current total cost at time $t$ as a
function of its current output $q_t(i)$ and its experience $x_t(i)$:
 $$ 
	C(q_t(i), x_t(i)),						 
$$
   where $x_t(i)$ is firm $i$'s cumulative output before time $t$, so
that $x_1(i) =  0 $  and  $x_2(i) = q_1(i)$.  Let
$$
 \Gamma(q_1,q_2) = C(q_1,0) + \delta C(q_2,q_1),
  $$
   where $\delta \in [0,1]$ is the discount factor, so $\Gamma$ represents a 
firm's
discounted sum of production costs across the two periods.

If amount $n$ of firms are active, industry output is $Q_t=\int_0^n
q_t(i)di$. The market demand function, $D(p)$, is the same in both
periods and is separable across time.  Let $P(Q)$ be the inverse
demand function.  Define $p_m$ as the minimum average cost at zero
experience.   
  $$ 
 p_m =\stackrel{\textstyle min}{\scriptstyle q \geq  0 } [C(q,0)/q], 
  $$
 and let $q_m$ be   the  corresponding minimum efficient scale, so
 $$
 q_m \in \stackrel{\textstyle argmin} {\scriptstyle q \geq  0 }
[C(q,0)/q] 
 $$
  
   
  We impose the following six  assumptions on  costs and demand: 
 \begin{itemize} \item[(A1)]
  $C(q,x)$ is continuously differentiable on ${\bf
R}^2_+$. 
  \item[(A2)]
   $C_q(q,x) > 0 $ for $q >0$ and $x \geq 0$;
$C_x(q,x) \leq 0 $ for all $(q,x) \in {\bf R}^2_+$ and $C_x(q,x) < 0
$ for all $(q,x) \in (0,K] \times [0,K]$, where $K$ is defined in
(A5).
   \item[(A3)]
  For any $ q > 0 $ and $x \geq 0 , C(q,x) > 0
$; Also, $C(0,0) > 0 $.
   \item[(A4)]
  $P$ is continuous and strictly decreasing;    $P(Q)\rightarrow 0 $ as 
$Q\rightarrow
+\infty$;  $P(Q)$ is integrable on any closed interval of  $ {\bf R}_+
$. 
 \item[(A5)]
  [Eventual Strong Decreasing Returns] There exists $K > 0 $ such
that the following holds: if either $q_1 > K$ or $ q_2 > K$ (or
both), then there exist $\alpha$  and $\beta \in [0,1]$ such that 
          $$
  \Gamma(q_1,q_2) > \Gamma(\alpha q_1,\beta q_2) + \Gamma((1-\alpha
)q_1,(1-\beta) q_2).
   $$
\item[(A6)] 
  $P(0) > p_m $.
 
 \end{itemize}


	Assumption (A1) guarantees the continuity of the marginal
cost and marginal benefit functions. 


Assumption (A2) says that the marginal cost is always positive, that
greater experience never increases the total cost, and that greater
experience strictly reduces the total cost of producing output in any period.  


  The total production cost  is
nonincreasing in the amount of accumulated experience. Figure 1 shows
one cost function that satisfies the assumptions--- the cost function
which will be  used  in Example 2 later in the article. Note the increasing
marginal costs for any level of learning, and the decreasing returns
to learning   for any level of output. 

  
 \epsfysize=3in 
   
\epsffile{Learn1.eps} 
   

   \begin{center} {\bf Figure 1: A Firm's Total Cost as a Function of
Output and Experience} \end{center}

Assumption (A3) says that  a firm with  no  experience incurs a fixed cost  of  
production,  a cost  which must be incurred even if output is zero.  


Assumption (A4)  is a standard assumption on the demand curve, and allows the 
social planner's problem to be well-defined.  
	   
Assumption (A5) says that if  output produced by a
firm is too large in any period , it is possible to have two firms produce the 
same
output vector at a lower total cost.  This prevents the industry from
being a natural monopoly.\footnote{ If one thinks in terms of
multiproduct firms, (A5) requires that the joint cost of production
is no longer subadditive if the firm produces an excessive amount of
the two goods (see Panzar [1989]). Note, incidentally, that the
crucial difference between a learning model and a static model of
joint production is time consistency: in our learning model we will
require that second-period profits be non-negative (or no firms would operate in 
the second period), whereas in static joint production, profits on either one of
the goods can be negative.} 

	Assumption (A6) places restrictions on the demand and cost
functions jointly to ensure existence of a nondegenerate equilibrium.
If $P(0)$ were allowed to take any value, no matter how small, then
the equilibrium might be at zero output for every firm. 

 
    
\bigskip
Only the  six assumptions listed above are   needed for our
main results, but with a little more structure   we can
strengthen the results further. We will do that in Section IV by
adding the following assumption, which is not implied by (A1)-(A6): 


 \begin{itemize}
\item[(A7)] [Convex Costs]
  $C$ is convex; for all $x \geq 0$, $C(q,x)$ is strictly convex in
$q$, and for all $q \geq 0$, if $ x_1 > x_2$, then $C_q(q, x_1) \leq
C_q(q, x_2)$. 
 \end{itemize}

Assumption (A7) requires that  the total cost function   be convex in $x$
and $q$, and that marginal cost be non-increasing  in experience.  For any level 
of experience, the current marginal cost is
strictly increasing in current output.  Assumption (A7) is sufficient
to ensure strict convexity of $\Gamma$ on ${\bf R}^2_+$. Part of this
assumption is that $C_x$ is nondecreasing in $x$; that is, there are
decreasing returns to learning at any given level of current output.
Note that this assumption allows for  a positive fixed cost,    incurred even 
when no output is produced.  

   Assumption (A7) is not necessary for existence, uniqueness, and
optimality of equilibrium prices and so  will not be used for
Propositions 1 and 2. 

\bigskip

 
	 Our specification of the cost function allows a firm to
accumulate experience on both its fixed and marginal costs. Each firm
maximizes its discounted stream of profits, taking prices as given.
An active firm exits the industry in the second period if its profits
from that time on would be negative.  A firm with no experience
enters the industry in the second period only if it can make positive
profits in that period. 


\bigskip
 \noindent
 {\bf III. Properties of the Competitive Equilibrium} 

 
	Let $p_t$ be the market price in period $t$. Denote firms
that stay in the market for both periods as {\bf staying or S-type
firms}, with output  $q_t$  and flow profit   $\pi_t$  in period $t$.  Denote
firms that exit at the end of the first period as {\bf exiting or
E-type firms}, with output $q_E$ and profit $\pi_E$.  Denote firms
that enter the industry at the beginning of the second period as {\bf
late-entering or L-type firms}, with output $q_L$ and profit $\pi_L$
in the second period ({\it not } discounted back to the first
period).  Finally, let $n_S, n_E,$ and $n_L$ be the measures of
active firms of each type.  


	A firm staying in the industry for both periods maximizes its
discounted sum of profits.  The first order conditions for the firm's
profit maximization problem are: 
   \begin{eqnarray} \label{e0c}
 p_1 = C_q(q_1,0) + \delta  C_x(q_2,q_1) \label{e0ca}	\\ 
    \label{e0cb}	 \hspace*{-64pt}   p_2 = C_q(q_2,q_1).  
 \end{eqnarray}
 
Equation (\ref{e0ca}) says that as long as learning still occurs, a
staying firm will choose output in the first period so that its
marginal cost is greater than the market price, since $C_x(q_2,q_1)$
is negative.  In other words, a staying firm overproduces in the
first period in order to reduce its cost in the second period.  As a
result, a staying firm makes losses initially in equilibrium, which
are counterbalanced by positive gains later. Equation (\ref{e0cb}) is
the standard ``price equals marginal cost'' condition. If the market
ends in the second period, any further learning is of no use to a
staying firm, so a firm maximizing profits from that time on chooses
output to equate price to its marginal cost.

	Marginal cost in the standard model is here replaced by what
we will  call the {\it Net Marginal Cost}: the increase in
lifetime discounted costs when current output increases, which for
the first period is $ C_q(q_1,0) + \delta C_x(q_2,q_1).  $   Note that the net 
marginal cost comes
arbitrarily close to the marginal cost of a non-experienced firm for
sufficiently large $q_1$, because if a firm produces too much today,
its marginal benefit from learning becomes almost zero.

   An equilibrium should be characterized by rational price-taking
behavior on the part of firms, but rationality and price taking do
not necessarily result in identical behavior by all firms.    Let 
$\pi_S(p_1,p_2)$ be the profit function of S-type firms. Then, 
 \begin{equation} \label{new1}
 \pi_S(p_1,p_2)=  \stackrel{Maximum}{ \{q_1, q_2 \geq 0\}}   [p_1q_1 + \delta 
p_2q_2 - C(q_1, 0) - \delta C(q_2, q_1)]. 
\end{equation}
   Similarly, let $\pi_E(p_1,p_2)$ and $\pi_L(p_1,p_2)$ denote the profit 
functions  of E  and L-type firms, so 
  \begin{equation} \label{new2}
 \pi_E(p_1,p_2)=  \stackrel{Maximum}{ \{q  \geq 0\}}   [p_1q    - C(q, 0) ]  
\end{equation}
 and 
 \begin{equation} \label{new3}
 \pi_L(p_1,p_2)=  \stackrel{Maximum}{ \{q  \geq 0\}}   [p_2q    - C(q, 0) ].  
\end{equation}
  For types $k = S,E,L$, let $J_k (p_1, p_2)$ denote the sets of solutions to 
the maximization problems  in  equations (\ref{new1}),  (\ref{new2}),  and 
(\ref{new2}).  


  We define
equilibrium as follows. 
 \begin{small}
 \begin{quotation}
\noindent
\underline{ Definition:} An equilibrium  is defined by:
 
 (a) Measures $(n_S, n_E,n_L)$ of type S, E and L firms who enter the
market. 

(b) Functions $q_1(i)$ and $ q_2(i)$, where $q_t:[0,n_S] \rightarrow
{\bf R}_+,\; j = 1,2, q_t(.)$ integrable (with respect to
     Lebesgue measure); $\; q_t(i)$ is the output produced by firm $i$ of
S-type in period  $t$.

(c) Functions $q_E:[0,n_E] \rightarrow {\bf R}_+$ and $q_L:[0,n_L]
\rightarrow{\bf R}_+$, integrable, where $q_E(i)$ and $q_L(j)$ are
the output produced by the i-th E and j-th L type firms in their
periods in the market.

(d)  Prices $p_1 \geq 0$ and $ p_2 \geq 0$. 



\vspace*{12pt}

\noindent
The variables defined in (a) - (d) must satisfy the following conditions to 
constitute an equilibrium: 
 
                       $ 
      (i) p_1 = P(Q_1 + Q_E), Q_1 =�\int_0^{n_S} q_1(i)di, Q_E
=\int_0^{n_E} q_E(i)di $ (Markets clear in the first period.) 


                   $ (ii) p_2 = P(Q_2 + Q_L), Q_2 =�\int_0^{n_S}
q_2(i)di, Q_L = \int_0^{n_L}�q_L(i)di$ (Markets clear in the second
period.) 

 $(iii)$     If  $n_S >0, (q_1(i), q_2(i)) \in  J_S(p_1, p_2); $ if  $n_E >0$, 
then $ q_E(i) \in J_E(p_1, p_2); $ if $n_L >0$, then $q_L(i) \in  J_L(p_1,p_2)$ 
(Active firms of all types maximize profit.) 

 $(iv)$ For $k = S,E,L$, 
 $$
  \begin{array}{ll}
  \pi_k(p_1, p_2) &  = 0 \;if\; n_k>0\\
  &    \leq 0  \;if\; n_k=0\\
 \end{array}
 $$
 (Every active  firm earns zero total profit over its period of stay, and 
further entry is not strictly profitable.) 

 \end{quotation}
\end{small}


      In equilibrium,  no firm can make
positive profit by behaving like some other type. No S-type firm can
do better by exiting at the end of period 1 nor can an E-type firm
make positive profit by staying on till period 2 (even if there are
no S-type firms in the market) and so forth.  This   ensures
sequential rationality on the part of the E-type and S-type firms,
who might otherwise find it advantageous to change their
second-period behavior halfway through the evolution of the industry.  



	In an equilibrium with exit, an exiting firm makes zero
profits in the first period, and in an equilibrium with late entry, a late 
entrant makes zero  profits in the second period.  A firm with no experience 
behaving
optimally during the single period in which it remains in the market
makes zero profits if and only if the market price equals its minimum
average cost.  In any equilibrium, no firm produces output in the
range of strongly diminishing returns.

  These requirements for rational and competitive behavior on the
part of the firms imply a number of restrictions on equilibrium
outcomes, which are summarized in Proposition 1. 
    
 

 \begin{small}
\noindent
   PROPOSITION 1.   {\it 1.  In any equilibrium, the price in each  
period is at most the minimum average cost for a firm with zero
experience.    2. A strictly positive measure of staying firms  produce in both 
periods.     3. All staying firms earn
strictly negative profits in the first period and strictly positive
profits in the second period.   4. If  there is exit in the equilibrium, the 
first-period
price exactly equals the minimum average cost, and all  exiting firms
earn zero profits in the first period.   5.  If there exist late-entering firms,
the second-period price is the minimum average cost for a firm with
zero experience.  6.   The output of every active firm in any time period is 
bounded above by  K.   7.  Late entry and exit cannot simultaneously occur in  a  
single   equilibrium. }

  \end{small}

\begin{footnotesize} 
{\it Proof}.  The proof of Proposition 1 is accomplished by showing the 
following:

\vspace*{-8pt}
      \begin{enumerate} 
 \item
  $p_j \leq p_m < P(0), j=1,2$.
 \item
 $n_S > 0 , Q_1 > 0 $ and $Q_2 > 0 $.
 \item
  For all $i \in [0,n_S], [p_1q_1(i) -C(q_1(i),0)] < 0 $ and
$[p_2q_2(i) - C(q_2(i),q_1(i))] > 0 $.  
 \item
 If $n_E > 0 $, then $Q_E > 0 ,$ $p_1 = p_m$ and for $i \in [0,n_E],
q_E(i) \in \{q: [C(q,0)/q] = p_m \}$. 
 \item
 If $n_L > 0 $, then $Q_L > 0 , p_2 = p_m$ and for $i \in [0,n_L],
q_L(i) \in \{q: [C(q,0)/q] = p_m \}$.  
 \item
  $q_1,q_2,q_E,q_L \leq K$.
    \item
 Either $n_E>0$ or $n_L>0$, but not both. 
	\end{enumerate}
 

    Recall that $p_m = Min \{[C(q,0)/q]: q \geq 0 \}$. Conditions
(iii)  and  (iv)  imply that if in an equilibrium we have $n_E
> 0 $, then $p_1 = p_m$ and $q_E(i) \in \{q: C(q,0)/q = p_m \}$.
Similarly, if $n_L > 0 $ then $p_2 = p_m$ and $q_L(i) \in� \{q:
C(q,0)/q = p_m \}$. 
 
     Condition  (iv)   also implies that $p_i \leq p_m , i =
1,2$. From assumption (A6), we have $P(0) > p_m$ and so in any
equilibrium it must be true that $p_i < P(0)$. It follows that $Q_1 +
Q_E = D(p_1) > 0 $ and $Q_2 + Q_L = D(p_2) > 0 $. 

   

 To prove part (6), it is sufficient to consider the
case of the staying firms. Suppose $q_t > K$ for some $t$.  In
equilibrium, a firm's lifetime profit is zero, so 
 \begin{equation} \label{Feb1} 
  0 = p_1q_1 + \delta p_2q_2-\Gamma(q_1,q_2) < p_1q_1 + p_2q_2- \Gamma(\alpha
q_1,\beta q_2)- \Gamma((1-\alpha )q_1,(1 - \beta )q_2), 
  \end{equation}
 for some $\alpha, \beta$ in $[0,1]$, using assumption (A5). The
rightmost expression, can be rewritten as
  \begin{equation}
\label{Feb2} 
 [ p_1 \alpha q_1+ p_2\beta q_2 - \Gamma(\alpha q_1,\beta q_2)] + [p_1(1 -
\alpha )q_1 + \delta p_2(1 - \beta )q_2 - \Gamma((1 - \alpha )q_1,(1 -
\beta )q_2)],
 \end{equation}
 which is either zero or negative.  In combination with the strong
inequality in (\ref{Feb1}), this yields a contradiction, so it must
be false that $q_t >K$ for some $t$. 


     Suppose $n_E > 0 $ and $n_L > 0 $. Then, $p_1 = p_2 = p_m $.
This violates condition (iv) of equilibrium,  since by  part (6) of Proposition 
1,  $C_x < 0 $, and facing those
prices a firm could produce $q_m$ in each period and earn $\pi_1=0$
and $\pi_2> 0$. Thus, $n_E > 0 $ and $n_L > 0 $ is impossible.

 Now, suppose there is an equilibrium where $n_S = 0 $.  Then, since
$D(p_t) > 0 , t =1,2$ in equilibrium implies that $n_E > 0 , n_L > 0
$, a contradiction. So, in equilibrium, we must have $n_S > 0 $.
This, in turn, can be used to show that $Q_1 > 0 $ and $ Q_2 > 0 $.
Suppose $Q_1 = Q_2 = 0 $. Then, $Q_E > 0 , Q_L > 0 $, i.e. $n_E > 0 ,
n_L > 0 $, a contradiction.  Suppose, $Q_1 = 0, Q_2 > 0 $. Then, $n_E
> 0 $, i.e. $p_1 = p_m$. Now if some S-type firm produces $q_1 = 0 $,
it earns a loss of $C(0,0)$.  On the other hand if it produces $q_1 =
q_m > 0 $ (where $C(q_m,0)/q_m = p_m$ ), then it has a lower cost
function in period 2 while the current loss is zero. So producing
$q_1 = 0 $ cannot be profit maximizing. Thus, $q_1(i) > 0 $ for
almost all $i \in [0,n_S]$, that is, $ Q_1 > 0 $, a contradiction.
Similarly, $Q_2 = 0 , Q_1 > 0 $ is ruled out. 

     From the first order conditions of profit maximization for
S-type firms it is clear that $p_1< C_q(q_1,0)$ so that $q_1$ does
not maximize period 1 profit at price $p_1$. Coupled with the fact that $p_1 <  
p_m$, this implies that 
  for all $i \in [0,n_S]$, $[p_1q_1(i) - C(q_1(i),0)] < 0
$,  so that (iv) implies $[p_2q_2(i) - C(q_2(i),q_1(i))] > 0 $. 

     Conditions  (iii) and (iv) imply that if $n_E > 0 $, then for $i \in
[0,n_E], q_E(i) > 0 $ and $[C(q_E(i),0)/q_E(i)] = p_m$. Similarly, if
$n_L > 0 $ then for $i \in [0,n_L], q_L(i) > 0 $ and
$[C(q_L(i),0)/q_L(i)] = p_m $.
 //
\end{footnotesize}

  

  In equilibrium, initially identical firms may
behave very differently, some staying, some exiting, and some
entering late.  A socially optimal allocation would solve the
following problem:
 
 
\begin{small} 
 \begin{quotation}
\noindent
\underline{The Social Planner's Problem (SPP*): }

Choose 

(a) $(n_S, n_E, n_L)$: the measures of type S, E, and L firms who
enter;

(b) Functions $q_1(i)$ and $q_2(i)$, where $q_t:[0,n_S]\rightarrow
{\bf R}_+, t = 1,2$, and $q_t(.)$ is integrable with respect to
Lebesque measure, $ q_t(i) $ being the output produced by staying
firm $i$ in period $t$;

(c) Integrable functions $q_E:[0,n_E]\rightarrow {\bf R}_+$ and
$q_L:[0,n_L] \rightarrow{\bf R}_+,$ where $q_E(i)$ and $q_L(j)$ are
the output produced by the $i$-th E and $j$-th L type firms,
respectively, in their periods of operation;

so as to maximize
    $$
 \begin{array}{l}
       \int_0^{Y_1} P(q)dq + \int_0^{Y_2} \delta P(q)dq
-   \int_0^{n_S}  [C(q_1(i),0) + 
 \delta C(q_2(i),q_1(i))]di 
 \\
          - \int_0^{n_E} [C(q_E(i),0)]di -\delta
\int_0^{n_L}[C(q_L(i),0)]di
 \end{array}
    $$
 where $Y_1 = Q_1 + Q_E$ and $Y_2 = Q_2 + Q_L$,  and 
    $$
     Q_1 = \int_0^{n_S}q_1(i)di,\;\;\;
  Q_2 = \int_0^{n_S}q_2(i)di, \;\;\;
Q_E
=\int_0^{n_E}q_E(i)di,\;\;\;
             Q_L =\int_0^{n_L}q_L(i)di.
   $$
 \end{quotation}
\end{small}

	The social planner's problem   includes choosing the entry and exit  of 
firms   as well as their output. 
The marginal social benefit  of   output in each period is exactly the demand 
price for that output.   Maximization of social surplus implies that the  net  
marginal cost of  each firm is equated to the demand price for total output 
produced.  Therefore, if the market price is equal to the demand price  
corresponding to the  socially optimal total output in each period,  then each 
active firm maximizes profit and the market clears in every period.  The 
complementary slackness condition for the social planner's problem with respect 
to  choice of entry of firms implies that every entering firm gets zero total 
profit and that no firm can make a strictly positive profit by entering the 
market.  Thus, the solution to the social planner's problem  corresponds to a 
competitive market outcome. 


   
 Under assumptions (A1) to (A6), not only does a competitive
equilibrium exist, but it is unique in prices and it is socially
optimal. 

PROPOSITION 2.{\it Under assumptions (A1) to (A6), an equilibrium
exists.  It is unique in prices and aggregate output, and it is
socially optimal.}

  One way to prove Proposition 2 would be by following the arguments in 
Hopenhayn  (1991),  which analyzes a  more general model of  dynamic competitive 
equilibrium with  stochastic shocks. We used a direct proof, an outline of which 
is contained in the Appendix,   in line with the approach of Jovanovic (1982). 
 

\bigskip
 
 Before  closing this section of the paper, one further  explanation may be 
useful.  
Our assumption that the  fixed cost of production is strictly positive is 
required to ensure the existence of a finite competitive equilibrium.
If the fixed cost of production is zero, (i.e. $C(0,x) = 0 $ for all
$x$),   and if costs are
convex, then a firm accumulates experience only in order to reduce
its marginal cost.\footnote{
 A referee has pointed out that the limiting behavior of the  competitive 
equilibrium in our model  as the fixed cost converges to zero does not 
necessarily coincide with the market outcome when fixed cost is zero and   
learning is absent.  Learning persists, as does its effect on prices and output, 
even as the  fixed cost  is reduced to zero.}    A well known result from 
standard price theory is
that a competitive industry with increasing marginal costs, free
entry, and no learning possibilities has no equilibrium if the fixed
cost of production is zero.  Loosely speaking, an infinite number of
firms operate in the market, each producing an infinitesimal amount
of output. This holds true even if firms are able to reduce their
costs by accumulating experience.\footnote{ Another way to understand this is 
through the social optimality of competitive equilibrium, an implication of 
which is that the competitive market structure and allocation of output 
minimizes the social cost of production.  If the firm-specific cost function is 
convex and the fixed cost is zero, social cost is minimized by having an 
indefinitely large number of firms, each producing infinitesimal output each 
period. 
      }

  


\bigskip
\noindent
 {\bf IV. Further Results: The Case of Convex Costs}
 
  
Let us now introduce assumption (A7),  convexity of the cost function,
noting that (A7) does not necessarily imply (A5), which must still be
retained, and that (A7) also requires that the marginal cost of production be 
weakly decreasing with learning. 

 Earlier we saw that the equilibrium is unique in prices.
When costs are convex, it is also unique in output and the number of
firms. 



 
PROPOSITION 3.{\it Under assumptions (A1)-(A7), the equilibrium     is unique in 
prices,
individual firms' outputs in each period, and the number of
firms. }
 

 \noindent
{\it  Proof:} See the Appendix. 



 
 Convexity also allows us to be more specific about the properties of
the equilibrium, as shown in the next   propositions.  In  an industry without 
the possibility of  learning,   identical firms produce the same output in 
equilibrium if
the marginal cost curve is upward sloping.   When the opportunity
for learning is added, identical firms could   behave differently in the same 
equilibrium. 

 
 
PROPOSITION 4.   {\it Under assumptions (A1)-(A7), the following is true
in equilibrium:
   \begin{itemize}
 \item[ (a)]
   Each of the staying firms behaves identically.
   \item[ (b)] If there is a positive measure of  exiting firms, they produce at 
the zero-experience minimum efficient scale, which is less than the output 
produced by staying firms in period 1. 
   \item[ (c)] 
 There exist no late-entering firms.
\end{itemize}

}%end of italics 
  


 \noindent
{\it  Proof:} See the Appendix. 



      

 Proposition 4 allows the unique equilibrium to take one of two
distinct forms, depending on the cost and demand parameters: (i) with
exit at the end of the first period, or (ii) without exit.
 
	In an equilibrium {\it with} exit, some firms, after
producing in the first period, decide to leave the industry. Thus,
two types of firms coexist, those staying for both
periods and those exiting at the end of the first period.
Furthermore, firms that are identical ex ante nonetheless produce different 
outputs even
in the first period.  For a given price in period 1, exiting firms
will produce less than staying firms, because overproducing to reduce
future costs has no value for a firm that plans to exit at the end of
the first period. 

 In an equilibrium {\it without} exit, all firms entering in the
first period stay in the industry both periods (i.e. all are staying
firms). Firms make losses today in order to accumulate experience,
while they earn profits tomorrow on their maturity. To break even,
the present value of the future profits must equal the losses today.

 It is perhaps surprising that assumption (A7) is needed to ensure
that there exist no late-entering firms in equilibrium. After all, a
late-entering firm must compete with staying firms that have lower
costs, and Proposition 1 showed that if late-entering firms do exist,
it must be the case that the price is $p_m$ in the second period, so
$p_2=p_m$ and the experienced firms are charging no more than than
inexperienced firms. Example 1, in which costs are nonconvex, shows
what can happen. 

     In the preceding section, Proposition  2 stated that the competitive 
equilibrium is efficient, and convex costs is only a special case of this.  This 
implies that when costs are convex, late entry is inefficient, even though the 
staying firms earn positive profit in the second period.   The reason is that 
late entrants, lacking experience, would have higher costs even if their profits 
were zero. 



\noindent
\underline{ Example 1: Nonconvex Costs and Late Entry}  

\noindent
 $D(p) = 40 - 3p$ \\
$\delta = 1 $ \\
 $
C(q,x) = \left\{   
 \begin{array}{ll} 
  q^2+ (4 -\frac{x}{100}) & for \; x < 3\\
 & \\
 q^2+ \frac{ 1}{(8/3) x - 7}   & for \; x  \geq 3\\ 
 \end{array} \right.
$ 
 
 In Example 1, the learning is entirely in the fixed cost. The
technology is nonconvex because the rate of learning increases at
$x=3$, but it does satisfy assumption (A5), because decreasing
returns set in at a large enough scale of operation.\footnote{ The
technology violates assumption (A1) because it is not continuous and
differentiable, but it should be clear that the cost function could
be smoothed without doing more than making the numbers less tidy. }

 In equilibrium, $n_S=10, n_E=0, n_L=4, q_1=3, q_2=2, q_L=2$,
$p_1=10/3$, and $p_2= 4$. 
 These prices clear the market, because
 $$
 D(p_1) = 40 - 3(10/3) = 30 =  n_Sq_1 + n_Eq_E = 10(3) +  0  
$$
 and
  $$
 D(p_2) = 40 - 3(4) = 28 =  n_Sq_2 + n_Lq_L = 10(2) +  4(2).
$$
   The prices yield zero profits for the late-entering firms because
$q_m =2$ and $p_m=4$.  They yield zero profits overall for the
staying firms because their profits are
  $$
 \begin{array}{ll}
   \pi_1+ \pi_2 & = [p_1q_1 - (q_1^2+4 -\frac{x_1}{100}) ]+ [p_2q_2 -
(q_2^2 +\frac{ 1}{(8/3) x_2 - 7})]\\
  & \\
  & = [(10/3)(3) - (3^2+4 -0) ]+ [(4)(2) - (2^2 +\frac{ 1}{(8/3) (3)
- 7})] = -3 + 3. 
 \end{array}
  $$

  Think of this from the point of view of a social planner. In the
first period, he decides to introduce just a few firms, so that all
of them can produce high output and acquire sufficient experience to
cross the threshold for effective learning. In the second period,
those firms cut back their output because further experience is not
so valuable, but this means that for the social planner to satisfy
demand he must introduce new firms. 
         
  Example 1 incidentally illustrates a point that will be generalized
in Proposition 6: learning can make prices {\it increase} over time,
 even though costs are falling.  This is because firms
overproduce in the first period, incidentally driving down the price,
in order to learn and save on their fixed costs later. 

 \bigskip
 

 The  discussion so far has shown that exit may
occur in equilibrium, which makes the question of   market efficiency  
  especially interesting.  A firm that exits
seems to waste its learning. Can it be socially efficient that some
firms   never make any use of their
first-period learning? Surprisingly enough, Propositions 2 and 3 tell
us that the answer is yes.
  The unique equilibrium may involve some firms entering in the first
period, producing a positive output and thereby reducing their costs,
but then exiting before the second period. Their learning is wasted.
Propositions 2 and 3 say that this is socially optimal--- a social
planner would also require that some firms exit, 
  rather than direct that there be fewer firms in period 1.
Social optimality therefore does not imply the kind of
``rationalization of industrial production'' that governments favor
when they try to consolidate firms in an industry. 


With a little thought, it becomes clear why this is so. Suppose
 the marginal cost curve initially slopes steeply upwards at some
production level $q'$,  but that after a firm
acquires experience, its marginal cost curve is closer to being
flat. In the first period, it would be very expensive to serve
market demand with firms producing much more than $q'$. Therefore,
the optimal plan is to have some firms produce only in the first
period, to keep output per firm low, but to have those firms
exit in the second period, because the diseconomies of scale then
become less severe.

  
A variable that will be important to the issue of exit is
$\theta(x)$, the ratio of the  minimum efficient scale  to the quantity demanded     
when the  price equals  minimum average cost.  Let us call
this the {\it natural concentration,} defined as 
   \begin{equation}
  \theta (x) =\frac{q_m(x)}{   D(p_m(x))   } ,	
 	  \end{equation}
  where
  $$
 q_m(x) = arg min_{ q } \{C(q,x)/q\}$$
  and
$$
 p_m(x) = C(q_m(x), x)/q_m(x).
 $$
 When the minimum efficient scale decreases with learning, the
natural concentration $\theta$ is falling in $x$: loosely speaking,  the market 
is  then able to sustain more firms when
firms are experienced than when they are not. 


\bigskip
	

 
 Proposition 5 lays out sufficient conditions under which exit does or does not 
occur   in a competitive equilibrium. 

PROPOSITION 5. {\it  Let  assumptions (A1)-(A7)  be true. 
  If $\theta (q_m) >\theta (0)$, where $q_m$ is the minimum efficient scale  for 
a firm with no learning, then there 
 exists $\delta_0
> 0 $ such that    exit occurs in equilibrium  for all   $\delta \in 
(0,\delta_0)$.  
 On the other hand,  if 
$\theta (q_m) < \theta (0)$, there exists $\delta_0 >
0$ such that no exit occurs in equilibrium  for all   $\delta \in (0,\delta_0)$. 
}

\noindent
 {\it Proof.}  See the Appendix. 

 
Firms  that overproduce initially in order to learn   suffer  losses
in period 1  which they are able to recover later as they become
inframarginal, with lower costs than potential entrants.  Suppose that $\delta 
=0$, so firms care only about first-period profits.  Then the equilibrium price 
would be $p_m(0) = p_m$ in  the first period, and each firm would produce 
$q_m(0) = q_m$.  These firms would find in period 2 that their experience level 
was $x=q_m$. As we are doing calculations for discount factors sufficiently near 
$\delta=0$, it is only this level of experience that we need consider.  
 If $\theta(q_m) < \theta(0)$, then the market can sustain more firms
with experience than without.  If   $\theta(q_m) > \theta(0)$, however,   then 
once
firms acquire experience,  the market cannot sustain as many of them, and some 
are  forced out
in the second period. 
	

 Proposition 5 has implications for the important special case in
which the marginal cost of production shifts down uniformly with
experience:
  $$
 C(q,x)= C_v(q) + q \phi(x) + F(x),  $$
  with $C_v(q)$ strictly convex in $q, \phi(x)<0$ and $F' (x)<0$.  (Note that 
this specification also allows the fixed cost to fall with learning.) The 
function $q_m(x)$ is nonincreasing in $x$ in this case. As a result,  
$\theta(x)< \theta (0)$ for all $x$, and Proposition 5 can be applied.
Exit will not occur in equilibrium, if discounting is sufficiently
heavy.\footnote{The proof of the fact that $q_m(x)$ is nonincreasing in
$x$ is as follows. $q_m(x)$ is defined by equating marginal to
average cost, i.e. $C_v'(q_m(x)) + \phi(x) = \frac{C_v (q_m(x))
}{q_m(x)} + \phi(x) + \frac{F(x)}{q_m(x)}.$ This yields 
 $C_v'(q_m(x)) q_m(x) - C_v (q_m(x)) = F(x).$ Since $C_v$ is strictly
convex, $C'_v(q)q - C_v(q) $ is strictly increasing in $q$. If $F$ is
nonincreasing in $x$, then $x_1 > x_2$ implies $q_m(x_1)  \leq
q_m(x_2)$ and then $\theta(x) < \theta(0)$.  }
  
  Suppose, on the other hand, that learning reduces only the fixed
cost.  Then the minimum efficient scale decreases with experience,
and so $\theta (x) < \theta(0)$ for all $x$, yielding Proposition 6.

  
PROPOSITION 6. {\it If learning reduces only the fixed cost, then in
equilibrium there is no exit, the price rises and each firm's output
falls over time: 
 $n_E = 0 , p_1 < p_2$, and $ q_1 > q_2$. }

 This is the price path illustrated in Example 1. Net  marginal
cost is always lower than the marginal cost of any experienced firm
if learning reduces fixed cost alone. Thus, if $p_1 \geq p_2,$ then $q_1
> q_2$, which contradicts the market equilibrium condition if later
entry is impossible.  Exit then does not occur because $p_1 < p_2
\leq p_m$. Thus, $n_E =0$.  As we have seen, we cannot draw general
conclusions about the properties of the price path, because it
depends on the initial costs, the type and intensity of learning, the
market demand, and the 
  discount rate. The same is true for the quantity path of staying
firms. 

  
 Environments in which the equilibrium has exit are fully  if less
intuitively characterized in Proposition 7.  

  PROPOSITION 7. {\it The following are necessary and sufficient
conditions for an equilibrium to have exit.  Let $(q_1^*,q_2^*)$ be
the solution to the following minimization problem:
 $$
 z = \stackrel{\textstyle Minimum}{\scriptstyle q_1, q_2 \geq 0 }\;\;
\frac{ C(q_1, 0) + \delta C( q_2, q_1) - p_mq_1}{ \delta q_2}.
  $$ 
  Under
assumptions (A1)-(A7), an equilibrium with $n_E > 0 $ exists if and
only if 
  $$
                   [D(p_m )/D(z)] > [q_1^{*}/q_2^{*}].  $$
    Furthermore, if there exists an equilibrium with exit then $p_2 =
z, q_1 = q_1^*, q_2 = q_2^*, n_E = [D(p_m ) -
n_S^*q_1^*]/q_m$.}

\begin{small}
  {\it Proof.} We know that if exit occurs in equilibrium
then $p_1 = p_m$ .  Consider the following minimization problem:
  $$
  \stackrel{\textstyle Minimize}{\scriptstyle q_1,q_2 \geq 0 }
(1/ \delta q_2)[C(q_1,0) + \delta C(q_2,q_1) - p_m q_1] 
 $$
  It can be checked that there is a unique interior solution, say,
$(q_1^*,q_2^*)$. Let $z$ be the value of the minimization problem.
Then, one can easily check that: $$ \begin{array}{l}
          p_m q_1 + z \delta q_2 - C(q_1,0) -  \delta C(q_2,q_1) \leq 0 
\;\;for\; all
\;\;(q_1,q_2) \\
          p_m q_1^* + z \delta q_2^* - C(q_1^*,0) -  \delta C(q_2^*,q_1^*) =  0.  
  \end{array}
$$
 Thus, the maximum profit earned by S-type firms is exactly zero if
$p_1 = p_m$ and $p_2 = z$. So, in equilibrium with exit, $p_2 = z$
and each firm produces $(q_1^*,q_2^*)$. Let $n_S^* = D(z)/q_2^*$. If
there is an equilibrium with exit, then $n_S^*q_1^* < D(p_m )$ and
$n_S^*q_2^* = D(z)$, so that 
                 \begin{equation} \label{ee42}
  D(p_m )/D(z) > q_1^*/q_2^*.
  \end{equation} Thus, (\ref{ee42}) is a necessary condition for an
equilibrium with exit. Now, suppose (\ref{ee42}) holds. Let $ n_E^* =
[D(p_m ) - n_S^*q_1^*]/q_m > 0 $.  It is easy to check that $(p_1 =
p_m , p_2 = z, n_S = n_S^*, n_E = n_E^* > 0 , q_1 = q_1^*, q_2 =
q_2^*, q_E = q_m)$ is an equilibrium.  // \end{small}

Consider any cost function $C$ and the minimization problem indicated
in   Proposition 7.    By definition of the minimum, it
must be true that at prices $p_1 = p_m$ and $p_2 = z$, the firm can
earn at most zero profit by producing in both periods. Obviously, $z
< p_m$.  Furthermore , check that the solution ($q_1^*, q_2^*$) to
this minimization problem is also a solution to the profit
maximization problem of a staying firm facing prices $ (p_1 = p_m,
p_2 = z$). The numbers $z, q_1^*$, and $q_2^*$ depend only on the
cost function and have nothing to do with market demand. The
proposition indicates that if $q_1^* \geq q_2^*$, there does not
exist any downward sloping market demand function for which exit
occurs in equilibrium. On the other hand, if $q_1^* < q_2^*$, exit
occurs in  equilibrium for any demand function $D$ which satisfies
 $$ 
\frac{ D(p_m)}{D(z)} > \frac{q_1^*}{q_2^*}.
$$
 This is a restriction on the behaviour of the demand function at
only two specific prices. Thus,  for such cost functions  the class of
demand functions for which exit occurs in equilibrium  is ``large.''
The lower $q_1^*$ is relative to $q_2^*$, the larger the class of
demand functions for which exit occurs.  


\bigskip 
 \noindent
 {\bf   V.  Examples and Implications}
 
   
Earlier we found two types of equilibria under convex costs: with and
without exit. 
	Under what cost and demand parameters will an equilibrium
with exit arise?  Example 2 helps develop some intuition for what may
happen. In it,  if the demand function is somewhat inelastic, then  after
the active firms reduce their costs in the first period by learning,
their potential second-period output is so great that the market is
saturated and some of them must exit.  

 
\noindent 
 \underline{Example 2: Industry Dynamics Under Different
Demand Parameters} 

\noindent
  $D(p) = 20 - bp$ \\
$ \delta =  0.9.$\\
  $ C(q,x) = q^2(1 + e^{-x}) + 10$\\

 Table 1 shows the equilibrium in Example 2 for two different values
of the demand   parameter, $b$. 

If $ b = 1.3$, demand is weaker, and more elastic for prices with
positive demand. In this case, there is no exit.  All
firms behave identically, producing higher output in the second
period than in the first because  costs fall enough with learning. Prices fall   
for the same reason. Overall profits are zero,
but they are negative in the first period and positive in the second.
  The losses in the first period can be seen as the cost of
learning, and the profits in the second period are quasi-rents on the
acquired learning. Even though second-period profits are positive, no
entry occurs, because an entrant would face higher costs, having
never learned how to produce cheaply. 

 
If $b = 1$, demand is stronger, and less elastic for prices below
20/1.3. In this case, there is exit.  The qualitative
features of the staying firms are the same as when $b=1.3$: output
rises, prices fall, and profits go from negative to positive over
time. When $b=1$, however, there are also exiting firms in the
market. These firms operate only in the first period, during which
they have zero profits, instead of the negative profits of the
staying firms. Their higher profits arise because their outputs are
smaller, but that means they acquire less learning than the staying
firms, and cannot compete profitably in the second period.   
``Shakeout''   has occurred. 
	
 
\begin{center}
  \begin{tabular}{ll|l|l} 
 \multicolumn{4}{c}{  TABLE 1: }\\
 \multicolumn{4}{c}{THE   EQUILIBRIUM IN EXAMPLE 2 }\\
 \multicolumn{4}{c}{ }\\   
 \hline 
\hline 
 \multicolumn{2}{r}{\rm Demand parameter $b$:} & 1 (strong, inelastic) & 1.3 
(weak, elastic)\\
 \hline Type of equilibrium & &{\it Exit} & {\it No Exit} \\ 
 Prices & $ (p_1, p_2) $ &   8.94. 6.63  &  8.76, 6.78 \\
 Industry output & $ (Q_1, Q_2)$ & (11.05, 13.38) & (8.60,  11.20)\\
Number of staying firms &  ($n_S$)  &  4.40 & 3.60 \\
Number of  exiting firms &  ($n_E$) &  0.18 &   0 \\
  Staying-firm  outputs & $(q_1, q_2)$ & (2.42, 3.04) &  (2.39, 3.11)\\
Exiting-firm outputs &  $(q_E,  0 )$ &  (2.24,  0 ) & ---\\
 Staying-firm profits &  $(\pi_1, \pi_2, \pi_S)$ & (-0.068,  0.076,  0 ) &
(-0.478,  0.531,  0  )\\
  Exiting-firm profits &  $( \pi_E)$ &0 & --- \\
         \hline 
  \end{tabular}  
 \end{center}    
 

  


   
\bigskip
 \noindent
 \underline{Learning and   Concentration}
  
  Does learning-by-doing increase    concentration?  In an
imperfectly competitive context, and without any learning in fixed
costs, Dasgupta \& Stiglitz (1988, p. 247) say that, ``...
firm-specific learning encourages the growth of industrial
concentration... Strong learning possibilities, coupled with vigorous
competition among rivals, ensures that history matters...''  Yet
Lieberman (1982, p. 886) could find no systematic relation between
learning by doing and industry concentration.  

  We have already found that history matters, even without the
Dasgupta-Stiglitz assumption of initial asymmetries, but Example 3
will show that the possibility of learning by doing can either
increase or reduce concentration, depending on the particular
industry.  Empirical predictions must take into account the type of
learning, not just its presence. 
 
 \newpage
\noindent
\underline{Example 3: Industry Concentration}


\noindent 
 $D(p) = 1/p $ \\
 $\delta = 0.9.$ \\ 
$ 
   C(q, x) = q^2(x + 1)^{-\kappa \lambda_{variable}} + (x+1)^{-\kappa 
\lambda_{fixed}}, 
$

    where $\kappa$ represents the speed of learning, for 
  $0 < \kappa < 1$, and $\lambda_{variable}$ and $\lambda_{fixed}$ each take the 
value
0 or 1 to represent whether learning occurs in variable costs or
fixed costs. 

 Let us denote as case (a) the case of learning in variable costs
alone, where $\lambda_{variable}= 1$ and $ \lambda_{fixed}=0$.  Let us denote as
case (b) the case of learning in fixed costs alone, where
$\lambda_{variable}= 0 $ and $ \lambda_{fixed}=1$. 


	In both cases (a) and (b), if $\kappa = 0 $ there is no learning.  
The cost function reduces to $C(q, x)=q^2 + 1$ and the
equilibrium number of firms is $n=0.5$. As $\kappa$ increases,
learning-by-doing becomes stronger.  The difference between the two
cost functions is that in (a) learning affects only marginal cost,
whereas in (b) it only affects the fixed cost. 

    Although $n=0.5$ when $\kappa=0$, when $ \kappa = 0.5$, the number of firms
is 0.469 in case (a) and 0.553 
 in case (b).   Since firms behave identically in this equilibrium,     if 
learning influences mainly the marginal cost,
it results in fewer and bigger firms, but if it reduces mainly the
fixed cost it results in more and smaller firms. These are the
results one would expect from  basic  price theory.  

	Further, as the speed of learning increases, industry
concentration increases in case (a) and decreases in case (b).
Finally, in both cases, consumers are enjoying lower prices as the
speed of learning increases. The number of firms is greater in an
industry with learning on fixed cost alone, however, than in an
industry with no learning possibilities.  Given that $ p_2 < p_m,$
and that $p_2$ equals marginal cost (the same for all experience
levels), we have that $q_2 < q_m$.  Hence, because no firm exits, the
number of firms in the industry is $n = D(p_2)/q_2 > D(p_m)/q_m$,
which is the number of firms in an industry with no learning
possibilities.

Antitrust authorities may learn an additional lesson from this model.
Consider the following scenario, which is possible for a wide range
of parameters.  In the first period, big firms and small firms
operate and charge high prices. In the second period, the big firms
all reduce their prices, the small firms go out of business, unable
to compete, and the big firms start earning strictly positive
profits.  An antitrust authority might look at this and infer predatory
pricing. That is wrong; the big firms earn zero profits viewed ex ante,
and the price drop is not strategic, but  a consequence of
falling costs, and the  exit of the small firms is socially optimal. 
\bigskip

  
  
 	
 \bigskip
\noindent
 {\bf VI. Concluding Remarks}
 
 
  When so much of the teaching that microeconomic theorists do
involves perfectly competitive partial equilibrium,  it is curious that
so much of our research has focussed on imperfect competition.
Perfectly competitive partial equilibrium is by no means a closed
subject, and there is more to be learned even about the models we
teach our beginning students and use in everyday analysis. In
particular, we still need a theory of endogenous market structure. Why do firms 
in an industry behave differently at different points in
its history, and why at any one time is heterogenous behavior
observed? 


 One line of research, exemplified by Hopenhayn (1992, 1993), looks
at the evolution of an industry in which firms encounter
heterogeneous productivity shocks.  Such shocks can explain why
industries evolve over time, and why so much heterogeneity is
observed even when firms are price takers and entry is free.  We have
come to the same general result that industries evolve and that firms
behave heterogeneously, but for different reasons:  in a fully
deterministic setting, but one with learning.  


 Our central purpose  has been to show that learning
and perfect competition are   compatible, and that learning has
curious implications for the evolution of a competitive industry.  We
have shown that in the presence of convex learning, firms must enter
at the beginning of an industry or never, and that the number of
firms may decline predictably over time. Firms may behave differently
even though they all begin with the same production opportunities.
Some firms may enter at a small scale knowing full well that they
will be forced to exit later; and these firms, in fact, will
initially be the most profitable. Whether the
equilibrium contains such firms or not, it will contain other firms
which make losses in the first period and profits in the second.
Viewing the situation from the second period, it may appear
oligopolistic, because these firms will then be earning positive
profits,  yet no entry will occur.  Viewed from the start of the
industry, however, these firms are merely reaping the returns to
their early investments in learning, investments which potential
entrants have not made.


This model has been quite general in some ways, but it is limited in
others, and opportunities abound for extending the model. The main
limitation of this model has been its restriction to two periods. By
this simplification, we have been able to employ general cost
  and demand functions. Allowing such general functions is
  important in this context, because industry evolution can be
  different depending on the curvature of these functions.  To
specify
  linear demand, marginal costs, and learning would be to run the
danger of missing important phenomena, something we conjecture is not
true of limiting the model to two periods.  The other limitation of
the model is the assumption of convex costs used for the later
propositions; in particular, the assumption that diminishing returns
in learning and static production are greater than the effect of
learning on marginal cost. This is certainly a reasonable case to
consider, but it is not the only case. Convexity was not needed,
however, to prove the existence, uniqueness, and optimality of
equilibrium prices. Moreover, it is remarkable that the industry
dynamics of entry and output are so rich even under convex costs. If  the cost 
functions were less constrained, we would of course expect
even more surprising results to be possible. 

 
  We have also shown that the competitive equilibrium is socially
optimal. Learning does not necessarily destroy this conclusion of
basic price theory.  Even if the equilibrium involves some firms
exiting early and not making use of the learning they acquired in the
first period, this is socially optimal. This contrasts sharply with
learning models which assume that marginal cost is constant in
current output, because in those models the social planner would
specify that the industry be a monopoly. Here, using standard
U-shaped cost curves, monopoly is not optimal and no intervention is
needed.  
 
  
\newpage
 \begin{small}

\noindent
 {\bf  Appendix: Proofs   }   (  The full proofs are available in the technical
version of this paper, Petrakis, Rasmusen \& Roy (1994),   from
Erasmuse@indiana.edu, or from the World Wide Web site,  
\\http://www.indiana.edu/$\sim$busecon/lrning.prf. )

 
\noindent 
 {\it Proof Outline for Propositions 2 and 3.}  Consider the social planner's 
problem (SPP)  defined  in the main text. The
problem can be decomposed into two stages:

(i) For any vector of total output to be produced by different S,E
and L type firms, the social planner decides on the minimum total
cost of producing this vector by choosing the measure of active
firms and their output. 

(ii) The social surplus from any total output vector can be written
as the area under the inverse demand curve and the social cost
corresponding to that output, where the social cost function is
defined in stage (i).

 One can use a result from Aumann and Perles (1965) to show existence
and characterize the social cost minimization problem in stage (i).
The minimand in this problem is not necessarily convex (unless we
assume [A7]) and there need not be a unique solution.  Using the
Lyapunov-Richter theorem, however, one can convexify the social cost
possibility set generated by using a continuum of firms even though
the individual firm's cost function is not necessarily convex.  The
social cost function (the value of the minimization problem) is
therefore convex and differentiable. This makes the problem in stage
(ii) a strictly concave maximization problem with a differentiable
maximand.

 Using a set of arguments based on the fact that $ P(Q) \rightarrow 0
$ as $Q \rightarrow + \infty $ and that the social marginal cost of
output is bounded above zero, we can show that there exists a
solution to the problem in stage (ii). As the maximand is strictly
concave, the solution is unique (in terms of total output produced by
different types of firms).  The way the production of this output
vector is organized depends on the cost minimization problem of stage
(i).  The inverse demand function generates a price in each period
such that demand is equal to total output. The first order conditions
for the social planner's maximization problem show that the price in
each period is equal to the social marginal cost of production if a
positive quantity is produced and the price is no greater than social
marginal cost otherwise. The social marginal cost (for each of the
types E,S and L) is the Lagrangean multiplier for the appropriate
social cost minimization problem in stage (i). One can show that in
any solution to the social cost minimization problem, each firm
produces output that maximizes its profit if the Lagrangean
multipliers are interpreted as prices. Furthermore, such profit is
zero if a positive quantity is produced and never exceeds zero. One
can then establish that every solution to the SPP is sustainable as a
competitive equilibrium. Also, the way the total output vector is
produced in equilibrium can be shown to minimize social cost.  Using
the concavity of the social surplus in problem (ii) and the first
order conditions of profit maximization, one can directly check that
the competitive allocation indeed satisfies all the conditions of
social optimality.  Hence, a production plan is socially optimal if
and only if it is sustainable as a competitive equilibrium. As there
exists a solution to the SPP, there exists a competitive equilibrium.
Furthermore, since the solution to the SPP is unique in total output
produced, the competitive equilibrium is unique in prices. 

 If, in addition, we assume (A7), the social cost minimization
problem in stage (i) becomes a convex problem, so it has a unique
solution in the measure of active firms of different types and their
output. So the competitive equilibrium allocation is unique in output
and measure of active firms under (A7), which is what Proposition 3
says.


 \bigskip

  
 \noindent
{\it Proof of Proposition 4}.  The first part of (a) and (b) follow immediately 
from
strict concavity of the profit function for each type of firm. (Note
that since the total amount $(Q_1,Q_2)$ produced by all S-type firms
is always strictly positive, $(q_1^*,q_2^*) >> 0 $.)  The second part
of (b) results from Proposition 1, because the negative first-period
profits of the staying firms result from their high production for
the sake of learning.  It remains to show that $n_L = 0 $ for part
(c). 
 
     Suppose that $n_L > 0 $. Then from Proposition 1, $n_E = 0 $ and $p_2
= p_m $. Under (A7), there exists a unique $q_m$ which minimizes
$[C(q,0)/q]$ over $q \geq 0 $. So, $q_L(i) = q_m$ and
     \begin{equation} \label{ee27}
     p_2 = p_m = C(q_m,0)/q_m = C_q(q_m,0).
  \end{equation} Furthermore,
     \begin{equation} \label{ee28}
     D(p_2) = D(p_m ) = n_Sq_2^* + n_Lq_m > n_Sq_2^*.   
 \end{equation}
From first order condition of profit maximization for firms which
produce in both periods we have that $C_q(q_2^*,q_1^*) = p_2 = p_m$
and, therefore (using (A7), (\ref{ee27})  and $q_1^* >  0 $)
       \begin{equation} \label{ee29}
       q_2^* \geq  q_m.                         
\end{equation}
Next we claim that the following inequality is true:
       \begin{equation} \label{ee30}
   C_w(q_2^*,q_1^*)q_1^* + C_q(q_2^*,q_1^*)q_2^* - C(q_2^*,q_1^*)
\geq 0.  \end{equation}
  By convexity of $C$ on ${\bf R}_+^2$,
  $$
     C(q_m,0) - C(q_2^*,q_1^*) \geq C_q(q_2^*,q_1^*)(q_m - q_2^*) +
C_w(q_2^*,q_1^*)(0 - q_1^*) 
 $$ 
 which implies that $ C_w(q_2^*,q_1^*)q_1^* +
C_q(q_2^*,q_1^*)q_2^* - C(q_2^*,q_1^*) \geq C_q(q_2^*,q_1^*)q_m - C(q_m,0) =
p_2q_m - C(q_m,0) = [p_2 - (C(q_m,0)/q_m)] =  0 $ (using (\ref{ee27}) ). 

From the first order conditions of profit maximization for S-type
firms and the fact that in equilibrium, the discounted sum of profits
is zero, we have: 
 $$
     C_q(q_1^*,0)q_1^* + \delta C_w(q_2^*,q_1^*)q_1^* + \delta
C_q(q_2^*,q_1^*)q_2^* - C(q_1^*,0) - \delta C(q_2^*,q_1^*) = 0.  $$
Using (\ref{ee30}) in the above equation we have:
  $$ 
 C_q(q_1^*,0)q_1^* - C(q_1^*,0) \leq 0 $$ which implies that 
        \begin{equation} \label{ee31}
  q_1^* \leq q_m                           
\end{equation}
so that, from (\ref{ee29}) , we have $ q_1^*  \leq q_2^*$. Thus,
        \begin{equation} \label{ee32}
       n_Sq_1^* \leq n_Sq_2^*.  
 \end{equation}
  From (\ref{ee28})  and (\ref{ee32})  we have
   $$
        D(p_2) > n_Sq_2^*  \geq n_Sq_1^* = D(p_1),  
$$
and so, $p_1 > p_2 = p_m$ , which violates Proposition 1 of the 
definition of equilibrium.  
  //



\bigskip

 

\noindent
  {\it Proof of Proposition 5.} We will first consider the case where  
$\theta(q_m) >  \theta (0)$ and show that Proposition 5 holds. 

   Suppose not. Then there exists sequence $\{\delta_t\}\rightarrow 0
$ such that for all $t$, if the discount factor $\delta = \delta_t$,
then no exit occurs in equilibrium. Let $(n_t, p_{1t}, p_{2t},
q_{1t}, q_{2t})$ be the equilibrium (with no exit) corresponding to
each $\delta_t$. Now, the sequences $\{p_{it}\},\{q_{it}\}, i =1,2$
are all bounded sequences (the prices lie in $[0,p_m ]$ and the
quantities in $[0,K]$). There exists a subsequence $\{t'�\}$ of
$\{t\}$ such that the sequences of prices and quantities described
above, converge to (say) $(p_i*,q_i*), i=1,2$. From first order and
zero profit conditions, we have that $$
    p_{1t'}� = C_q(q_{1t'}�,0) + \delta_t�C_x(q_{2t'}�,q_{1t'}�). 
 $$
  $$
     p_{2t'}� = C_q(q_{2t'}�,q_{1t'}�).              
$$
 $[p_{1t'}q_{1t'} - C(q_{1t'},0)] + \delta_t[p_{2t'}q_{2t'}�
-C(q_{2t'}�,q_{1t'}�)] = 0 $.
 
Taking limits as $t \rightarrow \infty$ yields
   \begin{equation} \label{ee33}
  p_1^* = C_q(q_1^*,0),                          
\end{equation}
   \begin{equation} \label{ee34}
  p_2^* = C_q(q_2^*,q_1^*),                        
 \end{equation}
   \begin{equation} \label{ee35}
  p_1^*q_1^* - C(q_1^*,0) = 0.
  \end{equation}
  From (\ref{ee33}) and (\ref{ee35}) we have that
    \begin{equation} \label{ee36}
      p_1^* = p_m , q_1^* = q_m.   
  \end{equation}
  By the definition of equilibrium, it must be true that firms earn
non-negative profit in period 2 so that for all $t'�, [p_{2t'}q_{2t'}
- C(q_{2t'},q_{1t'})] \geq 0 $. Taking limits, we have that
       \begin{equation} \label{ee37}
       p_2^*q_2^* - C(q_2^*,q_1^*) \geq   0.         
\end{equation}
Combining (\ref{ee34})  and (\ref{ee37}) , we can see that    
       \begin{equation} \label{ee38}
   p_2^* \geq p_m (q_1^*), q_2^* \geq q_m(q_1^*). 
  \end{equation}
  Since $D(p_{1t'})/q_{1t'} = D(p_{2t'})/q_{2t'}$, we have after
taking the limit as $t' \rightarrow � \infty $
      \begin{equation} \label{ee39}
   D(p_1^*)/q_1^* = D(p_2^*)/q_2^*. 
   \end{equation}
  From (\ref{ee36}) ,
         \begin{equation} \label{ee40}
  \frac{q_1^*  }{    D(p_1^*)}     =     \frac{q_m  }{    D(p_m )}   = �\theta 
(0)       
 \end{equation}
From (\ref{ee38})  
        \begin{equation} \label{ee41}
  \frac{q_2^*  }{   D(p_2^*)} \geq  \frac{q_m   (q_1^*)   }{   D(p_m    (q_1^*) 
)  } =   \theta (q_1^*)
  \end{equation}
  But
 $\theta (0) < \theta �(q_1^*)$,  as $q_1^* = q_m.$ Thus,
(\ref{ee40}) and (\ref{ee41}) contradict (\ref{ee39}).


 Let us now turn to the case where $\theta(q_m) < \theta(0)$. Suppose 
Proposition 5 is false in this case. Then there exists
a sequence $\{ \delta_i \} \rightarrow 0$ such that exit occurs in
equilibrium for all $i$. Let $(p_{1i}, p_{2i},q_{1i},q_{2i}, n_i)$ be
the associated equilibrium prices, outputs and numbers of staying
firms. Then $p_{1i}=p_m$.  Note that $\{(p_{1i}, p_{2i},q_{1i},q_{2i}
)\}$ is a bounded sequence, converging to, say $\{(p_{1},
p_{2},q_{1},q_{2} )\}$. Abusing notation, let this be the convergent
subsequence itself. Observe that
 $$
  p_{1i}= p_m = C_q(q_{1i}, 0) + \delta_i C_x(q_{2i}, q_{1i} ).  $$
  Since $C_x(q_{2i}, q_{1i} )$ stays bounded as $ i \rightarrow
\infty$, we have $p_2 = p_m (q_1) = p_m (q_m)$. (Note that $p_m(x)$
is continuous in $x$.) Observe that $p_{2i} = C_q(q_{2i}, q_{1i})$
and so, taking the limit, we have $p_2 = C_q(q_{2 }, q_{1 })=C_q(q_{2
}, q_m)$. Since $p_2 = p_m (q_m)$, we have $q_2 = q_m (q_m)$. Lastly,
note that for each $i$, 
 $$ \frac{D(p_{1i} )}{q_{1i}} \geq n_i = \frac{D(p_{2i} )}{q_{2i}},
$$
 so that taking the limit we have
 $$ 
 \frac{D(p_m )}{q_m} \geq \frac{D(p_m(q_m) )}{q_m(q_m)}, $$
 which is to say, $\theta(0) \leq \theta(q_m)$, a contradiction. 
 //
\end{small}





\newpage
 
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