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\begin{document}
\begin{Huge}
 %-------------------------------------------------------
  
\noindent
 April 8, 2017


\begin{center}


 {\sc  I will ask you to vote for one of these three titles: }
 
{\sc 
(1) The Hold-Up Problem Is Greatest When One Side of the Market Is Perfectly Competitive

\vfill

(2) Competitive Hold-Up: Monopoly Prices Are  Too High
To Maximize Profits when Retailers or Complements Are Perfectly Competitive
  \vfill

(3) A New Cost of Monopoly: Hold-Up of Perfectly Competitive Retailers or  Complementary Products

}

\end{center}

%---------------------------------------------------------------------------------
\newpage

%\vspace*{12pt}

\begin{center}

  {\bf  A New Cost of Monopoly: Hold-Up of Perfectly Competitive Retailers or  Complementary Products 
}  





\bigskip


   Eric   Rasmusen (BEPP, Kelley
School, Indiana University)

 

\end{center}

 
 If a monopolist  cannot commit to a wholesale price in advance, even competitive retailers will be reluctant to enter the market, knowing that once they have entered  the monopolist has incentive to choose a higher price and  reduce their quasi-rents.   Retailers earn zero profits in the long run, but this hurts the monopolist by shifting in the retailer short-run supply curve.     I call this   ``competitive hold-up''.  A similar problem occurs if the monopolist's product is sold directly to consumers but is complementary to a product sold by  a competitive industry. This is not double marginalization or the two-monopoly  complements externality.

 

{   I would like to thank \href{https://www.econjobrumors.com/topic/eric-rasmusen-wants-paper-comments-and-will-thank-ejmr-if-they%E2%80%99re-useful/page/4?replies=72#post-3429649}{  Economics Job Market Rumors},  Louis Kaplow, Qihong Liu,  Michael Rauh and  the BEPP Brown Bag Lunch  for
helpful comments. }


%---------------------------------------------------------------------------------
\newpage
\begin{center}
 {\sc   A Numerical Example of the Retailer Model}
\end{center}
 
The
 upstream  monopolist has constant marginal  cost $a=1$ which  he will sell at wholesale price $w$ per unit.

 A continuum of length $n$ of  competing retailers with identical cost curves enter at cost $ F=1$. Each retailer  chooses to sell $q(p)$ of the good at marginal cost    $c(q) +w$, with $c(q)=q$ here.

 Thus,   retailers have U-shaped cost curves. 

 
 

Let consumer  market demand be linear for this example: 
$
Q(p)    
= 1000 - 200 p.   
$

 
Since retailers compete, and since free entry will make their profits equal to zero, all retailers must produce at exactly the minimum average cost. This is found by minimizing $\frac{.5}{q} + .5q +w$, which yields   $q=1$ per  retailer. 

   
%---------------------------------------------------------------------------------
\newpage
\begin{center}
 {\sc   Social Optimum}
\end{center}

 The social marginal cost of serving consumers equals the minimum average cost  when the wholesale price is set to the wholesale marginal cost, so $w = a =1$, in which case the average cost is $\frac{.5}{1} + .5(1) +1 =2.$

 If we set $p=2$, then $Q=1000-200p =  600$. 

 Dividing  by $q=1$ yields the quantity  of retailers at the social optimum, $n=600$.

 This will yield zero profits to the monopolist  since $w=a$ and zero profits to the retailers since  retailer revenue of $pnq= (2)(6)(1) =12$ equals the retailer cost of $n(.5 + .5q^2 +wq) = (6)(.5 + .5 (1)^2 + 1((1)) = 12. $

 It will yield consumer surplus of $ .5  (5-2) (600)   = 900. $

  

%---------------------------------------------------------------------------------
\newpage
\begin{center}
 {  {\bf Proposition 1 }}
\end{center}

\noindent
 {  If the  upstream monopolist  cannot precommit to his wholesale price before retailers enter, his price will be higher, sales lower, and profits lower. 

As consumer surplus will also be lower, total surplus is lower than in a vertically integrated monopoly or a monopoly that can commit to wholesale prices.}

%---------------------------------------------------------------------------------
\newpage
\begin{center}
 {\sc   Retailer behavior in equilibrium}
\end{center}

 
 \hspace*{24pt} A retailer's profit is
\begin{equation} \label{rprofit}
 \pi_{retailer}= pq - \int_0^q c(x) dx - wq -F
\end{equation}


 Maximizing   with  respect to quantity yields   price equals marginal cost: 
\begin{equation} \label{e30b}
 p =   c(q) +w
\end{equation}

  
Because of free entry, retailer profits equal zero, so in the long-run,
\begin{equation} \label{e30c}
 \pi_{retailer}=  (q+w)q - .5q^2 - wq -F =0,
\end{equation}
which solves to $q=1$. Substituting  into   the market equilibrium condition   gives us the amount of retailers $n^*$ as a function of the wholesale price $w$: 
\begin{equation} \label{asst}
 n^* = 800 - 200w.
\end{equation}
 

%---------------------------------------------------------------------------------
\newpage
\begin{center}
 {\sc  Short Run Supply}
\end{center}

    In the short run, $n$ is fixed and profits are not necessarily zero.  Following the rule of marginal cost equals price,   $c(q) = q = p+w$ so the 
  individual  retailer short-run supply curve is
\begin{equation} \label{srsupply}
 q(p,w ) =   p-w.
\end{equation} 

   In equilibrium,  market supply  equals  market demand, so 
\begin{equation} \label{supplydemand}
Q_s = nq = Q_d = 1000-200p 
\end{equation}

Solving  for $p$ and substituting  yields $ q  =    5- n/200-w$, which, solving for $q$, leaves us with an expression for the individual retailer's output as a function of $n$ and $w$ when $p$ takes its resulting short-run equilibrium value: 
\begin{equation} \label{srsupply1}
q(n,w) = \frac{5- w}{1+.005n}. 
\end{equation} 

%---------------------------------------------------------------------------------
\newpage
\begin{center}
 {\sc   The Retailer Supply Curve}
\end{center}


As Figure 3 shows, the market supply curve swivels down as the number of retailers $n$ increases, since to increase market supply requires moving less far up the individual retailer's marginal cost curves. 

\begin{center}
 
 \includegraphics[width=7in]{supplyfigure.png}

\end{center}   
%---------------------------------------------------------------------------------
\newpage
\begin{center}
 {\sc  Monopolist behavior  with commitment to a wholesale price}
\end{center}

 
 \hspace*{24pt} 
The monopolist's profits equal the wholesale price minus the wholesale production cost times output.  Since the monopolist can commit to the price before retailers choose whether to enter, he takes into account his effect on $n$  and uses the retailer's long-run equilibrium condition   \eqref{asst} in choosing $w$: 
\begin{equation} 
\begin{array}{ll}
\pi_{monopolist} & =   (w-a) qn\\
& =   (w-1) (1) (800-200w) \\
& =  1000w -800 -200w^2.\\
\end{array}
\end{equation}
Solving for $w$  yields
$
w=2.5.$
 

 Using this $w$, in equilibrium $p=3.5, n= 300, q=1$  and $Q=300$.  The monopolist earns $\pi_{monopolist} =  (w-a)Q= 450,$ compared to 0 at the social optimum. Consumer surplus falls from 900  to .5(5-3.5)(300) = 225. 
  
%---------------------------------------------------------------------------------
\newpage
\begin{center}
 {\sc   Monopolist behavior  without  commitment to a wholesale price}
\end{center}
 
 
 \hspace*{24pt} 
  The  monopolist takes $n$ as given when he chooses $w$, and  uses  the retailers' short-run supply function   for his expectation as to the level of $q$ they will choose.    Thus,   
\begin{equation} 
\begin{array}{ll}
\pi_{monopolist}&=(w-a) n  q(n,w)  \\
&= \left(\frac{ n} {1+.005n} \right) (5w -5 -w^2  +w)  \\ 
\end{array}
\end{equation}

the monopolist will choose a wholesale price of $w=3$, regardless of the level of $n$ (special to the  example) because
\begin{equation} \label{nocommitment}
\frac{d\pi_{monopolist}}{dw} = \left(\frac{ n} {1+.005n} \right) ( 6-2w) =0  
\end{equation}

   The actual level of $n$ will anticipate $w$. For  zero profit, a retailer must  have minimum efficient scale of $q=1$ with marginal cost of 1, so  $p=w+1=4.$  Demand is 
  $Q_d  =  1000-200p  = 200$, so $n=200$. 

%---------------------------------------------------------------------------------
\newpage
\begin{center}
 {\sc  The case when the monopolist cannot commit  but retailers   are deceived.}
\end{center}

 
 \hspace*{24pt} 
  Suppose the monopolist promises   $w=2.5$ but  charges $w=3$.  As we have seen, if  they believe that $w=2.5$ then amount $n=300$ of retailers will enter.  The retailers will use their short-run supply curve  and choose
\begin{equation}
   q(n,w) = \frac{5-w}{1+.5n} =  \frac{5-3}{1+1.5} = .8,  
\end{equation}
 so output will be $Q= (300)(.8) = 240$ and $p =  5 - 2.4/2 = 3.8.$  

Monopoly profit will be $(w-a)Q = 480$.  Consumer surplus will be $.5(5-3.8)(240) =  144.$ Retailer profits will be negative.

   Industry profit is higher if the retailers are  deceived than if they are not---   426 instead of 400. 
 
%---------------------------------------------------------------------------------
\newpage
\begin{center}
{\sc TABLE 1: \\
Equilibrium Values in the Numerical Example}
\end{center}

 
 
\begin{center}


\begin{tabular}{llll l}
\hline
  & Monopoly   & Monopoly   & Monopoly &   Social   \\
  &   with  &  without   & with  & Optimum \\
 &     commitment &  commitment & deception    &     \\
\hline
 Wholesale price, $w$ &  2.5 &3 &3  &1  \\
 Retail price, $p$ & 3.5 & 4 & 3.8 &2 \\
Amount of retailers, $n$ & 300 &200 & 300 &600 \\
 Output per retailer, $q$ &1  &1 &.8  &1 \\
  &  & & &   \\
 Monopolist profit& 450& 400& 480 &0 \\
Retailer profit&        0&     0& -54 &0 \\
 Consumer surplus & 225 &  100 & 144 & 900  \\
Total surplus& 675 &  500  &   570 & 900\\
\hline
\end{tabular}
 \end{center}
 
 %---------------------------------------------------------------------------------
\newpage
\begin{center}
 {\sc Retailer Demand as a Function of Wholesale Price $w$ }
\end{center}
 
 

  
\begin{wrapfigure}{l}{0.7\textwidth}
\centering
\includegraphics[width=0.7\textwidth]{demandfigure.png}
%\caption{\label{fig:frog1}This is a figure caption.}
\end{wrapfigure} 
 
\begin{Large}

For a given   $Q$  there is no inefficiency in production as the result of hold-up: retailers operate at  $q^*$, though $n$ is  maybe too small.   If the retailers were  surprised by the high wholesale price,   too many would   enter and they would end up each producing at too small a scale. They foresee the high wholesale price, though, so just enough enter to  earn zero profit at the minimum efficient scale. But $Q$ is the result of an    inefficiently small industry. If it tried to produce the first-best output with $n$ too small, the cost would be   inefficiently high,    despite minimizing the cost of producing the second-best output. 
 
\end{Large}
%---------------------------------------------------------------------------------
\newpage
\begin{center}
{\bf Proposition  2 }
\end{center}

\noindent
 {  If the    monopolist  cannot precommit to his    output before  competitive firms enter the market for a complementary good, his output  and profit will be lower.  

 As consumer surplus will also be lower, total surplus is lower than in a vertically integrated monopoly or a monopoly that  could commit to  future output. }
 

%---------------------------------------------------------------------------------
\newpage
\begin{center}
 {\sc  Observations}
\end{center}

\noindent
 1. ``Monopoly'' here just means the supplier has downward-sloping demand, not literal monopoly. 

\noindent
 2. This doesn't depend on retailers being perfectly competitive. Monopolistic competition in retail  would work out the same way.  


%---------------------------------------------------------------------------------
\newpage
\begin{center}
 {\sc   The Complements Model}
\end{center}

 .  

Think of the benefits to the monopolist from successful deception. Suppose the monopolist promises to produce $Q^*$ once the secondary market opens up.  After the market does open up, the monopolist will   break his promise. He has already gotten the entry he wanted to stimulate to help his   own good would sell more, and the entrants will not leave (in the short run) even if their revenues do not cover their entry costs. So the monopolist will produce more than  $Q^*$. Foreseeing this, fewer firms would enter the competitive market. 


 %---------------------------------------------------------------------------------
 
%---------------------------------------------------------------------------------
\newpage
\begin{center}
 {\sc      This may look like double marginalization, but it is not. }
\end{center}


 Double marginalization  is the problem of successive monopolies, where the monopoly markups of the wholesaler and the retailer result in a price higher than what a vertically integrated monopolist would charge a subsidiary.

 Here, the retailers have no market power and would earn zero profits in any equilibrium. Unlike in double marginalization, two-part tariffs would not solve the monopolist's problem, but commitment to a one-part wholesale price would. As with double marginalization, however, vertical integration would solve the problem, reducing the wholesale and retail prices and increasing the combined profits of the sellers. 

%---------------------------------------------------------------------------------
\newpage
\begin{center}
 {\sc    This is not the usual problem of overpricing of complements. }
\end{center}

 It is well known that if two firms  that sell complementary goods each have market power, they can increase their profits by merging and reducing their prices. The problem they have is that they both mark up their prices above marginal cost, and when one of them increases his mark-up, he captures the entire gain but inflicts a negative externality on the other monopolist. 


 Here, in  the  competitive market  firms are charging a price equal to marginal cost, and price-taking to boot, so there is no margin over cost to adjust and make too high--- the profit   margin is zero. Rather, the  monopoly would like to commit to a low price to  get more firms to enter the other market.   When fewer enter,     it shifts in the supply curve for the competitive good   to the detriment of demand for the monopolist's   good. 
 
  

%---------------------------------------------------------------------------------
\newpage
\begin{center}
 {\sc  This is not  ``secret quantity expansion'':  the monopolist  secretly selling more to some retailers. Hart \& Tirole (1990),  O'Brien  \& Shaffer (1992).  }
\end{center}
 
 This would  ``cheat'' the other retailers who paid a  wholesale price based on their belief  that the retail price would be high because of the limited market quantity.  Foreseeing this opportunism, the retailers will not pay a high price in the first place, dealing a serious blow to the monopolist's profits, a problem  reminiscent of the Coase Conjecture's competition between  a monopolist present and future selves. 

With constant MC, the retail
  price would be above MC, making retailers 
eager to be allowed to buy and resell more of the good and creating the temptation
for secret quantity expansion by the supplier. 

Here, with rising MC, any retailer who buys more will have higher MC and 
  hence weaker demand than
the others. 

[explain better]


  

%---------------------------------------------------------------------------------
\newpage
\begin{center}
 {\sc Two other kinds of models of vertical relations}
\end{center}

  First is the opportunism problem created by unobservable actions. Telser (1960) pointed out that if retailers provide unobservable consumer services with positive externalities, the monopolist can use resale price maintenance to encourage competition in providing those services and discourage opportunistic low-service discount outlets  (see, too, the discussion in the survey by  Lafontaine  \&  Slade [2007]).   Here, neither monopolist nor retailer will be taking unobserved actions. 

Second,   common agency, the idea that upstream oligopolists can use contracts with retailers to soften competition between them. This literature began with Bernheim \& Whinston (1985) and Bonnano \& Vickers (1988); a recent example is Hunold \& Stahl (2016). Relaxing competition will be irrelevant in competitive holdup, however, since the model will contain only a single monopolist and retailers will be price-takers.  

  
 
  
 
 
 
 \end{Huge}
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