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\begin{LARGE} \begin{center} 
 {\bf The  Kleit Oligopoly Game:  A Classroom Game for  Chapter 14}
\footnote{6 February 2006.  Eric Rasmusen, Erasmuse@indiana.edu.
Http://www.rasmusen.org.    }
 \end{center} \end{LARGE}

  The  widget industry in  Smallsville  has $N$ firms.  Each firm
produces 150 widgets  per month.  All costs are fixed, because labor
is contracted for on a yearly basis, so we can ignore production cost
for the purposes of this case.   Widgets are perishable;   if they are
not sold within the month,  they explode in flames.

  There are two   markets for   widgets,   the  national  market, and
the local market.  The price in the national market is \$20 per
widget, with the customers paying for delivery, but the price in the
local market depends on how many are for sale there in a given month.
The price is given by the following market demand curve: $$ P = 100 -
\frac{Q}{N}, $$ where $Q$ is the total output of widgets sold in the
local market. If, however,  this equation would yield a negative
price, the price is just zero, since the excess widgets can be easily
destroyed.

    \$20 is the {\bf  opportunity cost } of  selling a widget
locally-- it is what the firm loses by making that decision.  The
benefit from the decision depends on what other firms do.  All firms
make their decisions at the sme time  on whether  to ship widgets out
of town to the national market. The train only comes to Smallsville
once a month, so firms cannot retract their decisions.   If a firm
delays making its decision till too late, then it misses the train,
and all its output will have to be sold in Smallsville.

\bigskip

 \noindent
 {\bf    General  Procedures }

  For the first seven months,  each of you will be a separate firm.
You will write down two things on an index card: (1) the number of the
month, and (2) your  LOCAL-market   sales  for that month.  Also
record your local and national market sales  on your Scoresheet.  The
instructor will collect the index cards and then announce the price
for that month.  You should then calculate your profit for the month
and add it to  your cumulative total, recording both numbers on your
Scoresheet.

 For the last five months, you will be  organized into five different
firms. Each firm has a capacity of 150, and submits a single index
card. The card should have the number of the firm on it, as well as
the month and the local output.  The instructor will then calculate
the market price, rounding it to the nearest dollar to make
computations easier.  Your own computations will be easier if you pick
round numbers for your output.

  If you do not turn in an index card by the deadline, you  have
missed the train and all  150 of your units must be sold locally. You
can change your decision up until the deadline  by handing in  a new
card noting both your old and your new output, e.g., ``I want to
change from 40 to 90.''

 \newpage

\noindent {\bf   Procedures Each Month}

1.   Each student is one firm.  No  talking.

2. Each student is one firm.  No talking.

3. Each student is one firm.  No talking.

4. Each student is one firm.  No talking.

5. Each student is one firm.  No talking.

6.  Each student is one firm. You can talk with each other, but then
you write down your own output and hand  all outputs in separately.

7.  Each student is one firm. You can talk with each other, but then
you write down your own output and hand  all outputs in separately.

\vspace*{12pt}

 

8.  You are organized into  Firms 1 through 5, so N=5.  People can
talk within the firms, but firms cannot talk to each other.  The
outputs of the firms are secret.

9.   You are organized into  Firms 1 through 5, so N=5.  People can
talk within the firms, but firms cannot talk to each other.  The
outputs of the firms are secret.

10. You are organized into  Firms 1 through 5, so N=5.   You can talk
to anyone you like, but when the talking is done,    each firm  writes
down its output secretly  and hands it in.

 11.  You are organized into  Firms 1 through 5, so N=5.   You can
talk to anyone you like, but when the talking is done,    each firm
writes down its output secretly  and hands it in. Write the number of
your firm with your output. This number will be made public once all
the outputs have been received.

\begin{LARGE} \begin{tabular} { | l  |  l | l| l | l| l  |   }
\hline 11   & {\bf $\;\;\;\;$ }  &{\bf $\;\;\;\;$ } & {\bf $\;\;\;\;$
} & {\bf $\;\;\;\;$ } & {\bf $\;\;\;\;$ }    \\
\hline \end{tabular} \end{LARGE}

 12. You are organized into  Firms 1 through 5, so N=5.   People can
talk with anyone they like,  and arrange to  submit outputs jointly if
they like.    Write the number of your firm with your output. This
number will be made public once all the outputs have been received.

\begin{LARGE} \begin{tabular} { | l  |  l | l| l | l| l |   }
\hline 
12   & {\bf $\;\;\;\;$ }  &{\bf $\;\;\;\;$ } & {\bf $\;\;\;\;$
} & {\bf $\;\;\;\;$ } & {\bf $\;\;\;\;$ }     \\
\hline \end{tabular} 
 \end{LARGE}

\bigskip 

\noindent {\bf   Winners}

   The instructor will congratulate  (a) whichever student has the
highest profits over  all the months; (b)  each member of the team
with the highest profits over the team   months;  and (c) each member
of the class  if the average  profit   over  all the months  exceeds
3,700 (an average price of 30).

 



\newpage

  \noindent
  {\bf   Analysis}

    If firm $i$  sells $Q_i$ locally, then local sales will be
$\sum_{i=1}^N Q_i$.   Industry revenue  will be the sum of local and
national-market  revenues:
\begin{equation} 
  {\displaystyle \Pi =      \left(  \sum_{i=1}^N Q_i   \right)
\left(   100- \frac{    \sum_{i=1}^N Q_i }{N}  \right) +  \left(
150-  \sum_{i=1}^N Q_i  \right) (20).}
 \end{equation}

     If this is maximized (which can be done using calculus), then the
average  sales per firm is 40  and the price is 60, for any number N
of firms.

 (Using calculus: Industry revenue if each firm produces $q$ is
$Nq(100- Nq/N) + N(150-q)(20)$, which equals  $100Nq -Nq^2  + 3000N -
20Nq$ .  Setting the derivative with respect to $q$ equal to zero
yields  $100N -2Nq  - 20N =0$, which  can be simplified to $q=40$).

   Suppose, however, that  $(N-1)$ of the firms are  selling 40 each,
but Firm N  is still making up its mind.  Firm N faces the following
demand curve, which substitutes 40 for the output of each of the other
firms: $$ P = 100 - \frac{40(N-1)}{N}    -\frac{ Q_N}{N} = 100-40 +
\frac{40 }{N}  -\frac{ Q_N}{N} =  60   -\frac{ Q_N-40}{N}. $$ If $N=
5$,  firm $N$ maximizes its profit by  choosing $Q_N=120$, something
which also needs calculus to calculate.  This means that  it may be
difficult to  create an environment in which each firm  sells just 40
locally.

Table 2  shows what happens at various levels of  local sales per firm
if all firms maintain the same level, given that the national price is
fixed at 20 and a firm's national sales are 150 minus its local sales.

 \begin{tabular}{l | llll} \hline & {\bf Local   }& {\bf  Local-Market
}& {\bf   National-Market } &  {\bf Total  }\\
 {\bf Local Sales  } & {\bf Price }& {\bf    Revenues}& {\bf Revenues}
&  {\bf  Revenues}\\ {\bf  per Firm  } &  & {\bf  per Firm  }& {\bf
per Firm  } &  {\bf per Firm  }\\
  \hline 
0 & 100 & 0 & 3,000 & 3,000\\ 10 & 90 & 900 & 2,800 & 3,700\\ 20 & 80
& 1,600 & 2,600 & 4,200\\ \hline 30  & 70 & 2,100& 2,400& 4,500\\ {\bf
40}  &{\bf  60 } &{\bf  2,400} &{\bf  2,200}  &{\bf 4,600} \\
 \hline 50 & 50 &2,500 & 2,000 & 4,500\\ 
60 & 40 & 2,400 & 1,800 & 4,200\\
 70 & 30& 2,100 & 1,600& 3,700\\ \hline 80& 20& 1,600 &1,400& 3,000\\
90 & 10& 900 &1,200& 2,100\\
 100 & 0 & 0 & 1,000 & 1,000\\
 \hline 
\end{tabular}
 \begin{center} TABLE 2:   PRICES AND OUTPUTS
  \end{center}

\newpage 

\begin{Large}

\noindent {\bf Cournot Equilibrium}

To find the Nash equilibrium (the Cournot equilibrium, this model),
set up the payoff function of the individual firm.   Suppose all the
other firms choose local sales of  $q$, but the first firm chooses
$q_1$.  Firm 1's payoff is then $$
\begin{array}{ll} \pi_1 & = Pq_1 + 20*(150-q_1) =  (100 - \frac{Q}{N})
q_1+ 20*(150- q_1) \\ &=    100 q_1 - \frac{((N-1)q + q_1)q_1}{N}+
20*(150-q_1)  \\ &=   100 q_1 -  \frac{ (N-1)q  q_1}{N}   - \frac{
q_1^2}{N} + 3000 - 20  q_1. \end{array}
$$

 Differentiating with respect to $q_1$ yields the first order
condition
 $$ \frac{d \pi_1}{dq_1}   =         100    - \frac{ (N-1)q  }{N}   -
\frac{   2q_1 }{N}  +0 -20= 0, $$ which can be solved to yield $80 N =
(N-1)q + 2q_1$.   If we furthermore   guess  that the equilibrium is
symmetric, so  $q= q_1$, then  we can write $80N = (N+1 )q  $, and $q
= \frac{80N}{N+1}$.

 If $N =1$,   the firm produces $q = \frac{80}{2} =  40$, and the
price is   60.

 If $N =2$,  each firm produces $q = \frac{160}{3} =  53\; 1/3$,
and the price is   $46 \;2/3$.\\
 If $N =3$,  each firm produces $q = \frac{240}{4} =  60$, and the
price is   $40$.\\
If $N =4$,   each firm produces $q = \frac{320}{5} =  64$, and the
price is   $36$.\\
 If $N =5$,   each firm produces $q = \frac{400}{6} =  66\; 2/3$,
and the price is   $33 \;1/3$.

  $N =20$:   each firm produces $q = \frac{1600}{21} =  76.19$;
  price =23.8.

  $N =40$:  each firm produces $q = \frac{3200}{41} =  78.05$;
  price = 21.95.

  $N =400$:   each firm produces $q = \frac{32000}{401} = 79.80$;
 price = 20.2.

\end{Large}

       \newpage

\begin{center}
 {\bf Instructor's Notes} 
\end{center}

 This game is adapted from  one developed by   Professor Andrew Kleit
of Louisiana State University.  A version was published as Meister,
Patrick, `` `Oligopoly'  - An In-Class Economic Game,''  {\it Journal
of Economic Education}, 30, 4:383-391  (Fall 1999).

  
 

  

  Equipment: \\
 1.  A buzzer (optional) \\ 
2. Index cards \\ 
3. A calculator or computer (Google will do calculations for you)\\

 Students are very unlikely to achieve an average revenue of 3,700 (which means an average price of 30). 

 Rather than just congratulating the students, you may wish to give real rewards--- a coffee mug  for the individual winner, cans of pop for the five-firm winners, and donuts for the entire class if they achieve over a threshold of 3,700. Since students often make arithmetic errors in adding up their scores, wait to give the prizes till the next class. 

 I can get through all the rounds in a 75-minute class, but it is a tight squeeze. 

  The first rounds can go by very quickly. They are so the students
will learn  how the demand curve works. Students will usually start
with cautiously low local outputs.    Allow more time per month for
the later months,  since the students will spend more time talking. 

You may wish to stop  and discuss strategies after the first 5 rounds, which are silent. 

  I like to leave an overhead   on the projector during the game with
what happens in each of the 12 months.  I write the prices on it month by month.

Make the point that in Month 12, cartels are legal, but not
inevitable. There is still  a holdout problem. If 4 firms agree to
produce 40 each, the 5th firm will hold out and produce 80.   Or,  you
might find that  total cartel output is 200, but one firm insists that
the others each produce 30 and it produce 80.

 Also make the point that the Nash equilibrium  price is not 20, but
33   1/3.  Explain that this is because if a firm deviates and sells
more, then it will drive the price down enough that its own profits
will fall too.  If, however, firms had unlimited capacities  {\it and}
they chose prices instead of quantities, the result would be
different.

\bigskip

\noindent
  Miscellaneous Notes.

 1. Professor John Maxwell says that in the Kleit game, he has found
prices falling with fewer firms. Variance of outputs falls too. The
reason:  The risk takers dominate their new teams.

2. This game works equally well, or better, if some students have
played it before. Most likely, the experienced students will all
choose high outputs in the first period, leading to low prices and
losses for them. You can make the point that a moderate amount of
sophistication can be worse than none at all.

3. Explain to the students that ordinarily just two local amount might
be optimal:  either 0 or 150. If a firm thinks that the price will be
more than 20 (even just 21) if it chooses 150, then it should  choose
150. If the firm thinks the price will be less than 20, it should
choose 0. Intermediate amounts are best responses only if the firm
thinks that its own sales would tip the price from being above 20 to
being below.

Humans have a psychological tendency to choose intermediate amounts
even when bang-bang solutions are optimal. This can also result from
risk aversion, but the  moderation  effect, called ``probability
matching''  is distinct, and well- known in psychology (see Herrnstein
\& Prelec [1991]).

4. One benefit from being able to talk is that students who understand
the situation can explain it to those who do not. Some students will
realize that the cartel optimum  is when each firm produces 40, but
others will get the  number  wrong, and not realize it till the open
discussion begins.

  R. J. Herrnstein and D. Prelec (1991) ``Melioration: A Theory of
Distributed Choice,'' {\it Journal of Economic Perspectives}, 5(3):
137-156  (1991).

Below are some pages you may wish to use for handouts or overhead slides. 



\titlepage

  \oddsidemargin -.8in 

\begin{Large}

 \vspace*{-64pt}

\noindent 
Your Name:\\ 
Your Firm: \\


 
 \begin{center} 
\begin{tabular} { | l ||  l| l  | l|| l    |l  | l|| l|   l|  }
 \hline 
 Month & Your& Local &Local    &Your &  Nat. &  National & Total &
Cumulative    \\
 & Local  &  Price & Revenue  &National   &Price  &  Revenue &
Revenue  & Revenue   \\
   & Sales &    &    &Sales  &   &    &     &    \\
    &   &    &    &   &   &    &     &    \\
 \hline 
\hline
  1  &   &    &    &   &   &    &     &    \\
\hline
 2  &   &    &    &   &   &    &     &    \\
\hline
 3  &   &    &    &   &   &    &     &    \\
\hline
 4 &   &    &    &   &   &    &     &    \\
\hline
 5  &   &    &    &   &   &    &     &    \\
\hline
 6  &   &    &    &   &   &    &     &    \\
\hline
 7  &   &    &    &   &   &    &     &    \\
\hline
 \hline
\hline
 8  &   &    &    &   &   &    &     &    \\
\hline
 9  &   &    &    &   &   &    &     &    \\
 \hline
 10  &   &    &    &   &   &    &     &    \\
\hline
 11  &   &    &    &   &   &    &     &    \\
 \hline
 12  &   &    &    &   &   &    &     &    \\
  \hline 
\hline
 \end{tabular}\\

\vspace*{24pt}

 Table 1: Scoresheet \end{center}

 \end{Large}

The local price depends on local output. Cumulative revenue is the sum
of that month's revenue plus every preceding month's revenue.

\titlepage

\begin{Large}

Table 2  shows what happens at various levels of  local sales per firm
if all firms maintain the same level, given that the national price is
fixed at 20 and a firm's national sales are 150 minus its local sales.

 \begin{tabular}{l | llll} \hline & {\bf Local   }& {\bf  Local-Market
}& {\bf   National-Market } &  {\bf Total  }\\
 {\bf Local Sales  } & {\bf Price }& {\bf    Revenues}& {\bf Revenues}
&  {\bf  Revenues}\\ {\bf  per Firm  } &  & {\bf  per Firm  }& {\bf
per Firm  } &  {\bf per Firm  }\\
  \hline 
0 & 100 & 0 & 3,000 & 3,000\\ 10 & 90 & 900 & 2,800 & 3,700\\ 20 & 80
& 1,600 & 2,600 & 4,200\\ \hline 30  & 70 & 2,100& 2,400& 4,500\\ {\bf
40}  &{\bf  60 } &{\bf  2,400} &{\bf  2,200}  &{\bf 4,600} \\
 \hline 50 & 50 &2,500 & 2,000 & 4,500\\ 
60 & 40 & 2,400 & 1,800 & 4,200\\
 70 & 30& 2,100 & 1,600& 3,700\\ \hline 80& 20& 1,600 &1,400& 3,000\\
90 & 10& 900 &1,200& 2,100\\
 100 & 0 & 0 & 1,000 & 1,000\\
 \hline 
\end{tabular}
 \begin{center} TABLE 2:   PRICES AND OUTPUTS 
  \end{center}

 \end{Large}

\end{document}
