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  \begin{center}
 \begin{Large}
    {\bf  The    Kleit Oligopoly Game } \\
             \end{Large}
           March 14, 2002\\
                                      Eric
Rasmusen\footnote{Eric
Rasmusen,   Indiana University,
Kelley School of Business,    BU 456,
  1309 E 10th Street,
  Bloomington, Indiana, 47405-1701.
  Office: (812) 855-9219.  Fax: 812-855-3354. Email:
Erasmuse@Indiana.edu.  Web: Php.indiana.edu/$\sim$erasmuse.
}
\end{center}

 This game is adapted from  one developed by   Professor
Andrew Kleit
of  Pennsylvania State University.


\noindent
 {\bf The Situation}


  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.




\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. This card should have the
number of the firm on it, as well as the month and the local
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.''

 The instructor will calculate the market price, rounding it
to the nearest dollar to make computations easier.  Note
that your own computations will be easier if you pick round
numbers for your output.





\bigskip

\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.

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.  (This
month counts for the   group bonus)

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.
(This month counts for the group bonus)

 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.  (This month counts for the   group bonus)

 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.   (This month
counts for the group bonus)


 \noindent
{\bf   Prizes}

 To add spice to the game, there will be three kinds of
bonuses.

First, whichever student has the highest profits over  all
the months  receives  4 extra points on the  next quiz.

Second, each member of the team with the highest profits
over the team   months receives 2 extra points on the  next
quiz.

 Third,  if the average  profit   over  all the  months
exceeds 4,000,  each member of the class will receive  3
extra points on the next quiz.


\begin{tabular} { | l  |  l | l| l | l| l  | l|l|}
\hline
  Month & Firm 1 & Firm 2 & Firm 3 & Firm 4 & Firm 5 &
Local Price \\
\hline
 1 &  -- &--& --&-- &--&  \\
 2 &  -- &--& --&-- &--&  \\
 3 &  -- &--& --&-- &--&  \\
4 &  -- &--& --&-- &--&  \\
 5 &  -- &--& --&-- &--&  \\
6 &  -- &--& --&-- &--&  \\
7 &  -- &--& --&-- &--&  \\
\hline
8   &   & &  &  &    &  \\
\hline
9   &   & &  &  &    &  \\
\hline
10   &   & &  &  &    &  \\
\hline
11   &   & &  &  &    &  \\
\hline
12   &   & &  &  &    &  \\
\hline
\end{tabular}\\
\begin{center}
  TABLE 1:  GAME RESULTS: OUTPUTS AND PRICES
\end{center}

\newpage
\begin{large}

\noindent
 Your Name:\\
 Your Firm: \\
Total Cumulative Revenue:\\
(months 1-7 plus 8-12)\\

 \vspace*{24pt}




\hspace*{-88pt}
\begin{tabular} { | l  |  l| l  | l| l    |l  | l|  l|}
\hline
  Month &  1 &  2 &   3 &  4 &  5 &  6 & 7   \\
\hline
\hline
Your Local Sales & \hspace*{48pt}  &\hspace*{48pt}
&\hspace*{48pt} &
\hspace*{48pt} &    \hspace*{48pt} & \hspace*{48pt}
&\hspace*{48pt}
\\
\hline
 Local Price &   &    &     &    &    &    &     \\
\hline
Local Revenue &   &    &     &    &    &    &     \\
\hline
 \hline
   Your National Sales  &   &    &     &    &
&   &   \\
 \hline
National Price &  20  &  20   & 20     &20     &20     & 20
& 20
   \\
\hline
National Revenue &   &    &     &    &    &    &
\\
\hline
 \hline
 Total Revenue&   &    &     &    &    &    &       \\
\hline
 \hline
Cumulative Revenue&   &    &     &    &    &    &
\\
\hline
 \hline
\end{tabular}\\

\vspace*{36pt}


\hspace*{-88pt}
\begin{tabular} { |l|    l |   l | l| l|  l| }
\hline
  Month & 8 & 9 & 10 & 11 & 12 \\
\hline
\hline
Your Local Sales &  \hspace*{60pt}&\hspace*{60pt}
&\hspace*{60pt}
&\hspace*{60pt}    &\hspace*{60pt}
\\
\hline
 Local Price     &   &   &   &   &   \\
\hline
Local Revenue    &   &   &   &   &   \\
\hline
 \hline
   Your National Sales       &   &   & &   &   \\
 \hline
National Price
& 20   & 20   &  20  & 20   & 20   \\
\hline
National Revenue    &   &   &   &   &
\\
\hline
 \hline
 Total Revenue   &   &   &   &   &   \\
\hline
 \hline
Cumulative Revenue   &   &   &   &   &
\\
\hline
 \hline
\end{tabular}\\

\begin{center}
  TABLE 2:  SCORESHEET
\end{center}

 \end{large}

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


\newpage


  \noindent
{\bf   Analysis}

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

     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.

The following table  shows what happens at various levels of
local
sales per firm if all firms maintain the same level.

 \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\\
 40 & 60 & 2,000& 2,200 &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 & 0 & 0\\
 \hline
\end{tabular}
\begin{center}
  TABLE 3:   PRICES AND OUTPUTS
\end{center}

\newpage
\noindent
 {\it 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 $N =5$, then each firm produces $q = \frac{400}{6} =
66.67
$, and
the price is   33.33. (I will round all decimals to two
places.)

 If  $N =20$, then each firm produces $q = \frac{1600}{21} =
76.19$,
and the price is  23.8.

 If $N =40$, then each firm produces $q = \frac{3200}{41} =
78.05$,
and the price is  21.95.

 If $N =400$, then each firm produces $q = \frac{32000}{401}
=
79.80$, and the price is  20.2.


       \newpage

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

  Equipment: \\
 1.  A buzzer \\
 2. Index cards \\
 3. A calculator\\

  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 there
will be discussion then.

  I like to leave an overhead   on the projector during the
game with what happens in each of the 12 months. While the
students are making their decisions, I write a table up on
the board to show the outputs and prices. I put up the
prices month by month, and then go back at the end of the
class and insert the individual firm outputs.

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.


\noindent
 Miscellaneous Notes.

 1. Professor  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.

  (Article from WSJ about how women individual  investors
make more money than men, because they churn less.)


      \end{document}










