section 6: monopoly, oligopoly, and game...
TRANSCRIPT
Section 6:
Monopoly, Oligopoly, and Game Theory
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Answer all of the following five questions.
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1. A monopolist faces an inverse demand curve given by p = 10 - y where p is the price of output and y is its output. Input prices are fixed throughout this problem atw .~ 1, and with these fixed input prices; , this firm's· cost function can be written simply as c(w, y) = eBY where a is' a positive parameter. Denote thefirm's,optimal output byy*. You do not have to calculate, ' the numerical value of y*. ,. (In fact, it is impossible to calculate the numerical vallie of y* exactly.) Instead, find, as a function of y* , 'how the firm's output changes when the parameter a changes by a small I II
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Answer all of the following five questions.
1. Suppose two firms form a Stackelberg duopoly. Suppose the market demand curve is p = 10 - Q where p is priCe and Q is mark~t quantity. Suppose (for simplicity's sake) that each firm's cost of production is zero.
(a) Find the profit earned by each firm in the Stacltelberg equilibrium.
(b) Find each firm's profit function ..
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. Question 3. Suppose two firms are Stackelberg duopolists, with the leader denoted by '1' and the follower by '2'. Let the output of firm i be qi, wh~re i equals 1 or 2. Suppose the cost function of each firm is given by c . qr where" c is a constant which is the same for both firms.
How will ql and q2 respond when crises?
(It may be a good idea not to start by trying to so~ve this problem in all its detail, but to "get a general, or abstract, idea of what is going on first. These' steps will earn you points even if you cannot solv~ the question in full detail.)
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... Question 2. Suppose an industry is a duopoly which fS in a (quantity-setting) Stackelberg equilibrium~
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Let the quantity of output produced by a Stackelbergieader be ql. Suppose, the government imposes a tax on the Stackelberg leader; the amount of ' tax
/,collected is rql. . .. '
·Will increases in r increase or decrease the profit of the StackelberK.follower:, (or can you not tell)? (Notice I asked about the ~,?llower, not the leader.)
How would your answer change if the industry 'were a (quantity-se~~ing) . Cournot duopoly instead of a (quantit:y:-setting) Stackelberg duop<?ly? ..
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5. The demand curve for a product produced by a duopoly is p(Y) _ 10 ~ Y where p is price and Y is aggregate output. The output '6f 'firm 1 is denoted Yl and its total costs are y~. The output of firm 2 is denoted Y2 and its total costs are y~. Assume the fir,ms in this duopoly play a quantity-setting game, not a price-setting game.
(a) Find each firm's reaction function.
(b) Find the Cournot equilibrium output of each firm.
(c) Find the Stackelberg equilibrium output of each firm.
(d) Find ~he cartel output of each firm. Denote these outputs by y{ and y~, where the "j" refers to "joint-profit-maximizing."
(e) Suppose firm 1 produces yf. If firm 2 wishes to cheat on the cartel agreement, how much will it produce? '
(f) Do not perform any mathematic,alcalculations in this part of the ques.tjon. I . . . "III
Denote th~ profits earned by the two firms in the Cournot equilibrium (part (b) above) by 1f} and 1f2 .
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6.A Stackelberg duopoly faces a market demand curve of the form p .= I, A ~ y where y is industry output, p is market price, and A is a given
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(b) . D~es the profit of the firm which is the' "follower" go up or down when' A increases?
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6710 Required Question
Suppose that the only way to produce commodity "q" is to use the single input "x" according to the production function "f." Suppose exactly two firms produce q. Suppose they are in a static Cournot equilibrium.
Assume that the price of x is "w." The firms in this problem cannot affect w. Assume that the inverse market demand curve for q is given by p(q).
(a) Suppose that w changes. What is the corresponding change in the amount of x which a firm buys? How is this related to
p" l' l' f + p' 1" f + 2p' f' l' + p 1" - p" f' l' f - p' f' l' ~~~~--~~~--~~--~~~~~~~~~~7 (p" f' f' f + p' f" f + 2p' f' f' + pf")2 - (p" f' f + p' f')2(1')2 .
(b) If w changes, how does the profit of each firm change?
You do not have to determine the sign of complicated expressions like the fraction given in part (a).
Hint 1. In answering part (a), I found it easier for most of the problem to express profit as something like 7r (though I did not use exactly 7r) instead of using something much more detailed.
Hint 2. Should dq be zero?
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3. The demand curve for a product produced by a symmetric duopoly is p(Y) = 14 - Y where p is price and Y is aggreg?te output. The output of firm 1 is denoted Yl and its total costs are 2Yl' The output of firm· 2 is denoted Y2 and its total costs are 2Y2. Assume the firms in this duopoly play a quantity-setting game, not &. price-setting game. Assume the game is played only once ,and that it is a simultaneous-ID?Ve game., ,
Fill in the following table describing the game. It is up to you to figure out what ','cooperate" and "defect" means in this situation, and to calculate what the appropriate payoffs are. Hint: half of the eight numbers in the table should be integers.
Firm 2 cooperate, defect
"
coo.perate 1 I' defect· 1-. ----+-----1. Firm 1
I I, '
r \"1\ Of ! f )(C\ V\f\
t C1C1 &
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:= )'1y, .~ 1,2 - y~ y, _ 2Yl .""" . I J y, _{,2. -/~ y, . J1~,~~-li:J
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3. Consider the following one-round (not repeated) game.
Row. top bottom
L----!-_-1----!--l
(a) Find the mixed strategy Nash Equilibrium.
(b ) Find the expected payoffs.
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L· R
0.) Row', f"'J,!....- T S H.. /. (t-t (wI. «e. It ~J r II a-..f/..Ll ;"j,~~; /.fIlet
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.of .fk f~b4h(k.f)t~ ft.k. cP{v,~'" Wl-t(pl'j' '-~d. R..:) :
~;t~ r -r (-3-rr.( + ~ iff<) +- r B (l:TTL Jr 47fR ) s. t. I' -r f P B ~ I , fr, p&
I
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fi/Y/t I e,xaW\ lCf96
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''1) _1 p_!:. IT ~ 3,' IS' 3.'
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::- fl ('3) + Fn,,{ 3 )
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Question _~. Consider the following ,game, called "Chicken." Qva I rRr~ EXllW\
,q~~
-- ROw stay swerve
Column stay swerve
-3,-3 6,1 1,6 I 4,4 Qu-t'5ti'rN' d.
Suppose this game is repeated an infinite'number'of-times. (!J) The 'Tit-for-tat;' strategy is defined as follows: swerve at t, = 1; then, play at date t whatever your opponent played at date t - 1. '
I
a) What is the cmtcolIl-e of the Tit-for-Tat strategy?
b) Is tbis outcome a Nash Equilibdum,? ,
c) If tbe outcome is a Nash, Equilibrium, is the N'a:sh Equilibrium sub game , - perfect?
111 1 , I
I
, ;
, I
Qvttl \'~vr~ EXC(ffl \qq¥
4f\~wer d.
: "Tit-for-Tat"
Str~tegy: "Tit-for-tat:" Swerve at t= 1; then, play at date t whatever " , your opponent played at date t"';' 1.
, '
" , Outcome: (sw, sw)
Is this a Nash equilibrium?
• If Row doesil~t deviate, Row gets: 4,4, 4, ~, ...
• If Row deviates, Row gets:
1. 6 (at, sw)'(Row deviates, Column plays i'tit-for-tat")
2. 1 (sw, st) (Row plays "tit-for-tat," Column plays "tit-for.;.tat")'
3. 6 (st, sw) (Row plays "tit-for-tat," Column plays "tit-for-tat")
4. 1 (sw, st) (Row plays "tit-for-tat," Column plays "tit-Jor-tat")
Etc. '
• Row,will not deviate when:
By symmetry, it is only necessary to compare the first two terms; . So Row will not deviate when: " ,
: 4 1 4+-->6+--
, l+r l+r
r < 1/2 = 0.5.,
• Column: Since the payoff inatrix is sy~metric and since both players play the same strategy, the calculations for Column's deviating would be the same as the above calculations for Row's deviating. '
• Conclusion: This is a Nash Equilibrium strategy whenever the players' discount rates satisfy r < 0.5.
Is the Nash equilibrium subgame perfect?
Suppose that at t = 1, Column deviated. Should Row deviate at t = 27
I "
t,
t 1 2 3 4
R C R gets I R C R gets
C deviated sw st qlAL~ F>((\tff\ tit-for-tat st sw 6 R deviate's sw sw 4 tit-for-tat ,sw st 1 tit-for-tat sw sw 4 lqqg: tit-for-tat st sw 6 tit-for-tat sw sw, 4
A(i~w.er 9- Uvit. ..
• Row will not deviate when:
1 6 l' 4 4 4 6+ 1 + r +(1 + r)2 + (1 +r)3 + .. > 4+ 1 + r + (1 + r)2 + (1 + r)3 + ....
As we just saw, this o'ccurs if and only if
r> 1/2 = 0.5.
• Column: Since the payoff matrix is'symmetric and since both players play the same strategy, the calculations for C~lumn would be the same as the above calculations for Row. "
I
• Conclusion: This condition for subgame perfection, r > 0.5, contradicts the condition needed for Nash Equilibrium, ~ < 0.5. Therefore, this equilibrium is notsubgame perfect.
, I
'111 1 , I
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Question 3 .. Conside! the following one-shot (that is, not repeatE}d) game. '
Column heads tails
, I
Sketch a graph of Row's expected utility (in equilibrium) versus' the vahie of (3. Consider both positive and negative values of: (3 .
. Sketch a graph of Column's expected ut~lity (in eqUilibrium) versus the value of (3. Consider both positive and negative values of (3:
Hint: mixed-strategy equilibria.
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~ It If T ~ 'l{ It I, "
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1lfJ\.J . T 4/~,..,~ It H -t. T I, t. II
) .' .'
. 0/,d /ril{ Col" ....... do rf R""p/~s J./? I .... f1..7 C,;,Je, {"lu ... ",,1A ~
o h'J flay,lt~ J.I tJ.".J (3 b:J pl"'"d T. /I ;3 < D, I' l~ .kJ II I~ ~ .. .;:;,.. Col .... " / -/L, ~s L."I< l:le H -r .".50 (II, H)
H CCQ)~ 0; e t .', ~ .
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., fht-b. J- 'G- .
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2. Suppose two players are playing the following game: C\)kS{,Oi!\ c1
Top Row Middle
Bottom
Column Left Right 5,1 4,0 6,0 3,1 6,4 4,4
(a) Identify all the pure-strategy Nash equilibria in the game. Ex-· tensive explanations are not necessary; if you are confident about your answer, it is enough to put arrows leading ~ut of each table entry which is not a Nash equilibrium, indicating which player will deviate and how. If you are not very confident about your answer, it might be a good idea to explain some more.
(b) Show that in this game, the equilibria which remain after "iterative deletion of weakly dominated strategies" depend on the
. order of deletion .. Hint: eliminate only one strategy per player in . each iteration.
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2. :T~epayo:ff matrix for a game called "Chicken" looks like this.
Column stay swerve
Row
(a) Find ~he:l:rruxed strategy equilibrium· .
. (b) Given' that .Row plays his equilibrium mixed strategy, find I the a off to Column If Column plays: p Y'
1. stay; .. 11.
iii.
.( c)
swerve; Column's. equilibrium ~xed strategy .
Consider the following gam~:
II;
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ColumIi left right.
'Row '. uP. x,x 2,0 down 0,2 1,1
~~d ~~~ fX4M l0i4l1
q lJ-t5+10v\ :l (!J)
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Ifx < 0 then this game is "Chicken" and not ~Prisoner's Dilemma"; . why? For what· values of x would this be aP;risoner's Dilemma game? '11'
Hint.: the following game is ,a .Prisoner's Dileinma game; does it have
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dominant strategies?
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Row
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cooperate I 3, 3 I 0, 4 I . defect 4,0 1,1
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2; In each ofthe three parts of this question you ar~ given one symmetric game in strategic form. Suppose each game is played sequentially instead of simultaneously. Find the extensive form of each game and
" determine whether it is better to" be the leader" or the follower in each
" game. " . (a)" ""
(b)
(c)
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Row
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Top Bottom
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a¥q;W\p;;;' 6. Consider the following game: aooo 0D
~~
Row up
down
(a) What is the Nash Equilibrium of the one-round game?
(b) Consider the following strategy for the infinitely-repeated game: if the number of past,rounds when (up, left) occurred is even, play (up, right), else play (down, left). "0" counts as an even number. I,
What is the outcome? Is it a Nash Equilibrium? Is it a subgame perfect Nash Equilibrium?
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2. Suppose two players play '~he following game infinitely many times.
. Row
Column cooperate defect
cooperate I 3,3 . I 0,4 I defect . 4, 0 . 1, 1 I _
Consider the following strategy: "Play Cooperate on the current move unless:
. (a) the other player defected on the last move, or
(b) you defected on the last move and the other player defected on . the next-to-Iast move, .
in which case play defect."
(a) What outcome will occur?
(b) Is th~s outcome a Nash equilibrium?
(c) Is this outcome a subgame-perfect Nash equilibrium?
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5. Suppose two players are involved in infinite repetitions of the following game, called "Chicken":
Column stay swerve
Row' stay 1-3, -31 6,1 swerve 1,6 4,4
Suppose both players play the "Unrelenting" strategy, which is: for t = 1, swerve; for t > 1, play stay if the other player has ever played stay at any time in the' past; otherwise, play swerve.
(a) What isthe outcome of the game?
(b) Is this an equilibriutn. outcome?
(c) Is this .. a subgame-perfect equilibrium outcome? I
If these answers depend on the discount rate, then describe this de- 1111 ,
pendence;
. You may "'fish to know that (1 + Vf1i)/4 ~ 2.14 .. I II
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3. Consider an indu'stry with two firms, each haying zero costs. The (in-., verse) demand curve facing this industry is
P(Y) = 100 - Y,
where Y,,= Yl + Y2 is total output ...
(a) Find the payoff to each firm if the two firms are in a symmetric duopoly and they produce the level of output that maximizes their, joint profits; You may wish to' call this payoff 7f j.
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(b) Find: the payoff to each firm if the two firms are in a symmetric duopoly and they produce the Cournot, level of output. You may wish to call this payoff 11" c.' , "
. . ~
( c ) Find the maximum payoff that one firm can get if the .o~her 'chooses the joint-profit maximizing output. You' may wish to call this payoff 11" d. " '
, (d), Let the discount rate be :T. Suppose the' firms are in an infinitely . repeated; symmetric duopoly. Suppo~e, the firms adopt the pun
ishment str.ategy of reverting to the Cournot game if either player ,defects,.fr()m~the j()int:-,profifir,t~ifu.i~iJ:lglevel of output.' . How .
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