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Lesson 2 HW — Transcript

by Russ deForest · 2,319 words · 334 segments · language en · Watch on YouTube

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0:00in problem one we revisit the
0:02stackelberg duopoly and we're looking at
0:04a case where the players have different
0:08costs so in this problem uh the market
0:12price was
0:14150 minus q1 minus Q2 and that's true as
0:19long as uh Q is less than or equal to
0:22150 Q is the sum q1 plus Q2 and the
0:27price is zero if Q is greater than
0:31150 and this is a linear demand function
0:35we're assuming that the the market price
0:38is linear in the total quantity Q This
0:40is a standard assumption even though in
0:44practice uh we only use this linearity
0:47assumption for small changes in P or
0:49small changes in Q and then we assumed
0:52that the firms have different costs so
0:54player one or firm one has the cost
0:56function C1 of q1 it is uh 9 *
1:00q1 and C2 of Q2 that is 4 *
1:06Q2 in part A we want to write out the
1:10payoff functions for each player and
1:12these are functions of both q1 and
1:15Q2 so we want to know what is pi 1 of q1
1:19Q2 and what is pi 2 of q1
1:26Q2 and both of these uh payoffs are the
1:29total profit so it's Revenue Minus cost
1:32for player one the revenue is whatever
1:34the price is times the quantity produced
1:37minus the
1:40cost this will be a piecewise function
1:44so it will be uh when we when we
1:46subtract this 9 q1 the cost will end up
1:48with 141 q1 - q1 ^ 2 - q1 *
1:54Q2 if the sum is less than or equal to 1
2:0050 and it will just
2:01be 9
2:03q1 if the sum q1 + Q2 is greater than
2:08150 in other words if the price is zero
2:11and then the payoff function for player
2:13two is similar I won't write it out for
2:16you here but it is also a pie wise
2:17function now in Part
2:22B we've asked for the strategy set for
2:26each firm uh recall that player one
2:30moves first chooses q1 and then player
2:33two observes that q1 before uh choosing
2:36Q2 so for player one the strategy set is
2:41just the interval from zero to Infinity
2:45but player two uh a strategy for player
2:48two needs to indicate an action for
2:50every node in the game
2:52tree player two has infinitely many
2:55nodes there's one node for each choice
2:59that uh player one makes and so we could
3:01view player 2's strategy set as a set of
3:05functions it's a uh each strategy is a
3:09function each
3:14strategy uh for player
3:18two is a
3:22function it should
3:24be it should take player One's Choice as
3:28an input
3:30and it should output a choice for player
3:35two so it should take q1 and it should
3:38give
3:39us Q2 and so the strategy set S2 is the
3:43set of all such functions these
3:46functions do not need to be continuous
3:47so any any function that takes q1 as the
3:51input and gives Q2 as the output is a
3:54strategy for player two now finally we
3:57want to look at part C we're going to
3:59use backward induction to to determine
4:02the the uh solution the backward
4:04induction solution for the game we start
4:07at the bottom of the tree so we start
4:09with player two player two's
4:12payoff uh if we assume the total uh
4:16quantity produced is less than
4:19150 then we can look at uh player two's
4:22payoff as 146 Q2 - Q2 ^ 2 - q1 * Q2
4:31that's if q1 + Q2 is less than 150 and
4:37if it's greater than
4:41zero then we can use calculus right this
4:45is uh a uh equation that's quadratic in
4:50Q2 and we have a negative squared term
4:54here so it's just a parabola we want to
4:57find uh the best choice of Q2 for a
5:01fixed q1 so we can take the derivative
5:05and set it equal to zero we take the
5:07derivative of Pi 2 with respect to Q2
5:10and set it equal to zero and we find
5:13that 146 - 2 Q2 - q1 = 0 so we're
5:20finding the vertex of a parabola
5:22although that parabula depends on player
5:25One's Choice q1
5:30that gives us Q2 = 73 - 12
5:35q1 but if q1 is large enough this would
5:40be negative so in fact uh to define the
5:43best
5:45response player 2's best response to q1
5:49is a pie wise function it will be 73
5:53minus 12
5:55q1 if uh q1 is less than or equ equal to
6:02146 if q1 is larger than 146 then uh
6:07this best response should be
6:13zero because Q2 cannot be negative and
6:1673 -2 q1 would be
6:19negative uh if q1 is greater than 146
6:24now uh because player one goes first and
6:27then player two observes player one can
6:30assume that player two will use their
6:33best response so we can look at player
6:36one's payoff now as a function of well
6:40it it is a function of q1 and Q2 we can
6:42rewrite this as the payoff of q1 and the
6:46best response to q1 so now it is just a
6:49function of q1 and if we continue to
6:53assume that q1 will be less than 146 we
6:57could write this out as 68 q1 - 12
7:06q12 we wrote player one's payoff
7:09function above now we're taking this
7:11part of the best response and plugging
7:13it in for Q2 we get 68 q1 - 12 q1
7:202 player 1's best response then can be
7:24found by uh taking the derivative of pi1
7:27with respect to q1
7:30and setting it equal to
7:33zero notice again that this is this is
7:36quadratic in q1 with a a negative
7:40coefficient on the squared term so this
7:41is just a a downward opening Parabola
7:44we're trying to find the vertex of that
7:46Parabola and one way is to find the
7:48critical
7:49point so if you take the derivative set
7:52it equal to zero you should find q1 =
7:5768 and then what is player 2's best
8:01response to 68 you can plug that in for
8:04q1 is certainly less than 146 and you
8:08find Q2 it's the best response to 68
8:13that is
8:1539 so by backward induction we find that
8:20uh player one chooses q1 = 68 player two
8:24responds by choosing Q2 equal to
8:2839 but player 2's strategy is actually a
8:32function this best response
8:35function now let's take a look at
8:37problem
8:42two we have another two-player game
8:45player one moves first player two
8:47observes player one before making their
8:49choice but now player one is a
8:50manufacturer player two is a retailer
8:53the retailer uh orders a product from
8:57the manufacturer
8:59at the wholesale price and sells it for
9:02the market price so the question is uh
9:05what price should the manufacturer set
9:07the wholesale price they sell the tablet
9:09to the retailer for and then how many uh
9:13how much of the product should the
9:15retailer order to sell in their store
9:18the retailer faces a market price we
9:21said that the objects are tablets but
9:23we're going to assume that uh uh the
9:27quantity is is INF L divisible um anyway
9:31the market price is 500 minus .1
9:37q and this is true as long as this price
9:41is non- negative so Q can be between
9:44zero and 5,000 and the market price is
9:48zero if Q is greater than
9:515,000 again a just a linear linearity
9:55assumption on uh the demand function
10:00now the payoff functions that we're
10:01writing they are functions of w and Q
10:05player one chooses W player two chooses
10:08the quantity Q for each player the
10:10payoff is the profit for player
10:13one the revenue is uh the price that
10:17they sell to the manufact the retailer
10:19for times the quantity ordered by the
10:21retailer minus the cost to the
10:26manufacturer so this is the revenue for
10:28the Manu manufacturer again it's uh this
10:31is the price that they're charging the
10:33retailer this is the quantity that the
10:35retailer wants to purchase and then the
10:38cost is $100 per tablet so uh 100 time
10:42Q the payoff for player
10:47two is the market price times the
10:50quantity minus the cost the cost for the
10:53retailer is the wholesale price times
10:55the the number they
10:57order so we could Factor out this Q here
11:00and write player 1's payoff is just W -
11:03100 * q but for player two writing this
11:07out involves the uh uh
11:11demand function the market price so we
11:14can write this out as
11:17500 Q -.1 q^2 minus
11:23WQ
11:25if Q is between 0 and 5,000 and will
11:29just be netive
11:31WQ if Q is greater than
11:345,000 and then as
11:37before uh we will um do the backward
11:40induction by assuming that Q is less
11:44than
11:455,000 and U finding player 2's optimal
11:50choice for a given W and then uh then by
11:54backward induction we'll uh let we'll
11:57we'll set up an optimization problem for
11:59player one they'll choose a particular
12:03W Part B uh is very similar to what we
12:07looked at in problem
12:09one player one has a strategy set that
12:12is an interval they're choosing
12:14wholesale price W from the interval 100
12:16to
12:17Infinity player 2's strategy set is a
12:20set of functions the domain of function
12:24is player One's Choice the range uh or
12:28the Cod domain is the um rather is the
12:32um the set of possible choices for
12:35player two so I I won't write out Part B
12:38but you should revisit that if you need
12:40to uh player 2's strategy set is a set
12:45of functions let's go ahead and and look
12:47at the backward induction so we're going
12:50to assume that we're in this situation
12:52this is the payoff for player two in
12:55this case and we would like to take the
12:59derivative with respect to q and again
13:02we're noting here that we have a
13:05quadratic in Q it's a downward opening
13:08Parabola because of this negative
13:12coefficient so we can take this
13:14derivative and set it equal to zero we
13:17see we have 500
13:19-2 Q - W = 0 so Q will be
13:272500 - 5 W that's true as long as this Q
13:33is non- negative so that's if uh W is
13:38less than or equal to 500 and
13:41Q equals 0 if W is greater than
13:45500 in other words this is a uh player
13:482's best response to W is this pie wise
13:52function
14:04now we can continue with the backward
14:06induction uh assuming that W will be
14:10less than
14:11500 uh player one then assumes that
14:14player two will play their best response
14:17so we write player one's payoff Pi 1 of
14:21WQ as just a function of w now it's
14:25w B2 of w player 2's best response to
14:38W again we're assuming that W will be
14:42less than 500 then we can take a
14:45derivative this is another uh uh
14:48equation that is quadratic in w a
14:50downward opening Parabola so all we need
14:52to do is find the vertex so we can take
14:56the derivative
14:59with respect to W and set it equal to
15:01zero and then solve and you should find
15:04that W equals
15:07300 once you've done that uh then you
15:10can show that Q it's the best response
15:13to
15:14300 that's
15:171,000 now uh in
15:21part uh e we're going to need to know
15:24what the total profit is we need to know
15:26the profit for each player so you can
15:29also calculate pi 1 of 300 1,000 you
15:35should uh see that this is
15:40200,000 and Pi 1 or Pi 2 rather player
15:452's payoff when w is 300 and Q is 1,000
15:49that's 100,000 so the total
15:53profit is
15:55300,000 200,000 of that goes to the
15:57manufacturer
15:59100,000 goes to the retailer in part D
16:03we want to change the problem and take
16:06the retailer completely out of the
16:08picture and ask what happens if the
16:11manufacturer just sells
16:13directly uh to the
16:17public so let's take a look at Part
16:24D so the manufacturer is trying to
16:26choose Q for themselves now they sell
16:28Direct directly to the Public Public so
16:31we're just going to write their payoff
16:33as a function of Q there's no
16:36intermediate retailer it should be the
16:38market price time Q minus their cost
16:41which is 100 q and as long as we assume
16:46that Q is less than 5,000 again we could
16:49write this as
16:53uh 400 Q -.1 q^2
16:59if Q
17:00is less than
17:025,000 so player one can choose the
17:05optimal
17:07Q we're just again finding the critical
17:10point by setting the first derivative
17:13equal to zero and we can see that uh Q
17:18equal
17:212,000 this is different from what we
17:23found in the game where the retailer
17:25ordered 1,000 tablets from the
17:27manufacturer in part e We compare the
17:30total profit in this case with the
17:33profit that we found in the
17:36game let's first take a look at the
17:39market price so the price was 500
17:43minus1 Q since Q is 2,000 the market
17:48price here is
17:51300 and the
17:53profit PQ minus 100 Q will be 400,00
18:00,000 so the total profit is larger than
18:03what we saw in the game the total profit
18:05there was 300,000 the price is also
18:07lower meaning the consumer is benefiting
18:09from a lower price this issue is called
18:12double marginalization I provided a link
18:14in the homework where you can read more
18:16about it but one thing we could conclude
18:19is that there are benefits to what's
18:21called vertical integration where the
18:23manufacturer uh sells directly to the to
18:26the public as long as the costs of that
18:28ver iCal integration are small

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