(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 12.3' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 133026, 3257] NotebookOptionsPosition[ 125913, 3153] NotebookOutlinePosition[ 126439, 3171] CellTagsIndexPosition[ 126396, 3168] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["\<\ Below is a program that will give you access to a nearly limitless number of \ practice problems for the Cournot, Quantity Leadership, and Price Leadership \ models. To start, click the new problem button below and it will provide you \ with a market demand function and cost functions for two firms. From there \ you can go onto solve the Cournot model, the Quantity Leadership model (firm \ 1 leader), and price leadership model (firm 1 leader). To help guide you in \ solving this models. Below is an example.\ \>", "Text", CellChangeTimes->{{3.860860227554003*^9, 3.860860232021385*^9}, { 3.860865605800578*^9, 3.8608657998754673`*^9}},ExpressionUUID->"eec5060c-4b41-41e4-a868-\ 17a6215f69fa"], Cell[CellGroupData[{ Cell["Cournot and Quantity Leadership (Stackelberg)", "Section", CellChangeTimes->{{3.8608658031450233`*^9, 3.860865804179702*^9}, { 3.888338687066223*^9, 3.888338698534359*^9}},ExpressionUUID->"67c44d70-82c9-42a4-85de-\ d3d304e74fb6"], Cell[TextData[{ "Inverse Market Demand: P=105-4\[Times]Q\nFirm 1 Cost Function: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["C", "1"], "(", SubscriptBox["q", "1"], ")"}], "=", RowBox[{"5", SubsuperscriptBox["q", "1", "2"]}]}], TraditionalForm]],ExpressionUUID-> "83895252-8f39-4e2d-8444-4b71a2e6b0de"], "\nFirm 2 Cost Function: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["C", "2"], "(", SubscriptBox["q", "2"], ")"}], "=", RowBox[{"4", SubsuperscriptBox["q", "2", "2"]}]}], TraditionalForm]],ExpressionUUID-> "c926e2d5-57ab-4539-83bc-f8fc607c7025"] }], "Text", CellChangeTimes->{{3.860865810224704*^9, 3.860865851202134*^9}, { 3.8608661071171837`*^9, 3.860866108557618*^9}},ExpressionUUID->"b0c2b3fe-03d6-4aa0-8d8c-\ ed1c5b231b9f"], Cell[CellGroupData[{ Cell["Example", "Subsection", CellChangeTimes->{{3.860865855972361*^9, 3.860865857607139*^9}, { 3.8883387188663*^9, 3.888338727937648*^9}},ExpressionUUID->"4d9eaccc-3b9a-4258-89dd-\ 61c75b6499bd"], Cell[CellGroupData[{ Cell["Cournot", "Subsubsection", CellChangeTimes->{{3.88833873059604*^9, 3.888338731394524*^9}},ExpressionUUID->"e5afbb83-3afc-43be-9557-\ d1d762f7819a"], Cell["\<\ Since we know that the firm\[CloseCurlyQuote]s goal is to maximize profits in \ every situation, we will start by finding their marginal revenue and marginal \ cost functions (because we know that profit maximization occurs when MR=MC). \ To find the marginal revenue function, we first need to find the total \ revenue function, because marginal revenue is the derivative of total \ revenue. The total revenue function of each firm is simply the quantity it \ produces multiplied by the price it receives for each good, therefore, for \ each firm:\ \>", "Text", CellChangeTimes->{{3.860865860081807*^9, 3.860866000023135*^9}},ExpressionUUID->"3886d893-f30b-42bb-b3cb-\ fb69aaad0c1f"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["TR", "1"], "=", RowBox[{ RowBox[{"P", "\[Times]", SubscriptBox["q", "1"]}], "=", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"105", "-", RowBox[{"4", "\[Times]", "Q"}]}], ")"}], SubscriptBox["q", "1"]}], "=", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"105", "-", RowBox[{"4", "\[Times]", RowBox[{"(", RowBox[{ SubscriptBox["q", "1"], "+", SubscriptBox["q", "2"]}], ")"}]}]}], ")"}], SubscriptBox["q", "1"]}], "=", RowBox[{ RowBox[{"105", SubscriptBox["q", "1"]}], "-", RowBox[{"4", SubsuperscriptBox["q", "1", "2"]}], "-", RowBox[{"4", SubscriptBox["q", "1"], SubscriptBox["q", "2"]}]}]}]}]}]}], TraditionalForm]],ExpressionUUID->"48d0d329-cb99-470d-8768-498ff6266265"]], \ "Item", CellChangeTimes->{{3.860866008378378*^9, 3.860866061798231*^9}, { 3.8638017585213423`*^9, 3.8638017594542713`*^9}},ExpressionUUID->"6e57153c-69ef-4c4d-a871-\ 01f7941a075a"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["TR", "2"], "=", RowBox[{ RowBox[{"P", "\[Times]", SubscriptBox["q", "2"]}], "=", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"105", "-", RowBox[{"4", "\[Times]", "Q"}]}], ")"}], SubscriptBox["q", "2"]}], "=", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"105", "-", RowBox[{"4", "\[Times]", RowBox[{"(", RowBox[{ SubscriptBox["q", "1"], "+", SubscriptBox["q", "2"]}], ")"}]}]}], ")"}], SubscriptBox["q", "2"]}], "=", RowBox[{ RowBox[{"105", SubscriptBox["q", "1"]}], "-", RowBox[{"4", SubscriptBox["q", "1"], SubscriptBox["q", "2"]}], "-", RowBox[{"4", SubsuperscriptBox["q", "2", "2"]}]}]}]}]}]}], TraditionalForm]],ExpressionUUID->"5ae55f54-c268-48eb-ab43-729e0b73fdaa"]], \ "Item", CellChangeTimes->{{3.860866008378378*^9, 3.860866081844507*^9}, { 3.8638017642355328`*^9, 3.863801765068137*^9}},ExpressionUUID->"19d9efa8-5d3d-4a36-b455-\ 628a371c47b1"], Cell["\<\ Note that for the price we substituted in the inverse market demand function. \ From here we can find the marginal revenue functions by taking the derivative \ of each total revenue function.\ \>", "Text", CellChangeTimes->{{3.860866089493445*^9, 3.860866141060425*^9}},ExpressionUUID->"d43e61aa-b274-474d-bbcc-\ e06092daa1e0"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["MR", "1"], "=", RowBox[{ FractionBox[ RowBox[{"\[PartialD]", SubscriptBox["TR", "1"]}], RowBox[{"\[PartialD]", SubscriptBox["q", "1"]}]], "=", RowBox[{"105", "-", RowBox[{"8", SubscriptBox["q", "1"]}], "-", RowBox[{"4", SubscriptBox["q", "2"]}]}]}]}], TraditionalForm]],ExpressionUUID->"37b31e54-cd96-421f-94dd-ed8cb19785b1"]], \ "Item", CellChangeTimes->{{3.860866147240242*^9, 3.86086618654803*^9}},ExpressionUUID->"3ca182af-49cc-44a3-b1e4-\ a158366bfbb3"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["MR", "2"], "=", RowBox[{ FractionBox[ RowBox[{"\[PartialD]", SubscriptBox["TR", "2"]}], RowBox[{"\[PartialD]", SubscriptBox["q", "2"]}]], "=", RowBox[{"105", "-", RowBox[{"4", SubscriptBox["q", "1"]}], "-", RowBox[{"8", SubscriptBox["q", "2"]}]}]}]}], TraditionalForm]],ExpressionUUID->"75fa5f73-3ea6-4ddc-a7d1-737b448e7334"]], \ "Item", CellChangeTimes->{{3.860866147240242*^9, 3.8608662012078867`*^9}},ExpressionUUID->"09eec8d3-ee80-46c0-9a10-\ 6e874fc94c6c"], Cell["\<\ As we are attempting to find the profit maximizing amount of output for each \ firm, we need to find the marginal cost curves as well. The marginal cost \ curve is the derivative of the total cost curves (given above).\ \>", "Text", CellChangeTimes->{{3.860866239304968*^9, 3.860866284182081*^9}},ExpressionUUID->"cb5592f3-7e69-4e28-931b-\ 031bc6487527"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["MC", "1"], "=", RowBox[{ FractionBox[ RowBox[{"\[PartialD]", RowBox[{ SubscriptBox["C", "1"], "(", SubscriptBox["q", "1"], ")"}]}], RowBox[{"\[PartialD]", SubscriptBox["q", "1"]}]], "=", RowBox[{"10", SubscriptBox["q", "1"]}]}]}], TraditionalForm]],ExpressionUUID->"997c3778-f0da-4049-98b7-93e9d10debdf"]], \ "Item", CellChangeTimes->{{3.860866289761587*^9, 3.860866317083858*^9}},ExpressionUUID->"5fb72ee8-76ee-48d2-8c45-\ 04fb656cb91b"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["MC", "2"], "=", RowBox[{ FractionBox[ RowBox[{"\[PartialD]", RowBox[{ SubscriptBox["C", "2"], "(", SubscriptBox["q", "2"], ")"}]}], RowBox[{"\[PartialD]", SubscriptBox["q", "2"]}]], "=", RowBox[{"8", SubscriptBox["q", "2"]}]}]}], TraditionalForm]],ExpressionUUID->"f00ebf79-1644-461f-945f-b7d53b5e8bc6"]], \ "Item", CellChangeTimes->{{3.860866289761587*^9, 3.860866332194713*^9}},ExpressionUUID->"f7aed38b-fadd-4179-9e2f-\ ed6fa4f5e799"], Cell["\<\ As we know that the profit maximizing condition for a firm is when MR=MC. \ Setting our equations equal to each other for each firm yields:\ \>", "Text", CellChangeTimes->{{3.860866334879262*^9, 3.8608663675625553`*^9}},ExpressionUUID->"23407d9a-905c-4b71-82cf-\ cd2ad3289dd9"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["MR", "1"], "=", RowBox[{ RowBox[{ SubscriptBox["MC", "1"], "\[RightArrow]", RowBox[{"105", "-", RowBox[{"8", SubscriptBox["q", "1"]}], "-", RowBox[{"4", SubscriptBox["q", "2"]}]}]}], "=", RowBox[{ RowBox[{ RowBox[{"10", SubscriptBox["q", "1"]}], "\[RightArrow]", RowBox[{"105", "-", RowBox[{"4", SubscriptBox["q", "2"]}]}]}], "=", RowBox[{ RowBox[{ RowBox[{"18", SubscriptBox["q", "1"]}], "\[RightArrow]", SubscriptBox["q", "1"]}], "=", RowBox[{"5.833", "-", RowBox[{"0.222", SubscriptBox["q", "2"]}]}]}]}]}]}], TraditionalForm]],ExpressionUUID->"3d04617f-ecbc-4b30-b4ca-bf2ef2ec58b7"]], \ "Item", CellChangeTimes->{{3.8608663759321823`*^9, 3.8608664475914783`*^9}},ExpressionUUID->"c789da2f-4656-43e1-b3b7-\ fa1bae083cc0"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["MR", "2"], "=", RowBox[{ RowBox[{ SubscriptBox["MC", "2"], "\[RightArrow]", RowBox[{"105", "-", RowBox[{"4", SubscriptBox["q", "1"]}], "-", RowBox[{"8", SubscriptBox["q", "2"]}]}]}], "=", RowBox[{ RowBox[{ RowBox[{"8", SubscriptBox["q", "2"]}], "\[RightArrow]", RowBox[{"105", "-", RowBox[{"4", SubscriptBox["q", "1"]}]}]}], "=", RowBox[{ RowBox[{ RowBox[{"16", SubscriptBox["q", "2"]}], "\[RightArrow]", SubscriptBox["q", "2"]}], "=", RowBox[{"6.5625", "-", RowBox[{"0.25", SubscriptBox["q", "1"]}]}]}]}]}]}], TraditionalForm]],ExpressionUUID->"549816d6-16b5-46ba-9899-8d7f247a7f56"]], \ "Item", CellChangeTimes->{{3.8608663759321823`*^9, 3.8608665087487793`*^9}},ExpressionUUID->"eb84e28d-3953-4a82-a62d-\ c552462be5d7"], Cell[TextData[{ "We should note that we have not solved for numbers, but rather equations \ for our profit maximizing conditions. Why is this? Since we are in a duopoly, \ the profits of one firm depend on the actions of the other firm. Therefore \ for firm 1 for instance, if firm 2 increases its quantity by 1 unit, the \ optimal thing for firm 1 to do is to reduce it\[CloseCurlyQuote]s production \ by 0.222 units. Since these equations tell us the best thing that one firm \ can do in response to the other firm, we call these best response functions. \ How much output is actually produced by both firms? In equilibrium, we would \ expect that each firm will do the best they possibly can given what the other \ firm is doing. This means that for firm 1\[CloseCurlyQuote]s best response \ function ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["q", "1"], "=", RowBox[{"5.833", "-", RowBox[{"0.222", SubscriptBox["q", "2"]}]}]}], ")"}], TraditionalForm]],ExpressionUUID-> "03fe9156-4bb7-4373-bf08-17e8e2ce5b2a"], ", the ", Cell[BoxData[ FormBox[ SubscriptBox["q", "2"], TraditionalForm]],ExpressionUUID-> "1f74f22f-5c6d-4d7f-b722-32e36e83ca13"], " in that equation is being determined by firm 2\[CloseCurlyQuote]s best \ response function. This means that firm 1 will know how much firm 2 will \ produce in response to its own choice of ", Cell[BoxData[ FormBox[ SubscriptBox["q", "2"], TraditionalForm]],ExpressionUUID-> "66f849cc-afb1-47a4-8b32-02f6220541dd"], ". Therefore we can find firm 1\[CloseCurlyQuote]s optimal quantity:" }], "Text", CellChangeTimes->{{3.860866520328115*^9, 3.8608666654747133`*^9}, { 3.8608667188877697`*^9, 3.860866868803726*^9}},ExpressionUUID->"a2575e7c-40b6-40a6-b9ef-\ 662550c99389"], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ { RowBox[{GridBox[{ { RowBox[{ RowBox[{ SubscriptBox["q", "1"], "=", RowBox[{"5.833", "-", RowBox[{"0.222", SubscriptBox["q", "2"]}]}]}], "\[LineSeparator]", RowBox[{ SubscriptBox["q", "1"], "=", RowBox[{"5.833", "-", RowBox[{"0.222", RowBox[{"(", RowBox[{"6.5625", "-", RowBox[{"0.25", SubscriptBox["q", "1"]}]}], ")"}]}]}]}]}]} }, GridBoxAlignment->{"Columns" -> {{"="}}}], "\[IndentingNewLine]", RowBox[{ SubscriptBox["q", "1"], "=", RowBox[{"5.833", "-", "1.457", "+", RowBox[{"0.0556", SubscriptBox["q", "1"]}]}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"0.9444", SubscriptBox["q", "1"]}], "=", "4.376`"}], "\[IndentingNewLine]", RowBox[{ SubscriptBox["q", "1"], "=", "4.634"}]}]} }, GridBoxItemSize->{"Columns" -> {{ Scaled[0.96]}}}], TraditionalForm]],ExpressionUUID->"632ab1c1-7bf2-4e1c-ba47-70955566c64e"]], \ "Text", CellChangeTimes->{{3.860866929645804*^9, 3.860867093265324*^9}, { 3.860867127980268*^9, 3.860867129518442*^9}},ExpressionUUID->"dede2a7d-d38c-4772-b54e-\ 2e5652a16668"], Cell["\<\ We can then plug this value back into the best response function for firm 2 \ to find their optimal quantity:\ \>", "Text", CellChangeTimes->{{3.8608671419075937`*^9, 3.860867167239867*^9}},ExpressionUUID->"d54e6283-f458-4c8e-8901-\ fe65acaafc41"], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {GridBox[{ { RowBox[{ RowBox[{ SubscriptBox["q", "2"], "=", RowBox[{"6.5625", "-", RowBox[{"0.25", SubscriptBox["q", "1"]}]}]}], "\[IndentingNewLine]", RowBox[{ SubscriptBox["q", "2"], "=", RowBox[{"6.5625", "-", RowBox[{"0.25", "\[Times]", "4.634"}]}]}], "\[IndentingNewLine]", RowBox[{ SubscriptBox["q", "2"], "=", "5.404"}]}]} }, GridBoxAlignment->{"Columns" -> {{"="}}}]} }, GridBoxItemSize->{"Columns" -> {{ Scaled[0.96]}}}], TraditionalForm]],ExpressionUUID->"c7344564-a383-42c8-a8d4-4645606e7719"]], \ "Text", CellChangeTimes->{{3.860867172340747*^9, 3.860867221582376*^9}},ExpressionUUID->"6c4c2df7-e0cd-4e4c-bf14-\ 3e5da67114cb"], Cell["\<\ Now that we know what each firm is going to produce (and therefore know the \ total market quantity), we can find the market price using the inverse market \ demand curve.\ \>", "Text", CellChangeTimes->{{3.860867238666986*^9, 3.8608672736299047`*^9}},ExpressionUUID->"05654518-3380-4db3-86ef-\ b3daf20fb1b8"], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {GridBox[{ { RowBox[{ RowBox[{"P", "=", RowBox[{"105", "-", RowBox[{"4", "\[Times]", "Q"}]}]}], "\[IndentingNewLine]", RowBox[{"P", "=", RowBox[{"105", "-", RowBox[{"4", "\[Times]", RowBox[{"(", RowBox[{ SubscriptBox["q", "1"], "+", SubscriptBox["q", "2"]}], ")"}]}]}]}], "\[IndentingNewLine]", RowBox[{"P", "=", RowBox[{"105", "-", RowBox[{"4", "\[Times]", RowBox[{"(", RowBox[{"4.634", "+", "5.404"}], ")"}]}]}]}], "\[IndentingNewLine]", RowBox[{"P", "=", "64.848"}]}]} }, GridBoxAlignment->{"Columns" -> {{"="}}}]} }, GridBoxItemSize->{"Columns" -> {{ Scaled[0.96]}}}], TraditionalForm]],ExpressionUUID->"58997eca-2a87-427c-a296-0f4c51e64e8e"]], \ "Text", CellChangeTimes->{{3.860867283088571*^9, 3.860867332847083*^9}},ExpressionUUID->"cd67c081-7e9e-4012-b9be-\ 56c3045a20a8"], Cell["\<\ Finally, we can determine each of the firms\[CloseCurlyQuote] profits (\ \[CapitalPi]=TR-TC):\ \>", "Text", CellChangeTimes->{{3.860867352089718*^9, 3.860867371724571*^9}},ExpressionUUID->"d408afdb-facc-4d8b-98b7-\ 82ef645b979b"], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {GridBox[{ { RowBox[{ RowBox[{ SubscriptBox["\[CapitalPi]", "1"], "=", RowBox[{"TR", "-", "TC"}]}], "\[LineSeparator]", RowBox[{ SubscriptBox["\[CapitalPi]", "1"], "=", RowBox[{ RowBox[{"P", "\[Times]", SubscriptBox["q", "1"]}], "-", RowBox[{ SubscriptBox["C", "1"], "(", SubscriptBox["q", "1"], ")"}]}]}], "\[IndentingNewLine]", RowBox[{ SubscriptBox["\[CapitalPi]", "1"], "=", RowBox[{ RowBox[{ RowBox[{"64.848", "\[Times]", "4.634"}], "-", RowBox[{"5", "\[Times]", SuperscriptBox["4.634", "2"]}]}], "=", "193.136"}]}], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ SubscriptBox["\[CapitalPi]", "2"], "=", RowBox[{ RowBox[{"P", "\[Times]", SubscriptBox["q", "2"]}], "-", RowBox[{ SubscriptBox["C", "2"], "(", SubscriptBox["q", "2"], ")"}]}]}], "\[LineSeparator]", RowBox[{ SubscriptBox["\[CapitalPi]", "2"], "=", RowBox[{ RowBox[{ RowBox[{"64.848", "\[Times]", "5.404"}], "-", RowBox[{"4", "\[Times]", SuperscriptBox["5.404", "2"]}]}], "=", "233.62572799999998`"}]}]}]} }, GridBoxAlignment->{"Columns" -> {{"="}}}]} }, GridBoxItemSize->{"Columns" -> {{ Scaled[0.96]}}}], TraditionalForm]],ExpressionUUID->"ab3dc135-2bb7-434f-8ab9-dfeec0d2f4a4"]], \ "Text", CellChangeTimes->{{3.86086738169554*^9, 3.8608675193326902`*^9}},ExpressionUUID->"688b3b29-2650-45aa-b736-\ 1728edea52c1"] }, Open ]], Cell[CellGroupData[{ Cell["Quantity Leadership", "Subsubsection", CellChangeTimes->{{3.8608676195546837`*^9, 3.860867623109939*^9}},ExpressionUUID->"e834b559-3a45-4301-adad-\ 92e8b295cb5d"], Cell["\<\ Quantity leadership has many similarities with the Cournot model. In fact, we \ are going to use the same best response functions that were derived in the \ Cournot model to solve the quantity leadership model. The most significant \ difference between these two models is that in the quantity leadership model, \ one firm gets to pick its quantity first and then the other firm has to \ respond to that choice (in the Cournot model, both firms chooses their \ quantity simultaneously). You can imagine, that this yields quite the \ advantage to the leading firm. We know based upon their best response \ functions, that increasing your quantity results in the other firm reducing \ their quantity. This means that in the quantity leadership model, the leading \ firm (in this example firm 1), might be able to take some sales away from the \ other firm by choosing a higher quantity than they would in the Cournot \ model. \ \>", "Text", CellChangeTimes->{{3.860867625373728*^9, 3.86086785552429*^9}},ExpressionUUID->"e182cab8-35fb-41b9-95cf-\ 52942f455ae3"], Cell[TextData[{ "How do firms choose their quantity in the quantity leadership model? The \ leading firm (firm 1) gets to choose their quantity and then the following \ firm, chooses their optimal quantity in response. Now the leading firm cannot \ just choose any quantity it wants, because it knows that the following firm \ will respond with its own output. Therefore, in making its decision, firm 1 \ will take into account how its choice of quantity will effect firm 2\ \[CloseCurlyQuote]s choice. It does that by incorporating firm 2\ \[CloseCurlyQuote]s best response function into its decision problem. \n\n\ From the Cournot model, we know that firm 2\[CloseCurlyQuote]s best response \ function is ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["q", "2"], "=", RowBox[{"6.5625", "-", RowBox[{"0.25", SubscriptBox["q", "1"]}]}]}], TraditionalForm]],ExpressionUUID-> "10833968-0554-4525-b5a2-a8d616ba5818"], ". This means that the demand facing firm 1 is:" }], "Text", CellChangeTimes->{{3.860867856991137*^9, 3.860868074424259*^9}},ExpressionUUID->"71c625e8-2419-47f0-b921-\ 6d94b7c80340"], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {GridBox[{ { RowBox[{ RowBox[{"P", "=", RowBox[{ RowBox[{"105", "-", RowBox[{"4", "\[Times]", "Q"}]}], "=", RowBox[{"105", "-", RowBox[{"4", RowBox[{"(", RowBox[{ SubscriptBox["q", "1"], "+", SubscriptBox["q", "2"]}], ")"}]}]}]}]}], "\[IndentingNewLine]", RowBox[{"P", "=", RowBox[{"105", "-", RowBox[{"4", RowBox[{"(", RowBox[{ SubscriptBox["q", "1"], "+", "6.5625", "-", RowBox[{"0.25", SubscriptBox["q", "1"]}]}], ")"}]}]}]}], "\[IndentingNewLine]", RowBox[{"P", "=", RowBox[{"105", "-", RowBox[{"4", SubscriptBox["q", "1"]}], "-", "26.25`", "+", SubscriptBox["q", "1"]}]}], "\[IndentingNewLine]", RowBox[{"P", "=", RowBox[{"78.75", "-", RowBox[{"3", SubscriptBox["q", "1"]}]}]}]}]} }, GridBoxAlignment->{"Columns" -> {{"="}}}]} }, GridBoxItemSize->{"Columns" -> {{ Scaled[0.96]}}}], TraditionalForm]],ExpressionUUID->"302793eb-70e5-4067-85a4-a68550031cfd"]], \ "Text", CellChangeTimes->{{3.860868079991849*^9, 3.860868166144886*^9}},ExpressionUUID->"0abc32d9-5f4e-43db-ba87-\ d09a8ba3abd4"], Cell["\<\ where the second line substitutes in firm 2\[CloseCurlyQuote]s best response \ function. Notice that the final line is the demand available to firm 1, once \ we have taken into account firm 2\[CloseCurlyQuote]s output. From here we can \ find firm 1 marginal revenue function:\ \>", "Text", CellChangeTimes->{{3.8608681749177027`*^9, 3.860868238590858*^9}},ExpressionUUID->"5af41ffc-a427-4943-bff3-\ 80e64481dcb4"], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {GridBox[{ { RowBox[{ RowBox[{ SubscriptBox["TR", "1"], "=", RowBox[{ RowBox[{"P", "\[Times]", SubscriptBox["q", "1"]}], "=", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"78.75", "-", RowBox[{"3", SubscriptBox["q", "1"]}]}], ")"}], SubscriptBox["q", "1"]}], "=", RowBox[{ RowBox[{"78.75", SubscriptBox["q", "1"]}], "-", RowBox[{"3", SubsuperscriptBox["q", "1", "2"]}]}]}]}]}], "\[IndentingNewLine]", RowBox[{ SubscriptBox["MR", "1"], "=", RowBox[{ FractionBox[ RowBox[{"\[PartialD]", SubscriptBox["TR", "1"]}], RowBox[{"\[PartialD]", SubscriptBox["q", "1"]}]], "=", RowBox[{"78.75", "-", RowBox[{"6", SubscriptBox["q", "1"]}]}]}]}]}]} }, GridBoxAlignment->{"Columns" -> {{"="}}}]} }, GridBoxItemSize->{"Columns" -> {{ Scaled[0.96]}}}], TraditionalForm]],ExpressionUUID->"d9b4b026-48f7-4b24-bdf9-dc7d4adaa2d2"]], \ "Text", CellChangeTimes->{{3.860868247536746*^9, 3.860868297028532*^9}},ExpressionUUID->"126a253f-a8cf-4232-8f05-\ e2b54c18d5b1"], Cell["\<\ Firm 1\[CloseCurlyQuote]s goal is still to maximize profits, which occurs \ when MR=MC. Therefore the optimal quantity of firm 1 is:\ \>", "Text", CellChangeTimes->{{3.860868308906784*^9, 3.86086834129014*^9}},ExpressionUUID->"1283e8f3-4918-40db-8985-\ 9761ddaf943a"], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {GridBox[{ { RowBox[{ RowBox[{"MR", "=", "MC"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"78.75", "-", RowBox[{"6", SubscriptBox["q", "1"]}]}], "=", RowBox[{"10", SubscriptBox["q", "1"]}]}], "\[IndentingNewLine]", RowBox[{"78.75", "=", RowBox[{"16", SubscriptBox["q", "1"]}]}], "\[IndentingNewLine]", RowBox[{ SubscriptBox["q", "1"], "=", "4.922"}]}]} }, GridBoxAlignment->{"Columns" -> {{"="}}}]} }, GridBoxItemSize->{"Columns" -> {{ Scaled[0.96]}}}], TraditionalForm]],ExpressionUUID->"eb8675e3-78ae-4f7b-a86d-ddfa14a0d4c4"]], \ "Text", CellChangeTimes->{{3.860868345095446*^9, 3.860868385178055*^9}},ExpressionUUID->"4703fd19-9a22-4be6-861c-\ 8302b11a7aa2"], Cell["\<\ Plugging this value into the best response function for firm 2, yields\ \>", "Text", 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$CellContext`STEP13$$ = 3, $CellContext`STEP14$$ = 3, $CellContext`STEP15$$ = 3, $CellContext`STEP2$$ = 3, $CellContext`STEP3$$ = 3, $CellContext`STEP4$$ = 3, $CellContext`STEP5$$ = 3, $CellContext`STEP6$$ = 3, $CellContext`STEP7$$ = 3, $CellContext`STEP8$$ = 3, $CellContext`STEP9$$ = 3, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[ Button[ "New Problem", {$CellContext`c10 = RandomInteger[{1, 5}], $CellContext`c11 = RandomInteger[{1, 5}], $CellContext`c12 = RandomInteger[{1, 5}], $CellContext`c20 = RandomInteger[{1, 5}], $CellContext`c21 = RandomInteger[{1, 5}], $CellContext`c22 = RandomInteger[{1, 5}], $CellContext`A = RandomInteger[{20, 200}], $CellContext`b = RandomInteger[{1, 5}], $CellContext`c1 = RandomInteger[{1, 5}], $CellContext`c2 = RandomInteger[{1, 5}], $CellContext`COSTIND = RandomInteger[{1, 2}]}]], 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Hold[$CellContext`STEP9$$], 1, "Step 1"}, { 1 -> "Instruction", 2 -> "Formula", 3 -> "Answer"}}, {{ Hold[$CellContext`STEP10$$], 1, "Step 2"}, { 1 -> "Instruction", 2 -> "Formula", 3 -> "Answer"}}, {{ Hold[$CellContext`STEP11$$], 1, "Step 3"}, { 1 -> "Instruction", 2 -> "Formula", 3 -> "Answer"}}, {{ Hold[$CellContext`STEP12$$], 1, "Step 4"}, { 1 -> "Instruction", 2 -> "Formula", 3 -> "Answer"}}, {{ Hold[$CellContext`STEP13$$], 1, "Step 5"}, { 1 -> "Instruction", 2 -> "Formula", 3 -> "Answer"}}, {{ Hold[$CellContext`STEP14$$], 1, "Step 6"}, { 1 -> "Instruction", 2 -> "Formula", 3 -> "Answer"}}, {{ Hold[$CellContext`STEP15$$], 1, "Step 7"}, { 1 -> "Instruction", 2 -> "Formula", 3 -> "Answer"}}}, Typeset`size$$ = { 911., {471.134033203125, 476.865966796875}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`STEP1$$ = 1, $CellContext`STEP10$$ = 1, $CellContext`STEP11$$ = 1, $CellContext`STEP12$$ = 1, $CellContext`STEP13$$ = 1, $CellContext`STEP14$$ = 1, $CellContext`STEP15$$ = 1, $CellContext`STEP2$$ = 1, $CellContext`STEP3$$ = 1, $CellContext`STEP4$$ = 1, $CellContext`STEP5$$ = 1, $CellContext`STEP6$$ = 1, $CellContext`STEP7$$ = 1, $CellContext`STEP8$$ = 1, $CellContext`STEP9$$ = 1}, "ControllerVariables" :> {}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Module[{$CellContext`q1BRC$, $CellContext`q1BRP$, $CellContext`q2BRC$, \ $CellContext`q2BRP$, $CellContext`CQ1S$, $CellContext`CQ2S$, \ $CellContext`CMPS$, $CellContext`C\[CapitalPi]1$, \ $CellContext`C\[CapitalPi]2$, $CellContext`QLF1RDC$, $CellContext`QLF1RDP$, \ $CellContext`QLQ1S$, $CellContext`QLQ2S$, $CellContext`QLMPS$, \ $CellContext`QL\[CapitalPi]1$, $CellContext`QL\[CapitalPi]2$, \ $CellContext`PLMR1$, $CellContext`PLQ2S$, $CellContext`PL\[CapitalPi]1$, \ $CellContext`PL\[CapitalPi]2$, $CellContext`PLMPS$, $CellContext`AK$, \ $CellContext`LTC1$, $CellContext`LTC2$, $CellContext`MC1$, $CellContext`MC2$, \ $CellContext`TR1$, $CellContext`TR2$, $CellContext`MR1$, $CellContext`MR2$, \ $CellContext`BR1$, $CellContext`BR2$, $CellContext`QS1$, $CellContext`QS2$, \ $CellContext`PS$, $CellContext`\[CapitalPi]1$, $CellContext`\[CapitalPi]2$, \ $CellContext`RD1$, $CellContext`TR1QL$, $CellContext`MR1QL$, \ $CellContext`QS1QL$, $CellContext`QS2QL$, $CellContext`PSQL$, $CellContext`\ \[CapitalPi]1QL$, $CellContext`\[CapitalPi]2QL$}, $CellContext`LTC1$ = Boole[$CellContext`COSTIND == 1] ($CellContext`c10 + $CellContext`c11 $CellContext`q1) + Boole[$CellContext`COSTIND == 2] ($CellContext`c10 + $CellContext`c11 $CellContext`q1 + \ $CellContext`c12 $CellContext`q1^2); $CellContext`LTC2$ = Boole[$CellContext`COSTIND == 1] ($CellContext`c20 + $CellContext`c21 $CellContext`q2) + Boole[$CellContext`COSTIND == 2] ($CellContext`c20 + $CellContext`c21 $CellContext`q2 + \ $CellContext`c22 $CellContext`q2^2); $CellContext`MC1$ = D[$CellContext`LTC1$, $CellContext`q1]; $CellContext`MC2$ = D[$CellContext`LTC2$, $CellContext`q2]; $CellContext`TR1$ = \ ($CellContext`A - $CellContext`b ($CellContext`q1 + $CellContext`q2)) \ $CellContext`q1; $CellContext`TR2$ = ($CellContext`A - $CellContext`b \ ($CellContext`q1 + $CellContext`q2)) $CellContext`q2; $CellContext`MR1$ = D[$CellContext`TR1$, $CellContext`q1]; $CellContext`MR2$ = D[$CellContext`TR2$, $CellContext`q2]; $CellContext`BR1$ = ReplaceAll[$CellContext`q1, Last[ Solve[$CellContext`MC1$ == $CellContext`MR1$, $CellContext`q1]]]; \ $CellContext`BR2$ = ReplaceAll[$CellContext`q2, Last[ Solve[$CellContext`MC2$ == $CellContext`MR2$, $CellContext`q2]]]; \ $CellContext`QS1$ = N[ ReplaceAll[$CellContext`q1, Last[ Solve[ ReplaceAll[$CellContext`BR1$, $CellContext`q2 -> \ $CellContext`BR2$] == $CellContext`q1, $CellContext`q1]]]]; $CellContext`QS2$ = ReplaceAll[$CellContext`BR2$, $CellContext`q1 -> $CellContext`QS1$]; \ $CellContext`PS$ = $CellContext`A - $CellContext`b ($CellContext`QS1$ + \ $CellContext`QS2$); $CellContext`\[CapitalPi]1$ = $CellContext`PS$ \ $CellContext`QS1$ - ReplaceAll[$CellContext`LTC1$, $CellContext`q1 -> \ $CellContext`QS1$]; $CellContext`\[CapitalPi]2$ = $CellContext`PS$ \ $CellContext`QS2$ - ReplaceAll[$CellContext`LTC2$, $CellContext`q2 -> \ $CellContext`QS2$]; $CellContext`RD1$ = FullSimplify[$CellContext`A - $CellContext`b ($CellContext`q1 + \ $CellContext`BR2$)]; $CellContext`TR1QL$ = $CellContext`RD1$ $CellContext`q1; \ $CellContext`MR1QL$ = D[$CellContext`TR1QL$, $CellContext`q1]; $CellContext`QS1QL$ = ReplaceAll[$CellContext`q1, Last[ Solve[$CellContext`MR1QL$ == $CellContext`MC1$, \ $CellContext`q1]]]; $CellContext`QS2QL$ = ReplaceAll[$CellContext`BR2$, $CellContext`q1 -> \ $CellContext`QS1QL$]; $CellContext`PSQL$ = $CellContext`A - $CellContext`b \ ($CellContext`QS1QL$ + $CellContext`QS2QL$); $CellContext`\[CapitalPi]1QL$ = \ $CellContext`PSQL$ $CellContext`QS1QL$ - ReplaceAll[$CellContext`LTC1$, $CellContext`q1 -> \ $CellContext`QS1QL$]; $CellContext`\[CapitalPi]2QL$ = $CellContext`PSQL$ \ $CellContext`QS2QL$ - ReplaceAll[$CellContext`LTC2$, $CellContext`q2 -> \ $CellContext`QS2QL$]; $CellContext`QLF1RDC$ = N[($CellContext`b + 2 $CellContext`c2) $CellContext`A/( 2 ($CellContext`b + $CellContext`c2))]; $CellContext`QLF1RDP$ = N[($CellContext`b + 2 $CellContext`c2) $CellContext`b/( 2 ($CellContext`b + $CellContext`c2))]; $CellContext`QLQ1S$ = ReplaceAll[$CellContext`q1, Last[ Solve[$CellContext`QLF1RDC$ - 2 $CellContext`QLF1RDP$ $CellContext`q1 == 2 $CellContext`c1 $CellContext`q1, $CellContext`q1]]]; \ $CellContext`QLQ2S$ = $CellContext`q2BRC$ - $CellContext`q2BRP$ \ $CellContext`QLQ1S$; $CellContext`QLMPS$ = $CellContext`A - $CellContext`b \ ($CellContext`QLQ1S$ + $CellContext`QLQ2S$); $CellContext`QL\[CapitalPi]1$ = \ $CellContext`QLMPS$ $CellContext`QLQ1S$ - $CellContext`c1 \ $CellContext`QLQ1S$^2; $CellContext`QL\[CapitalPi]2$ = $CellContext`QLMPS$ \ $CellContext`QLQ2S$ - $CellContext`c2 $CellContext`QLQ2S$^2; $CellContext`AK$[ Pattern[$CellContext`IND, Blank[]], Pattern[$CellContext`STEP, Blank[]], Pattern[$CellContext`HINT, Blank[]], Pattern[$CellContext`FORMULA, Blank[]], Pattern[$CellContext`ANSWER, Blank[]]] := If[$CellContext`IND == 0, { StringJoin["Step ", ToString[$CellContext`STEP], ":"], "", ""}, If[$CellContext`IND == 1, { StringJoin["Step ", ToString[$CellContext`STEP], ":", $CellContext`HINT], "", ""}, If[$CellContext`IND == 2, { StringJoin["Step ", ToString[$CellContext`STEP], ":", $CellContext`HINT], $CellContext`FORMULA, ""}, { StringJoin["Step ", ToString[$CellContext`STEP], ":", $CellContext`HINT], $CellContext`FORMULA, \ $CellContext`ANSWER}]]]; Grid[{{ Style["Problem", Bold], SpanFromLeft}, {"Market Demand:", StringJoin["P=", ToString[$CellContext`A], "-", ToString[$CellContext`b], "\[Times]Q"]}, { "Firm 1 Cost Function:", StringJoin["\!\(\*SubscriptBox[\(LTC\), \(1\)]\)=", ToString[$CellContext`LTC1$, StandardForm]]}, { "Firm 2 Cost Function:", StringJoin["\!\(\*SubscriptBox[\(LTC\), \(2\)]\)=", ToString[$CellContext`LTC2$, StandardForm]]}, {}, { Style["Answers", Bold], SpanFromLeft}, { "Instruction", "Formula", "Answer"}, { Style["Cournot Model:", Bold], ""}, $CellContext`AK$[$CellContext`STEP1$$, 1, "Find the total revenue for each firm\n", "TR=P[\!\(\*SubscriptBox[\(q\), \ \(1\)]\)+\!\(\*SubscriptBox[\(q\), \(2\)]\)]*\!\(\*SubscriptBox[\(q\), \(i\)]\ \)", StringJoin["\!\(\*SubscriptBox[\(TR\), \(1\)]\)=", ToString[ Expand[$CellContext`TR1$], StandardForm], "\n", "\!\(\*SubscriptBox[\(TR\), \(2\)]\)=", ToString[ Expand[$CellContext`TR2$], StandardForm]]], $CellContext`AK$[$CellContext`STEP2$$, 2, "Find the marginal revenue for each firm\n", "\!\(\*SubscriptBox[\(MR\), \(i\)]\)=\!\(\*FractionBox[\(\ \[PartialD]\*SubscriptBox[\(TR\), \(i\)]\), \ \(\[PartialD]\*SubscriptBox[\(q\), \(i\)]\)]\)", StringJoin["\!\(\*SubscriptBox[\(MR\), \(1\)]\)=", ToString[ Expand[$CellContext`MR1$], StandardForm], "\n", "\!\(\*SubscriptBox[\(MR\), \(2\)]\)=", ToString[ Expand[$CellContext`MR2$], StandardForm]]], $CellContext`AK$[$CellContext`STEP3$$, 3, "Find the marginal cost for each firm\n", "\!\(\*SubscriptBox[\(MC\), \(i\)]\)=\!\(\*FractionBox[\(\ \[PartialD]\*SubscriptBox[\(LTC\), \(i\)]\), \ \(\[PartialD]\*SubscriptBox[\(q\), \(i\)]\)]\)", StringJoin["\!\(\*SubscriptBox[\(MC\), \(1\)]\)=", ToString[ Expand[$CellContext`MC1$], StandardForm], "\n", "\!\(\*SubscriptBox[\(MC\), \(2\)]\)=", ToString[ Expand[$CellContext`MC2$], StandardForm]]], $CellContext`AK$[$CellContext`STEP4$$, 4, "Find the best response function for each firm\n", "\!\(\*SubscriptBox[\(MC\), \(i\)]\)=\!\(\*SubscriptBox[\(MR\), \ \(i\)]\)", StringJoin["\!\(\*SubscriptBox[\(q\), \(1\)]\)=", ToString[ Expand[$CellContext`BR1$], StandardForm], "\n", "\!\(\*SubscriptBox[\(q\), \(2\)]\)=", ToString[ Expand[$CellContext`BR2$], StandardForm]]], $CellContext`AK$[$CellContext`STEP5$$, 5, "Find the optimal quantity to produce for firm 1\n", "Plug \!\(\*SubscriptBox[\(BR\), \(2\)]\) into \ \!\(\*SubscriptBox[\(BR\), \(1\)]\)", StringJoin["\!\(\*SubsuperscriptBox[\(q\), \(1\), \(*\)]\)=", ToString[ Expand[ Round[$CellContext`QS1$, 0.001]], StandardForm]]], $CellContext`AK$[$CellContext`STEP6$$, 6, "Find the optimal quantity to produce for firm 2\n", "Plug \!\(\*SubsuperscriptBox[\(q\), \(1\), \(*\)]\) into \ \!\(\*SubscriptBox[\(BR\), \(2\)]\)", StringJoin["\!\(\*SubsuperscriptBox[\(q\), \(2\), \(*\)]\)=", ToString[ Expand[ Round[$CellContext`QS2$, 0.001]], StandardForm]]], $CellContext`AK$[$CellContext`STEP7$$, 7, "Find the market price\n", "Plug \!\(\*SubsuperscriptBox[\(q\), \(1\), \ \(*\)]\)+\!\(\*SubsuperscriptBox[\(q\), \(2\), \(*\)]\) into the inverse \ demand curve", StringJoin["\!\(\*SuperscriptBox[\(P\), \(*\)]\)=", ToString[ Expand[ Round[$CellContext`A - $CellContext`b ($CellContext`QS1$ + \ $CellContext`QS2$), 0.001]], StandardForm]]], $CellContext`AK$[$CellContext`STEP8$$, 8, "Find the profit of each firm\n", "\!\(\*SubscriptBox[\(\[CapitalPi]\), \ \(i\)]\)=\!\(\*SubscriptBox[\(TR\), \(i\)]\)-\!\(\*SubscriptBox[\(TC\), \ \(i\)]\)", StringJoin["\!\(\*SubscriptBox[\(\[CapitalPi]\), \(1\)]\)=", ToString[ Expand[ Round[$CellContext`\[CapitalPi]1$, 0.001]], StandardForm], "\n", "\!\(\*SubscriptBox[\(\[CapitalPi]\), \(2\)]\)=", ToString[ Expand[ Round[$CellContext`\[CapitalPi]2$, 0.001]], StandardForm]]], {}, { Style["Quantity Leadership (Firm 1 Leader):", Bold, Medium]}, { Style[ "The problem is the same as with the cournot model above, so we \ already know the best response function of firm 2", Italic, Medium], SpanFromLeft}, $CellContext`AK$[$CellContext`STEP9$$, 1, "Find firm 1's residual demand using firm 2's best response \ function\n", "\!\(\*SubscriptBox[\(P\), \(1\)]\)=A-b(\!\(\*SubscriptBox[\(q\), \ \(1\)]\)+\!\(\*SubscriptBox[\(BR\), \(2\)]\))", StringJoin["\!\(\*SubscriptBox[\(P\), \(1\)]\)=", ToString[ Expand[$CellContext`RD1$], StandardForm]]], $CellContext`AK$[$CellContext`STEP10$$, 2, "Find firm 1's TR using it's residual demand curve\n", "\!\(\*SubscriptBox[\(TR\), \(1\)]\)=\!\(\*SubscriptBox[\(P\), \ \(1\)]\)\!\(\*SubscriptBox[\(q\), \(1\)]\)", StringJoin["\!\(\*SubscriptBox[\(TR\), \(1\)]\)=", ToString[ Expand[$CellContext`TR1QL$], StandardForm]]], $CellContext`AK$[$CellContext`STEP11$$, 3, "Find firm 1's MR\n", "\!\(\*SubscriptBox[\(MR\), \(1\)]\)=\!\(\*FractionBox[\(\ \[PartialD]\*SubscriptBox[\(TR\), \(1\)]\), \ \(\[PartialD]\*SubscriptBox[\(q\), \(1\)]\)]\)", StringJoin["\!\(\*SubscriptBox[\(MR\), \(1\)]\)=", ToString[ Expand[$CellContext`MR1QL$], StandardForm]]], $CellContext`AK$[$CellContext`STEP12$$, 4, "Find firm 1's optimal quantity\n", "\!\(\*SubscriptBox[\(MR\), \(1\)]\)=\!\(\*SubscriptBox[\(MC\), \ \(1\)]\)", StringJoin["\!\(\*SubsuperscriptBox[\(q\), \(1\), \(*\)]\)=", ToString[ Expand[ Round[$CellContext`QS1QL$, 0.001]], StandardForm]]], $CellContext`AK$[$CellContext`STEP13$$, 5, "Find firm 2's optimal quantity\n", "Plug \!\(\*SubsuperscriptBox[\(q\), \(1\), \(*\)]\) into \ \!\(\*SubscriptBox[\(BR\), \(2\)]\)", StringJoin["\!\(\*SubsuperscriptBox[\(q\), \(2\), \(*\)]\)=", ToString[ Expand[ Round[$CellContext`QS2QL$, 0.001]], StandardForm]]], $CellContext`AK$[$CellContext`STEP14$$, 6, "Find the market price\n", "Plug \!\(\*SubsuperscriptBox[\(q\), \(1\), \ \(*\)]\)+\!\(\*SubsuperscriptBox[\(q\), \(2\), \(*\)]\) into the inverse \ demand function", StringJoin["\!\(\*SuperscriptBox[\(P\), \(*\)]\)=", ToString[ Expand[ Round[$CellContext`PSQL$, 0.001]], StandardForm]]], $CellContext`AK$[$CellContext`STEP15$$, 7, "Find the profit of each firm\n", "\!\(\*SubscriptBox[\(\[CapitalPi]\), \ \(i\)]\)=\!\(\*SubscriptBox[\(TR\), \(i\)]\)-\!\(\*SubscriptBox[\(TC\), \ \(i\)]\)", StringJoin["\!\(\*SubscriptBox[\(\[CapitalPi]\), \(1\)]\)=", ToString[ 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"1"}], ",", RowBox[{"STEP12", "=", "1"}], ",", RowBox[{"STEP13", "=", "1"}], ",", RowBox[{"STEP14", "=", "1"}], ",", RowBox[{"STEP15", "=", "1"}]}]], "Input", CellChangeTimes->{ 3.892038944121603*^9},ExpressionUUID->"9941f0d7-4a79-4b34-b3cf-\ bf576895b866"], Cell[BoxData["1"], "Output", CellChangeTimes->{{3.8891849285903873`*^9, 3.889184936695945*^9}}, CellLabel->"Out[7]=",ExpressionUUID->"acb7b286-ee9a-4d89-94e8-36c0adbe7f9d"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Bertrand Competition with differentiated products", "Section", CellChangeTimes->{{3.888338774194317*^9, 3.888338786165969*^9}},ExpressionUUID->"44580ae3-30e0-49bf-9b62-\ 5994bad9089d"], Cell[CellGroupData[{ Cell["Example", "Subsection", CellChangeTimes->{{3.888341233456565*^9, 3.888341234415372*^9}},ExpressionUUID->"6f830754-de0f-4215-9cda-\ aebd8c5f7f09"], Cell["\<\ Suppose that you have two firms: 1 and 2. Each firm produces a different \ version of a product and they compete on price. The firm\[CloseCurlyQuote]s \ demand curves are given by:\ \>", "Text", CellChangeTimes->{{3.888341244398924*^9, 3.8883412731642923`*^9}, { 3.888341320236671*^9, 3.888341327811644*^9}},ExpressionUUID->"6fa53462-81c4-46ab-87d5-\ 09b78beeb57b"], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {GridBox[{ { RowBox[{ RowBox[{ SubscriptBox["q", "1"], "=", RowBox[{"408", "-", RowBox[{"10", SubscriptBox["p", "1"]}], "+", RowBox[{"4", SubscriptBox["p", "2"]}]}]}], "\[LineSeparator]", RowBox[{ SubscriptBox["q", "2"], "=", RowBox[{"367", "+", RowBox[{"3", SubscriptBox["p", "1"]}], "-", RowBox[{"7", SubscriptBox["p", "2"]}]}]}]}]} }, GridBoxAlignment->{"Columns" -> {{"="}}}]} }, GridBoxItemSize->{"Columns" -> {{ Scaled[0.96]}}}], TraditionalForm]],ExpressionUUID->"c6c7aa92-19d4-4086-9e1c-d178e5c173e3"]], \ "Text", CellChangeTimes->{{3.8883413975138903`*^9, 3.888341424986788*^9}},ExpressionUUID->"642f98c4-21ca-4839-a403-\ 02d067ad3da0"], Cell["Additionally, assume each firm\[CloseCurlyQuote]s marginal cost is:", \ "Text", CellChangeTimes->{{3.888341427478322*^9, 3.8883414400227423`*^9}},ExpressionUUID->"58a556b7-9c58-41ea-a5a5-\ 44e4ecf19c4d"], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {GridBox[{ { RowBox[{ RowBox[{ SubscriptBox["MC", "1"], "=", "30"}], "\[LineSeparator]", RowBox[{ SubscriptBox["MC", "2"], "=", "96"}]}]} }, GridBoxAlignment->{"Columns" -> {{"="}}}]} }, GridBoxItemSize->{"Columns" -> {{ Scaled[0.96]}}}], TraditionalForm]],ExpressionUUID->"26ef9321-dc5f-4c31-8a78-18d4ebed61d8"]], \ "Text", CellChangeTimes->{{3.8883414430110683`*^9, 3.8883414547853813`*^9}},ExpressionUUID->"8d4cfc22-f900-4a0a-9596-\ 108ebbe07c78"], Cell["\<\ What will each firm charge for their version of the product in equilibrium?\ \>", "Text", CellChangeTimes->{{3.888341464856353*^9, 3.8883414791681137`*^9}},ExpressionUUID->"8d1b8e40-9d58-4dcb-ac20-\ 9495ea984cfa"], Cell[TextData[StyleBox["Solution", FontWeight->"Bold"]], "Text", CellChangeTimes->{{3.888341482362803*^9, 3.888341484366269*^9}},ExpressionUUID->"d27e6fea-2285-44bc-b1b4-\ 1ee13aaa434a"], Cell["Start by finding each firm\[CloseCurlyQuote]s total revenue equation", \ "Item", CellChangeTimes->{{3.888341489788804*^9, 3.888341507676957*^9}},ExpressionUUID->"eb4b9f96-5a82-49b3-a516-\ e25a578be510"], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ { RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{GridBox[{ { RowBox[{ SubscriptBox["TR", "1"], "=", RowBox[{ RowBox[{ SubscriptBox["p", "1"], SubscriptBox["q", "1"]}], "=", RowBox[{ SubscriptBox["p", "1"], "("}]}]}]} }, GridBoxAlignment->{"Columns" -> {{"="}}}], "408"}], "-", RowBox[{"10", SubscriptBox["p", "1"]}], "+", RowBox[{"4", SubscriptBox["p", "2"]}]}], ")"}], "=", RowBox[{ RowBox[{"408", SubscriptBox["p", "1"]}], "-", RowBox[{"10", SubsuperscriptBox["p", "1", "2"]}], "+", RowBox[{"4", SubscriptBox["p", "1"], SubscriptBox["p", "2"]}]}]}], "\[LineSeparator]", RowBox[{ SubscriptBox["TR", "2"], "=", RowBox[{ RowBox[{ SubscriptBox["p", "2"], SubscriptBox["q", "2"]}], "=", RowBox[{ RowBox[{ SubscriptBox["p", "2"], "(", RowBox[{"367", "+", RowBox[{"3", SubscriptBox["p", "1"]}], "-", RowBox[{"7", SubscriptBox["p", "2"]}]}], ")"}], "=", RowBox[{ RowBox[{"367", SubscriptBox["p", "2"]}], "+", RowBox[{"3", SubscriptBox["p", "1"], SubscriptBox["p", "2"]}], "-", RowBox[{"7", SubsuperscriptBox["p", "2", "2"]}]}]}]}]}]}]} }, GridBoxItemSize->{"Columns" -> {{ Scaled[0.96]}}}], TraditionalForm]],ExpressionUUID->"aad45f04-9e0c-405e-b578-4bb1a51d4116"]], \ "Text", CellChangeTimes->{{3.888341512063055*^9, 3.888341578542495*^9}},ExpressionUUID->"d556428b-d657-47c7-acf1-\ d6a73a6628bf"], Cell["Next find the marginal revenue equation for each firm:", "Item", CellChangeTimes->{{3.888341489788804*^9, 3.8883415066195803`*^9}, { 3.8883415926235123`*^9, 3.888341602878482*^9}},ExpressionUUID->"0aed8c86-a07a-4572-b130-\ 3720291795d0"], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {GridBox[{ { RowBox[{ RowBox[{ SubscriptBox["MR", "1"], "=", 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