Use Wolfram Mathematica to solve the following problems: Conside...

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Use Wolfram Mathematica to solve the following problems:

Consider an industry consisting of two firms, competing as Cournot duopolists. The two firms produce an homogenous good with demand given by p = A − Q, where Q = q1 + q2 is the sum of individual outputs. Each firm’s marginal costofproductionisgivenbyci =C−xi−βxj,i,j=1,2andj̸=i,wherexi andxj
are the firms’ R&D investment. The term β ∈ [0, 1] is the R&D spillover parameter, which determines the size of the spillover from the R&D investment made by the rival firm. The cost of R&D is given by γx2i /2. The decision marking sequence is as follows: (1) firms choose R&D effort, then (2) firms choose their output.

1. Suppose that the firms behave non-cooperatively at both stages, find the equilibrium R&D and output.

2. Suppose that the firms only compete in the final goods market, but choose to cooperate when choosing their R&D investment, find find equilibrium R&D and output.

3. Compare the R&D investment and output levels in the two models for the entire ranges of R&D spillover, when does the non-cooperative R&D model dominate the cooperative model? Are there any parameter values for which they are equal? In which model is welfare greater and is this true for all values of γ.
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      • eloquent.retrieved: App\Models\SubjectCat (65μs)
      • Illuminate\Cache\Events\KeyWritten (316μs)
      • Illuminate\Cache\Events\CacheHit (176μs)
      • Illuminate\Cache\Events\CacheHit (197μs)
      • Illuminate\Cache\Events\CacheHit (202μs)
      • Illuminate\Cache\Events\CacheHit (264μs)
      • Illuminate\Cache\Events\CacheHit (346μs)
      • Illuminate\Cache\Events\CacheHit (204μs)
      • Illuminate\Cache\Events\CacheHit (171μs)
      • Illuminate\Cache\Events\CacheHit (198μs)
      • Illuminate\Cache\Events\CacheHit (155μs)
      • Illuminate\Cache\Events\CacheHit (551μs)
      • Illuminate\Cache\Events\CacheHit (460μs)
      • Illuminate\Cache\Events\CacheHit (1.38ms)
      • Illuminate\Cache\Events\CacheHit (190μs)
      • Illuminate\Cache\Events\CacheHit (242μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (248μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • creating: site.layouts.app (517μs)
      • composing: site.layouts.app (25μs)
      • creating: components.canonical (1.15ms)
      • composing: components.canonical (115μs)
      • creating: components.open-graph (239μs)
      • composing: components.open-graph (83μs)
      • creating: site.headers.header (318μs)
      • composing: site.headers.header (303μs)
      • Illuminate\Cache\Events\CacheHit (3.44ms)
      • creating: components.footer (80μs)
      • composing: components.footer (96μs)
      • Illuminate\Cache\Events\CacheHit (3.18ms)
      • Illuminate\Cache\Events\CacheMissed (399μs)
      • Illuminate\Database\Events\StatementPrepared (1.18ms)
      • Illuminate\Database\Events\QueryExecuted (1.41ms)
      • eloquent.retrieved: App\Models\SubjectCat (68μs)
      • Illuminate\Cache\Events\KeyWritten (431μs)
      • Illuminate\Cache\Events\CacheHit (504μs)
      • Illuminate\Cache\Events\CacheHit (222μs)
      • Illuminate\Cache\Events\CacheHit (142μs)
      • Illuminate\Cache\Events\CacheHit (113μs)
      • Illuminate\Cache\Events\CacheHit (171μs)
      • Illuminate\Cache\Events\CacheHit (325μs)
      • Illuminate\Cache\Events\CacheHit (333μs)
      • Illuminate\Cache\Events\CacheHit (262μs)
      • Illuminate\Cache\Events\CacheHit (193μs)
      • Illuminate\Cache\Events\CacheHit (193μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (177μs)
      • creating: components.forms.contact-us (410μs)
      • composing: components.forms.contact-us (249μs)
      • creating: components.forms.get-started (199μs)
      • composing: components.forms.get-started (85μs)
      • creating: components.forms.free-tool-download (150μs)
      • composing: components.forms.free-tool-download (78μs)
      • creating: components.forms.claim-free-worksheet (240μs)
      • composing: components.forms.claim-free-worksheet (203μs)
      • creating: components.forms.tutor-subscription-waitlist (223μs)
      • composing: components.forms.tutor-subscription-waitlist (74μs)
      • creating: components.forms.tutor-subscription-join (122μs)
      • composing: components.forms.tutor-subscription-join (118μs)
      • creating: components.forms.tutor-support (178μs)
      • composing: components.forms.tutor-support (215μs)
      • 314 x Illuminate\Cache\Events\CacheHit (18.04%)
        117ms
        17 x Illuminate\Database\Events\QueryExecuted (7.26%)
        46.98ms
        17 x Illuminate\Database\Events\StatementPrepared (7.07%)
        45.73ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (1.57%)
        10.15ms
        8 x Illuminate\Cache\Events\KeyWritten (1.23%)
        7.95ms
        8 x Illuminate\Cache\Events\CacheMissed (0.71%)
        4.59ms
        1 x Illuminate\Routing\Events\Routing (0.33%)
        2.15ms
        1 x Illuminate\Routing\Events\RouteMatched (0.19%)
        1.22ms
        9 x eloquent.retrieved: App\Models\SubjectCat (0.18%)
        1.19ms
        1 x Illuminate\Database\Events\ConnectionEstablished (0.18%)
        1.16ms
        1 x creating: components.canonical (0.18%)
        1.15ms
        2 x eloquent.retrieved: App\Models\Subject (0.1%)
        659μs
        1 x eloquent.booting: App\Models\SubjectCat (0.1%)
        625μs
        1 x creating: components.breadcrumbs (0.09%)
        573μs
        1 x creating: site.layouts.app (0.08%)
        517μs
        1 x composing: components.breadcrumbs (0.07%)
        439μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.06%)
        410μs
        1 x creating: components.forms.contact-us (0.06%)
        410μs
        14 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.05%)
        354μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.05%)
        349μs
        1 x creating: site.headers.header (0.05%)
        318μs
        1 x composing: site.headers.header (0.05%)
        303μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.04%)
        291μs
        1 x creating: homework.show (0.04%)
        262μs
        1 x composing: components.forms.contact-us (0.04%)
        249μs
        1 x creating: components.forms.claim-free-worksheet (0.04%)
        240μs
        1 x creating: components.open-graph (0.04%)
        239μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.03%)
        223μs
        1 x composing: components.forms.tutor-support (0.03%)
        215μs
        1 x composing: components.forms.claim-free-worksheet (0.03%)
        203μs
        1 x creating: components.forms.get-started (0.03%)
        199μs
        1 x creating: components.forms.tutor-support (0.03%)
        178μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.03%)
        163μs
        1 x eloquent.booting: App\Models\Subject (0.02%)
        158μs
        1 x creating: components.forms.free-tool-download (0.02%)
        150μs
        1 x creating: components.forms.tutor-subscription-join (0.02%)
        122μs
        1 x composing: components.forms.tutor-subscription-join (0.02%)
        118μs
        1 x composing: components.canonical (0.02%)
        115μs
        1 x composing: homework.show (0.02%)
        100μs
        1 x composing: components.footer (0.01%)
        96μs
        1 x composing: components.forms.get-started (0.01%)
        85μs
        1 x composing: components.open-graph (0.01%)
        83μs
        1 x creating: components.footer (0.01%)
        80μs
        1 x composing: components.forms.free-tool-download (0.01%)
        78μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.01%)
        74μs
        1 x eloquent.booted: App\Models\SubjectCat (0.01%)
        64μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        62μs
        1 x eloquent.booted: App\Models\Subject (0.01%)
        61μs
        1 x composing: site.layouts.app (0%)
        25μs
      14 templates were rendered
      • 1x homework.showshow.blade.phpblade
      • 1x components.breadcrumbsbreadcrumbs.blade.phpblade
      • 1x site.layouts.appapp.blade.phpblade
      • 1x components.canonicalcanonical.blade.phpblade
      • 1x components.open-graphopen-graph.blade.phpblade
      • 1x site.headers.headerheader.blade.phpblade
      • 1x components.footerfooter.blade.phpblade
      • 1x components.forms.contact-uscontact-us.blade.phpblade
      • 1x components.forms.get-startedget-started.blade.phpblade
      • 1x components.forms.free-tool-downloadfree-tool-download.blade.phpblade
      • 1x components.forms.claim-free-worksheetclaim-free-worksheet.blade.phpblade
      • 1x components.forms.tutor-subscription-waitlisttutor-subscription-waitlist.blade.phpblade
      • 1x components.forms.tutor-subscription-jointutor-subscription-join.blade.phpblade
      • 1x components.forms.tutor-supporttutor-support.blade.phpblade
      uri
      GET college-homework-library/{category}/{subject}/{id}
      middleware
      web, utm.parameters
      controller
      App\Http\Controllers\HomeworkLibraryController@show
      namespace
      where
      as
      homework.show
      file
      app/Http/Controllers/HomeworkLibraryController.php:79-176
      17 statements were executed, 5 of which were duplicates, 12 unique. Show only duplicated70.24ms
      • Connection Establishedtwenty4_siteHomeworkLibraryController.php#91
        Backtrace
        • 13. app/Http/Controllers/HomeworkLibraryController.php:91
        • 14. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 15. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 16. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary` where `status` = 'published' and `price` > 0 and `solutionslibrary`.`id` = '15440' limit 1
        26.14mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
        • 0: published
        • 1: 0
        • 2: 15440
        Backtrace
        • 16. app/Http/Controllers/HomeworkLibraryController.php:97
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 19. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 20. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subjects` where `subjects`.`id` in (20)
        4.57mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` in (15440)
        2.62mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subject_cats` where `subject_cats`.`id` = 4 limit 1
        1.24mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 4
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
        • 27. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 28. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 29. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
      • select * from `solutionslibrary` where `id` <> 15440 and `subject` = 20 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        18.01mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 15440
        • 1: 20
        • 2: published
        • 3: 0
        Backtrace
        • 14. app/Repositories/HomeworkLibraryRepository.php:30
        • 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 18. app/Repositories/HomeworkLibraryRepository.php:39
        • 19. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `subjects` where `subjects`.`id` in (20)
        1.54mstwenty4_siteHomeworkLibraryRepository.php#30
        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
        • 20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 23. app/Repositories/HomeworkLibraryRepository.php:39
        • 24. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 15440 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        2.02mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 15440
        • 1: question
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:58
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 15440 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        1.28mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 15440
        • 1: teaser
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:69
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 15440 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'solution' order by `order` asc, `id` asc
        1.74mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 15440
        • 1: solution
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:80
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `subject_cats` where `subject_cats`.`id` = 4 limit 1
        1.39mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 4
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 32. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 33. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 34. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
        • 35. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
      • select * from `subject_cats` where `subject_cats`.`id` = 4 limit 1
        1.27mstwenty4_siteSubject.php#100
        Bindings
        • 0: 4
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 10 limit 1
        1.49mstwenty4_siteSubject.php#100
        Bindings
        • 0: 10
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 33 limit 1
        1.09mstwenty4_siteSubject.php#100
        Bindings
        • 0: 33
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 5 limit 1
        1.19mstwenty4_siteSubject.php#100
        Bindings
        • 0: 5
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1
        1.55mstwenty4_siteSubject.php#100
        Bindings
        • 0: 34
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 31 limit 1
        1.23mstwenty4_siteSubject.php#100
        Bindings
        • 0: 31
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 36 limit 1
        1.87mstwenty4_siteSubject.php#100
        Bindings
        • 0: 36
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 28. view::components.footer:170
      App\Models\HomeworkLibrary\HomeworkLibraryFile
      14HomeworkLibraryFile.php
      App\Models\SubjectCat
      9SubjectCat.php
      App\Models\HomeworkLibrary\HomeworkLibrary
      7HomeworkLibrary.php
      App\Models\Subject
      2Subject.php
          _token
          tTycfy4h46VugR5OYgmhXGMX5iCQGWUf7EIrEMHq
          utm_source
          direct
          redirectUrl
          /college-homework-library/Business/Economics/15440
          _previous
          array:1 [ "url" => "https://staging.dev.24houranswers.com/college-homework-library/Business/Econom...
          _flash
          array:2 [ "old" => [] "new" => [] ]
          PHPDEBUGBAR_STACK_DATA
          []
          path_info
          /college-homework-library/Business/Economics/15440
          status_code
          200
          
          status_text
          OK
          format
          html
          content_type
          text/html; charset=UTF-8
          request_query
          []
          
          request_request
          []
          
          request_headers
          0 of 0
          array:21 [ "priority" => array:1 [ 0 => "u=0, i" ] "accept-encoding" => array:1 [ 0 => "gzip, deflate, br, zstd" ] "sec-fetch-dest" => array:1 [ 0 => "document" ] "sec-fetch-user" => array:1 [ 0 => "?1" ] "sec-fetch-mode" => array:1 [ 0 => "navigate" ] "sec-fetch-site" => array:1 [ 0 => "none" ] "accept" => array:1 [ 0 => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" ] "user-agent" => array:1 [ 0 => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" ] "upgrade-insecure-requests" => array:1 [ 0 => "1" ] "sec-ch-ua-platform" => array:1 [ 0 => ""Windows"" ] "sec-ch-ua-mobile" => array:1 [ 0 => "?0" ] "sec-ch-ua" => array:1 [ 0 => ""HeadlessChrome";v="129", "Not=A?Brand";v="8", "Chromium";v="129"" ] "cache-control" => array:1 [ 0 => "no-cache" ] "pragma" => array:1 [ 0 => "no-cache" ] "x-amzn-trace-id" => array:1 [ 0 => "Root=1-6805ad65-6bce8f1a22974c02359b5d3e" ] "host" => array:1 [ 0 => "staging.dev.24houranswers.com" ] "x-forwarded-port" => array:1 [ 0 => "443" ] "x-forwarded-proto" => array:1 [ 0 => "https" ] "x-forwarded-for" => array:1 [ 0 => "18.223.134.71" ] "content-length" => array:1 [ 0 => "" ] "content-type" => array:1 [ 0 => "" ] ]
          request_cookies
          []
          
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