This week’s problem asks you to dig into the problem of public good...

Question
This week’s problem asks you to dig into the problem of public goods. In ordinary markets with private goods, it’s easy to show that the market solution where supply and demand intersect is also the most efficient solution. However, in markets where goods are “public” in the economic sense of non-rivalry and non-excludability, it is hard for producers and consumers to come naturally to the efficient quantity through the price system.

Consider the town of Springfield with only three residents, Sophia, Amber, and Cedric. The three residents are trying to determine how large, in acres, they should build the public park. The table below shows each resident’s willingness to pay for each acre of the park.

Acres Sophia Amber Cedric
1 $10 $24 $6
2 8 18 5
3 6 14 4
4 3 8 3
5 1 6 2
6 0 4 1
7 0 2 0

We’ll start by searching for the efficient outcome in one scenario (I’ll walk you through it), and then you’ll be asked to think about how things change when our assumptions change.

Example 1: Suppose the cost to build the park is $9 per acre. How many acres should the park be to maximize total surplus from the park in Springfield?

To do this question, consider how much value the first acre would provide. Sophia gets $10 in benefits, Amber gets $24 in benefits, and Cedric Gets $6. That’s $40, so the benefit exceeds the $9 cost of building the first acre. The first acre should get built. Keep checking this for each acre.
For the second acre, Sophia gets $8, Amber gets $18, and Cedric gets $5, for a total of $31. Again, $31>$9, so keep building.
If you do this, you’ll see that five acres is the total that should get built. (You can assume that if the costs of an acre are exactly equal to the benefits, that acre will get built.)
Also, think about this from the perspective of each of the residents. When 5 acres are built, Sophia gets $28 in benefits. She gets $10 in benefits from the first acre, $8 from the second, and so on. ($10+8+6+3+1=$28). If she was asked to vote “Yes” or “No” for building a 5-acre park in exchange for a $15 increase in property taxes, say, she would vote YES.


QUESTIONS: Refer to the table above. Show your work for each of your answers.

1. Suppose the cost to build the park is $24 per acre. How large should the park be to maximize total surplus from the park in Springfield?

2. Suppose the cost to build the park is $24 per acre and that the residents have agreed to split the cost of building the park equally. That is, they each have to pay $8 per acre. To maximize his own surplus, how many acres would Cedric like Springfield to build? HINT: To find Cedric’s surplus, add up how much benefit he gets from each acre that is built, then compare it to the amount he has to pay.

3. Suppose the cost to build the park is $24 per acre and that the residents have agreed to split the cost of building the park equally. If the residents vote to determine the size of park to build, basing their decision solely on their own willingness to pay (and trying to maximize their own surplus), what is the largest park size for which the majority of residents would vote “yes?”

4. Suppose the cost to build the park is $24 per acre and that the residents have agreed to split the cost of building the park equally. If the residents decide to build a park with size equal to the number of acres that maximizes total surplus from the park, how much total surplus will Sophia receive?
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1. Suppose the cost to build the park is $24 per acre. How large should the park be to maximize total surplus from the park in Springfield?

The total benefits of first acre is $40 for the three residents (10+24+6=40)
The total benefits of the second acre is $31 for the three residents (8+18+5=31)
The total benefits of the third acre is $24 for the three residents (6+14+4 = 24)
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      • Illuminate\Cache\Events\CacheHit (272μs)
      • Illuminate\Cache\Events\CacheHit (258μs)
      • Illuminate\Cache\Events\CacheHit (196μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (161μs)
      • Illuminate\Cache\Events\CacheHit (184μs)
      • Illuminate\Cache\Events\CacheHit (8.38ms)
      • Illuminate\Cache\Events\CacheHit (380μs)
      • Illuminate\Cache\Events\CacheHit (168μs)
      • Illuminate\Cache\Events\CacheHit (206μs)
      • Illuminate\Cache\Events\CacheHit (175μs)
      • Illuminate\Cache\Events\CacheHit (180μs)
      • Illuminate\Cache\Events\CacheHit (157μs)
      • Illuminate\Cache\Events\CacheHit (175μs)
      • Illuminate\Cache\Events\CacheHit (155μs)
      • Illuminate\Cache\Events\CacheHit (206μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (191μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (245μs)
      • Illuminate\Cache\Events\CacheHit (507μs)
      • Illuminate\Cache\Events\CacheHit (198μs)
      • Illuminate\Cache\Events\CacheHit (151μs)
      • Illuminate\Cache\Events\CacheHit (181μs)
      • Illuminate\Cache\Events\CacheHit (381μs)
      • Illuminate\Cache\Events\CacheHit (215μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (190μs)
      • Illuminate\Cache\Events\CacheHit (158μs)
      • Illuminate\Cache\Events\CacheMissed (290μs)
      • Illuminate\Database\Events\StatementPrepared (6.98ms)
      • Illuminate\Database\Events\QueryExecuted (1.23ms)
      • eloquent.retrieved: App\Models\SubjectCat (137μs)
      • Illuminate\Cache\Events\KeyWritten (493μs)
      • Illuminate\Cache\Events\CacheHit (366μs)
      • Illuminate\Cache\Events\CacheHit (456μs)
      • Illuminate\Cache\Events\CacheHit (218μs)
      • Illuminate\Cache\Events\CacheHit (212μs)
      • Illuminate\Cache\Events\CacheHit (247μs)
      • Illuminate\Cache\Events\CacheHit (336μs)
      • Illuminate\Cache\Events\CacheHit (213μs)
      • Illuminate\Cache\Events\CacheHit (203μs)
      • Illuminate\Cache\Events\CacheHit (261μs)
      • Illuminate\Cache\Events\CacheHit (180μs)
      • Illuminate\Cache\Events\CacheHit (278μs)
      • Illuminate\Cache\Events\CacheHit (198μs)
      • Illuminate\Cache\Events\CacheHit (203μs)
      • Illuminate\Cache\Events\CacheHit (175μs)
      • Illuminate\Cache\Events\CacheHit (158μs)
      • Illuminate\Cache\Events\CacheHit (305μs)
      • Illuminate\Cache\Events\CacheHit (179μs)
      • creating: site.layouts.app (1.01ms)
      • composing: site.layouts.app (26μs)
      • creating: components.canonical (8.84ms)
      • composing: components.canonical (99μs)
      • creating: components.open-graph (8.07ms)
      • composing: components.open-graph (85μs)
      • creating: site.headers.header (265μs)
      • composing: site.headers.header (160μs)
      • Illuminate\Cache\Events\CacheHit (1.73ms)
      • creating: components.footer (140μs)
      • composing: components.footer (95μs)
      • Illuminate\Cache\Events\CacheHit (1.04ms)
      • Illuminate\Cache\Events\CacheHit (321μs)
      • Illuminate\Cache\Events\CacheHit (256μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (184μs)
      • Illuminate\Cache\Events\CacheHit (268μs)
      • Illuminate\Cache\Events\CacheHit (189μs)
      • Illuminate\Cache\Events\CacheHit (375μs)
      • Illuminate\Cache\Events\CacheHit (471μs)
      • Illuminate\Cache\Events\CacheHit (205μs)
      • Illuminate\Cache\Events\CacheHit (286μs)
      • Illuminate\Cache\Events\CacheHit (316μs)
      • Illuminate\Cache\Events\CacheHit (481μs)
      • Illuminate\Cache\Events\CacheHit (559μs)
      • creating: components.forms.contact-us (8.65ms)
      • composing: components.forms.contact-us (200μs)
      • creating: components.forms.get-started (244μs)
      • composing: components.forms.get-started (91μs)
      • creating: components.forms.free-tool-download (219μs)
      • composing: components.forms.free-tool-download (86μs)
      • creating: components.forms.claim-free-worksheet (972μs)
      • composing: components.forms.claim-free-worksheet (170μs)
      • creating: components.forms.tutor-subscription-waitlist (444μs)
      • composing: components.forms.tutor-subscription-waitlist (84μs)
      • creating: components.forms.tutor-subscription-join (9.11ms)
      • composing: components.forms.tutor-subscription-join (132μs)
      • creating: components.forms.tutor-support (198μs)
      • composing: components.forms.tutor-support (86μs)
      • 312 x Illuminate\Cache\Events\CacheHit (16.95%)
        191ms
        19 x Illuminate\Database\Events\StatementPrepared (8.34%)
        93.97ms
        19 x Illuminate\Database\Events\QueryExecuted (5.57%)
        62.80ms
        10 x Illuminate\Cache\Events\CacheMissed (1.32%)
        14.93ms
        1 x Illuminate\Routing\Events\Routing (0.86%)
        9.74ms
        1 x creating: components.forms.tutor-subscription-join (0.81%)
        9.11ms
        1 x creating: components.canonical (0.78%)
        8.84ms
        1 x creating: components.forms.contact-us (0.77%)
        8.65ms
        10 x Illuminate\Cache\Events\KeyWritten (0.73%)
        8.20ms
        1 x creating: components.open-graph (0.72%)
        8.07ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (0.47%)
        5.29ms
        11 x eloquent.retrieved: App\Models\SubjectCat (0.12%)
        1.36ms
        1 x Illuminate\Database\Events\ConnectionEstablished (0.1%)
        1.16ms
        1 x creating: site.layouts.app (0.09%)
        1.01ms
        1 x creating: components.forms.claim-free-worksheet (0.09%)
        972μs
        1 x Illuminate\Routing\Events\RouteMatched (0.07%)
        784μs
        1 x creating: components.breadcrumbs (0.05%)
        563μs
        1 x eloquent.booting: App\Models\SubjectCat (0.05%)
        548μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.04%)
        444μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.04%)
        402μs
        1 x creating: homework.show (0.04%)
        396μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.03%)
        347μs
        1 x creating: site.headers.header (0.02%)
        265μs
        1 x creating: components.forms.get-started (0.02%)
        244μs
        2 x eloquent.retrieved: App\Models\Subject (0.02%)
        234μs
        1 x creating: components.forms.free-tool-download (0.02%)
        219μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.02%)
        207μs
        4 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.02%)
        201μs
        1 x composing: components.breadcrumbs (0.02%)
        200μs
        1 x composing: components.forms.contact-us (0.02%)
        200μs
        1 x creating: components.forms.tutor-support (0.02%)
        198μs
        1 x composing: components.forms.claim-free-worksheet (0.02%)
        170μs
        1 x composing: site.headers.header (0.01%)
        160μs
        1 x composing: homework.show (0.01%)
        155μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        143μs
        1 x creating: components.footer (0.01%)
        140μs
        1 x composing: components.forms.tutor-subscription-join (0.01%)
        132μs
        1 x eloquent.booting: App\Models\Subject (0.01%)
        118μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        109μs
        1 x composing: components.canonical (0.01%)
        99μs
        1 x composing: components.footer (0.01%)
        95μs
        1 x composing: components.forms.get-started (0.01%)
        91μs
        1 x composing: components.forms.free-tool-download (0.01%)
        86μs
        1 x composing: components.forms.tutor-support (0.01%)
        86μs
        1 x composing: components.open-graph (0.01%)
        85μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.01%)
        84μs
        1 x eloquent.booted: App\Models\SubjectCat (0.01%)
        59μs
        1 x eloquent.booted: App\Models\Subject (0%)
        55μs
        1 x composing: site.layouts.app (0%)
        26μs
      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
      19 statements were executed, 5 of which were duplicates, 14 unique. Show only duplicated100ms
      • 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` = '20805' limit 1
        14.15mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
        • 0: published
        • 1: 0
        • 2: 20805
        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)
        920μstwenty4_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 (20805)
        1.12mstwenty4_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
        5.06mstwenty4_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` <> 20805 and `subject` = 20 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        17.77mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 20805
        • 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)
        1.31mstwenty4_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` = 20805 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        900μstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 20805
        • 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` = 20805 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        4.78mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 20805
        • 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` = 20805 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'solution' order by `order` asc, `id` asc
        1.64mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 20805
        • 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
        9.75mstwenty4_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` = 1 limit 1
        3.12mstwenty4_siteSubject.php#100
        Bindings
        • 0: 1
        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` = 3 limit 1
        1.6mstwenty4_siteSubject.php#100
        Bindings
        • 0: 3
        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` = 4 limit 1
        5.33mstwenty4_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
        2.27mstwenty4_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
        7.71mstwenty4_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` = 11 limit 1
        7.78mstwenty4_siteSubject.php#100
        Bindings
        • 0: 11
        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
        3.44mstwenty4_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
        4.12mstwenty4_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
        7.41mstwenty4_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
      App\Models\SubjectCat
      11SubjectCat.php
      App\Models\HomeworkLibrary\HomeworkLibrary
      7HomeworkLibrary.php
      App\Models\HomeworkLibrary\HomeworkLibraryFile
      4HomeworkLibraryFile.php
      App\Models\Subject
      2Subject.php
          _token
          0HR5YuICt0RJ7Rf9MUtzBPeBNEJyVK9ZbR80uCYr
          utm_source
          direct
          redirectUrl
          /college-homework-library/Business/Economics/20805
          _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/20805
          status_code
          200
          
          status_text
          OK
          format
          html
          content_type
          text/html; charset=UTF-8
          request_query
          []
          
          request_request
          []
          
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