Matematika Bisnis (menghitung variabel surplus konsumen)

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Fransisikus Asisi Verry Kristanto

 

D → q = √( 52 – p ) , S → p = 7 + ¼ q²

D → q² = 52 – p

S → p = 7 + ¼ q²

         ¼q² = p – 7 

            q² = 4p – 28

E → D = S

52 – p = 4p – 28

-5p = -80

p = 16

q² = 4p – 28

    = 4 (16) – 28

    = 36

q =  √36

   = 6

Maka titik E = (6 , 16)

D → q = √( 52 – p ) → q² = 52 – p → p = 52 – q²

S → p = 7 + ¼ q²

            ₆

Sk =    ∫ ( 52 – q² ) dq – p.q

⁰                       ₆                     

     = 52q – ⅓ (q³) ]  - p . q

                              ⁰

     = [ 52 (6) – ⅓ (6³)  ] – 0 – p . q

     = ( 312 – 72 ) – 0 – (16) (6)

     = 240 – 0 – 96

     = 144

Maka Surplus Konsumen: 144

                      ₆

Sp =    p . q - ∫ ( 7 + ¼ q² ) dq

              ⁰                                          ₆

     = (16) (6) – ( 7q – 1/12 (q³) ]                         

                                                  ⁰

     = 96 – [ 7 (6) – 1/12 (6³) ] – 0

     = 96 – ( 42 – 18 )

     = 36

Maka Surplus Produsen: 36

D → 3q² =  75 – p  →  p = 75 - 3q²

S → 6q = p – 3 → p = 6q + 3

E → D = S

75 – 3q² = 6q + 3

-3q² - 6q = 3 – 75

-3q² - 6q = -72

-3q² - 6q + 72 = 0

q² + 2q - 24 = 0

( q + 6 ) ( q – 4 )

q = -6 atau q = 4

q = -6 (tidak memenuhi) jadi q = 4

p = 6q + 3

   = 6 (4) + 3

   = 27

Maka titik E = ( 4 , 27 )

            ₄

Sk =    ∫ ( 75 - 3q² ) dq – p.q

⁰                  ₄                      

     = ( 75q – q³ ) ]  - p . q

                           ⁰

     = [ 75 (4) – 4³ ] – 0 – p . q

     = 300 – 48 – 0 – ( 27 . 4 )

     = 144

Maka Surplus Konsumen: 144

                      ₄

Sp =    p . q - ∫ ( 6q + 3 ) dq

              ⁰                                 ₄

     = ( 27 . 4 ) – ( 3q² + 3q ) ]                        

                                             ⁰

     = ( 27 . 4 ) – [ 3 (4)² + 3 (4) ] – 0

     = 108 –  ( 48 + 12 ) – 0

     = 108 – 60 – 0

     = 48

Maka Surplus Produsen: 48

D → p – 100 + 2q = 0 → p = 100 – 2q

S → p – 20 – 2q = 0 → p = 20 + 2q

E → D = S

100 – 2q = 20 + 2q

-4q = -80

    q = 20

p = 100 – 2q

   = 100 – 2(20)

   = 60

Maka titik E = ( 20 , 60 )

           ₂₀

Sk =    ∫ ( 100 – 2q ) dq – p.q

⁰              ₂₀                     

     = 100q – q² ]  - p . q

                         ⁰

     = [ 100 (20) – (20)² ] – 0 – (60) (20)

     = ( 2000 – 400 ) – 0 – 1200

     = 1600 – 1200

     = 400

Maka Surplus Konsumen: 400

                     ₂₀

Sp =    p . q - ∫ ( 20 + 2q ) dq

              ⁰                      ₂₀

     = p . q – ( 20q + q² ) ]                        

                                      ⁰

     = (60) (20) – [ 20(20) + (20)² - 0]

     = 1200 – ( 400 + 400 )

     = 400

Maka Surplus Produsen: 400

D → q = √100 – p → p = 10 – q

S → q = p – 2 → p = q + 2

E → D = S

10 – q = q + 2

-2q = -8

    q = 4

p = 10 – q

   = 10 – 4

   = 6

Maka titik E = ( 4 , 6 )

            ₄

Sk =    ∫ ( 10 – q ) dq – p.q

⁰                  ₄                     

     = 10q – ½ q² ]  - p . q

                           ⁰

     = [ 10 (4) – ½ (4)² ] – 0 – (6) (4)

     = ( 40 – 8 ) – 0 – 24 

     = 32 – 24

     = 8

Maka Surplus Konsumen: 8

                      ₄

Sp =    p . q - ∫ ( q + 2 ) dq

              ⁰                           ₄

     = p . q – ( ½ q² + 2q ) ]                        

                                         ⁰

     = (6) (4) – [ ½ (4)² + 2 (4) ] – 0

     = 24 – ( 8 + 8 )

     = 8

Maka Surplus Produsen: 8