Beranda > Economics, Microeconomics > Production Function (Lanjutan Perilaku Produsen)

Production Function (Lanjutan Perilaku Produsen)


Fungsi Produksi (Contoh Soal)

 

Bila diketahui bahwa fungsi produksi barang X dapat dirumuskan dalam format Cobb-Douglas Function Q = 5.K1/4L3/4 di mana Q0 = 1.000 unit

Diminta:

  1. Hitunglah least-cost combination penggunaan input K dan L bila r = Rp. 10.000 per machine hour, dan w = Rp 5.000 per man hour.
  2. Hitung kembali jawaban soal (a) bila Q’ = 1.100 dan Q” = 1.500
  3. Buatlah rekapitulasi yang menggambarkan Cost Function
  4. Susun kembali rekapitulasi tersebut pada soal c bila cost of capital naik 10% dan labor cost naik 5%
  5. Dapatkah saudara menggambarkannya dalam bentuk grafik

 

Jawaban :

  1. C = r.K + w.L

    Meminimumkan biaya dengan kendala fungsi produksi

    Dengan SOP optimasi:

    minimumkan    : C = r.K + w.L

    yang memenuhi :

    kendala         : Q = 5.K1/4L3/4

    fungsi sasaran    : G = r.K + w.L – λ(5.K1/4L3/4 – Q)

    Syarat primer:

    = 0    ; = 0

     

    Sehingga:

    = r – λ.5.(1/4).K-3/4L3/4 = 0

    r – λ.(5/4).K-3/4L3/4 = 0

    λ.(5/4).K-3/4L3/4 = r

    λ = r/((5/4).K-3/4L3/4)

     

    = w – λ.5.(3/4).K1/4L-1/4 = 0

    w – λ.(15/4).K1/4L-1/4 = 0

    λ.(15/4).K1/4L-1/4 = w

    λ = w/((15/4).K1/4L-1/4)

     

    λ = λ

     

    r/((5/4).K-3/4L3/4) = w/((15/4).K1/4L-1/4)

    r.(15/4).K1/4L-1/4 = w.(5/4).K-3/4L3/4

    kedua ruas dikali 4

    r.15.K1/4L-1/4 = w.5.K-3/4L3/4

    kedua ruas dibagi 5

    r.3.K1/4L-1/4 = w.1.K-3/4L3/4

    kedua ruas dikali K3/4.L1/4

    r.3.K1L0 = w.K0L1

    r.3.K1.1 = w.1.L1

    r.3.K = w.L

    r.3.K/w = L

    L = r.3.K/w

     

    Diketahui r = 10.000 per machine hour, w = Rp.5.000 per man hour

     

    L = 10.000.3.K/5.000

    L = 30.000K/5.000

    L = 6K

     

    Diketahui Q0 = 1.000 unit

    Q = 5.K1/4L3/4

    Q = 5. K1/4(6K)3/4

    Q = 5. K1/4.63/4K3/4

    Q = 5.63/4.K

    Q = 19,17K

    K = Q/19,17

    K = 1.000/19.17

    K = 52,17 machine hour

    L = 6.K

    L = 6. 52.17

    L = 313,02 man hour

     

  2. Bila Q’ = 1.100 unit

    K = Q/19,17

    K = 1.100/19.17

    K = 57,38 machine hour

    L = 6.K

    L = 344,28 man hour

    Bila Q” = 1.500

    K = Q/19,17

    K = 1.500/19.17

    K = 78,25 machine hour

    L = 6.K

    L = 469,5 man hour

     

  3. Rekapitulasi Cost schedule

    r = Rp.10.000 per machine hour; w = Rp.5.000 per man hour

Q (unit) 

C (rupiah) = r.K + w.L 

Keterangan 

1.000 

2.086.800 

K = 52,17; L = 313,02

1.100 

2.295.200 

K = 57,38; L = 344,28

1.500 

3.130.000 

K = 78,25; L = 469,5 

 

  1. Bila cost of capital naik 10%, labor cost naik 5%

    r = 10.000 (1 + 10%)

    r = 10.000 (1,1)

    r = 11.000

    w = 5.000 (1+5%)

    w = 5.000 (1.05)

    w = 5.250

     

    Karena adanya perubahan r dan w, maka kombinasi penggunaan input optimal berubah menjadi:

    L = r.3.K/w

    L = 11.000.3.K/5.250

    L = 33.000K/5.250

    L = 6,29K

     

    Sehingga

    Q = 5.K1/4L3/4

    Q = 5. K1/4(6,2857K)3/4

    Q = 5. K1/4.6,28573/4K3/4

    Q = 5.6,28573/4.K

    Q = 19,85K

    K = Q/19,85

     

    Bika Q0 = 1.000

    K = 1.000/19,85

    K = 50,38

    L = 6,29K

    L = 6,29. 50,38

    L = 316,18

     

    Bila Q’ = 1.100

    K = 1.100/19,85

    K = 55,42

    L = 6,29K

    L = 6,29. 55,42

    L = 348,59

     

    Bila Q’ = 1.500

    K = 1.500/19,85

    K = 75,57

    L = 6,29K

    L = 6,29. 75,57

    L = 475,34

    r = Rp.11.000 per machine hour; w = Rp.5.250 per man hour

     

Q (unit)

C (rupiah) = r.K + w.L 

Keterangan 

1.000 

2.214.125

K = 50,38; L = 316,18

1.100 

2.439.718

K = 55,42; L = 348,59

1.500 

3.326.805

K = 75,57; L = 475,34 

 

  1. Grafik Cost Function

 

Pada r = 10.000, w = 5.000

C = r.K + w.L

Q = 5.K1/4L3/4

L = 6.K

 

Q = 5.K1/4L3/4

Q = 5.K1/4.(6K)3/4

Q = 5.63/4.K1/4.K3/4

Q = 5.63/4.K

Q = 19,1682K

K = Q/19,1682

K = 0.0521.Q

 

C = r.K + w.L

C = 10.000.K + 5.000.6.K

C = 10.000.K + 30.000K

C = 40.000K

C = 40.000.0.0521.Q

C = 2084Q

 

Pada r = 11.000, w = 5.200

C = r.K + w.L

Q = 5.K1/4L3/4

L = 6,29K

Q = 5.K1/4L3/4

Q = 5.K1/4.(6,29K)3/4

Q = 5.6,293/4.K1/4.K3/4

Q = 5.63/4.K

Q = 19,8590K

K = Q/19,8590

K = 0.05036.Q

 

C = r.K + w.L

C = 11.000.K + 5.250.6,29.K

C = 11.000.K + 33.022,5K

C = 44.022,5K

C = 44.022,5. 0.05036.Q

C = 2.216,97Q

 

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