Pembahasan Soal Ujian Profesi Aktuaris
SOAL
Jika fungsi permintaan dinyatakan dalam persamaan \({Q_d} = 180 – 2P\), berapakah besarnya surplus konsumen (consumer surplus) pada harga Rp. 65?
- 525
- 470
- 510
- 625
- 375
| Diketahui | \({Q_d} = 180 – 2P\) \(2P = 180 – {Q_d}\) \(P = 90 – 0,5{Q_d}\) |
| Step 1 | Masukkan P=0 dan Q=0
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| Step 2 | Ketika \({P_d}\) (surplus konsumen) Rp. 65, \({P_d} = 90 – 0,5{Q_d}\) \(65 = 90 – 0,5{Q_d}\) \(0,5{Q_d} = 90 – 65\) \({Q_d} = \frac{{25}}{{0,5}} = 50\) |
| Rumus |
Consumer Surplus ada 2
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| Step 3 |
Dengan rumus 1 \(Cs = \int\limits_0^{{Q_d}} {f\left( Q \right)\,dQ} – \left( {{Q_d} \cdot {P_d}} \right)\) \(= \int\limits_0^{50} {\left( {90 – 0,5Q} \right)dQ} – \left( {50 \cdot 65} \right)\) \(= \left[ {\left. {90Q – \frac{{0,5{Q^2}}}{2}\,} \right|_0^{50}} \right] – 3250\) \(= \left[ {\left( {90 \cdot 50} \right) – \frac{{0,5\left( {{{50}^2}} \right)}}{2}\,\, – \,0} \right] – 3250\) \(= 3875 – 3250 = 625\) |
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Dengan rumus 2 \(Cs = \int\limits_{{P_d}}^P {f\left( P \right)\,dP} \) \(= \int\limits_{65}^{90} {180 – 2P\,\,dP} \) \(= \left. {180P – \frac{{2{P^2}}}{2}\,\,} \right|_{65}^{90}\) \(= \left[ {180\left( {90} \right) – \frac{{2\left( {{{90}^2}} \right)}}{2}} \right] – \left[ {180\left( {65} \right) – \frac{{2\left( {{{65}^2}} \right)}}{2}} \right]\) \(= 8100 – 7475 = 625\) |
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| Jawaban | d. 625 |