Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Ekonomi |
| Periode Ujian |
: |
November 2018 |
| Nomor Soal |
: |
6 |
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
- Ketika P = 0
\(P = 90 – 0,5{Q_d}\)
\(0 = 90 – 0,5{Q_d}\)
\(0,5{Q_d} = 90\)
\({Q_d} = 180\)
- Ketika Q = 0
\(P = 90 – 0,5{Q_d}\)
\(P = 90 – 0,5(0)\)
\(P = 90\)
- Sehingga \(\left( {P,Q} \right) = \left( {90\,,\,\,180} \right)\)
|
| 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
- \(Cs = \int\limits_0^{{Q_d}} {f\left( Q \right)\,dQ} – \left( {{Q_d} \cdot {P_d}} \right)\)
- \(Cs = \int\limits_{{P_d}}^P {f\left( P \right)\,dP} \)
|
| Step 3 |
Dengan rumus 1
\(Cs = \int\limits_0^{{Q_d}} {f\left( Q \right)\,dQ} – \left( {{Q_d} \cdot {P_d}} \right)\)
\(= \int\limits_0^{50} {\left( {90 – 0,5Q} \right)dQ} – \left( {50 \cdot 65} \right)\)
\(= \left[ {\left. {90Q – \frac{{0,5{Q^2}}}{2}\,} \right|_0^{50}} \right] – 3250\)
\(= \left[ {\left( {90 \cdot 50} \right) – \frac{{0,5\left( {{{50}^2}} \right)}}{2}\,\, – \,0} \right] – 3250\)
\(= 3875 – 3250 = 625\)
|
|
Dengan rumus 2
\(Cs = \int\limits_{{P_d}}^P {f\left( P \right)\,dP} \)
\(= \int\limits_{65}^{90} {180 – 2P\,\,dP} \)
\(= \left. {180P – \frac{{2{P^2}}}{2}\,\,} \right|_{65}^{90}\)
\(= \left[ {180\left( {90} \right) – \frac{{2\left( {{{90}^2}} \right)}}{2}} \right] – \left[ {180\left( {65} \right) – \frac{{2\left( {{{65}^2}} \right)}}{2}} \right]\)
\(= 8100 – 7475 = 625\)
|
| Jawaban |
d. 625 |
