1. **Problem Statement:** Calculate optimal prices, demands, and profits for different customer segments with given demand functions and cost $c=10$. --- ### Exercise 01 2. **Given:** - $Qd_1 = 5000 - 20p$ - $Qd_2 = 5000 - 40p$ - Cost per unit $c = 10$ 3. **Step 1: Calculate optimal price for each segment** - Formula: $p = \frac{a}{2b} + \frac{c}{2}$ - For Segment 1: $a=5000$, $b=20$ $$p_1 = \frac{5000}{2 \times 20} + \frac{10}{2} = \frac{5000}{40} + 5 = 125 + 5 = 130$$ - For Segment 2: $a=5000$, $b=40$ $$p_2 = \frac{5000}{2 \times 40} + \frac{10}{2} = \frac{5000}{80} + 5 = 62.5 + 5 = 67.5$$ 4. **Step 2: Calculate demand and profit for each segment** - Demand: $Qd = a - b p$ - Profit per segment: $\pi = (p - c) \times Qd$ - Segment 1: $$Qd_1 = 5000 - 20 \times 130 = 5000 - 2600 = 2400$$ $$\pi_1 = (130 - 10) \times 2400 = 120 \times 2400 = 288000$$ - Segment 2: $$Qd_2 = 5000 - 40 \times 67.5 = 5000 - 2700 = 2300$$ $$\pi_2 = (67.5 - 10) \times 2300 = 57.5 \times 2300 = 132250$$ 5. **Step 3: Calculate optimal single price for both segments combined** - Sum $a$: $5000 + 5000 = 10000$ - Sum $b$: $20 + 40 = 60$ $$p = \frac{10000}{2 \times 60} + \frac{10}{2} = \frac{10000}{120} + 5 = 83.33 + 5 = 88.33$$ 6. **Step 4: Calculate demand and profit at single price** - Segment 1: $$Qd_1 = 5000 - 20 \times 88.33 = 5000 - 1766.6 = 3233.4$$ $$\pi_1 = (88.33 - 10) \times 3233.4 = 78.33 \times 3233.4 \approx 253344$$ - Segment 2: $$Qd_2 = 5000 - 40 \times 88.33 = 5000 - 3533.2 = 1466.8$$ $$\pi_2 = (88.33 - 10) \times 1466.8 = 78.33 \times 1466.8 \approx 114885$$ - Total profit single price: $$\pi = 253344 + 114885 = 368229$$ 7. **Profit comparison:** - Differential pricing profit: $288000 + 132250 = 420250$ - Single price profit: $368229$ - Increase in profit: $$420250 - 368229 = 52021$$ --- ### Exercise 02 8. **Given:** - $Qd_1 = 3600 - 80p$ - $Qd_2 = 3600 - 60p$ - $Qd_3 = 3600 - 20p$ - Cost $c=10$ 9. **Step 1: Calculate optimal price for each segment** - Segment 1: $$p_1 = \frac{3600}{2 \times 80} + \frac{10}{2} = \frac{3600}{160} + 5 = 22.5 + 5 = 27.5$$ - Segment 2: $$p_2 = \frac{3600}{2 \times 60} + 5 = \frac{3600}{120} + 5 = 30 + 5 = 35$$ - Segment 3: $$p_3 = \frac{3600}{2 \times 20} + 5 = \frac{3600}{40} + 5 = 90 + 5 = 95$$ 10. **Step 2: Calculate demand and profit for each segment** - Segment 1: $$Qd_1 = 3600 - 80 \times 27.5 = 3600 - 2200 = 1400$$ $$\pi_1 = (27.5 - 10) \times 1400 = 17.5 \times 1400 = 24500$$ - Segment 2: $$Qd_2 = 3600 - 60 \times 35 = 3600 - 2100 = 1500$$ $$\pi_2 = (35 - 10) \times 1500 = 25 \times 1500 = 37500$$ - Segment 3: $$Qd_3 = 3600 - 20 \times 95 = 3600 - 1900 = 1700$$ $$\pi_3 = (95 - 10) \times 1700 = 85 \times 1700 = 144500$$ - Total profit differential pricing: $$24500 + 37500 + 144500 = 206500$$ 11. **Step 3: Calculate single optimal price for all segments combined** - Sum $a = 3600 + 3600 + 3600 = 10800$ - Sum $b = 80 + 60 + 20 = 160$ $$p = \frac{10800}{2 \times 160} + 5 = \frac{10800}{320} + 5 = 33.75 + 5 = 38.75$$ 12. **Step 4: Calculate demand and profit at single price** - Segment 1: $$Qd_1 = 3600 - 80 \times 38.75 = 3600 - 3100 = 500$$ $$\pi_1 = (38.75 - 10) \times 500 = 28.75 \times 500 = 14375$$ - Segment 2: $$Qd_2 = 3600 - 60 \times 38.75 = 3600 - 2325 = 1275$$ $$\pi_2 = 28.75 \times 1275 = 36656.25$$ - Segment 3: $$Qd_3 = 3600 - 20 \times 38.75 = 3600 - 775 = 2825$$ $$\pi_3 = 28.75 \times 2825 = 81187.5$$ - Total profit single price: $$14375 + 36656.25 + 81187.5 = 132218.75$$ 13. **Step 5: Profit difference** - Increase by differential pricing: $$206500 - 132218.75 = 74281.25$$ --- **Final answers:** - Exercise 01: - $p_1=130$, $p_2=67.5$ - Profit differential pricing = 420250 - Single price $p=88.33$, profit = 368229 - Profit increase = 52021 - Exercise 02: - $p_1=27.5$, $p_2=35$, $p_3=95$ - Profit differential pricing = 206500 - Single price $p=38.75$, profit = 132218.75 - Profit increase = 74281.25