1. **State the problem:** Two firms A and B form a cartel to maximize joint profit. Their cost functions are: $$C_1 = 20x_1 + 2x_1^2$$ $$C_2 = 48 + 16x_2 + 2x_2^2$$ The market demand is given by: $$x = 50 - 0.5p$$ where $$x = x_1 + x_2$$ is total output. 2. **Express price as a function of total output:** From demand, $$x = 50 - 0.5p \implies 0.5p = 50 - x \implies p = 100 - 2x$$ 3. **Write the total revenue (TR) for the cartel:** $$TR = p \times x = (100 - 2x) x = 100x - 2x^2$$ where $$x = x_1 + x_2$$. 4. **Write the total cost (TC) for both firms:** $$TC = C_1 + C_2 = (20x_1 + 2x_1^2) + (48 + 16x_2 + 2x_2^2) = 48 + 20x_1 + 16x_2 + 2x_1^2 + 2x_2^2$$ 5. **Write the joint profit function $$\pi$$:** $$\pi = TR - TC = (100(x_1 + x_2) - 2(x_1 + x_2)^2) - (48 + 20x_1 + 16x_2 + 2x_1^2 + 2x_2^2)$$ 6. **Expand and simplify profit:** $$\pi = 100x_1 + 100x_2 - 2(x_1^2 + 2x_1x_2 + x_2^2) - 48 - 20x_1 - 16x_2 - 2x_1^2 - 2x_2^2$$ $$= 100x_1 + 100x_2 - 2x_1^2 - 4x_1x_2 - 2x_2^2 - 48 - 20x_1 - 16x_2 - 2x_1^2 - 2x_2^2$$ $$= (100x_1 - 20x_1) + (100x_2 - 16x_2) - (2x_1^2 + 2x_1^2) - 4x_1x_2 - (2x_2^2 + 2x_2^2) - 48$$ $$= 80x_1 + 84x_2 - 4x_1^2 - 4x_2^2 - 4x_1x_2 - 48$$ 7. **Maximize profit by setting partial derivatives to zero:** $$\frac{\partial \pi}{\partial x_1} = 80 - 8x_1 - 4x_2 = 0$$ $$\frac{\partial \pi}{\partial x_2} = 84 - 8x_2 - 4x_1 = 0$$ 8. **Solve the system:** From first equation: $$80 = 8x_1 + 4x_2$$ From second equation: $$84 = 8x_2 + 4x_1$$ Multiply first by 2: $$160 = 16x_1 + 8x_2$$ Multiply second by 1: $$84 = 8x_2 + 4x_1$$ Multiply second by 2: $$168 = 16x_2 + 8x_1$$ Rewrite system: $$160 = 16x_1 + 8x_2$$ $$168 = 8x_1 + 16x_2$$ Multiply second equation by 2 and subtract first: $$168 = 8x_1 + 16x_2$$ Multiply by 2: $$336 = 16x_1 + 32x_2$$ Subtract first: $$336 - 160 = (16x_1 + 32x_2) - (16x_1 + 8x_2)$$ $$176 = 24x_2 \implies x_2 = \frac{176}{24} = \frac{22}{3} \approx 7.33$$ Substitute back to first: $$80 = 8x_1 + 4 \times \frac{22}{3} = 8x_1 + \frac{88}{3}$$ $$8x_1 = 80 - \frac{88}{3} = \frac{240 - 88}{3} = \frac{152}{3}$$ $$x_1 = \frac{152}{24} = \frac{19}{3} \approx 6.33$$ 9. **Calculate total output:** $$x = x_1 + x_2 = \frac{19}{3} + \frac{22}{3} = \frac{41}{3} \approx 13.67$$ 10. **Calculate price:** $$p = 100 - 2x = 100 - 2 \times \frac{41}{3} = 100 - \frac{82}{3} = \frac{300 - 82}{3} = \frac{218}{3} \approx 72.67$$ 11. **Calculate maximum joint profit:** $$\pi = 80x_1 + 84x_2 - 4x_1^2 - 4x_2^2 - 4x_1x_2 - 48$$ Substitute values: $$x_1 = \frac{19}{3}, x_2 = \frac{22}{3}$$ Calculate each term: $$80x_1 = 80 \times \frac{19}{3} = \frac{1520}{3}$$ $$84x_2 = 84 \times \frac{22}{3} = \frac{1848}{3}$$ $$4x_1^2 = 4 \times \left(\frac{19}{3}\right)^2 = 4 \times \frac{361}{9} = \frac{1444}{9}$$ $$4x_2^2 = 4 \times \left(\frac{22}{3}\right)^2 = 4 \times \frac{484}{9} = \frac{1936}{9}$$ $$4x_1x_2 = 4 \times \frac{19}{3} \times \frac{22}{3} = 4 \times \frac{418}{9} = \frac{1672}{9}$$ Sum terms: $$80x_1 + 84x_2 = \frac{1520}{3} + \frac{1848}{3} = \frac{3368}{3}$$ Sum squared terms: $$4x_1^2 + 4x_2^2 + 4x_1x_2 = \frac{1444}{9} + \frac{1936}{9} + \frac{1672}{9} = \frac{5052}{9}$$ Convert $$\frac{3368}{3}$$ to ninths: $$\frac{3368}{3} = \frac{3368 \times 3}{9} = \frac{10104}{9}$$ Calculate profit: $$\pi = \frac{10104}{9} - \frac{5052}{9} - 48 = \frac{5052}{9} - 48$$ Convert 48 to ninths: $$48 = \frac{432}{9}$$ Final profit: $$\pi = \frac{5052}{9} - \frac{432}{9} = \frac{4620}{9} = 513.33$$ **Final answers:** - Firm A produces $$x_1 = \frac{19}{3} \approx 6.33$$ units - Firm B produces $$x_2 = \frac{22}{3} \approx 7.33$$ units - Maximum joint profit is approximately 513.33 - Price charged is approximately 72.67