1. **Problem statement:** Calculate the consumer surplus for the demand functions given the market prices. 2. **Consumer Surplus (CS) definition:** It is the area between the demand curve and the market price line, up to the quantity demanded at that price. 3. **Part a) Demand function:** $P = 70 - 2Q$, market price $P_0 = 14$. 4. **Find quantity demanded at $P_0$:** $$14 = 70 - 2Q \implies 2Q = 70 - 14 = 56 \implies Q = 28$$ 5. **i) Geometric method:** - The consumer surplus is the area of the triangle with height $(70 - 14) = 56$ and base $28$. - Area = $\frac{1}{2} \times 28 \times 56 = 784$ 6. **ii) Integration method:** - Consumer surplus = $\int_0^{28} (70 - 2Q) dQ - 14 \times 28$ - Calculate integral: $$\int_0^{28} (70 - 2Q) dQ = [70Q - Q^2]_0^{28} = 70 \times 28 - 28^2 = 1960 - 784 = 1176$$ - Subtract total expenditure: $$14 \times 28 = 392$$ - Consumer surplus = $1176 - 392 = 784$ 7. **Part b) Demand function:** $P = \frac{125}{Q+2}$, market price $P_0 = 25$. 8. **Find quantity demanded at $P_0$:** $$25 = \frac{125}{Q+2} \implies Q+2 = \frac{125}{25} = 5 \implies Q = 3$$ 9. **Consumer surplus by integration:** $$CS = \int_0^3 \frac{125}{Q+2} dQ - 25 \times 3$$ - Calculate integral: $$\int_0^3 \frac{125}{Q+2} dQ = 125 \int_0^3 \frac{1}{Q+2} dQ = 125 [\ln(Q+2)]_0^3 = 125 (\ln 5 - \ln 2) = 125 \ln \frac{5}{2}$$ - Calculate total expenditure: $$25 \times 3 = 75$$ - Consumer surplus: $$CS = 125 \ln \frac{5}{2} - 75$$ **Final answers:** - a) i) Geometric: $784$ - a) ii) Integration: $784$ - b) Integration: $125 \ln \frac{5}{2} - 75$