1. **State the Problem:** We need to analyze the Milk Tea market using demand and supply functions, find the market equilibrium, and explore how external changes affect this equilibrium. 2. **Demand and Supply Functions:** Given parameters are: - Demand: $$ Q_d = 100 - 0.5P $$ where $P$ is price, $Q_d$ is quantity demanded. - Supply: $$ Q_s = 30 + 0.5P $$ where $Q_s$ is quantity supplied. 3. **Find Market Equilibrium:** Market equilibrium occurs where demand equals supply: $$ Q_d = Q_s $$ $$ 100 - 0.5P = 30 + 0.5P $$ Add $0.5P$ to both sides: $$ 100 = 30 + P $$ Subtract 30: $$ 70 = P $$ So, equilibrium price is $P=70$. Find equilibrium quantity by substituting $P=70$ into either equation: Demand side: $$ Q_d = 100 - 0.5 \times 70 = 100 - 35 = 65 $$ Supply side: $$ Q_s = 30 + 0.5 \times 70 = 30 + 35 = 65 $$ So, equilibrium quantity is 65 cups. 4. **Interpretation:** At price 70, buyers demand 65 cups and sellers supply 65 cups — the market clears. 5. **Scenario A - Sugar Price Hike (Supply Decreases):** Supply decreases means the supply curve shifts left. This can be modeled by lowering the intercept $a$ in supply function from 30 to say 20: $$ Q_s' = 20 + 0.5P $$ Find new equilibrium where $Q_d = Q_s'$: $$ 100 - 0.5P = 20 + 0.5P $$ $$ 100 - 20 = 0.5P + 0.5P $$ $$ 80 = P $$ New price is $P=80$. Quantity: $$ Q = 100 - 0.5 \times 80 = 100 - 40 = 60 $$ Quantity demanded and supplied is 60 cups. This shows price rises, quantity falls due to decreased supply. 6. **Scenario B - Increase in Demand (K-pop endorsement):** Demand increases, shift demand curve right by increasing intercept from 100 to 110: $$ Q_d' = 110 - 0.5P $$ Set equal to original supply: $$ 110 - 0.5P = 30 + 0.5P $$ $$ 110 - 30 = P $$ $$ 80 = P $$ New equilibrium price is 80. Quantity: $$ Q = 110 - 0.5 \times 80 = 110 - 40 = 70 $$ Quantity demanded and supplied is 70 cups. Price and quantity both increase. 7. **Summary:** Market equilibrium is where quantity demanded equals quantity supplied. External events shift demand or supply curves, changing price and quantity.