1. **Problem 1:** Given consumption function $C = a + bY$, with $C=60$ when $Y=40$ and $C=90$ when $Y=80$, find $a$, $b$, and the consumption function. 2. **Problem 2:** Demand schedule $P = a - bQ$, find $a$, $b$, and the demand function given data (not explicitly provided, so we use points from the graph: intercepts $a=24$, $b=\frac{24}{30}=0.8$). 3. **Problem 3:** Linear demand slope $-\frac{3}{10}$, price $P=6$ at quantity $Q=20$. Find intercepts and prices at $Q=10$ and quantity at $P=3$. 4. **Problem 4:** Demand slope $-0.8$, price $P=10000$ at quantity $Q=100000$. Find demand function $P=f(Q)$ and quantity demanded at $P=12000$. 5. **Problem 5:** Demand linear, sells 40 units at price 34 and 30 units at price 36. Find demand function, predict quantity at price 2, and price at quantity 20. 6. **Problem 6:** Quantity falls from 120 to 100 when price rises from 40 to 50. Find demand function and price when quantity is zero. 7. **Problem 7:** Supply function linear, price rises from 1000 to 1200, supply rises from 2000 to 2400. Find supply function $P=f(Q)$, estimate quantity at price 800, and price at quantity 3000. --- ### Step-by-step solutions: **1. Consumption function:** Given $C = a + bY$. From $C=60$ at $Y=40$: $$60 = a + 40b$$ From $C=90$ at $Y=80$: $$90 = a + 80b$$ Subtract equations: $$90 - 60 = (a + 80b) - (a + 40b) \Rightarrow 30 = 40b \Rightarrow b = \frac{30}{40} = 0.75$$ Substitute $b=0.75$ into first equation: $$60 = a + 40 \times 0.75 = a + 30 \Rightarrow a = 60 - 30 = 30$$ **Consumption function:** $$C = 30 + 0.75Y$$ **2. Demand function:** Given $P = a - bQ$. From graph points: intercepts at $P=24$ when $Q=0$ (point B), and $Q=30$ when $P=0$ (point A). So $a = 24$ (price intercept). Slope $b = \frac{a}{Q_{intercept}} = \frac{24}{30} = 0.8$ Demand function: $$P = 24 - 0.8Q$$ **3. Demand schedule slope $-\frac{3}{10}$:** Given slope $m = -\frac{3}{10} = -0.3$. At $Q=20$, $P=6$: Use point-slope form: $$P - 6 = -0.3(Q - 20)$$ Simplify: $$P = 6 - 0.3Q + 6 = 12 - 0.3Q$$ Intercepts: Price intercept ($Q=0$): $$P = 12 - 0.3 \times 0 = 12$$ Quantity intercept ($P=0$): $$0 = 12 - 0.3Q \Rightarrow Q = \frac{12}{0.3} = 40$$ Price at $Q=10$: $$P = 12 - 0.3 \times 10 = 12 - 3 = 9$$ Quantity at $P=3$: $$3 = 12 - 0.3Q \Rightarrow 0.3Q = 9 \Rightarrow Q = 30$$ **4. Demand function with slope $-0.8$:** Given $P = a - 0.8Q$, and $P=10000$ at $Q=100000$. Find $a$: $$10000 = a - 0.8 \times 100000 = a - 80000 \Rightarrow a = 90000$$ Demand function: $$P = 90000 - 0.8Q$$ Quantity demanded at $P=12000$: $$12000 = 90000 - 0.8Q \Rightarrow 0.8Q = 90000 - 12000 = 78000 \Rightarrow Q = \frac{78000}{0.8} = 97500$$ **5. Demand function from two points:** Points: $(Q_1=40, P_1=34)$ and $(Q_2=30, P_2=36)$. Slope: $$m = \frac{36 - 34}{30 - 40} = \frac{2}{-10} = -0.2$$ Equation: $$P - 34 = -0.2(Q - 40)$$ Simplify: $$P = 34 - 0.2Q + 8 = 42 - 0.2Q$$ Predict quantity at $P=2$: $$2 = 42 - 0.2Q \Rightarrow 0.2Q = 40 \Rightarrow Q = 200$$ Price at $Q=20$: $$P = 42 - 0.2 \times 20 = 42 - 4 = 38$$ **6. Demand function from quantity and price changes:** Points: $(Q_1=120, P_1=40)$ and $(Q_2=100, P_2=50)$. Slope: $$m = \frac{50 - 40}{100 - 120} = \frac{10}{-20} = -0.5$$ Equation: $$P - 40 = -0.5(Q - 120)$$ Simplify: $$P = 40 - 0.5Q + 60 = 100 - 0.5Q$$ Price when quantity sold is zero: $$P = 100 - 0.5 \times 0 = 100$$ **7. Supply function:** Price rises from 1000 to 1200, quantity from 2000 to 2400. Slope: $$m = \frac{1200 - 1000}{2400 - 2000} = \frac{200}{400} = 0.5$$ Equation: $$P - 1000 = 0.5(Q - 2000)$$ Simplify: $$P = 1000 + 0.5Q - 1000 = 0.5Q$$ Estimate quantity at $P=800$: $$800 = 0.5Q \Rightarrow Q = 1600$$ Price at $Q=3000$: $$P = 0.5 \times 3000 = 1500$$