Consumer Surplus - Business Calculus

1. The demand for a particular brand of gourmet coffee is given by the equation $p = 150 - 2q$, and the supply is given by $p = 30 + q$, where $p$ is the price in dollars per pound and $q$ is the quantity in thousands of pounds. What is the consumer surplus at the market equilibrium?
   Answer: $1,600 First, find the equilibrium point by setting demand equal to supply: $150 - 2q = 30 + q$, which gives $120 = 3q$, so the equilibrium quantity is $q_0 = 40$ (or 40,000 pounds). The equilibrium price is $p_0 = 30 + 40 = 70$. Consumer surplus (CS) is calculated as $CS = \int_{0}^{q_0} D(q) \,dq - p_0 q_0$. The integral part is $\int_{0}^{40} (150 - 2q) \,dq = [150q - q^2]_0^{40} = (150(40) - 40^2) - 0 = 6000 - 1600 = 4400$. The total expenditure is $p_0 q_0 = 70 \times 40 = 2800$. Therefore, $CS = 4400 - 2800 = 1600$.
2. For a certain product, the demand function is $p = 30 - 2q$. If the total consumer surplus is $100, what is the market price?
   Answer: $10 Consumer surplus (CS) for a linear demand curve forms a triangle. The area is also given by the integral $CS = \int_{0}^{q_0} (D(q) - p_0) \,dq$. With $D(q) = 30 - 2q$ and $p_0 = 30 - 2q_0$, the integral becomes $\int_{0}^{q_0} ((30 - 2q) - (30 - 2q_0)) \,dq = \int_{0}^{q_0} (2q_0 - 2q) \,dq = [2q_0q - q^2]_0^{q_0} = 2q_0^2 - q_0^2 = q_0^2$. Given $CS = 100$, we have $q_0^2 = 100$, so $q_0 = 10$. The market price is $p_0 = D(q_0) = 30 - 2(10) = 10$.
3. The demand for a product is given by $p = 120 - 0.5q$. Due to a new government subsidy, the effective market price for consumers drops from $50 to $40. What is the resulting increase in consumer surplus?
   Answer: $1,500 First, calculate the consumer surplus at each price. For a linear demand curve, CS is the area of a triangle: $CS = \frac{1}{2} \times q_0 \times (D(0) - p_0)$. The p-intercept is $D(0)=120$. Case 1: $p_1 = 50$. Quantity $q_1$ is found from $50 = 120 - 0.5q$, so $0.5q_1 = 70$, $q_1 = 140$. $CS_1 = \frac{1}{2} \times 140 \times (120 - 50) = 70 \times 70 = 4900$. Case 2: $p_2 = 40$. Quantity $q_2$ is found from $40 = 120 - 0.5q$, so $0.5q_2 = 80$, $q_2 = 160$. $CS_2 = \frac{1}{2} \times 160 \times (120 - 40) = 80 \times 80 = 6400$. The increase in consumer surplus is $CS_2 - CS_1 = 6400 - 4900 = 1500$.
4. The relationship between the price $p$ of a product and the quantity $q$ that consumers will buy is given by the equation $q = 36 - p^2$. If the market price is $p_0 = 4$, what is the consumer surplus?
   Answer: $64/3 This problem provides the demand function as $q(p)$. It is often easier to calculate consumer surplus by integrating with respect to $p$ using the formula $CS = \int_{p_0}^{p_{max}} q(p) \,dp$. Here, $p_0 = 4$. The maximum price $p_{max}$ occurs when quantity $q=0$, so $0 = 36 - p_{max}^2$, which gives $p_{max} = 6$. The integral is $CS = \int_{4}^{6} (36 - p^2) \,dp = [36p - \frac{p^3}{3}]_4^6$. Evaluating at the limits: At $p=6$, we have $36(6) - \frac{6^3}{3} = 216 - 72 = 144$. At $p=4$, we have $36(4) - \frac{4^3}{3} = 144 - \frac{64}{3} = \frac{432-64}{3} = \frac{368}{3}$. The consumer surplus is $144 - \frac{368}{3} = \frac{432 - 368}{3} = \frac{64}{3}$.
5. A concert venue has the demand function $p = 250 - 5q - q^2$ for ticket sales. Due to fire safety regulations, the venue can only accommodate 8 attendees maximum. If tickets are sold at the market clearing price for this constraint, what percentage of the potential consumer surplus (if there were no capacity constraint) is actually realized?
   Answer: 61.3% With capacity limited to 8 attendees, the price is $p = 250 - 5(8) - 8^2 = 250 - 40 - 64 = 146$. The actual consumer surplus is $CS_{actual} = \int_0^8 (250 - 5q - q^2 - 146) dq = \int_0^8 (104 - 5q - q^2) dq = [104q - 2.5q^2 - \frac{q^3}{3}]_0^8 = 104(8) - 2.5(64) - \frac{512}{3} = 832 - 160 - 170.67 = 501.33$. Without capacity constraints, the optimal quantity occurs where $p = 0$: $0 = 250 - 5q - q^2$, so $q^2 + 5q - 250 = 0$. Using the quadratic formula: $q = \frac{-5 + \sqrt{25 + 1000}}{2} = \frac{-5 + \sqrt{1025}}{2} ≈ 13.66$. The potential consumer surplus is $CS_{potential} = \int_0^{13.66} (250 - 5q - q^2) dq = [250q - 2.5q^2 - \frac{q^3}{3}]_0^{13.66} ≈ 818.24$. Therefore, the percentage realized is $\frac{501.33}{818.24} ≈ 61.3\%$.
6. The demand for a product is given by $p = A - q^2$, where $A$ is a positive constant representing the maximum possible price. The market price is fixed at $p_0$. If the value of $A$ increases while $p_0$ remains constant (and $A > p_0$), how is the consumer surplus affected?
   Answer: Consumer surplus increases. First, find the quantity sold, $q_0$, by setting $p_0 = A - q_0^2$, which gives $q_0 = \sqrt{A - p_0}$. Consumer surplus is $CS = \int_{0}^{q_0} (A - q^2 - p_0) \,dq$. Since $A-p_0 = q_0^2$, this is $CS = \int_{0}^{q_0} (q_0^2 - q^2) \,dq = [q_0^2q - \frac{q^3}{3}]_0^{q_0} = q_0^3 - \frac{q_0^3}{3} = \frac{2}{3}q_0^3$. Substituting $q_0 = \sqrt{A - p_0}$ gives $CS = \frac{2}{3}(A - p_0)^{3/2}$. To see how CS changes with $A$, we can take the derivative: $\frac{d(CS)}{dA} = \frac{2}{3} \cdot \frac{3}{2}(A - p_0)^{1/2} = \sqrt{A - p_0}$. Since $A > p_0$, this derivative is positive, meaning that as $A$ increases, consumer surplus increases.
7. A luxury item has the demand function $p = \frac{1000}{q + 5}$. If the government imposes a minimum price of $50, what is the change in consumer surplus compared to the free market equilibrium where the price would be $40?
   Answer: Consumer surplus decreases by $136.44 At $40: $40 = \frac{1000}{q + 5}$, so $q = 20$. At $50: $50 = \frac{1000}{q + 5}$, so $q = 15$. Original CS: $\int_0^{20} (\frac{1000}{q + 5} - 40) dq = [1000\ln(q + 5) - 40q]_0^{20} = 1000\ln(25) - 800 - 1000\ln(5) ≈ 386.44$. New CS: $\int_0^{15} (\frac{1000}{q + 5} - 50) dq = [1000\ln(q + 5) - 50q]_0^{15} = 1000\ln(20) - 750 - 1000\ln(5) ≈ 250$. Change: $250 - 386.44 = -136.44$.
8. A company faces two market segments with demand functions $p_1 = 100 - 2q_1$ and $p_2 = 80 - q_2$. If the company sets a uniform price of $40 across both segments, what is the total consumer surplus across both markets?
   Answer: $1,700 For market 1: At $40, $40 = 100 - 2q_1$, so $q_1 = 30$. $CS_1 = \int_0^{30} (100 - 2q_1 - 40) dq_1 = \int_0^{30} (60 - 2q_1) dq_1 = [60q_1 - q_1^2]_0^{30} = 1800 - 900 = 900$. For market 2: At $40, $40 = 80 - q_2$, so $q_2 = 40$. $CS_2 = \int_0^{40} (80 - q_2 - 40) dq_2 = \int_0^{40} (40 - q_2) dq_2 = [40q_2 - \frac{q_2^2}{2}]_0^{40} = 1600 - 800 = 800$. Total CS = $900 + 800 = 1700$.
9. The demand function for a specialized electronic component is modeled by $p = 45e^{-0.1q}$, where $p$ is the price in dollars and $q$ is the quantity in thousands. If 10,000 units are sold (so $q_0=10$), what is the consumer surplus?
   Answer: $450 - 900e^{-1}$ First, find the market price $p_0$ when $q_0 = 10$: $p_0 = 45e^{-0.1(10)} = 45e^{-1}$. Consumer surplus is $CS = \int_{0}^{10} 45e^{-0.1q} \,dq - p_0 q_0$. The integral is $\int 45e^{-0.1q} \,dq = \frac{45}{-0.1}e^{-0.1q} = -450e^{-0.1q}$. Evaluating from 0 to 10 gives $[-450e^{-0.1q}]_0^{10} = (-450e^{-1}) - (-450e^0) = 450 - 450e^{-1}$. The total expenditure is $p_0 q_0 = (45e^{-1})(10) = 450e^{-1}$. Therefore, $CS = (450 - 450e^{-1}) - 450e^{-1} = 450 - 900e^{-1}$.
10. Two distinct products, A and B, are sold at the same equilibrium price and equilibrium quantity. The demand for Product A is highly price-elastic, while the demand for Product B is highly price-inelastic. Which statement accurately compares the consumer surplus for the two products?
    Answer: Product B has a larger consumer surplus due to its steeper demand curve. Price-inelastic demand (Product B) is represented by a steeper demand curve, while price-elastic demand (Product A) is represented by a flatter one. Consumer surplus is the area under the demand curve and above the equilibrium price line. Since both products have the same equilibrium point $(q_0, p_0)$, the steeper demand curve for Product B will be higher than the demand curve for Product A for all quantities less than $q_0$. This means the area representing consumer surplus for Product B is larger than that for Product A.
11. In the context of calculating consumer surplus for the smart speaker, what is the economic interpretation of the quantity given by the definite integral $\int_{0}^{q_0} D(q) \,dq$?
    Answer: The theoretical maximum amount that consumers would have been willing to pay for the $q_0$ units. The definite integral $\int_{0}^{q_0} D(q) \,dq$ represents the total area under the demand curve from $q=0$ to $q=q_0$. This area corresponds to the sum of the prices that consumers would have been willing to pay for each individual unit up to the quantity $q_0$. Therefore, it represents the total theoretical value or willingness to pay. Total revenue is $p_0 q_0$, and the net benefit is the consumer surplus, which is $\int_{0}^{q_0} D(q) \,dq - p_0 q_0$.
12. If the market price for this item is $50, what is the total consumer surplus?
    Answer: $1,300 First, determine the equilibrium quantity $q_0$ at the price $p_0 = 50$. For $0 \le q \le 40$, $p$ ranges from $100$ down to $60$. For $q > 40$, $p$ is less than $60$. Since $p_0=50$ is in this second range, we use the second demand function: $50 = 80 - 0.5q$, which gives $0.5q = 30$, so $q_0 = 60$. The consumer surplus integral must be split into two parts: $CS = [\int_{0}^{40} (100 - q) \,dq + \int_{40}^{60} (80 - 0.5q) \,dq] - p_0 q_0$. Part 1: $[100q - \frac{q^2}{2}]_0^{40} = 4000 - 800 = 3200$. Part 2: $[80q - \frac{0.5q^2}{2}]_{40}^{60} = [80q - 0.25q^2]_{40}^{60} = (4800 - 900) - (3200 - 400) = 3900 - 2800 = 1100$. Total willingness to pay is $3200 + 1100 = 4300$. Total expenditure is $p_0 q_0 = 50 \times 60 = 3000$. Consumer surplus is $4300 - 3000 = 1300$.
13. The demand for an artisanal cheese is given by $p = 60 - 4\sqrt{q}$, where $p$ is the price per kilogram. The supply is perfectly elastic at a price of $20 per kilogram. Calculate the consumer surplus at market equilibrium.
    Answer: $4000/3 Market equilibrium occurs where demand price equals supply price. The supply price is fixed at $p_0=20$. Set the demand function equal to this price: $20 = 60 - 4\sqrt{q}$. This simplifies to $4\sqrt{q} = 40$, so $\sqrt{q} = 10$, and the equilibrium quantity is $q_0 = 100$. Consumer surplus is $CS = \int_{0}^{100} (60 - 4q^{1/2}) \,dq - (20)(100)$. The integral evaluates to $[60q - 4(\frac{q^{3/2}}{3/2})]_0^{100} = [60q - \frac{8}{3}q^{3/2}]_0^{100} = 60(100) - \frac{8}{3}(100)^{3/2} = 6000 - \frac{8}{3}(1000) = 6000 - \frac{8000}{3} = \frac{18000-8000}{3} = \frac{10000}{3}$. The total expenditure is $p_0 q_0 = 20 \times 100 = 2000$. So, $CS = \frac{10000}{3} - 2000 = \frac{10000}{3} - \frac{6000}{3} = \frac{4000}{3}$.
14. The demand function for a digital subscription service is $p = \frac{100}{q+1}$, where $p$ is the monthly price and $q$ is the number of subscribers in millions. If the market price is set at $10 per month, what is the consumer surplus?
    Answer: $100 \ln(10) - 90$ First, find the quantity demanded at the market price $p_0 = 10$. Set $10 = \frac{100}{q+1}$, which gives $10(q+1) = 100$, so $q+1 = 10$, and $q_0 = 9$. Consumer surplus is $CS = \int_{0}^{9} \frac{100}{q+1} \,dq - (10)(9)$. The integral is $100[\ln|q+1|]_0^{9} = 100(\ln(10) - \ln(1)) = 100\ln(10)$. The total expenditure is $p_0 q_0 = 10 \times 9 = 90$. Thus, $CS = 100\ln(10) - 90$.

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