Business Calculus Quiz: Consumer Surplus
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Consumer SurplusQuestion 1 of 14

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?

Product A has a larger consumer surplus due to its flatter demand curve.
Product B has a larger consumer surplus due to its steeper demand curve.
The consumer surplus is identical since they share the same equilibrium point.
The relationship cannot be determined without the specific demand functions.
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Business Calculus Quiz

Business Calculus Quiz: Consumer Surplus

Practice Consumer Surplus in Business Calculus with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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This quiz focuses on Consumer Surplus, giving you a quick way to practice the rules, question types, and explanations that matter most for Business Calculus.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

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?

  1. Product A has a larger consumer surplus due to its flatter demand curve.
  2. Product B has a larger consumer surplus due to its steeper demand curve. (correct answer)
  3. The consumer surplus is identical since they share the same equilibrium point.
  4. The relationship cannot be determined without the specific demand functions.
Explanation: 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 (q0,p0)(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 q0q_0. This means the area representing consumer surplus for Product B is larger than that for Product A.

Question 2

The demand function for a specialized electronic component is modeled by p=45e−0.1qp = 45e^{-0.1q}, where pp is the price in dollars and qq is the quantity in thousands. If 10,000 units are sold (so q0=10q_0=10), what is the consumer surplus?

  1. 450−900e−1450 - 900e^{-1} (correct answer)
  2. 450−450e−1450 - 450e^{-1}
  3. 450e−1450e^{-1}
  4. 450450
Explanation: First, find the market price p0p_0 when q0=10q_0 = 10: p0=45e−0.1(10)=45e−1p_0 = 45e^{-0.1(10)} = 45e^{-1}. Consumer surplus is CS=∫01045e−0.1q dq−p0q0CS = \int_{0}^{10} 45e^{-0.1q} \,dq - p_0 q_0. The integral is ∫45e−0.1q dq=45−0.1e−0.1q=−450e−0.1q\int 45e^{-0.1q} \,dq = \frac{45}{-0.1}e^{-0.1q} = -450e^{-0.1q}. Evaluating from 0 to 10 gives [−450e−0.1q]010=(−450e−1)−(−450e0)=450−450e−1[-450e^{-0.1q}]_0^{10} = (-450e^{-1}) - (-450e^0) = 450 - 450e^{-1}. The total expenditure is p0q0=(45e−1)(10)=450e−1p_0 q_0 = (45e^{-1})(10) = 450e^{-1}. Therefore, CS=(450−450e−1)−450e−1=450−900e−1CS = (450 - 450e^{-1}) - 450e^{-1} = 450 - 900e^{-1}.

Question 3

The demand for a product is given by p=120−0.5qp = 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?

  1. $1,400
  2. $1,500 (correct answer)
  3. $4,900
  4. $6,400
Explanation: First, calculate the consumer surplus at each price. For a linear demand curve, CS is the area of a triangle: CS=12×q0×(D(0)−p0)CS = \frac{1}{2} \times q_0 \times (D(0) - p_0). The p-intercept is D(0)=120D(0)=120. Case 1: p1=50p_1 = 50. Quantity q1q_1 is found from 50=120−0.5q50 = 120 - 0.5q, so 0.5q1=700.5q_1 = 70, q1=140q_1 = 140. CS1=12×140×(120−50)=70×70=4900CS_1 = \frac{1}{2} \times 140 \times (120 - 50) = 70 \times 70 = 4900. Case 2: p2=40p_2 = 40. Quantity q2q_2 is found from 40=120−0.5q40 = 120 - 0.5q, so 0.5q2=800.5q_2 = 80, q2=160q_2 = 160. CS2=12×160×(120−40)=80×80=6400CS_2 = \frac{1}{2} \times 160 \times (120 - 40) = 80 \times 80 = 6400. The increase in consumer surplus is CS2−CS1=6400−4900=1500CS_2 - CS_1 = 6400 - 4900 = 1500.

Question 4

The demand for a particular brand of gourmet coffee is given by the equation p=150−2qp = 150 - 2q, and the supply is given by p=30+qp = 30 + q, where pp is the price in dollars per pound and qq is the quantity in thousands of pounds. What is the consumer surplus at the market equilibrium?

  1. $800
  2. $1,600 (correct answer)
  3. $2,800
  4. $4,400
Explanation: First, find the equilibrium point by setting demand equal to supply: 150−2q=30+q150 - 2q = 30 + q, which gives 120=3q120 = 3q, so the equilibrium quantity is q0=40q_0 = 40 (or 40,000 pounds). The equilibrium price is p0=30+40=70p_0 = 30 + 40 = 70. Consumer surplus (CS) is calculated as CS=∫0q0D(q) dq−p0q0CS = \int_{0}^{q_0} D(q) \,dq - p_0 q_0. The integral part is ∫040(150−2q) dq=[150q−q2]040=(150(40)−402)−0=6000−1600=4400\int_{0}^{40} (150 - 2q) \,dq = [150q - q^2]_0^{40} = (150(40) - 40^2) - 0 = 6000 - 1600 = 4400. The total expenditure is p0q0=70×40=2800p_0 q_0 = 70 \times 40 = 2800. Therefore, CS=4400−2800=1600CS = 4400 - 2800 = 1600.

Question 5

The demand function for a digital subscription service is p=100q+1p = \frac{100}{q+1}, where pp is the monthly price and qq is the number of subscribers in millions. If the market price is set at $10 per month, what is the consumer surplus?

  1. 100ln⁡(10)−100100 \ln(10) - 100
  2. 100ln⁡(10)−90100 \ln(10) - 90 (correct answer)
  3. 100ln⁡(9)−90100 \ln(9) - 90
  4. 100ln⁡(10)100 \ln(10)
Explanation: First, find the quantity demanded at the market price p0=10p_0 = 10. Set 10=100q+110 = \frac{100}{q+1}, which gives 10(q+1)=10010(q+1) = 100, so q+1=10q+1 = 10, and q0=9q_0 = 9. Consumer surplus is CS=∫09100q+1 dq−(10)(9)CS = \int_{0}^{9} \frac{100}{q+1} \,dq - (10)(9). The integral is 100[ln⁡∣q+1∣]09=100(ln⁡(10)−ln⁡(1))=100ln⁡(10)100[\ln|q+1|]_0^{9} = 100(\ln(10) - \ln(1)) = 100\ln(10). The total expenditure is p0q0=10×9=90p_0 q_0 = 10 \times 9 = 90. Thus, CS=100ln⁡(10)−90CS = 100\ln(10) - 90.

Question 6

The demand for a product is given by p=A−q2p = A - q^2, where AA is a positive constant representing the maximum possible price. The market price is fixed at p0p_0. If the value of AA increases while p0p_0 remains constant (and A>p0A > p_0), how is the consumer surplus affected?

  1. Consumer surplus increases. (correct answer)
  2. Consumer surplus decreases.
  3. Consumer surplus remains unchanged because the market price is fixed.
  4. The effect cannot be determined without knowing the specific values of AA and p0p_0.
Explanation: First, find the quantity sold, q0q_0, by setting p0=A−q02p_0 = A - q_0^2, which gives q0=A−p0q_0 = \sqrt{A - p_0}. Consumer surplus is CS=∫0q0(A−q2−p0) dqCS = \int_{0}^{q_0} (A - q^2 - p_0) \,dq. Since A−p0=q02A-p_0 = q_0^2, this is CS=∫0q0(q02−q2) dq=[q02q−q33]0q0=q03−q033=23q03CS = \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 q0=A−p0q_0 = \sqrt{A - p_0} gives CS=23(A−p0)3/2CS = \frac{2}{3}(A - p_0)^{3/2}. To see how CS changes with AA, we can take the derivative: d(CS)dA=23⋅32(A−p0)1/2=A−p0\frac{d(CS)}{dA} = \frac{2}{3} \cdot \frac{3}{2}(A - p_0)^{1/2} = \sqrt{A - p_0}. Since A>p0A > p_0, this derivative is positive, meaning that as AA increases, consumer surplus increases.

Question 7

The demand for a particular item is described by a piecewise function. For the first 40 units (0≤q≤400 \le q \le 40), the demand is p=100−qp = 100 - q. For any additional units (q>40q > 40), the demand is p=80−0.5qp = 80 - 0.5q.

If the market price for this item is $50, what is the total consumer surplus?

  1. $900
  2. $1,250
  3. $1,300 (correct answer)
  4. $4,300
Explanation: First, determine the equilibrium quantity q0q_0 at the price p0=50p_0 = 50. For 0≤q≤400 \le q \le 40, pp ranges from 100100 down to 6060. For q>40q > 40, pp is less than 6060. Since p0=50p_0=50 is in this second range, we use the second demand function: 50=80−0.5q50 = 80 - 0.5q, which gives 0.5q=300.5q = 30, so q0=60q_0 = 60. The consumer surplus integral must be split into two parts: CS=[∫040(100−q) dq+∫4060(80−0.5q) dq]−p0q0CS = [\int_{0}^{40} (100 - q) \,dq + \int_{40}^{60} (80 - 0.5q) \,dq] - p_0 q_0. Part 1: [100q−q22]040=4000−800=3200[100q - \frac{q^2}{2}]_0^{40} = 4000 - 800 = 3200. Part 2: [80q−0.5q22]4060=[80q−0.25q2]4060=(4800−900)−(3200−400)=3900−2800=1100[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=43003200 + 1100 = 4300. Total expenditure is p0q0=50×60=3000p_0 q_0 = 50 \times 60 = 3000. Consumer surplus is 4300−3000=13004300 - 3000 = 1300.

Question 8

The relationship between the price pp of a product and the quantity qq that consumers will buy is given by the equation q=36−p2q = 36 - p^2. If the market price is p0=4p_0 = 4, what is the consumer surplus?

  1. $64/3 (correct answer)
  2. $80
  3. $304/3
  4. $368/3
Explanation: This problem provides the demand function as q(p)q(p). It is often easier to calculate consumer surplus by integrating with respect to pp using the formula CS=∫p0pmaxq(p) dpCS = \int_{p_0}^{p_{max}} q(p) \,dp. Here, p0=4p_0 = 4. The maximum price pmaxp_{max} occurs when quantity q=0q=0, so 0=36−pmax20 = 36 - p_{max}^2, which gives pmax=6p_{max} = 6. The integral is CS=∫46(36−p2) dp=[36p−p33]46CS = \int_{4}^{6} (36 - p^2) \,dp = [36p - \frac{p^3}{3}]_4^6. Evaluating at the limits: At p=6p=6, we have 36(6)−633=216−72=14436(6) - \frac{6^3}{3} = 216 - 72 = 144. At p=4p=4, we have 36(4)−433=144−643=432−643=368336(4) - \frac{4^3}{3} = 144 - \frac{64}{3} = \frac{432-64}{3} = \frac{368}{3}. The consumer surplus is 144−3683=432−3683=643144 - \frac{368}{3} = \frac{432 - 368}{3} = \frac{64}{3}.

Question 9

A company faces two market segments with demand functions p1=100−2q1p_1 = 100 - 2q_1 and p2=80−q2p_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?

  1. $1,400
  2. $1,700 (correct answer)
  3. $2,100
  4. $2,500
Explanation: For market 1: At $40, 40=100−2q140 = 100 - 2q_1, so q1=30q_1 = 30. CS1=∫030(100−2q1−40)dq1=∫030(60−2q1)dq1=[60q1−q12]030=1800−900=900CS_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−q240 = 80 - q_2, so q2=40q_2 = 40. CS2=∫040(80−q2−40)dq2=∫040(40−q2)dq2=[40q2−q222]040=1600−800=800CS_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=1700900 + 800 = 1700.

Question 10

The demand for an artisanal cheese is given by p=60−4qp = 60 - 4\sqrt{q}, where pp is the price per kilogram. The supply is perfectly elastic at a price of $20 per kilogram. Calculate the consumer surplus at market equilibrium.

  1. $0
  2. $2000
  3. $4000/3 (correct answer)
  4. $10000/3
Explanation: Market equilibrium occurs where demand price equals supply price. The supply price is fixed at p0=20p_0=20. Set the demand function equal to this price: 20=60−4q20 = 60 - 4\sqrt{q}. This simplifies to 4q=404\sqrt{q} = 40, so q=10\sqrt{q} = 10, and the equilibrium quantity is q0=100q_0 = 100. Consumer surplus is CS=∫0100(60−4q1/2) dq−(20)(100)CS = \int_{0}^{100} (60 - 4q^{1/2}) \,dq - (20)(100). The integral evaluates to [60q−4(q3/23/2)]0100=[60q−83q3/2]0100=60(100)−83(100)3/2=6000−83(1000)=6000−80003=18000−80003=100003[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 p0q0=20×100=2000p_0 q_0 = 20 \times 100 = 2000. So, CS=100003−2000=100003−60003=40003CS = \frac{10000}{3} - 2000 = \frac{10000}{3} - \frac{6000}{3} = \frac{4000}{3}.

Question 11

A company's market research department models the demand for its new smart speaker with the function p=D(q)p = D(q), where pp is the price in dollars and qq is the quantity in thousands of units. The equilibrium price is p0p_0 and the equilibrium quantity is q0q_0.

In the context of calculating consumer surplus for the smart speaker, what is the economic interpretation of the quantity given by the definite integral ∫0q0D(q) dq\int_{0}^{q_0} D(q) \,dq?

  1. The total revenue received by the company at the equilibrium price.
  2. The total amount that consumers actually spent to purchase the q0q_0 units.
  3. The theoretical maximum amount that consumers would have been willing to pay for the q0q_0 units. (correct answer)
  4. The net economic benefit, or surplus value, that consumers receive from purchasing the product.
Explanation: The definite integral ∫0q0D(q) dq\int_{0}^{q_0} D(q) \,dq represents the total area under the demand curve from q=0q=0 to q=q0q=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 q0q_0. Therefore, it represents the total theoretical value or willingness to pay. Total revenue is p0q0p_0 q_0, and the net benefit is the consumer surplus, which is ∫0q0D(q) dq−p0q0\int_{0}^{q_0} D(q) \,dq - p_0 q_0.

Question 12

A luxury item has the demand function p=1000q+5p = \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?

  1. Consumer surplus decreases by $86.64
  2. Consumer surplus decreases by $136.44 (correct answer)
  3. Consumer surplus decreases by $173.29
  4. Consumer surplus decreases by $220.15
Explanation: At $40: 40=1000q+540 = \frac{1000}{q + 5}, so q=20q = 20. At $50: 50=1000q+550 = \frac{1000}{q + 5}, so q=15q = 15. Original CS: ∫020(1000q+5−40)dq=[1000ln⁡(q+5)−40q]020=1000ln⁡(25)−800−1000ln⁡(5)≈386.44\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: ∫015(1000q+5−50)dq=[1000ln⁡(q+5)−50q]015=1000ln⁡(20)−750−1000ln⁡(5)≈250\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=−136.44250 - 386.44 = -136.44.

Question 13

A concert venue has the demand function p=250−5q−q2p = 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?

  1. 45.2%
  2. 52.8%
  3. 61.3% (correct answer)
  4. 68.7%
Explanation: With capacity limited to 8 attendees, the price is p=250−5(8)−82=250−40−64=146p = 250 - 5(8) - 8^2 = 250 - 40 - 64 = 146. The actual consumer surplus is CSactual=∫08(250−5q−q2−146)dq=∫08(104−5q−q2)dq=[104q−2.5q2−q33]08=104(8)−2.5(64)−5123=832−160−170.67=501.33CS_{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=0p = 0: 0=250−5q−q20 = 250 - 5q - q^2, so q2+5q−250=0q^2 + 5q - 250 = 0. Using the quadratic formula: q=−5+25+10002=−5+10252≈13.66q = \frac{-5 + \sqrt{25 + 1000}}{2} = \frac{-5 + \sqrt{1025}}{2} ≈ 13.66. The potential consumer surplus is CSpotential=∫013.66(250−5q−q2)dq=[250q−2.5q2−q33]013.66≈818.24CS_{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 501.33818.24≈61.3%\frac{501.33}{818.24} ≈ 61.3\%.

Question 14

For a certain product, the demand function is p=30−2qp = 30 - 2q. If the total consumer surplus is $100, what is the market price?

  1. $10 (correct answer)
  2. $15
  3. $20
  4. $30
Explanation: Consumer surplus (CS) for a linear demand curve forms a triangle. The area is also given by the integral CS=∫0q0(D(q)−p0) dqCS = \int_{0}^{q_0} (D(q) - p_0) \,dq. With D(q)=30−2qD(q) = 30 - 2q and p0=30−2q0p_0 = 30 - 2q_0, the integral becomes ∫0q0((30−2q)−(30−2q0)) dq=∫0q0(2q0−2q) dq=[2q0q−q2]0q0=2q02−q02=q02\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=100CS = 100, we have q02=100q_0^2 = 100, so q0=10q_0 = 10. The market price is p0=D(q0)=30−2(10)=10p_0 = D(q_0) = 30 - 2(10) = 10.