Business Calculus Quiz: Producer Surplus

10 questions · exam conditions

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Producer SurplusQuestion 1 of 10

The market for a product has a supply function $S(q) = 20 + 0.5q$ and a demand function $D(q) = 110 - q$ . If the government imposes a per-unit tax of $3 on the producers, what is the new producer surplus?

$841

$899

$900

$1,015

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Business Calculus Quiz

Business Calculus Quiz: Producer Surplus

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

What this quiz covers

This quiz focuses on Producer 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

$841 (correct answer)
$899
$900
$1,015

Explanation: The tax shifts the supply curve up by $3. The new supply curve is $S_T(q) = (20 + 0.5q) + 3 = 23 + 0.5q

. Find the new equilibrium:

23 + 0.5q = 110 - q \Rightarrow 1.5q = 87 \Rightarrow q_1 = 58

. The new market price paid by consumers is

p_1 = D(58) = 110 - 58 = 52

. The price producers receive is

p_T = p_1 - 3 = 49

. The new producer surplus is calculated using the price producers receive (

p_T

), the new quantity (

q_1

), and the original supply curve

S(q)

:

PS_2 = p_T q_1 - \int_0^{q_1} S(q) ,dq = (49)(58) - \int_0^{58} (20 + 0.5q) ,dq = 2842 - [20q + 0.25q^2]_0^{58} = 2842 - (20(58) + 0.25(

58^2

)) = 2842 - (1160 + 841) = 2842 - 2001 = 841$.

Question 2

The supply function for a product is $S(q) = 15 + \sqrt{2q+25}$ . If the market price is fixed at $24, what is the producer surplus?

$50.67 (correct answer)
$150.67
$621.33
$672.00

Explanation: First, find the quantity supplied at

p_0 = 24

24 = 15 + \sqrt{2q+25} \Rightarrow 9 = \sqrt{2q+25} \Rightarrow 81 = 2q+25 \Rightarrow 56 = 2q \Rightarrow q_0 = 28

. Total revenue is

p_0 q_0 = (24)(28) = 672

. The producer surplus is

PS = 672 - \int_0^{28} (15 + \sqrt{2q+25}) \,dq

. The integral is

\int_0^{28} (15 + (2q+25)^{1/2}) \,dq = [15q + \frac{1}{2} \cdot \frac{2}{3}(2q+25)^{3/2}]_0^{28} = [15q + \frac{1}{3}(2q+25)^{3/2}]_0^{28} = (15(28) + \frac{1}{3}(81)^{3/2}) - (0 + \frac{1}{3}(25)^{3/2}) = (420 + \frac{1}{3}(729)) - (\frac{1}{3}(125)) = (420 + 243) - \frac{125}{3} = 663 - \frac{125}{3} = \frac{1989-125}{3} = \frac{1864}{3} \approx 621.33

. Then,

PS = 672 - \frac{1864}{3} = \frac{2016-1864}{3} = \frac{152}{3} \approx 50.67

Question 3

Initially, the supply function for a product is $S_1(q) = 40 + q$ . Due to a technological improvement, the supply function changes to $S_2(q) = 20 + q$ . The demand function remains $D(q) = 100 - 2q$ . What is the increase in producer surplus resulting from the technological improvement?

$155.56 (correct answer)
$177.78
$200.00
$355.56

Explanation: First, calculate the original producer surplus (

PS_1

). Equilibrium:

40+q = 100-2q \Rightarrow 3q=60 \Rightarrow q_1=20, p_1=60

PS_1 = (60)(20) - \int_0^{20} (40+q) \,dq = 1200 - [40q + \frac{q^2}{2}]_0^{20} = 1200 - (800+200) = 200

. Next, calculate the new producer surplus (

PS_2

). Equilibrium:

20+q = 100-2q \Rightarrow 3q=80 \Rightarrow q_2=80/3, p_2=140/3

PS_2 = (\frac{140}{3})(\frac{80}{3}) - \int_0^{80/3} (20+q) \,dq = \frac{11200}{9} - [20q + \frac{q^2}{2}]_0^{80/3} = \frac{11200}{9} - (20(\frac{80}{3}) + \frac{(80/3)^2}{2}) = \frac{11200}{9} - (\frac{1600}{3} + \frac{3200}{9}) = \frac{11200 - 4800 - 3200}{9} = \frac{3200}{9} \approx 355.56

. The increase is

PS_2 - PS_1 = \frac{3200}{9} - 200 = \frac{3200-1800}{9} = \frac{1400}{9} \approx 155.56

Question 4

A manufacturer's supply function is piecewise, defined as $S(q) = 20$ for $0 \le q \le 10$ and $S(q) = 10 + q$ for $q > 10$ . If the product sells at a market price of $40, what is the producer surplus?

$200
$400 (correct answer)
$450
$600

Explanation: At a price of

p_0 = 40

, the quantity supplied is found from the second piece of the function:

40 = 10 + q \Rightarrow q_0 = 30

. Producer surplus is

PS = p_0 q_0 - \int_0^{q_0} S(q) \,dq

. The integral must be split according to the piecewise definition:

PS = (40)(30) - \left( \int_0^{10} 20 \,dq + \int_{10}^{30} (10+q) \,dq \right) = 1200 - \left( [20q]_0^{10} + [10q + \frac{q^2}{2}]_{10}^{30} \right) = 1200 - \left( 200 + ((300+\frac{900}{2}) - (100+\frac{100}{2})) \right) = 1200 - (200 + (750 - 150)) = 1200 - (200 + 600) = 1200 - 800 = 400

Question 5

The producer surplus for a market in equilibrium at $(q_0, p_0)$ is given by $PS = p_0q_0 - \int_0^{q_0} S(q)\,dq$ . Which of the following represents the total amount that producers would have been willing to accept for supplying $q_0$ units of the product?

$p_0q_0$
$\int_0^{q_0} S(q)\,dq$ (correct answer)
$p_0q_0 - \int_0^{q_0} S(q)\,dq$
$S(q_0) - S(0)$

Explanation: The supply curve

S(q)

represents the marginal cost of producing the

q

-th unit. Integrating the marginal cost function from

0

q_0

gives the total variable cost of producing

q_0

units. This total variable cost is the minimum total amount that producers would require to be willing to supply

q_0

units of the product. The term

p_0q_0

is the total revenue. The full expression

p_0q_0 - \int_0^{q_0} S(q)\,dq

is the producer surplus itself, which is the extra benefit producers receive above their minimum willingness to accept.

Question 6

A company's marginal cost function is $MC(q) = 2q + 5$ dollars per unit. If the company operates under perfect competition with a market price of $25 per unit, and the company has fixed costs of $50, what is the producer surplus when the company produces at the profit-maximizing quantity?

$75
$100 (correct answer)
$125
$150

Explanation: Under perfect competition, the firm produces where price equals marginal cost:

25 = 2q + 5

, so

q = 10

. The supply curve is the marginal cost curve above minimum average variable cost. Since MC = 2q + 5 and there are no variable costs mentioned other than those represented by MC, the supply curve is p = 2q + 5. Producer surplus =

\int_0^{10} (25 - (2q + 5)) dq = \int_0^{10} (20 - 2q) dq = [20q - q^2]_0^{10} = 200 - 100 = 100

. Choice A subtracts fixed costs incorrectly. Choice C adds fixed costs incorrectly. Choice D assumes different pricing.

Question 7

The supply curve for a product is $S(q) = 2q + 8$ . At market equilibrium, the total revenue for all producers is $1,450. What is the producer surplus?

$625 (correct answer)
$725
$825
$1,450

Explanation: Total revenue is

TR = p_0 q_0 = 1450

. At equilibrium,

p_0 = S(q_0) = 2q_0 + 8

. Substitute this into the revenue equation:

(2q_0 + 8)q_0 = 1450 \Rightarrow 2q_0^2 + 8q_0 - 1450 = 0 \Rightarrow q_0^2 + 4q_0 - 725 = 0

. Using the quadratic formula,

q_0 = \frac{-4 + \sqrt{16 - 4(1)(-725)}}{2} = \frac{-4 + \sqrt{16+2900}}{2} = \frac{-4 + \sqrt{2916}}{2} = \frac{-4+54}{2} = 25

. So,

q_0=25

. The producer surplus is

PS = p_0 q_0 - \int_0^{q_0} S(q)\,dq = 1450 - \int_0^{25} (2q+8)\,dq = 1450 - [q^2+8q]_0^{25} = 1450 - (25^2 + 8(25)) = 1450 - (625 + 200) = 1450 - 825 = 625

Question 8

The supply function for a commodity is given by $S(q) = 10 + \frac{q^2}{40}$ . If the market equilibrium price is $20, what is the producer surplus expressed as a percentage of total revenue?

25.0%
33.3% (correct answer)
50.0%
66.7%

Explanation: First find the equilibrium quantity

q_0

20 = 10 + \frac{q_0^2}{40} \Rightarrow 10 = \frac{q_0^2}{40} \Rightarrow q_0^2 = 400 \Rightarrow q_0 = 20

. Total Revenue (TR) is

p_0 q_0 = (20)(20) = 400

. Producer Surplus (PS) is

PS = p_0 q_0 - \int_0^{q_0} S(q) \,dq = 400 - \int_0^{20} (10 + \frac{q^2}{40}) \,dq = 400 - [10q + \frac{q^3}{120}]_0^{20} = 400 - (10(20) + \frac{20^3}{120}) = 400 - (200 + \frac{8000}{120}) = 400 - (200 + \frac{200}{3}) = 200 - \frac{200}{3} = \frac{600-200}{3} = \frac{400}{3}

. The percentage is

\frac{PS}{TR} \times 100 = \frac{400/3}{400} \times 100 = \frac{1}{3} \times 100 \approx 33.3\%

Question 9

The supply function for a commodity is given by $S(q) = 5q$ , where $q$ is the number of units. If the producer surplus is $640, what is the equilibrium price $p_0$?

$16
$57
$80 (correct answer)
$1,280

Explanation: Producer surplus (PS) is given by

PS = p_0 q_0 - \int_0^{q_0} S(q) \,dq

. We know

p_0 = S(q_0) = 5q_0

. Substituting this into the PS formula gives:

PS = (5q_0)q_0 - \int_0^{q_0} 5q \,dq = 5q_0^2 - [\frac{5}{2}q^2]_0^{q_0} = 5q_0^2 - \frac{5}{2}q_0^2 = \frac{5}{2}q_0^2

. We are given

PS = 640

. So,

640 = \frac{5}{2}q_0^2 \Rightarrow 1280 = 5q_0^2 \Rightarrow 256 = q_0^2 \Rightarrow q_0 = 16

. The question asks for the equilibrium price

p_0

, not the quantity.

p_0 = S(16) = 5(16) = 80

Question 10

In a market with supply function $S(q) = 15 + q$ and demand function $D(q) = 75 - 2q$ , the government imposes a price floor of $45. What is the producer surplus under this policy?

$200.00
$187.50
$337.50 (correct answer)
$450.00

Explanation: First, find the equilibrium price:

15+q = 75-2q \Rightarrow 3q=60 \Rightarrow q_0=20

, so

p_0=15+20=35

. Since the price floor of $45 is above the equilibrium price of $35, it is binding. At a price of $45, the quantity demanded is

45 = 75 - 2q_d \Rightarrow q_d=15

. The quantity supplied is

45 = 15 + q_s \Rightarrow q_s=30

. The quantity traded is the minimum of these, so

q_{traded}=15

. Producer surplus is calculated using the floor price (

p_f=45

) and the quantity traded (

q_{traded}=15

PS = p_f q_{traded} - \int_0^{q_{traded}} S(q) \,dq = (45)(15) - \int_0^{15} (15+q) \,dq = 675 - [15q + \frac{q^2}{2}]_0^{15} = 675 - (15(15) + \frac{15^2}{2}) = 675 - (225 + 112.5) = 675 - 337.5 = 337.50