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Producer Surplus - Business Calculus

  1. The supply function for a product is S(q)=15+2q+25S(q) = 15 + \sqrt{2q+25}. If the market price is fixed at $24, what is the producer surplus?
    Answer: $50.67 First, find the quantity supplied at p0=24p_0 = 24: 24=15+2q+25⇒9=2q+25⇒81=2q+25⇒56=2q⇒q0=2824 = 15 + \sqrt{2q+25} \Rightarrow 9 = \sqrt{2q+25} \Rightarrow 81 = 2q+25 \Rightarrow 56 = 2q \Rightarrow q_0 = 28. Total revenue is p0q0=(24)(28)=672p_0 q_0 = (24)(28) = 672. The producer surplus is PS=672−∫028(15+2q+25) dqPS = 672 - \int_0^{28} (15 + \sqrt{2q+25}) \,dq. The integral is ∫028(15+(2q+25)1/2) dq=[15q+12⋅23(2q+25)3/2]028=[15q+13(2q+25)3/2]028=(15(28)+13(81)3/2)−(0+13(25)3/2)=(420+13(729))−(13(125))=(420+243)−1253=663−1253=1989−1253=18643≈621.33\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=672−18643=2016−18643=1523≈50.67PS = 672 - \frac{1864}{3} = \frac{2016-1864}{3} = \frac{152}{3} \approx 50.67.
  2. In a market with supply function S(q)=15+qS(q) = 15 + q and demand function D(q)=75−2qD(q) = 75 - 2q, the government imposes a price floor of $45. What is the producer surplus under this policy?
    Answer: $337.50 First, find the equilibrium price: 15+q=75−2q⇒3q=60⇒q0=2015+q = 75-2q \Rightarrow 3q=60 \Rightarrow q_0=20, so p0=15+20=35p_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=75−2qd⇒qd=1545 = 75 - 2q_d \Rightarrow q_d=15. The quantity supplied is 45=15+qs⇒qs=3045 = 15 + q_s \Rightarrow q_s=30. The quantity traded is the minimum of these, so qtraded=15q_{traded}=15. Producer surplus is calculated using the floor price (pf=45p_f=45) and the quantity traded (qtraded=15q_{traded}=15): PS=pfqtraded−∫0qtradedS(q) dq=(45)(15)−∫015(15+q) dq=675−[15q+q22]015=675−(15(15)+1522)=675−(225+112.5)=675−337.5=337.50PS = 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.
  3. The supply curve for a product is S(q)=2q+8S(q) = 2q + 8. At market equilibrium, the total revenue for all producers is $1,450. What is the producer surplus?
    Answer: $625 Total revenue is TR=p0q0=1450TR = p_0 q_0 = 1450. At equilibrium, p0=S(q0)=2q0+8p_0 = S(q_0) = 2q_0 + 8. Substitute this into the revenue equation: (2q0+8)q0=1450⇒2q02+8q0−1450=0⇒q02+4q0−725=0(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, q0=−4+16−4(1)(−725)2=−4+16+29002=−4+29162=−4+542=25q_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, q0=25q_0=25. The producer surplus is PS=p0q0−∫0q0S(q) dq=1450−∫025(2q+8) dq=1450−[q2+8q]025=1450−(252+8(25))=1450−(625+200)=1450−825=625PS = 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.
  4. The market for a product has a supply function S(q)=20+0.5qS(q) = 20 + 0.5q and a demand function D(q)=110−qD(q) = 110 - q. If the government imposes a per-unit tax of $3 on the producers, what is the new producer surplus?
    Answer: $841 The tax shifts the supply curve up by $3. The new supply curve is ST(q)=(20+0.5q)+3=23+0.5qS_T(q) = (20 + 0.5q) + 3 = 23 + 0.5q. Find the new equilibrium: 23+0.5q=110−q⇒1.5q=87⇒q1=5823 + 0.5q = 110 - q \Rightarrow 1.5q = 87 \Rightarrow q_1 = 58. The new market price paid by consumers is p1=D(58)=110−58=52p_1 = D(58) = 110 - 58 = 52. The price producers receive is pT=p1−3=49p_T = p_1 - 3 = 49. The new producer surplus is calculated using the price producers receive (pTp_T), the new quantity (q1q_1), and the original supply curve S(q)S(q): PS2=pTq1−∫0q1S(q) dq=(49)(58)−∫058(20+0.5q) dq=2842−[20q+0.25q2]058=2842−(20(58)+0.25(582))=2842−(1160+841)=2842−2001=841PS_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.
  5. A manufacturer's supply function is piecewise, defined as S(q)=20S(q) = 20 for 0≤q≤100 \le q \le 10 and S(q)=10+qS(q) = 10 + q for q>10q > 10. If the product sells at a market price of $40, what is the producer surplus?
    Answer: $400 At a price of p0=40p_0 = 40, the quantity supplied is found from the second piece of the function: 40=10+q⇒q0=3040 = 10 + q \Rightarrow q_0 = 30. Producer surplus is PS=p0q0−∫0q0S(q) dqPS = p_0 q_0 - \int_0^{q_0} S(q) \,dq. The integral must be split according to the piecewise definition: PS=(40)(30)−(∫01020 dq+∫1030(10+q) dq)=1200−([20q]010+[10q+q22]1030)=1200−(200+((300+9002)−(100+1002)))=1200−(200+(750−150))=1200−(200+600)=1200−800=400PS = (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.
  6. The supply function for a commodity is given by S(q)=5qS(q) = 5q, where qq is the number of units. If the producer surplus is $640, what is the equilibrium price p0p_0?
    Answer: $80 Producer surplus (PS) is given by PS=p0q0−∫0q0S(q) dqPS = p_0 q_0 - \int_0^{q_0} S(q) \,dq. We know p0=S(q0)=5q0p_0 = S(q_0) = 5q_0. Substituting this into the PS formula gives: PS=(5q0)q0−∫0q05q dq=5q02−[52q2]0q0=5q02−52q02=52q02PS = (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=640PS = 640. So, 640=52q02⇒1280=5q02⇒256=q02⇒q0=16640 = \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 p0p_0, not the quantity. p0=S(16)=5(16)=80p_0 = S(16) = 5(16) = 80.
  7. A company's marginal cost function is MC(q)=2q+5MC(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?
    Answer: $100 Under perfect competition, the firm produces where price equals marginal cost: 25=2q+525 = 2q + 5, so q=10q = 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 = ∫010(25−(2q+5))dq=∫010(20−2q)dq=[20q−q2]010=200−100=100\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.
  8. Initially, the supply function for a product is S1(q)=40+qS_1(q) = 40 + q. Due to a technological improvement, the supply function changes to S2(q)=20+qS_2(q) = 20 + q. The demand function remains D(q)=100−2qD(q) = 100 - 2q. What is the increase in producer surplus resulting from the technological improvement?
    Answer: $155.56 First, calculate the original producer surplus (PS1PS_1). Equilibrium: 40+q=100−2q⇒3q=60⇒q1=20,p1=6040+q = 100-2q \Rightarrow 3q=60 \Rightarrow q_1=20, p_1=60. PS1=(60)(20)−∫020(40+q) dq=1200−[40q+q22]020=1200−(800+200)=200PS_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 (PS2PS_2). Equilibrium: 20+q=100−2q⇒3q=80⇒q2=80/3,p2=140/320+q = 100-2q \Rightarrow 3q=80 \Rightarrow q_2=80/3, p_2=140/3. PS2=(1403)(803)−∫080/3(20+q) dq=112009−[20q+q22]080/3=112009−(20(803)+(80/3)22)=112009−(16003+32009)=11200−4800−32009=32009≈355.56PS_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 PS2−PS1=32009−200=3200−18009=14009≈155.56PS_2 - PS_1 = \frac{3200}{9} - 200 = \frac{3200-1800}{9} = \frac{1400}{9} \approx 155.56.
  9. The supply function for a commodity is given by S(q)=10+q240S(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?
    Answer: 33.3% First find the equilibrium quantity q0q_0: 20=10+q0240⇒10=q0240⇒q02=400⇒q0=2020 = 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 p0q0=(20)(20)=400p_0 q_0 = (20)(20) = 400. Producer Surplus (PS) is PS=p0q0−∫0q0S(q) dq=400−∫020(10+q240) dq=400−[10q+q3120]020=400−(10(20)+203120)=400−(200+8000120)=400−(200+2003)=200−2003=600−2003=4003PS = 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 PSTR×100=400/3400×100=13×100≈33.3%\frac{PS}{TR} \times 100 = \frac{400/3}{400} \times 100 = \frac{1}{3} \times 100 \approx 33.3\%.
  10. The producer surplus for a market in equilibrium at (q0,p0)(q_0, p_0) is given by PS=p0q0−∫0q0S(q) dqPS = 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 q0q_0 units of the product?
    Answer: ∫0q0S(q) dq\int_0^{q_0} S(q)\,dq The supply curve S(q)S(q) represents the marginal cost of producing the qq-th unit. Integrating the marginal cost function from 00 to q0q_0 gives the total variable cost of producing q0q_0 units. This total variable cost is the minimum total amount that producers would require to be willing to supply q0q_0 units of the product. The term p0q0p_0q_0 is the total revenue. The full expression p0q0−∫0q0S(q) dqp_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.

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