Math Problem Statement

from here: The consumer surplus is the area between the demand curve and the equilibrium price, from 𝑄

0 Q=0 to the equilibrium quantity 𝑄

3.446 Q=3.446. It is calculated as:

Consumer Surplus

∫ 0 𝑄 𝑒 𝑃 demand ( 𝑄 )   𝑑 𝑄 βˆ’ 𝑃 𝑒 β‹… 𝑄 𝑒 , Consumer Surplus=∫ 0 Q e ​

​ P demand ​ (Q)dQβˆ’P e ​ β‹…Q e ​ , where:

𝑃 demand ( 𝑄 )

βˆ’ 𝑄 2 βˆ’ 3 𝑄 + 30 P demand ​ (Q)=βˆ’Q 2 βˆ’3Q+30 (demand curve), 𝑄 𝑒

3.446 Q e ​ =3.446 (equilibrium quantity), 𝑃 𝑒

7.789 P e ​ =7.789 (equilibrium price). Step 1: Compute the integral of 𝑃 demand ( 𝑄 ) P demand ​ (Q) from 0 0 to 3.446 3.446 The demand curve is:

𝑃 demand ( 𝑄 )

βˆ’ 𝑄 2 βˆ’ 3 𝑄 + 30. P demand ​ (Q)=βˆ’Q 2 βˆ’3Q+30. The integral of 𝑃 demand ( 𝑄 ) P demand ​ (Q) is:

∫ ( βˆ’ 𝑄 2 βˆ’ 3 𝑄 + 30 )   𝑑 𝑄

βˆ’ 𝑄 3 3 βˆ’ 3 𝑄 2 2 + 30 𝑄 + 𝐢 . ∫(βˆ’Q 2 βˆ’3Q+30)dQ=βˆ’ 3 Q 3

​ βˆ’ 2 3Q 2

​ +30Q+C. Evaluate from 𝑄

0 Q=0 to 𝑄

3.446 Q=3.446:

At 𝑄

3.446 Q=3.446:

( βˆ’ ( 3.446 ) 3 3 βˆ’ 3 ( 3.446 ) 2 2 + 30 ( 3.446 ) ) , (βˆ’ 3 (3.446) 3

​ βˆ’ 2 3(3.446) 2

​ +30(3.446)), ( 3.446 ) 3

40.891 (3.446) 3 =40.891, βˆ’ 40.891 3

βˆ’ 13.630 βˆ’ 3 40.891 ​ =βˆ’13.630, ( 3.446 ) 2

11.873 (3.446) 2 =11.873, 3 ( 11.873 ) 2

17.810 2 3(11.873) ​ =17.810, 30 ( 3.446 )

103.380 30(3.446)=103.380, Total: βˆ’ 13.630 βˆ’ 17.810 + 103.380

71.940 βˆ’13.630βˆ’17.810+103.380=71.940. At 𝑄

0 Q=0:

( βˆ’ ( 0 ) 3 3 βˆ’ 3 ( 0 ) 2 2 + 30 ( 0 ) )

(βˆ’ 3 (0) 3

​ βˆ’ 2 3(0) 2

​ +30(0))=0. Thus:

∫ 0 3.446 𝑃 demand ( 𝑄 )   𝑑 𝑄

71.940. ∫ 0 3.446 ​ P demand ​ (Q)dQ=71.940. Step 2: Subtract the rectangle area The rectangle area is:

𝑃 𝑒 β‹… 𝑄 𝑒

7.789 β‹… 3.446

26.833. P e ​ β‹…Q e ​ =7.789β‹…3.446=26.833. Step 3: Compute consumer surplus Consumer Surplus

71.940 βˆ’ 26.833

45.107. Consumer Surplus=71.940βˆ’26.833=45.107. Final Answer: The consumer surplus is:

45.107 . What is the producer surplus at equilibrium condition? Use 3 decimal places

Solution

To compute the Producer Surplus, we use the formula:

Producer Surplus=Peβ‹…Qeβˆ’βˆ«0QePsupply(Q) dQ\text{Producer Surplus} = P_e \cdot Q_e - \int_0^{Q_e} P_{\text{supply}}(Q) \, dQ

where:

  • Pe=7.789P_e = 7.789 (equilibrium price),
  • Qe=3.446Q_e = 3.446 (equilibrium quantity),
  • Psupply(Q)P_{\text{supply}}(Q) is the supply curve.

Step 1: Identify the supply curve

If the supply curve is not explicitly given, it is typically assumed to be linear and starts at the origin (perfect competition) or given explicitly. Please confirm or clarify the supply curve equation.

Assume Psupply(Q)=aQ+bP_{\text{supply}}(Q) = aQ + b where aa and bb can be determined from the context or explicitly provided. Let me know if the supply curve is available or needs to be approximated.

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Math Problem Analysis

Mathematical Concepts

Integration
Economic Surplus
Equilibrium Analysis

Formulas

Consumer Surplus = ∫ Pdemand(Q)dQ - Pe β‹… Qe
Producer Surplus = Pe β‹… Qe - ∫ Psupply(Q)dQ

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Undergraduate Economics or Calculus Course

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