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3.6 · Q1

Q.If the demand function is p=35−2x−x2p = 35 - 2x - x^2 and the demand x0x_0 is 3, find the consumers' surplus.

CBSENCERTSubjective· 3mImportance★★★★★
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✓ Free question

At x0=3x_0=3 the market price is p0=20p_0=20; the consumers' surplus is the area between the demand curve and the price line, =∫03p dx−p0x0=₹27=\int_0^{3}p\,dx-p_0x_0=\text{₹}27.

Consumers' surplus: CS=∫0x0p dx−p0x0\displaystyle CS=\int_0^{x_0}p\,dx-p_0x_0, where p=f(x)p=f(x) is the demand price, x0x_0 the equilibrium quantity and p0p_0 the equilibrium price.

  1. Given demand p=35−2x−x2p=35-2x-x^2 and x0=3x_0=3.
  2. Equilibrium price: p0=35−2(3)−(3)2=35−6−9=₹20p_0=35-2(3)-(3)^2=35-6-9=\text{₹}20.
  3. ∫03(35−2x−x2) dx=[35x−x2−x33]03=105−9−9=87\displaystyle\int_0^3(35-2x-x^2)\,dx=\Big[35x-x^2-\tfrac{x^3}{3}\Big]_0^3=105-9-9=87.
  4. CS=87−p0x0=87−(20)(3)=87−60=27CS=87-p_0x_0=87-(20)(3)=87-60=27.
✓Final answer

Consumers' surplus =₹27=\text{₹}27.

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