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3.6 · Q4

Q.The demand function for a commodity is p=80−3x−x2p = 80 - 3x - x^2. Find the consumers' surplus for p=40p = 40.

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With p0=40p_0=40 the demand p=80−3x−x2p=80-3x-x^2 gives x0=5x_0=5; the consumers' surplus is ∫05p dx−p0x0=7256≈₹120.83\int_0^{5}p\,dx-p_0x_0=\dfrac{725}{6}\approx\text{₹}120.83.

Consumers' surplus: CS=∫0x0p dx−p0x0\displaystyle CS=\int_0^{x_0}p\,dx-p_0x_0.

  1. Given p=80−3x−x2p=80-3x-x^2, p0=40p_0=40.
  2. Equilibrium quantity: 40=80−3x−x2⇒x2+3x−40=0⇒(x+8)(x−5)=0⇒x0=540=80-3x-x^2\Rightarrow x^2+3x-40=0\Rightarrow(x+8)(x-5)=0\Rightarrow x_0=5 (positive root). …

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