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Worked Examples · Example 18

Q.The demand function for a commodity is p=10x+1p = \frac{10}{x+1}. Find the consumers' surplus when the prevailing market price is 5.

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✓ Free question

At p0=5p_0=5 the quantity is x0=1x_0=1; CS=∫0110x+1 dx−5=10ln⁡2−5≈₹1.93CS=\int_0^1\frac{10}{x+1}\,dx-5=10\ln2-5\approx ₹1.93.

Consumer's surplus CS=∫0x0f(x) dx−p0x0CS=\displaystyle\int_0^{x_0} f(x)\,dx - p_0x_0, with p=f(x)p=f(x) the demand curve and ∫dxx+1=ln⁡∣x+1∣\int\frac{dx}{x+1}=\ln|x+1|.

  1. Demand: p=10x+1p=\dfrac{10}{x+1}. Set p=p0=5p=p_0=5: 5=10x+1⇒x+1=2⇒x0=1.5=\dfrac{10}{x+1}\Rightarrow x+1=2\Rightarrow x_0=1.
  2. ∫0110x+1 dx=10[ln⁡(x+1)]01=10(ln⁡2−ln⁡1)=10ln⁡2.\displaystyle\int_0^{1}\frac{10}{x+1}\,dx=10\big[\ln(x+1)\big]_0^1=10(\ln2-\ln1)=10\ln2.
  3. p0x0=5×1=5.p_0x_0=5\times1=5.
  4. CS=10ln⁡2−5.CS=10\ln2-5. Numerically ln⁡2≈0.6931⇒10(0.6931)−5=6.931−5=1.931.\ln2\approx0.6931\Rightarrow 10(0.6931)-5=6.931-5=1.931.
✓Final answer

Consumer's surplus =10log⁡e2−5≈₹1.93=10\log_e 2-5\approx ₹1.93.

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