The demand function for a commodity is p = 20 e-x/10. Find the co… - Vidyadip

Question

The demand function for a commodity is p = 20 e^-x/10. Find the consumer's surplus at equilibrium price p = 2. (Given log₁₀ e = 0.4343)

✓

Answer

Given, the demand function is
$
p=20 e^{-x / 10} ...(i)
$
and the equilibrium price $p _0=2$.
Substituting this value of $p _0=2$ in (i), we get
$\begin{array}{l}
2=20 e^{-x_0 / 10} \Rightarrow e^{-x_0 / 10}=\frac{1}{10} \quad \ldots ...(ii) \\
\Rightarrow e^{x_0 / 10}=10 \Rightarrow \log _e 10=\frac{x_0}{10} \\
\Rightarrow x_0=10 \log _{e} 10=\frac{10}{\log _{10} e}=\frac{10}{0.4343}=\frac{100000}{4343} \\
\Rightarrow x_0=23.03 \ldots(iii) \\
\therefore CS=\int_0^{x_0} 20 e^{-x / 10} d x-x_0 \times p_0 \\
=20\left[\frac{e^{-x / 10}}{-\frac{1}{10}}\right]_0^{x_0}-23.03 \times 2 \text { (using (iii)) } \\
=-200\left[\left[e^{-x_0 / 10}-e^0\right]-46.06\right. \\
=-200\left[\frac{1}{10}-1\right]-46.06 \text { (using (ii)) } \\
=180-46.06=133.94
\end{array}$
Hence, consumer's surplus is 133.94

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