Answer to Question #175143 in Math for ruhi

Question

supply price = 50+q/2 demand price 150-q/5 find producer and consumer surplus

Answer 1

Find the equilibrium point

$$50+\dfrac{q}{2}=150-\dfrac{q}{5}$$

$$q=\dfrac{1000}{7}, p=50+\dfrac{500}{7}=\dfrac{850}{7}$$

The equilibrium point $(\dfrac{1000}{7}, \dfrac{850}{7})$

The consumer surplus is

$$\displaystyle\int_{0}^{1000/7}(150-\dfrac{q}{5})dq-\dfrac{1000}{7}(\dfrac{850}{7})$$

$$=[150q-\dfrac{q^2}{10}]\begin{matrix} 1000/7 \\ 0 \end{matrix}-\dfrac{850000}{49}=$$

$$=\dfrac{100000}{49}\approx2040.82$$

The producer surplus is

$$\dfrac{1000}{7}(\dfrac{850}{7})-\displaystyle\int_{0}^{1000/7}(50+\dfrac{q}{2})dq$$

$$=\dfrac{850000}{49}-[50q+\dfrac{q^2}{4}]\begin{matrix} 1000/7 \\ 0 \end{matrix}$$

$$=\dfrac{250000}{49}\approx5102.04$$