Suppose the demand and supply curves of a salt are given by
qD = 1000 – P
qS = 700⁴ + 2P

(a) Find the equilibrium price and quantity.

(b) Now suppose that the price of an inputs of an input used to produce salt has increased so that the new supply curve is qS = 400 + 2P. How does the equilibrium price and quantity change?

(c) Suppose the government has imposed a tax of Rs 3 per unit on salt of salt. How does it affect the equilibrium price and quantity?

Dear Student

a) To find out equilibrium price and quantity :

$$\begin{gathered} qD=qS \\ \Rightarrow 1000-P=700+2P \\ \Rightarrow 3P=300 \\ \Rightarrow P=100units \end{gathered}$$

Now , substituting the value of P in any one equation we get :
 $$\begin{gathered} qD=1000-P \\ \Rightarrow qD=900units \end{gathered}$$

Therefore , equilibrium price = 100 units and equilibrium quantity = 900 units 

b) New supply curve :
$$\begin{gathered} qS\text{'}=400+2P \\ So, new equilibrium price : \\ qD=qS\text{'} \\ \Rightarrow 1000-P=400+2P \\ \Rightarrow 3P=600 \\ \Rightarrow P=200units \\ Equilibrium quantity : \\ qS\text{'}=400+2\times 200 \\ \Rightarrow qS\text{'}=800units \end{gathered}$$
So due to a change in the price of inputs the supply curve shifts leftward . The equilibrium price rises from 100 to 200 and the equilibrium quantity falls from 900 to 800 .

c) The imposition of tax of Rs.3 per unit of salt sold will increase the cost of producing salt .
This will shift the supply curve left .
New quantity supplied equation :

$qS\text{'}\text{'}=700+2\left(P-3\right)$
Now at equilibrium :

$$\begin{gathered} qD=qS\text{'}\text{'} \\ \Rightarrow 1000-P=700+2(P-3) \\ \Rightarrow 3P=306 \\ \Rightarrow P=306/3 \\ \Rightarrow P=102units \end{gathered}$$

Putting this value in equation of qS'' :

$$\begin{gathered} qS\text{'}\text{'}=700+2(102-3) \\ \Rightarrow qS\text{'}\text{'}=700+198 \\ \Rightarrow qS\text{'}\text{'}=898units \end{gathered}$$
So after tax price rises from Rs100 to Rs102 and quantity falls from 900 units to 898 units .

Regards .