Question

The demand equation for a market is given by p(q + 3) = 15 and, for some constant α, the supply equation is q = αp − 1 where p is the price and q is the quantity. Given that the equilibrium price for this market is three, determine the equilibrium quantity and the value of α. Find the consumer and producer surpluses?

Answer 1

Demand equation is given as

p(q+3)=15

Given the equilibrium price is 3, we would substitute the value of p in the equation above as 3.

3(q+3)=15

q+3= 15/3=5

q= 5-3=2

So equilibrium quantity would be equal to 2.

Given the equilibrium quantity and price, we could simply substitute them in equation of supply to get the value of .

q= p - 1

2= 3-1

3= 3

= 1

So the value of would be one.

Now the consumer and the producer surplus

Consumer surplus would be the area of triangle between the maximum price at Q=0 and the equilibrium price and quantity ie

1/2×(P|q=0- P*=3)×(equilibrium quantity at P=3)

=1/2×(5-3)×2. [When quantity is 0, price is maximum. So p(0+3)=15 => P=15/3=5]

=1/2×2×2

=2$

Consumer surplus is $2

Producer surplus would be the

PS= 1/2×(P*- P|q=0) × Q*

= 1/2(3-1)×2. [Putting Q= 0 in supply equation, we get 0=p-1 =>P=1)

=1/2×2×2

=$2

Producer surplus would be hence $2.

(You can comment for doubts)