Answer to Question #224667 in Microeconomics for Ishaq

Question

Question 3
A Professor of Economics from the Department of Economics Education of the Faculty of Social Sciences Education of the University of Education, Winneba, has estimated the demand and supply functions for students’ accommodation on campus to be respectively:
10P + 3Q = 60 and
P = Q – 0.5
Further, the professor claims that the average cost of Hostel operators is
AC = 20/Q+1+0.2Q
If the hostel operators decide to maximize sales revenue instead of profit, show how this will affect consumer’s surplus. What is the producer’s surplus under pure competition at the equilibrium point?

Answer 1

Solution:

To maximize revenue, first derive the revenue function from the demand function:

Demand function: 10P + 3Q = 60

$P = 6 - \frac{3}{10}Q$

Revenue function = P $\times$ Q

Revenue function = $(6 - \frac{3}{10}Q) \times Q = 6Q - \frac{3}{10}Q^{2}$

To derive the maximizing revenue quantity, find the first derivative and set it to zero:

First derivative:

$\frac{\partial R} {\partial Q} = 6 - \frac{6}{10}Q$

$6 - \frac{6}{10}Q = 0$

$6 =\frac{6}{10}Q$

Q = 10

Revenue maximizing quantity = 10

Substitute in the price function to determine maximizing price:

$P = 6 - \frac{3}{10}Q$

$P = 6 - \frac{3}{10}(10)$

P = 6 – 3 = 3

P = 3

Revenue maximizing price = 3

Maximizing Revenue = P $\times$ Q = 3 x 10 = 30

Consumer surplus = (0.5) (10) (6-3)

Consumer surplus = 15

Consumer surplus will be much higher when the hostel operators decide to maximize sales revenue instead of profit

 

Calculating producer surplus at equilibrium:

At equilibrium: Qd = Qs

Qd: 10P + 3Q = 60

$Q = 20 - \frac{10}{3}P$

Qs: P = Q – 0.5

Q = P + 0.5

Set Qd = Qs

$20 - \frac{10}{3}P = P + 0.5$

Multiply both sides by 3:

60 – 10P = 3P + 1.5

60 – 1.5 = 3P + 10P

58.5 = 13P

P = 4.5

Equilibrium Price = 4.5

Substitute in either demand or supply equation to derive equilibrium quantity:

Demand function:

$Q = 20 - \frac{10}{3}P$

$Q = 20 - \frac{10}{3}(4.5)$

Q = 20 – 15

Q = 5

Supply function:

Q = P + 0.5

Q = 4.5 + 0.5

Q = 5

Equilibrium Quantity = 5

Producer surplus at equilibrium:

½ $\times$ (5) $\times$ (4.5 – 0.5)

½ $\times$20 = 10

Producer surplus at equilibrium = 10