the demand function for a firm is given as P=60-0.5p and the supply function is p= 10+Q

Question

1. find the equilibrium price and quantity.

2.calculate the consumer surplus and producer surplus at equlibrium

Answer 1

The supply curve for a product shows the price p per unit at which the manufacturer 

will sell (or supply) q units.  

The demand curve for a product indicates the price p per unit at which consumers will 

purchase (or demand) q units.

The point (q₀, p₀) where the two curves intersect is called the point of equilibrium with

p₀ the price per unit at which consumers will buy the same quantity q₀ of a product that 

producers seek to sell at p₀.   

That is, p₀ is the price at which stability in the producer-consumer relationship occurs.    

I will assume here that the statement of the problem above is meant to read:

Demand Equation is p = 60 − 0.5q;

Supply Equation is p = 10 + q.

Equate 60 − 0.5q to 10 + q to gain 1.5q =50 or q₀ = 100/3.

Then p₀ can be taken from 10 + q₀ as (30 + 100)/3 or 130/3.

Equilibrium Quantity and Price are then (q₀, p₀) equal to (100/3, 130/3).

Construct Consumers' Surplus as CS = ∫(from 0 to q₀)[f(q) − p₀]dq or

∫(from 0 to 100/3)[60 − 0.5q −130/3]dq. Integrate to [50q/3 − 0.25q²]

and evaluate from 0 to 100/3 to obtain 5000/9.

Build Producers' Surplus as PS = ∫(from 0 to q₀) [p₀ − g(q)]dq or

∫(from 0 to 100/3) [130/3 − (10 + q)]dq. Integrate to [100q/3 − q²/2]

and evaluate from 0 to 100/3 to obtain 5000/9.