The demand curve for labor is  Q=100-5P. The supply curve is Q=5P. Helpful hint. The indirect (inverse) demand curve is P=20-0.2Q a) What is the equilibrium price and quantity. b) What is the...

In order to answer these questions, we must do a bit of simple algebra. 

First, we will find the equilibrium price and quantity using the equations given.  The demand curve’s equation was Q = 100 – 5P and the supply curve’s equation was Q = 5P.  To find the equilibrium quantity, we set these equations to be equal to one another.  That gives us

100 – 5P = 5P

100 = 10P

P = 10

This means that the equilibrium price is $10.  Now that we have the equilibrium price, we can find the equilibrium quantity by plugging the price in to either of the demand or supply curve equations.

Q = 100 – 5(10) or Q = 5 (10)

Q = 100 – 50 = 50 or Q = 50

Whichever way we figure it, the equilibrium quantity is 50.

Now we need to figure the consumer surplus.  First, we find the y-intercept of the demand curve.  This is the price where the quantity demanded (Q) will be 0.  We do this using the equation for the demand curve. 

0 = 100 – 5P

-100 = -5P

20 = P

This tells us that consumers will buy zero hours of labor when the price is $20.  We can now use this to find the consumer surplus.  We can find the consumer surplus by finding the area of a triangle whose vertices are the equilibrium quantity, the maximum price (y-intercept of the demand curve) and the equilibrium price.  The formula for the area of a triangle is ½(b*h).

The base of the triangle is found by subtracting the equilibrium price from the maximum price.  In this case, that is 20 – 10  = 10.  The height of the triangle is found by finding how far the equilibrium quantity is from the y-axis (50).  To visualize this, please follow the link and scroll down to the graph.

Using the formula for the area of a triangle, we have

Consumer surplus = ½ (10*50)

= ½ (500)

= 250

This tells us that the consumer surplus is 250.

To find the producer surplus, we find the area of the triangle whose vertices are the equilibrium quantity, the minimum price (y-intercept of the supply curve) and the equilibrium price.

First, we find the y-intercept of the supply curve.  This is the price where the quantity supplied (Q) will be 0.  We do this using the equation for the supply curve.

Q = 5P

0 = 5P

P = 0

This tells us that no labor will be supplied when the price is zero.

We can now find the producer surplus by finding the area of the triangle.  The base of the triangle is found by subtracting the minimum price from the equilibrium.  In this case, that is 10 - 0  = 10.  The height of the triangle is found by finding how far the equilibrium quantity is from the y-axis (50).

Using the formula for the area of a triangle, we have

Producer surplus = ½ (10*50)

= ½ (500)

= 250

So the producer surplus is 250.

To find the total surplus, we simply add the producer surplus and the consumer surplus.

250 + 250 = 500

The total surplus is 500.

To recap, then:

• equilibrium price is $10
• equilibrium quantity is 50
• consumer surplus is 250
• producer surplus is 250
• total surplus is 500