Question

A small organic grocer will buy 100 avocados per month if the price is $1 each and 60 avocados per month if the price is $3 each. A farmer is willing to supply the grocer with 60 avocados per month if the price is $2 each and 80 avocados per month if the price is $3 each. Assume that the supply and demand functions are both linear. Use the letter q for the quantity variable in the first two questions below. The demand function is p = The supply function is p = The market equilibrium point occurs where p = and q = Therefore, the equilibrium price for each avocado is __________

Answer 1

The demand function is P=6-0.05Q

The supply function is P=0.05-1

The market equilibrium point occurs where p = 2.5 and q = 70. Therefore, the equilibrium price for each avocado is $2.5

Explanation:

We have a linear function for the demand and the supply. We have two points for each one and we have to write the equation from them.

Demand function

1. 100 avocados per month at $1
2. 60 avocados per month at $3

The linear function can be written as:

`P=mQ+b`

where Q is the quantity and P is the price.

The parameter m can be calculated as:

`m=(P_1-P_2)/(Q_1-Q_2)=(1-3)/(100-60)=(-2)/(40)=-0.05`

We can then calculate b as:

`P_1=mQ_1+b\\\\b=P_1-mQ_1=1-(-0.05)*100=1+5=6`

Then, the demand function is
`P=6-0.05QP`.

Supply function

1. 60 avocados per month at $2
2. 80 avocados per month at $3

The linear function can be written as:

`Q=mP+b`

where Q is the quantity and P is the price.

The parameter m can be calculated as:

`m=(P_1-P_2)/(Q_1-Q_2)=(2-3)/(60-80)=(-1)/(-20)=0.05`

We can then calculate b as:

`P_1=mQ_1+b\\\\b=P_1-mQ_1=2-(0.05)*60=2-3=-1`

Then, the supply function is
`Q_s=0.05P-1`

The market equilibrium will happen when both functions intersect.

That is:

`P_s=P_d\\\\6-0.05Q=0.05Q-1\\\\6+1=0.10Q\\\\Q=7/0.10=70`

The quantity of equilibrium is 70 avocados.

Replacing this quantity in any of both functions we can calculate the price of equilibrium:

`P_s*=0.05Q_(eq)-1=0.05*70-1=3.5-1=2.5`