Question

The quantity demanded x each month of Russo Espresso Makers is 250 when the unit price p is $138. The quantity demanded each month is 1000 when the unit price is $108. The suppliers will market 700 espresso makers when the unit price is $72. At a unit price of $102, they are willing to market 2200 units. Both the supply and demand equations are known to be linear.

(a) Find the demand equation.
p =



(b) Find the supply equation.
p =



(c) Find the equilibrium quantity and the equilibrium price.

units
$

Answer 1

(a)
`D(q)=(-1)/(25) q+148`

(b)
`S(q)=(1)/(50)q+58`

(c)
`p_(*) =88\\\\q_(*) =1500`

Explanation:

(a) For the demand equation D(q) we have

P1: 138 Q1: 250

P2: 108 Q2: 1000

We can find m, which is the slope of the demand equation,

`m=(p_(2) -p_(1) )/(q_(2) -q_(1) )=(108-38)/(1000-250) =(-30)/(750)=(-1)/(25)`

and then we find b, which is the point where the curve intersects the y axis.

We can do it by plugging one point and the slope into the line equation form:

`y=mx+b\\\\D(q)=mq+b\\\\138=(-1)/(25)(250) +b\\\\138=-10+b\\\\138+10=b=148`

With b: 148 and m: -1/25 we can write our demand equation D(q)

`D(q)=(-1)/(25) q+148`

(b) to find the supply equation S(q) we have

P1: 102 Q1: 2200

P2: 102 Q2: 700

Similarly we find m, and b

`m=(p_(2) -p_(1) )/(q_(2) -q_(1) )=(72-102)/(700-2200) =(-30)/(-1500)=(1)/(50)`

`y=mx+b\\\\D(q)=mq+b\\\\72=(1)/(50)(700) +b\\\\72=14+b\\\\72-14=b=58\\`

And we can write our Supply equation S(q):

`S(q)=(1)/(50)q+58`

(c) Now we may find the equilibrium quantity q* and the equilibrium price p* by writing D(q)=S(q), which means the demand equals the supply in equilibrium:

`D(q)=S(q)\\\\(-1)/(25)q+148=(1)/(50)q+58\\\\`

`148-58=(1q)/(50) +(1q)/(25) \\\\90= (1q)/(50) +(2q)/(50)\\\\90=(3q)/(50)\\ \\q=1500\\\\`

We plug 1500 as q into any equation, in this case S(q) and we get:

`S(q)=(1)/(50)q+58\\\\S(1500)=(1)/(50)(1500)+58\\\\S(1500)=30+58\\\\S(1500)=88`

Which is the equilibrium price p*.