Question

Stutz Department Store will buy 10 pairs of sunglasses if the price is $93 per pair and 30 pairs if the price is $43. The supplier of the sunglasses is willing to provide 35 pairs if the price is $98 per pair but only 5 pairs if the price is $38. Assuming that the supply and demand functions for the sunglasses are linear, find the market equilibrium point.

Answer 1

$68 for 20 pairs

Step-by-step explanation:

Demand

$93 per pair for 10 pairs

$43 per pair for 30 pairs

Supply

$98 per pair for 35 pairs

$38 per pair for 5 pairs

Let x denotes price and y denotes a pair of sunglasses

x1, y1 = (93, 10)

x2, y2 = (43, 30)

x3 y3 = (98, 35)

x4, y4 = (38, 5)

Slope intercept form is given by

y = mx + b

m = y2 - y1/x2 - x1 = 30 - 10/43 - 93 = 20/-50 = -0.4

Now put any one x, y point from demand into slope equation to find b

30 = -0.4(43) + b

30 = -17.2 + b

b = 47.2

so the demand equation becomes

y = -0.4x + 47.2 eq. 1

Now for the supply equation

m = y4 - y3/x4 - x3 = 5 - 35/38 -98 = -30/-60 = 0.5

Now put any one x, y point from supply into slope equation to find b

5 = 0.5(38) + b

5 = 19 + b

b = -14

so the supply equation becomes

y = 0.5x - 14 eq. 2

Now we want to find the intersection of these two lines so that we can get the equilibrium point

-0.4x + 47.2 = 0.5x - 14

-0.4x - 0.5x + 47.2 + 14 = 0

-0.9x + 61.2 = 0

0.9x = 61.2

x = 61.2/0.9

x = 68

Now put it into eq. 1 or eq. 2

y = 0.5(68) - 14

y = 34 - 14

y = 20

So at market equilibrium the price of sunglasses is $68 for 20 pairs.