Question

A group of retailers will buy 104 televisions from a wholesaler if the price is $300 and 144 if the price is $250. The wholesaler is willing to supply 88 if the price is $225 and 168 if the price is $315. Assuming the resulting supply and demand functions are linear, find the equilibrium point for the market.

Answer 1

The equilibrium point for the market is approximately $270.03 with a quantity of 123.97.

Step-by-step explanation:

To find the equilibrium point for the market, we need to determine the price and quantity at which the quantity demanded equals the quantity supplied. Let's assume the demand function is linear and can be expressed as Qd = a - bP, where Qd is the quantity demanded and P is the price. Using the given information, we can find the slope (b) of the demand function:

144 - 104 / 250 - 300 = b => 40 / -50 = b => b = -0.8

The demand function becomes Qd = a - 0.8P. We can use one of the given price-quantity pairs to find the intercept (a). Let's use the pair (250, 144):

144 = a - 0.8(250) => 144 = a - 200 => a = 344

So the demand function is Qd = 344 - 0.8P.

To find the supply function, we can use a similar approach. Let's assume the supply function is linear and can be expressed as Qs = cP + d, where Qs is the quantity supplied and P is the price. Using the given information, we can find the slope (c) of the supply function:

168 - 88 / 315 - 225 = c => 80 / 90 = c => c = 0.8889

The supply function becomes Qs = 0.8889P + d. We can use one of the given price-quantity pairs to find the intercept (d). Let's use the pair (225, 88):

88 = 0.8889(225) + d => 88 = 200 + d => d = -112

So the supply function is Qs = 0.8889P - 112.

To find the equilibrium point, we set the quantity demanded equal to the quantity supplied and solve for the price:

344 - 0.8P = 0.8889P - 112

344 + 112 = 0.8P + 0.8889P

456 = 1.6889P

P = 456 / 1.6889

P ≈ 270.03

Substituting this price back into either the demand or supply function gives the equilibrium quantity:

Qd = 344 - 0.8(270.03) ≈ 123.97

So the equilibrium point for the market is approximately (270.03, 123.97).

Answer 2

(275,122)

When the price will be $275, the quantity will be 122 televisions.

Step-by-step explanation:

We have been given that a group of retailers will buy 104 televisions from a wholesaler if the price is $300 and 144 if the price is $250.

As the quantity of televisions depends on price of televisions, so our demand curve will pass through points (300,104) and (250,144).

Let us find slope of demand line using slope formula.

`m=(y_2-y_1)/(x_2-x_1)`, where,

`m =\text{Slope of line}`,

`y_2-y_1=\text{Difference between two y-coordinates}`,

`x_2-x_1=\text{Difference between same x-coordinates of two y-coordinates}`

Upon substituting coordinates of our given points in slope formula we will get,

`m=(104-144)/(300-250)`

`m=(-40)/(50)`

`m=-(4)/(5)`

Let us substitute
`m=-(4)/(5)` coordinates of point (250,144) in slope intercept form of equation (
`y=mx+b`).

`144=-(4)/(5)*250+b`

`144=-4*50+b`

`144=-200+b`

`144+200=-200+200+b`

`344=b`

Upon substituting
`b=344` and
`m=-(4)/(5)` we will get equation of our demand line as:

`y=-(4)/(5)x+344`

Similarly we will find the equation of supply line using points (225,88) and (315,168).

`m=(168-88)/(315-225)`

`m=(80)/(90)`

`m=(8)/(9)`

Let us substitute
`m=(8)/(9)` and coordinates of point (225,88) in slope intercept form of equation (
`y=mx+b`).

`88=(8)/(9)*225+b`

`88=8*25+b`

`88=200+b`

`88-200=200-200+b`

`-112=b`

Upon substituting
`b=-112` and
`m=(8)/(9)` we will get equation of our supply line as:

`y=(8)/(9)x-122`

Let us equate both lines to find the point where both lines intersect.

`-(4)/(5)x+344=(8)/(9)x-122`

`-(4)/(5)x-(8)/(9)x+344=(8)/(9)x-(8)/(9)x-122`

`-(4)/(5)x-(8)/(9)x+344=-122`

`-(4)/(5)x-(8)/(9)x+344-344=-122-344`

`-(4)/(5)x-(8)/(9)x=-466`

Let us have a common denominator.

`-(4*9)/(5*9)x-(8*5)/(9*5)x=-466`

`-(36)/(45)x-(40)/(45)x=-466`

`(-36-40)/(45)x=-466`

`(-76)/(45)x=-466`

`(-76)/(45)*(45)/(-76)x=-466*(45)/(-76)`

`x=466*(45)/(76)`

`x=6.1315789473684211*45`

`x=275.9210526315789495\approx 275`

Let us substitute x=275 in any of our equation to solve for y.

`y=-(4)/(5)*275+344`

`y=-4*55+344`

`y=-220+344`

`y=122`

Therefore, the equilibrium point for the market will be (275,122).