Question

Based upon market research, the Hawthorne Company has determined that consumers are willing to purchase 132 units of their portable media player each week when the price is set at $62.40 per unit. At a unit price of $7.80, consumers are willing to buy 405 units per week. (a) Determine the weekly demand equation for this product, assuming price, p, and quantity, x, are linearly related. p = (b) Determine the weekly revenue function. R(x) = (C) Determine the number of units consumers will demand weekly when the price is $37.60 per portable media player. -256 units etermine the number of units consumers will demand weekly when the revenue is maximized. units (e) Determine the price of each unit when the revenue is maximized. dollars

Answer 1

At P = $62.4, demand was 132 goods.

At P = $7.8, demand was 405 goods.

(a). It is known at prices and quantity are negatively correlated in the demand function. That is slope will be negative.

From the data given above, the equation can be written as,

62.4 = a - 132b

7.8 = a - 405b

Now, subtracting these equations to get,

62.4 - 7.8 = 405b - 132b

54.6 = 273b

b = 54.6/273 = 0.2

Putting this value of b in any of the equation to get the value of a,

62.4 = a - 132(0.2)

62.4 = a - 26.4

a = 88.8

Thus, the weekly demand function would be,

P = 88.8 - 0.2Q

(b). The weekly revenue function would be,

PQ = R = 88.8Q - 0.2(Q^2)

(c). P = $37.6, then

P = 88.8 - 0.2Q

0.2Q = 88.8 - 37.6

Q = 51.2/0.2 = 256

(d). When revenue is maximised then,

dR/dQ = 88.8 - 0.4Q = 0

0.4Q = 88.8

Q = 88.8/0.4 = 222

Consumers will demand 222 goods weekly when the revenue is maximised.

(e). When Q = 222, then

P = 88.8 - 0.2*222

P = 88.8 - 44.4

P = $44.4