3 The demand for widgets X is given by 8 marks total 4 marks

3. The demand for widgets (X) is given by: (8 marks total; 4 marks each) X-75-5PX The production of widgets has the following average variable costs: AVC = 2.5X-39 Fixed costs are 10. Determine: The output level that minimizes the average total cost (ATC) of widgets a. b. The output level that maximizes profits.

Solution

Question 3

(a)

AVC = 2.5X - 39

TVC = AVC * X = (2.5X - 39) * X = 2.5X² - 39X

MC = dTVC/dX = d(2.5X² - 39X)/dX = 5X - 39

TFC = 10

AFC = 10/X

ATC = AVC + AFC = 2.5X - 39 + (10/X)

ATC is at its minimum when it is equal to MC

ATC = MC

2.5X - 39 + (10/X) = 5X - 39

2.5X + (10/X) = 5X

X = 2

Thus,

The output level that minimizes the average total cost (ATC) of widgets is 2 widgets.

(b)

Demand for widgets is as follows -

X = 75 - 5Px

5Px = 75 - X

Px = 15 - 0.20X

TR = Px * X = (15 - 0.20X) * X = 15X - 0.20X²

MR = dTR/dX = d(15X - 0.20X²)/dX = 15 - 0.40X

Profit-maximizing level of output is that level of output corresponding to which MR equals MC.

MR = MC

15 - 0.40X = 5X - 39

-5.40X = -54

X = 10

The output level that maximizes profits is 10 widgets.