Question

Suppose that the cost function of a firm is c(y) = 3y²+ 75.

(a) What are the fixed costs and variable costs?

(b) Determine the minimum average cost. Derive and sketch in a diagram the average variable cost (AVC), marginal cost (MC), and average cost (AC).

(c) Derive the supply curve.

(d) When the market price is 45, what are the quantity supplied and profit / loans?

(e) When the market price s 15, what ane the quantity supplied and profit / loss?

Answer 1

c(y) = 3y²+ 75

Marginal Cost = dC/dy = 6y

Fixed Cost = 75 (the constant value in the cost function)

Variable Cost = = 3y² (The part of the cost function dependent on output "y")

AVC = 3y, min AVC = dAVC/dy = 3

Supply curve is the marginal cost curve above the min AVC

Average Cost = C/y = 3y + 75/y

min AC= dAVC/dq = 3-75/y^2 =0

75/y^2 = 3 or y^2 = 25 or y=5

at y=5 the average cost is minimum.

at y=5, the average cost = 3y + 75/y = 3*5+75/5 = 30 (So in the long run price will be equal to the average cost P=ATC=30)

Below P=ATC=MC=$30, the firm will shut down in the long run

AT P=$15, the output determined at P=MC condition, 15=6y or y = 2.5 units (Quantity supplied)

VC at this level = 3*2.5 = $7.5 and ATC = 3y + 75/y = 3*2.5+75/2.5= 37.5

The variable cost is less than the price level, so the firm will continue producing despite suffering losses

LOSS = (P-ATC)*Q = (15-37.5)*2.5 = -56.25

AT P=$45, the output determined at P=MC condition, 45=6y or y = 7.5 units (Quantity supplied)

VC at this level = 3*7.5 = $22.5 and ATC = 3y + 75/y = 3*7.5+75/7.5= 32.5

The variable cost is less than the price level, so the firm will continue producing despite suffering losses

PROFITS = (P-ATC)*Q = (45-32.5)*7.5 = 93.75