Question

Consider the following cost functions.

      i.    LTC(q) = 3q³ – 6q²,

      ii.   STC(q) = 8q² + 800.

      For each function, answer the following

      a.   What is its total variable cost and total fixed cost?

      b.   At what output is average cost minimized?

      c.   Graph the marginal and average cost curves.

Answer 1

LTC(q) = 3q³ – 6q², ATC = TC/q = 3q^2 - 6q

also MC = dTC/dq = 9q^2 - 12q

all of this is total variable cost as there is no constant term to indicate fixed costs of production

so TVC = 3q³ – 6q² and TFC = 0

Average cost is minimized at dATC/dq = 0

= 6q - 6 = 0

6q = 6 or q = 1, the average total costs are minimized at q=1

STC(q) = 8q² + 800, ATC = TC/q = 8q + 800/q

also MC = dTC/dq = 16q

so TVC = 8q² and TFC = 800

Average cost is minimized at dATC/dq = 0

= 8-800/q^2 = 0

800/q^2 = 8

q^2 = 100

q = +-10, negative value ignored, so q = 10

, the average total costs are minimized at q=10