A firm has determined that its variable costs are given by the following relationship:
VC = .05Q3 - 5Q2 + 500Q
where Q is the quantity of output produced.
(a) Determine the output level where average variable costs are minimized.
(b) Determine the output level where marginal costs are minimized.
a) Average variable costs are variable costs divided by number of units produced: AVC = VC/Q = 0.05Q2 - 5Q + 500 In order to find the output level where average variable costs are minimized, we should differentiate the AVC function: dAVC/dQ = 0.1Q - 5 0.1Q - 5 = 0 Q = 50 With the output level of 50 items the firm minimizes its average variable costs.
b) Marginal cost is the change in total cost as a result of a small change in output. As we know, that fixed costs relatively are constant when the level of output is changed. That`s why we can differentiate the variable cost function therefore we are attempting to find the change in total cost as a result of an infinitesimally small change in one or more of the factors affecting total cost. Marginal Cost (MC) is the derivative of total cost (in our case, variable cost) with respect to output (dVC/dQ). Marginal costs are:MC = d(VC)/d(Q) = 0.15Q2 - 10Q + 500 To find out the output level where marginal costs are minimized, we differentiate the MC function: dMC/dQ = 0.3Q - 10 0.3Q - 10 = 0 Q = 33.3 When the firm produces the quantity of 33.3, it minimizes it`s marginal costs.