Average cost and marginal cost of production

Question

Problem 1. Determine a total cost function of transport services (e.g. road freight) as a function of volume of production. How you can derive now the average cost and the marginal cost of production? Why it is important to know and to be able to formulate average and marginal cost functions when we are interested in evaluating the natural monopoly?

Let Q = volume of production
TC = Total Cost

TC = 300Q + 10000

You can derive the Average cost (AC) by dividing the TC by Quantity.

AC = [300Q + 10000]/Q

You can determine the Marginal Cost (MC) by taking the derivative of TC with respect to Quantity

MC = 300

It is important to know what your average cost is because if the price is below AC, you are losing money. It is important to know the MC if you are a natural monopoly because the profit maximizing quantity is at the point at which MC = MR. Prior to that quantity point, any additional production produces more revenue than it costs, so you should increase production. Any quantity above that MC>MR and so you are losing money on the individual products above the optimal quantity.

Problem 2. Cost functions of private car use (use of road infrastructure)

Road’s average speed of traffic (v, km/h) depends on traffic volume (q, vehicles/h) as follows:

v = 50 – q/100

Average travel cost (c, EUR/h) is a function of average speed v such as

c = 0,5(3 + 200/v)

a) What is the function of total cost C for road use?

b) Derive the marginal cost function

c) What is the average cost c and marginal cost MC at traffic volume q = 500?

Requirement a

v = 50 – q/100

c = 0,5(3 + 200/v) → c = 1.5 + 100/v → c – 1.5 = 100/v → v = 100/(c – 1.5)

50 – q/100 = 100/(c – 1.5)
(5000 – q)/100 = 100/(c – 1.5)
5000 – q = 10000/(c – 1.5)
c – 1.5 = 10000/(5000 – q)
c = 1.5 + 10000/(5000 – q)

Requirement b

MC = first derivative of c = 1.5 + 10000/(5000 – q)
MC = d(1.5 + 10000/(5000 – q))

d(1.5) + d/dx[(10000)/(5000-q)] = 0 + [(5000-q)(0)-10000(-1)]/[5000-q]2 = 0 + [0 - (0 + 100000)/(5000-q)²]

= 10000/(5000-q)²

MC = 10000/(5000-q)²

Requirement c

Average cost = [1.5 + 10000/(5000-500)]/500 = 3.7222
MC = 10000/(5000-500)²
MC = 10000/45002
MC = 0.0005