Question

The short run cost function for Crunchies has Marginal Cost MC = 5+.4Q (where Q is measured in tonnes/period and MC is in $/tonne) and Fixed Costs of $720/period. Give the level of Q at which ATC reaches a minimum Numeric Answer:

Answer 1

Solution:

Average total cost (ATC) reaches the minimum where it equals the marginal cost (that is, marginal cost curve cuts the average total cost curve at the minimum of average total cost curve).

Total cost = integration of marginal cost

Total cost = 5*Q + 0.4*Q² /2 + C, where C is a constant term

Total cost = 5Q + 0.2Q² + C

C here is also indication for total fixed cost, which is already given to us as $720 per period, so

Total cost = 5Q + 0.2Q² + 720

With the total cost function, we can find average total cost function as follows:

ATC = TC/Q

ATC = (5Q + 0.2Q² + 720)/Q

ATC = 5 + 0.2Q + 720/Q

Now, we need to find the value of Q where ATC = MC

So, 5 + 0.2Q + 720/Q = 5 + 0.4Q

720/Q = 0.4Q - 0.2Q

0.2Q*Q = 720

Q² = 720/0.2 = 3600

So, required value of Q = (3600)^1/2 = 60 units

Thus, ATC reaches its minimum at 60 units of output.