Finding AC, MC, and Natural Monopoly from a Nonlinear Cost Function | Econ Step-by-Step
In this problem, we’ll analyze a cost function to explore how it affects a firm’s average and marginal costs.
We’ll derive these costs, interpret and graph their relationship, and assess whether this cost structure reflects a natural monopoly. If you’re looking for a full cost minimization problem, check out this article.
To start, for this problem, all we’re given is a nonlinear cost function (nonlinear due to the exponent):
- C(Q) = 2Q^(1/2)
We know that is Q is raised to a smaller power (exponent) than 1, then it will feature diminishing marginal costs. Intuitively, this means that producing more lowers marginal cost, or the cost per additional unit of output produced. From this cost function, we also know that there’s no fixed cost and only variable cost—cost that varies with output.
We’ll now derive average cost and marginal cost. Average cost is cost divided by quantity, or C/Q:
- AC = C/Q = (2Q^(1/2))/Q = 2Q^(-1/2) = 2/(Q^(1/2))
Marginal cost is the derivative of the cost function:
dC/dQ = d/dq(2Q^(1/2)) = (1/2)(2)(Q^(-1/2)) = 1(Q^(-1/2)) = Q^(-1/2)
Hence, we can think of Q^(-1/2) as 1/Q^(1/2)) to easily see that AC is double MC. This means that AC will always be larger than MC—for any value we insert for Q, AC is simply a larger fraction of that value (2/x versus 1/x).
Let’s plot the marginal cost and average cost in relation to quantity:
Note that average cost curves can only intersect marginal cost curves when the two costs are equal. Logically, if your marginal cost (your cost to produce one additional unit) is under your average cost, then your cost will go down, while if your marginal cost is greater than your average cost, then your verage cost will increase with that additional unit of output. This intuitively explains why the only point where average cost stops falling and starts rising is where MC = AC.
As the final linguistic portion of this question: do we know whether this cost function represents a natural monopoly?
Natural monopolies occur when the cheapest way to produce is one firm. Mathematically, this translates to three criteria:
- Does AC decrease at all levels of output?
- Is MC always smaller than AC?
- Are there economies of scale over the full range of output?
If these are met, we know we’re dealing with a natural monopoly since average cost always falls as production increases, and therefore one firm producing all output in a market will maintain the lowest average cost.
In our problem, note that AC = 2/(Q^(1/2)) and MC = 1/(Q^(1/2). Therefore, 2/(Q^(1/2)) = 2(1/(Q^(1/2))), or AC = 2MC, so MC = 1/2AC. In this equation, we can plug in any value for Q and MC will always be 1/2 the value of AC. So, our three criteria are met.
Those are the five parts of this problem! If you’re a student in econ 101 or 201, you’ll find these articles useful (exact problem walkthroughs like the ones you’ll see on exams).
Else, hope you found this helpful, and leave any questions in the comments!
