Difficulty: Intermediate | Prerequisites: basic calculus (derivatives), understanding of cost function components (fixed vs variable costs)
This material sits at the heart of the perfectly competitive firm's decision-making in ECN 312. You are learning how a firm uses its cost structure to decide how much to produce and at what price it will start producing at all. The tools here, deriving marginal cost, finding the shutdown price, and building a supply function, are the building blocks for understanding market supply and long-run equilibrium in later chapters. If you are comfortable taking derivatives and you understand what fixed and variable costs are, you are ready for this.
A perfectly competitive firm's supply curve is the portion of its marginal cost curve that sits above average cost. To find it, you derive MC from the total cost function, set MC equal to AC to find the minimum price at which the firm will produce, and then invert the MC function to express quantity as a function of price.
Short-run cost function, C(q)
The total cost of producing q units when at least one input (typically capital) is fixed. Includes both fixed costs and variable costs.
In simple terms, this is everything you spend to make q items, given that some costs (like rent or equipment) cannot be changed right now.
Marginal cost, MC(q)
The derivative of the total cost function with respect to quantity: MC(q) = dC/dq. It tells you the additional cost of producing one more unit.
Think of it as: "How much does it cost me to make the next t-shirt?"
Average cost, AC(q)
Total cost divided by quantity: AC(q) = C(q)/q. It tells you the per-unit cost of production at a given output level.
Think of it as: "If I spread all my costs evenly across every unit, what does each one cost me?"
Shutdown price (minimum of AC)
The lowest price at which a firm will produce a positive quantity in the short run (when no costs are sunk). Found where MC = AC, because MC crosses AC at AC's minimum.
In simple terms, this is the break-even price. Below this price, every unit you sell loses money on average, so you produce nothing.
Short-run supply function, q(P)
The quantity a competitive firm will produce at each price. It is the MC curve, but only for prices at or above the shutdown price. Below that price, quantity supplied is zero.
Think of it as: the firm's "menu" telling the market how much it will sell at any given price.
Given a short-run cost function, MC is simply its derivative with respect to q. Constants (fixed costs) drop out.
Start with the cost function: C(q) = 5.4 + (W + t + 0.25)q + 0.10q³
Substitute the given values (W = 0.50, t = 0.25): C(q) = 5.4 + 1·q + 0.10q³
Differentiate with respect to q:
The fixed cost 5.4 drops out (derivative of a constant is zero)
The linear term 1·q becomes 1
The cubic term 0.10q³ becomes 3 × 0.10 × q² = 0.30q²
Result: MC(q) = 1 + 0.3q²
Notice that MC is increasing in q. This is typical: rising marginal cost is what gives the supply curve its upward slope.
The shutdown price is the minimum of the average cost curve. MC intersects AC at this minimum, so setting MC = AC gives you both the efficient scale (the output at minimum AC) and the shutdown price.
Write out AC: AC(q) = C(q)/q = 5.4/q + 1 + 0.10q²
Set AC = MC: 5.4/q + 1 + 0.10q² = 1 + 0.30q²
Simplify: the constant terms (1) cancel, leaving 5.4/q = 0.20q²
Multiply both sides by q: 5.4 = 0.20q³
Solve: q³ = 27, so q = 3
Plug q = 3 back into MC to find the price: MC(3) = 1 + 0.3(9) = 3.70
The shutdown price is P = 3.70. Below this price, the firm produces nothing. At or above this price, the firm is at least covering all its costs (fixed and variable) on average.
Why the MC = AC trick works: MC pulls AC up when MC > AC and drags it down when MC < AC. The crossover point is where AC is neither rising nor falling, which is its minimum.
In perfect competition, the firm is a price-taker: it sets P = MC and solves for q. The supply function is this relationship, but only for prices at or above the shutdown price.
Start from the profit-maximisation rule: P = MC(q) = 1 + 0.3q²
Rearrange for q²: q² = (P - 1) / 0.3
Take the square root: q = √((P - 1) / 0.3)
Apply the shutdown condition: this holds only for P ≥ 3.70; for P < 3.70, q = 0
The complete short-run supply function is:
q(P) = √((P - 1) / 0.3) for P ≥ 3.70
q(P) = 0 for P < 3.70
Graphically, the supply curve is the MC curve from the point (q = 3, P = 3.70) upward and to the right. Below P = 3.70, the supply curve sits on the vertical axis at q = 0.
C(q) = 5.4 + (W + t + 0.25)q + 0.10q^3MC(q) = \frac{\partial C(q)}{\partial q} = 1 + 0.3q^2AC(q) = \frac{C(q)}{q} = \frac{5.4}{q} + 1 + 0.10q^2q(P) = \sqrt{\frac{P - 1}{0.3}} \quad \text{for } P \geq 3.70This is how any small business owner thinks about pricing. A food-truck operator, for example, faces fixed costs (the truck, the permits) and rising marginal costs (overtime labour, perishable ingredients bought at premium in larger quantities). The shutdown price tells them the lowest price at which it is worth opening the window at all. Below that, they are better off staying closed for the day.
Students often think the shutdown price equals marginal cost at q = 0. It does not. The shutdown price is MC (and AC) at the quantity where AC is at its minimum, not at zero output.
Students sometimes confuse the shutdown condition with the break-even condition. In this problem (with no sunk costs) they happen to be the same, but when some fixed costs are sunk, the shutdown price drops to the minimum of average variable cost, not average total cost.
A common error is forgetting to apply the price floor. The supply function is not just q = √((P - 1)/0.3) for all P. For P < 3.70, q = 0. Missing this loses marks.
Students frequently differentiate the cost function incorrectly by forgetting the chain rule or the power rule on the cubic term. Always double-check: d/dq of 0.10q³ is 0.30q², not 0.10q².
⚠️ Deriving MC and AC from a cost function and finding their intersection is one of the most commonly tested skills in intermediate micro. Expect to see it on the exam with different cost function forms.
⚠️ You must know when to use minimum AC (no sunk costs) vs minimum AVC (some costs sunk) as the shutdown condition. The problem will usually tell you whether costs are sunk.
⚠️ Writing the supply function as a piecewise function (with the price floor) is required for full marks. A bare inverse-MC expression without the cutoff is incomplete.
True or false: The marginal cost curve always intersects the average cost curve at the average cost curve's minimum. True.
Fill in the blank: If no costs are sunk, a firm will produce a positive quantity only if P ≥ ______. minimum AC (the shutdown price).
True or false: A competitive firm's short-run supply curve is its entire MC curve. False. It is only the portion of MC at or above the shutdown price.
Fill in the blank: For the cost function C(q) = 5.4 + q + 0.10q³, the fixed cost is ______. 5.4.
True or false: If P = 2.00, a firm with shutdown price 3.70 would produce a small positive quantity. False. It would produce zero.
Q: Given a cost function C(q) = F + cq + dq³, what is the general formula for MC(q)?
A: MC(q) = c + 3dq². You differentiate each term: F drops out, cq becomes c, and dq³ becomes 3dq².
Q: How do you find the shutdown price when no costs are sunk?
A: Set MC = AC and solve for q. This gives you the efficient scale. Plug that q back into MC (or AC, they are equal at that point) to get the shutdown price.
Q: If the cost function is C(q) = 10 + 2q + 0.5q², what is the short-run supply function (assuming no sunk costs)?
A: MC(q) = 2 + q. AC(q) = 10/q + 2 + 0.5q. Set MC = AC: 2 + q = 10/q + 2 + 0.5q, so 0.5q = 10/q, giving q² = 20, q = √20 ≈ 4.47. Shutdown price = MC(√20) = 2 + √20 ≈ 6.47. Supply: q(P) = P - 2 for P ≥ 6.47, and q = 0 otherwise.
Q: Why is the supply curve the MC curve above min AC, rather than the entire MC curve?
A: Below min AC, the firm cannot cover its average costs at any output level. Producing would mean losing money on every unit on average, so the firm is better off producing nothing.
This material connects directly to market supply and equilibrium (Problem 2 in this set), where individual supply curves are summed horizontally to get the market supply curve. It also feeds into long-run analysis: in the long run, free entry and exit drive the price to the minimum of long-run average cost, which is the long-run analogue of the shutdown price you learned here. Understanding sunk costs vs non-sunk costs also ties back to the producer theory foundations from earlier chapters.
short-run supply, marginal cost function, average cost, shutdown price, break-even price, minimum efficient scale, competitive firm supply, price-taker, P = MC, supply curve derivation, cost function derivative, fixed cost, variable cost, sunk cost, ECN 312, intermediate microeconomics, perfect competition supply