Cost curves, efficient scale and shutdown point
From the cost function you get everything a firm needs to choose its output: marginal cost, average costs and the two break-even prices.
Formulas and rules
- Marginal cost MC = C′(x); fixed cost disappears when you differentiate
- Average total cost ATC = C(x)/x, average variable cost AVC = C_v(x)/x
- Efficient scale: minimum of ATC → long-run break-even price
- Shutdown point: minimum of AVC → short-run shutdown price
- MC crosses ATC and AVC at their minimum; in perfect competition supply is p = MC
Your turn
Three problems on this topic. Type numbers like 12.5 or 1,250 and elasticities with their sign.
Task
A firm has the cost function C(x) = 1,550 + 195x − 24x² + x³ (C in €). Find the shutdown point (minimum of average variable cost), the short-run shutdown price and marginal cost at that point. Should the firm produce in the short run at a market price of €70.00?
Shutdown output x
Short-run shutdown price in €
Marginal cost at that output in €
Produce at €70.00?
Numbers like 12.5 or 1,250; elasticities with their sign (−0.75); percentages as a number (12.5). Round to 2 decimal places.
Worked example
A firm has the cost function C(x) = 714 + 11x + 2x² (C in €). At an output of x = 21, find marginal cost, average total cost and average variable cost.
Worked solution
- 1.Marginal cost = derivative: MC(x) = 11 + 4x → MC(21) = 11 + 4 × 21 = €95.00
- 2.Total cost C(21) = 714 + 231 + 882 = 1,827; ATC = 1,827 / 21 = €87.00
- 3.Variable cost without the 714 fixed cost: AVC = 11 + 2 × 21 = €53.00
Typical mistake: Including fixed cost in marginal cost or average variable cost. Fixed cost disappears when you take the derivative and only belongs in average total cost.
A firm in perfect competition has the cost function C(x) = 648 + 13x + 2x² (C in €). Find the efficient scale (minimum of average total cost) and the long-run break-even price. The market price is €133.00: what quantity does the firm supply and what profit does it make?
Worked solution
- 1.ATC(x) = 648/x + 13 + 2x; minimum: ATC′(x) = −648/x² + 2 = 0 → x² = 324 → x = 18
- 2.ATC(18) = 36 + 13 + 36 = €85.00 (MC crosses ATC there)
- 3.Supply: p = MC → 133 = 13 + 4x → x = 30
- 4.Profit = 133 × 30 − C(30) = 3,990 − 2,838 = €1,152.00
Typical mistake: Finding the profit-maximising output from p = ATC instead of p = MC. In perfect competition price equals marginal cost; p = ATC only gives the output at which profit is zero.
Common questions
What is the difference between efficient scale and the shutdown point?
Efficient scale is the minimum of average total cost: below that price the firm makes a loss in the long run. The shutdown point is the minimum of average variable cost: below that price even short-run production does not pay.
Why does a firm keep producing at a loss?
As long as the price is above AVC, each unit covers its variable cost plus part of the fixed cost. Fixed cost has to be paid anyway, so the loss is smaller with production than without.