Source: Lecture 13, Intermediate Microeconomic Theory, Arizona State University
Difficulty: Intermediate. Prerequisites: Part 1 of these notes (average and marginal cost curves), profit maximisation via first-order conditions, basic calculus.
Once you know a firm's cost curves, the next question is: how much should it produce, and when should it stop producing entirely? This section connects marginal cost to the firm's supply decision and introduces the shutdown condition. These are the tools that let you go from a production function all the way to a supply curve, which is one of the central skills tested in intermediate micro.
A profit-maximising firm sets price equal to marginal cost, which means the supply curve is the upward-sloping portion of the MC curve. The firm shuts down when price falls below average variable cost, because at that point it loses more by producing than by producing nothing. Combining these two results gives the firm's complete supply curve.
Supply curve
The relationship between the market price of a good and the quantity a firm chooses to produce. For a price-taking firm, this is the upward-sloping portion of the MC curve above AVC.
In simple terms, it tells you how much a firm will make at each possible price.
Inverse supply
The supply relationship written as price as a function of quantity: p = MC(y). It is the first-order condition of the profit maximisation problem.
Think of it as reading the supply curve the other way round: instead of "at this price, how much?", it says "for this quantity, what price is needed?"
Profit function
Profit equals total revenue minus total cost: pi = py - c(y). The firm maximises this by choosing the output level where marginal revenue (which equals price for a price-taker) equals marginal cost.
Shutdown condition
The firm shuts down (produces zero) when price falls below average variable cost: p < AVC. At that point, revenue does not even cover variable costs, so the firm loses less by shutting down and paying only its fixed costs.
In simple terms, if you cannot even pay the bills that go up with production, you are better off not producing at all.
First-order condition (FOC)
The equation obtained by setting the derivative of profit with respect to output equal to zero: dp/dy = p - MC(y) = 0, which gives p = MC(y).
This is the mathematical statement of the profit-maximisation rule.
Start with the profit function for a price-taking firm:
pi = py - c(y)
Take the first-order condition by differentiating with respect to y and setting equal to zero:
dpi/dy = p - MC(y) = 0, so p = MC(y)
This means the firm produces where price equals marginal cost. Solving for y as a function of p gives the supply curve. Solving for p as a function of y gives the inverse supply curve.
Using c(y) = (1/5)(y - 5)^3 + 50:
Profit: pi = py - (1/5)(y - 5)^3 - 50
FOC: p - (3/5)(y - 5)^2 = 0
Supply: y = 5 + sqrt(5p/3)
Inverse supply: p = (3/5)(y - 5)^2
The inverse supply function is simply the MC curve.
Because the MC curve is U-shaped, a horizontal line at any price intersects it at two points:
y = 5 + sqrt(5p/3) and y = 5 - sqrt(5p/3)
Only the point on the upward-sloping portion of MC is a profit maximum. The point on the downward-sloping portion is a profit minimum (the second-order condition fails there). Always take the solution on the rising segment.
A firm shuts down when producing makes it worse off than producing nothing. If the firm produces nothing, its loss equals its fixed costs (it still pays F). The firm should shut down when:
py - c(y) < -F
Substituting c(y) = c_v(y) + F and simplifying:
py - c_v(y) - F < -F
py - c_v(y) < 0
py < c_v(y)
p < c_v(y)/y
p < AVC
So the shutdown rule is: produce nothing if price is below average variable cost.
Using c(y) = (1/5)(y - 5)^3 + 50 and y = 5 + sqrt(5p/3):
At p = 10: y is approximately 9, and pi = 10(9) - (1/5)(4)^3 - 50 = 90 - 12.8 - 50 = 27.2. Positive profit. The firm produces.
At p = 5: y is approximately 8, and pi = 5(8) - (1/5)(3)^3 - 50 = 40 - 5.4 - 50 = -15.4. A loss, but compare with shutdown.
At p = 1: y is approximately 6, and pi = 1(6) - (1/5)(1)^3 - 50 = 6 - 0.2 - 50 = -44.2. A larger loss.
If the firm produces nothing: pi = 0 - (1/5)(0 - 5)^3 - 50 = 25 - 50 = -25.
So at p = 5, the firm loses 15.4 by producing, which is better than losing 25 by shutting down. It keeps producing. At p = 1, the firm loses 44.2, which is worse than losing 25. It shuts down.
The firm's supply curve is the portion of the MC curve that lies above the AVC curve. Below the minimum of AVC, the firm produces zero. At prices above min(AVC), it produces where p = MC on the upward-sloping segment.
Plugging y = 0 into a complicated cost function is a fast way to identify fixed costs, because all variable terms vanish.
This example walks through the entire chain, which is the type of problem that appears on exams.
Production function: y = k^(1/6) * l^(1/6)
Price of capital: r = 5
Price of labour: w = 10
Substitute the constraint l = y^6 / k into the cost expression:
min 5k + 10(y^6 / k)
Take the derivative with respect to k and set it to zero:
5 - 10(y^6 / k^2) = 0, which gives 1/2 = y^6 / k^2
Solving: k = sqrt(2) * y^3
Substitute back to find l: l = y^6 / (sqrt(2) * y^3) = y^3 / sqrt(2)
c = 5 * sqrt(2) * y^3 + (10 / sqrt(2)) * y^3 = (20 / sqrt(2)) * y^3
Note there are no fixed costs here (c(0) = 0), so this firm never faces a shutdown decision, only a break-even one.
Profit: pi = py - (20 / sqrt(2)) * y^3
FOC: p - (60 / sqrt(2)) * y^2 = 0
Inverse supply: p = (60 / sqrt(2)) * y^2
Solving for y: y = sqrt(p * sqrt(2) / 60)
At p = 20: 20 = (60 / sqrt(2)) * y^2, so y^2 = 20 * sqrt(2) / 60 = sqrt(2) / 3
Therefore y = (1/3)^(1/2) * 2^(1/4)
This is approximately 0.69 units of output. The exact form is fine for an exam answer.
Concept | Formula | When to use |
|---|---|---|
Profit | pi = py - c(y) | Starting point for supply derivation |
FOC (price-taker) | p = MC(y) | Profit-maximisation condition |
Supply curve | Solve p = MC(y) for y | Upward-sloping portion only |
Inverse supply | p = MC(y) directly | Same as the MC curve |
Shutdown condition | p < AVC(y) | Produce zero if this holds |
Shutdown profit | pi = -F | Loss when producing nothing |
For the Cobb-Douglas example (y = k^(1/6) * l^(1/6), r = 5, w = 10):
Cost function: c(y) = (20/sqrt(2)) * y^3
MC(y) = (60/sqrt(2)) * y^2
Supply curve: p = (60/sqrt(2)) * y^2
Students often think a firm making negative profit should always shut down. That is wrong. A firm should shut down only if its loss from producing exceeds its loss from producing nothing (which is -F). The test is p < AVC, not p < AC.
Students forget that the MC curve has two intersections with any horizontal price line (for a U-shaped MC). Only the intersection on the upward-sloping segment is the profit-maximising quantity. The other is a profit minimum.
A common error in the full derivation is forgetting to check the second-order condition. Make sure d^2pi/dy^2 < 0 at the chosen output level.
Students sometimes confuse "supply is marginal cost" with "supply is the entire MC curve." The supply curve is only the portion of MC above AVC. Below min(AVC), quantity supplied is zero.
⚠️ The production-function-to-supply-curve derivation (cost minimisation, then profit maximisation) is a classic multi-step exam problem. Practise the full chain until each step is routine.
⚠️ Be ready to evaluate profit at specific prices and determine whether the firm operates or shuts down, as in the p = 1, 5, 10 example.
⚠️ Know how to find fixed costs by plugging y = 0 into the cost function.
True or false: A firm's supply curve is its entire marginal cost curve. (Answer: False. Only the upward-sloping portion above AVC.)
Fill in the blank: The firm shuts down when p < ______. (Answer: AVC)
True or false: If a firm is making negative profit, it should always shut down. (Answer: False. It should shut down only if its loss exceeds its fixed costs, i.e. only if p < AVC.)
Fill in the blank: Inverse supply is the same as the ______ curve. (Answer: marginal cost)
True or false: A cost function with no fixed costs means the firm never faces a shutdown decision. (Answer: True. With F = 0, the firm shuts down only at p = 0.)
Q: Given c(y) = (1/5)(y - 5)^3 + 50, derive the firm's supply curve.
A: Write the profit function pi = py - (1/5)(y - 5)^3 - 50. Take the FOC: p - (3/5)(y - 5)^2 = 0. Solve for y: y = 5 + sqrt(5p/3). This is the supply curve (taking only the positive root for the upward-sloping segment).
Q: At what price does the firm in the above example shut down?
A: The firm shuts down when p < min(AVC). Evaluate profit at zero output: pi(0) = -25. The firm shuts down when its profit from producing is worse than -25. From the lecture's calculations, p = 1 yields pi = -44.2 (worse than -25, so shut down), while p = 5 yields pi = -15.4 (better than -25, so keep producing). The shutdown price is at the minimum of AVC.
Q: Starting from y = k^(1/6) * l^(1/6) with r = 5, w = 10, derive the cost function.
A: Substitute the production constraint to get l = y^6/k. Minimise 5k + 10(y^6/k) over k: the FOC gives k = sqrt(2) * y^3 and l = y^3/sqrt(2). Total cost is c(y) = (20/sqrt(2)) * y^3.
Q: Why do we take only the upward-sloping solution when MC has two intersections with a price line?
A: The downward-sloping intersection fails the second-order condition for profit maximisation. At that point, a small increase in output raises profit (MC is falling, so revenue grows faster than cost), meaning the firm has not yet reached its optimum. Only on the upward-sloping segment does a further increase in output reduce profit, confirming a maximum.
Q: A firm has c(y) = 4y^2 + 100. What is its supply curve? At what price does it shut down?
A: MC = 8y, so the inverse supply is p = 8y, giving y = p/8. AVC = 4y. The firm shuts down when p < AVC = 4y. Substituting y = p/8: p < 4(p/8) = p/2, which is never true for positive p. So this firm never shuts down (its AVC starts at zero and always lies below MC). This makes sense because the fixed cost of 100 is sunk, and the firm always covers its variable costs.
The supply curve derived here is for a single firm. Aggregating these across all firms in a market gives the market supply curve, which you will use in equilibrium analysis. The shutdown condition connects to the distinction between short-run and long-run supply: in the long run, all costs are variable, so the long-run supply curve is the entire MC curve above min(AC). The cost-minimisation step in the full example connects back to isoquants and isocost lines from earlier in the course.
Supply curve, inverse supply, shutdown condition, shutdown price, profit maximisation, first-order condition, FOC, marginal cost equals price, price-taker, competitive firm supply, average variable cost shutdown rule, cost minimisation, Cobb-Douglas production function, ECN 312, intermediate microeconomics, producer theory, Arizona State University