Introduction

🔁 Where We Came From

Sessions 2 and 3 built the buyer: preferences, the constrained optimum, market demand and its elasticity.

Today we build the other side. Supply is not assumed either: it comes out of costs and the objective of maximizing profit. 🏭

And then we ask what changes when the firm is large enough to face the whole demand curve by itself. 👑

🗺️ Today’s Map

  1. 🏭 Where costs come from, and why \(MC\) rises
  2. 🧱 Fixed, variable, average and marginal cost
  3. 🎯 Profit maximization: \(MR = MC\)
  4. 🏁 Perfect competition: \(P = MC\), and supply is marginal cost
  5. ❤️ Producer surplus and the shutdown condition
  6. 👑 Monopoly, the markup, and the link back to elasticity

By the end we hold a supply curve. Session 5 puts it against demand. 🔧

Part 1 · Production and Costs

🏭 Where Costs Come From

Costs are not primitive. They come from a production function and the prices of inputs.

\[q = f(L), \qquad MP_L = f'(L)\]

Diminishing marginal returns: \(f'' < 0\). Adding workers to a fixed plant raises output by less and less.

🔗 Which Is Exactly Why MC Rises

To make one more unit you need \(1/MP_L\) extra workers, each costing the wage \(w\). So

\[MC = \frac{w}{MP_L}\]

\(MP_L\) falls, therefore \(MC\) rises. The U shape of the cost curve is diminishing returns, seen from the cost side. 🔑

And it tells you what shifts \(MC\): the wage and productivity. Hold that list; it is exactly what shifts supply in session 5. 🔁

🧮 From a Production Function to a Cost Curve

Take \(q = \sqrt{L}\), so \(L = q^{2}\), with wage \(w\) and a fixed cost \(F\).

\[TC(q) = F + w\,q^{2}, \qquad MC(q) = 2wq, \qquad AC(q) = \frac{F}{q} + wq\]

\(MC\) rises because \(f'' < 0\); \(AC\) falls first (the fixed cost is spread over more units) and rises later (marginal cost drags it up).

Everything on the next few slides is already visible here. The shapes are not conventions, they are consequences. 📐

🧱 Types of Cost

Fixed cost (FC): does not depend on quantity (rent, insurance).

Variable cost (VC): grows with output (materials, energy).

Total cost: \(TC = FC + VC\). Average cost: \(AC = TC/q\). Average variable cost: \(AVC = VC/q\).

➕ Marginal Cost

Marginal cost (MC): the cost of producing one more unit.

\[ MC = \frac{d\,TC}{dq}. \]

Typically U shaped: it falls first (gains from specialization), then rises (diminishing returns).

📊 The Cost Curves

Note: MC crosses AC at its minimum, and it has to. \(AC' = (MC - AC)/q\), so \(AC\) falls exactly while \(MC < AC\) and rises once \(MC > AC\). 📐

🎯 The Golden Rule of Profit

The firm chooses \(q\) to maximize profit \(\pi(q) = R(q) - TC(q)\).

First order condition: \(\pi'(q) = 0\), that is \[ MR = MC. \]

Produce up to the point where marginal revenue equals marginal cost. No more, no less.

Check the second order condition too: \(\pi'' = MR' - MC' < 0\), so MC must cut MR from below. Of the two solutions, only the rising one is a maximum. ⚠️

📝 Review · Part 1

Two multiple choice questions and one exercise. ✍️

❓ Multiple Choice 1

A firm’s marginal cost curve is U shaped and it faces a given price. Profit is maximized where \(P = MC\):

A. At either of the two quantities where the curves cross.

B. At the lower of the two, where MC is falling.

C. Only if the firm is also minimizing average cost.

D. At the higher of the two, where MC is rising.

✅ D. The second order condition needs \(MC' > MR' = 0\). On the falling branch, \(P = MC\) is a profit minimum, and producing one more unit strictly improves things.

❓ Multiple Choice 2

Marginal cost is currently below average cost. Then average cost is:

A. Falling.

B. Rising.

C. At its minimum.

D. Constant.

✅ A. From \(AC' = (MC - AC)/q\), the sign of \(AC'\) is the sign of \(MC - AC\). It is the same arithmetic as a grade average: a mark below your average pulls it down.

🧮 Numerical Exercise

A firm has \(q = \sqrt{L}\), a wage of \(w = 2\) euros and a fixed cost of \(F = 18\) euros.

a) Write \(TC(q)\), \(MC(q)\) and \(AC(q)\).
b) At which quantity is average cost minimized?
c) Check that \(MC\) equals \(AC\) there.
d) What is the minimum average cost?

✅ Solution

a) \(L = q^{2}\), so \(TC = 18 + 2q^{2}\), \(MC = 4q\) and \(AC = 18/q + 2q\).

b) \(AC' = -18/q^{2} + 2 = 0 \Rightarrow q^{2} = 9\), so \(q = 3\).

c) \(MC(3) = 12\) and \(AC(3) = 6 + 6 = 12\). They meet, exactly as \(AC' = (MC - AC)/q\) requires. ✅

d) \(12\) euros per unit. Note this is the long run break-even price: below it the firm cannot cover its full cost at any quantity.

Part 2 · Competitive Supply

🏁 Perfect Competition

Many small firms, a homogeneous product. Each one is a price taker.

It faces the polar case from session 3: a perfectly elastic demand for its own output. Charge a cent more and it sells nothing.

If it is a price taker, selling one more unit always brings in \(P\): hence \(MR = P\), and the profit rule becomes \(P = MC\). 📈

🏭 The Seller’s Side, Same Logic

Read \(P = MC\) the way we read demand in session 3. A seller hands over the good and receives the price, so they sell when

\[p - c \;\ge\; 0\]

where \(c\) is the cost of producing that unit.

Reservation price of a seller: the lowest price at which they still sell. It equals their marginal cost \(c\).

Sort those costs and you get the supply curve: whoever can produce below the price sells. 🏭

🧾 A Warning About That Cost

\(c\) is the cost of producing one more unit.

Costs the firm pays whether or not it produces that unit (the rent on the factory, the licence, the loan already taken) are not in \(c\).

So the supply curve is built out of variable cost only. We need this in about three slides. 🧠

➕ From the Firm to Market Supply

Same aggregation as demand, in the other direction: at each price, ask every firm how much it wants to produce and add the quantities.

\[Q_s(p) = \sum_{j=1}^{m} q_j^{*}(p), \qquad \text{where } p = MC_j(q_j^{*})\]

Each firm’s supply is its rising marginal cost, so market supply slopes up. Horizontal aggregation again. 📈

❤️ Producer Surplus, Carefully

Producer surplus (PS): revenue minus the variable cost of what was produced. The area below the price and above the supply curve.

The supply curve is built from marginal costs, and marginal cost contains only variable cost. Fixed costs never entered it, so they are not in PS either.

\[\text{PS} = p\,q - VC(q), \qquad \text{profit} = p\,q - VC(q) - FC = \text{PS} - FC\]

🚨 So PS Is Not Profit

A firm can have a perfectly healthy producer surplus and still lose money, once the fixed costs are paid.

That gap is exactly why a firm keeps operating at a loss in the short run: as long as \(\text{PS} > 0\), shutting down would lose even more.

Two different questions: “is this trade worth doing?” (PS) and “was this business worth starting?” (profit). 🧠

🛑 Should It Produce at All?

\(P = MC\) says how much. It does not say whether. For that, compare with average cost.

🕐 Short run: fixed costs are sunk, so ignore them (session 1). Produce if \(P \ge AVC\); shut down if not.

🕰️ Long run: everything is variable. Stay in the market only if \(P \ge AC\); otherwise exit.

Between \(AVC\) and \(AC\) the firm makes a loss and still operates, because it covers its variable cost and contributes something toward the fixed one.

That gap is exactly the difference between producer surplus and profit from two slides ago. Now you know where it lives. 🔁

⚖️ In the Long Run, Entry Eats the Profit

Suppose price sits above minimum average cost, so firms are making money.

Nothing stops new firms entering. Market supply expands, and the price falls.

It stops falling only where profit is zero, that is at \(P = \min AC\). Which is why long run competitive price equals minimum average cost. 🎯

“Zero profit” sounds like failure and is not: it means capital earns exactly what it would earn elsewhere. The opportunity cost is already inside the cost. 💡

✨ The Efficiency of Competition

Under perfect competition, \(P = MC\): the price equals the cost of producing the last unit.

The buyer’s value of the last unit equals its cost. No trade worth doing is left undone, and none that destroys value is done.

Hold on to this landmark. Everything in Part 3 is measured against it. 🎯

📝 Review · Part 2

Two multiple choice questions and one exercise. ✍️

❓ Multiple Choice 3

In the short run a competitive firm finds \(AVC < P < AC\). It should:

A. Keep producing, even though it is making a loss.

B. Shut down at once, since it is losing money.

C. Raise its price until it covers AC.

D. Produce where \(P = AC\) instead.

✅ A. Fixed costs are sunk in the short run, so they should not enter the decision. Since the price covers variable cost, operating loses less than shutting down. Option C is not available: the firm is a price taker.

❓ Multiple Choice 4

Producer surplus is:

A. Revenue minus total cost, including fixed cost.

B. Revenue minus the variable cost of what was produced.

C. The firm’s accounting profit for the year.

D. The area above the demand curve and below the price.

✅ B. The supply curve is made of marginal costs, which contain no fixed cost. Subtract fixed cost from PS and only then do you have profit.

🧮 Numerical Exercise

Total cost \(TC = q^{2} + 10\) (euros), in a competitive market at a price of 12 euros.

a) Find the optimal quantity.
b) Compute profit.
c) Compute producer surplus, and reconcile it with (b).
d) Below which price would the firm shut down in the short run?

✅ Solution

a) \(MC = 2q\), and \(P = MC\) gives \(12 = 2q\), so \(q^{*} = 6\). MC is rising, so it really is the maximum.

b) \(\pi = 12 \times 6 - (36 + 10) = 72 - 46 = 26\) euros.

c) \(VC = q^{2} = 36\), so \(\text{PS} = 72 - 36 = 36\) euros. And \(\text{PS} - FC = 36 - 10 = 26\), the profit. ✅

d) \(AVC = q^{2}/q = q\), which is minimized as \(q \to 0\). So \(AVC \le P\) holds for any positive price: this firm never shuts down in the short run. The fixed cost of 10 is sunk and irrelevant to that call.

Part 3 · Market Power

👑 Monopoly

A single seller. The firm is not a price taker: it faces the entire demand curve.

To sell more, it has to cut the price on all units. That is why \(MR < P\).

The profit rule (\(MR = MC\)) still holds, but now \(MR\) lies below the price.

🧱 Where Market Power Comes From

A monopoly is not an accident. Something has to keep rivals out.

🏗️ Costs: a natural monopoly, where average cost falls over the whole relevant range, so one firm serves the market more cheaply than two.

📜 Law: patents, licences, concessions. Deliberately granted, usually to pay for the invention.

🔒 Control of an input, or a network that gets more valuable the more users it has.

📉 The Monopolist’s Choice

It picks the \(q\) where MR = MC, and charges the price \(P_m\) on the demand curve (above MC).

💸 The Inefficiency of Monopoly

The monopolist produces less and charges more than competition would.

\(P_m > MC\): between the monopoly quantity and the competitive one, buyers value the good above what it costs to make, and those trades do not happen.

Deadweight loss: the surplus destroyed by producing below the efficient level.

Note what is not the problem: the monopolist’s profit is a transfer from buyers, not a loss. What is lost is the triangle of trades nobody gets. 🎯

📐 Markup and Elasticity

Recall the formula from session 3: \(MR = P\left(1 + \frac{1}{\varepsilon}\right)\). With \(MR = MC\):

\[ \frac{P - MC}{P} = -\frac{1}{\varepsilon} = \frac{1}{|\varepsilon|}. \]

Lerner index: the margin over cost is larger the more inelastic demand is. Market power is the power to set price above cost. 💰

Sanity check: perfect competition is \(|\varepsilon| \to \infty\), giving a margin of zero and \(P = MC\). Competition is the limit case, not a different model. 🎯

🚫 A Monopolist Never Prices in the Inelastic Region

A result worth deriving, because it surprises people:

\(MC \ge 0\) and \(MC = MR = P\left(1 + \frac{1}{\varepsilon}\right)\), with \(P > 0\). So \(1 + 1/\varepsilon \ge 0\), which forces \(|\varepsilon| \ge 1\).

So a profit-maximizing monopolist always sits on the elastic part of demand. Never the inelastic part.

The intuition from session 3: where demand is inelastic, raising the price raises revenue and cuts output, so it raises revenue and cuts cost at the same time. No firm leaves that on the table. 💰

🏢 Financial Applications

📊 Understanding market power is essential for valuing firms: high margins signal competitive advantage, and the Lerner index says where they come from.

Sector analysis (how concentrated the market is, how elastic demand is) feeds directly into asset valuation. 💼

A margin that no barrier explains is a margin that competition will take away. That is a forecast, not a judgement. 🧠

📝 Review · Part 3

Two multiple choice questions and one exercise. ✍️

❓ Multiple Choice 5

Under monopoly, at the optimum:

A. P = MC.

B. MR > P.

C. P > MC.

D. P = MR.

✅ C. Since MR < P and MR = MC at the optimum, it follows that P > MC: there is a margin over cost.

❓ Multiple Choice 6

A profit-maximizing monopolist with \(MC \ge 0\) always operates where demand is:

A. Inelastic, since that is where the margin is largest.

B. Unit elastic, since that maximizes revenue.

C. At whatever elasticity the market happens to have.

D. Elastic, since \(MR = MC \ge 0\) requires \(|\varepsilon| \ge 1\).

✅ D. From \(MC = P(1 + 1/\varepsilon) \ge 0\). Option A confuses the Lerner index, which says a less elastic demand allows a bigger margin, with the claim that the firm ends up in the inelastic region. It never does.

🧮 Numerical Exercise

A monopoly with demand \(P = 20 - Q\) and constant \(MC = 4\) euros. Note \(MR = 20 - 2Q\).

a) Find the monopoly quantity and price.
b) Compute the Lerner index and the elasticity at that point.
c) What would a competitive industry with the same costs produce?
d) Compute the deadweight loss.

✅ Solution

a) \(20 - 2Q = 4 \Rightarrow Q_m = 8\), and \(P_m = 20 - 8 = 12\) euros.

b) Lerner \(= (12 - 4)/12 = 2/3\), so \(|\varepsilon| = 3/2 > 1\): elastic, as it must be.

c) \(P = MC\) gives \(20 - Q = 4\), so \(Q_c = 16\) at a price of 4 euros. The monopolist produces half as much.

d) The lost triangle between \(Q_m\) and \(Q_c\): \(\tfrac{1}{2}(12 - 4)(16 - 8) = 32\) euros of surplus that simply disappears. ✅

Wrap-Up

🎯 What to Take From This Session

🏭 Costs come from technology. Diminishing returns is why \(MC\) rises: \(MC = w/MP_L\).

📐 \(MC\) cuts \(AC\) at its minimum, because \(AC' = (MC - AC)/q\). Not a drawing convention, an identity.

🎯 The firm produces where \(MR = MC\), on the rising branch. Under price taking that is \(P = MC\), so supply is marginal cost.

❤️ PS is not profit: it leaves out fixed cost, which is what makes the short run shutdown rule \(P \ge AVC\) rather than \(P \ge AC\).

👑 Market power is \(P > MC\), and the Lerner index ties the margin to \(1/|\varepsilon|\). Competition is the case \(|\varepsilon| \to \infty\).

👋 Next Session

Market Equilibrium and Taxes.

We now have both curves. Next they meet: what sets the price, why that outcome maximizes surplus, and what a tax does to it. ⚖️

See you next week. 🙌