In Lecture 8, we learned that consumer surplus measures the net benefit buyers get from a market — the gap between what they’re willing to pay and what they actually pay.

💡 Producer surplus is not the same as profit! Profit = TR − TC = TR − VC − FC. Producer surplus = TR − VC. The difference is fixed costs:

\[PS = \pi + FC\]

. . .

Example: A hotel earns TR = €50,000, has VC = €30,000 and FC = €25,000.

• \(PS = 50{,}000 - 30{,}000 = €20{,}000\) ➡️ the hotel gains €20,000 from operating
• \(\pi = 50{,}000 - 55{,}000 = -€5{,}000\) ➡️ the hotel makes a loss!
• But \(PS > 0\) means the hotel is still better off operating (it covers VC and contributes €20,000 toward FC)

👉 This is exactly the shutdown logic from Lecture 12!

Left: PS = area above supply, below price = triangle. Right: CS = area below demand, above price = triangle (from Lecture 8). They are mirror images!

Example (from the graph): Supply is \(P = 10 + 0.5Q\), market price \(P^* = €30\).

At \(P^* = 30\): \(Q^* = \frac{30 - 10}{0.5} = 40\)

\[PS = \frac{1}{2} \times 40 \times (30 - 10) = \frac{1}{2} \times 40 \times 20 = €400\]

👉 Compare with CS formula: \(CS = \frac{1}{2} \times Q^* \times (b - P^*)\). Same triangle logic, just flipped!

The process:

1. At a given price \(P\), each firm \(i\) supplies \(q_i\) (where \(P = MC_i\))
2. Market quantity at that price: \(Q_S = q_1 + q_2 + q_3 + \ldots\)
3. Repeat for every price
4. Connect the dots → market supply curve

👉 If firms have different cost structures, the market supply curve may have kinks (just like market demand in Lecture 8!)

• For \(P < 10\): no firm produces → \(Q_S = 0\)
• For \(10 \leq P < 20\): only Firm A produces → \(Q_S = q_A\)
• For \(P \geq 20\): both firms produce → \(Q_S = q_A + q_B\) (kink at P = 20!)

👉 The kink occurs when Firm B’s supply “turns on” — same logic as the demand kink in Lecture 8.

From the previous example:

Firm A: \(P = 10 + 2q_A\) → \(q_A = \frac{P - 10}{2}\) for \(P \geq 10\)

Firm B: \(P = 20 + q_B\) → \(q_B = P - 20\) for \(P \geq 20\)

Market supply (horizontal sum = add quantities at each price):

For \(10 \leq P < 20\) (only A): \(Q_S = q_A = \frac{P - 10}{2}\)

For \(P \geq 20\) (A + B): \(Q_S = \frac{P - 10}{2} + (P - 20) = \frac{P - 10 + 2P - 40}{2} = \frac{3P - 50}{2}\)

. . .

Interpreting the parameters:

• \(c\) = the minimum price needed for any production (supply “turns on” at \(P = c\))
• \(d = \frac{\Delta P}{\Delta Q}\) (positive!) — how much price must rise to induce one more unit of output
• Direct form: \(Q = \frac{P - c}{d}\) (useful for horizontal summation)

👉 Compare with demand: \(P = b + mQ\) where \(m < 0\). Supply has \(d > 0\) — that’s the upward slope!

Supply: \(P = 5 + 0.25Q\). At \(P^* = €30\): \(Q^* = \frac{30 - 5}{0.25} = 100\). Slope = \(\frac{\Delta P}{\Delta Q} = \frac{10}{40} = 0.25\).

\(PS = \frac{1}{2} \times 100 \times (30 - 5) = €1{,}250\).

👉 Total welfare = CS + PS. At the competitive equilibrium, total welfare is maximized — there’s no way to make one side better off without making the other worse off. We’ll explore this more in Lecture 17 (market equilibrium).

Imagine the market for guided tours in Lisbon:

. . .

👉 Total welfare = €800 + €800 = €1,600. This is the total gain from trade in this market!

Today’s Key Takeaways:

1. Producer surplus = \(TR - VC\) = area above supply, below price. It measures the gain to producers from trading
2. PS ≠ profit! \(\pi = PS - FC\). A firm can have positive PS but negative profit (and still rationally operate)
3. PS for linear supply: \(PS = \frac{1}{2} \times Q^* \times (P^* - c)\) — a triangle, mirroring CS
4. Market supply = horizontal sum of individual supply curves (add quantities at each price)
5. With different firms, market supply may have kinks (when new firms “turn on” at higher prices)
6. Linear supply: \(P = c + dQ\) — intercept \(c\) is the minimum supply price, slope \(d > 0\)
7. Total welfare = CS + PS, maximized at the competitive equilibrium
8. Direct form \(Q = \frac{P - c}{d}\) is useful for horizontal summation

Connection: We now have both sides of the market: demand (Lectures 5–9) and supply (Lectures 10–14). Next lecture: supply elasticity — how sensitive is quantity supplied to price?

Next (Lecture 15, April 10): Calculation and Determinants of Supply Elasticity

Question: A hotel has TR = €80,000/month, VC = €50,000/month, and FC = €35,000/month. What is its producer surplus and economic profit?

A. PS = €30,000, Profit = €30,000

B. PS = €30,000, Profit = −€5,000

C. PS = −€5,000, Profit = −€5,000

D. PS = €45,000, Profit = €10,000

Question: The market supply for beach umbrella rentals in Cascais is \(P = 2 + 0.1Q\) (€ per rental). At a market price of €12, what is the producer surplus?

A. €500

B. €600

C. €1,000

D. €1,200

The market for walking tours in Porto has two types of operators:

• Professional guides (20 identical firms): individual supply \(P = 8 + 2q_P\) (€ per tour)
• Freelance guides (30 identical firms): individual supply \(P = 15 + q_F\) (€ per tour)

a) Write the individual direct supply function (\(q\) as a function of \(P\)) for each type of guide.

b) At what price does each type “enter” the market (start supplying)?

c) Write the market supply function for each price range. (Hint: there will be a kink!)

d) At a market price of €25 per tour, how many tours does each type of guide offer? What is total market supply?

e) Calculate the total producer surplus at \(P = €25\). (Calculate PS for each group separately, then sum.)

a) Direct supply functions (solve for \(q\)):

Professional: \(P = 8 + 2q_P\) → \(q_P = \frac{P - 8}{2}\) for \(P \geq 8\)

Freelance: \(P = 15 + q_F\) → \(q_F = P - 15\) for \(P \geq 15\)

b) Each type “enters” at the vertical intercept of their supply:

• Professionals enter at \(P = €8\) (lower costs, enter first)
• Freelancers enter at \(P = €15\) (higher costs, enter later)

👉 The kink in market supply occurs at \(P = €15\), where freelancers join the professionals.

c) Market supply (horizontal sum):

For \(8 \leq P < 15\) (only professionals, 20 firms):

\[Q_S = 20 \times \frac{P - 8}{2} = 10(P - 8) = 10P - 80\]

For \(P \geq 15\) (professionals + freelancers, 20 + 30 firms):

\[Q_S = 20 \times \frac{P - 8}{2} + 30 \times (P - 15) = 10P - 80 + 30P - 450 = 40P - 530\]

d) At \(P = €25\):

• Each professional: \(q_P = \frac{25 - 8}{2} = 8.5\) tours → 20 firms × 8.5 = 170 tours
• Each freelancer: \(q_F = 25 - 15 = 10\) tours → 30 firms × 10 = 300 tours
• Total market supply: \(Q_S = 170 + 300 = 470\) tours

✅ Check: \(Q_S = 40(25) - 530 = 1000 - 530 = 470\) ✅

e) Producer surplus at \(P = €25\):

Professionals (20 firms): Each firm’s PS = \(\frac{1}{2} \times q_P \times (P - c_P) = \frac{1}{2} \times 8.5 \times (25 - 8) = \frac{1}{2} \times 8.5 \times 17 = €72.25\)

Total PS for professionals: \(20 \times 72.25 = €1{,}445\)

Freelancers (30 firms): Each firm’s PS = \(\frac{1}{2} \times q_F \times (P - c_F) = \frac{1}{2} \times 10 \times (25 - 15) = \frac{1}{2} \times 10 \times 10 = €50\)

Total PS for freelancers: \(30 \times 50 = €1{,}500\)

Total market PS = €1,445 + €1,500 = €2,945

💡 Professionals have a higher PS per firm (€72.25 vs €50) because their costs are lower — they capture more surplus from each sale. But freelancers contribute more in aggregate because there are more of them.