Principles of Economics and Management I
Session 5 · Market Equilibrium and Taxes
Introduction
🔁 Where We Came From
Three sessions of building. Session 2 and 3 gave us demand and its elasticity; session 4 gave us supply as marginal cost.
Today the two curves finally meet, and a single number comes out of it: the market price. 💶
Then we ask the question that pays for the whole apparatus: what does a tax do, and who really pays it? 🧾
🗺️ Today’s Map
- ⚖️ Equilibrium, and why nobody has to know the price in advance
- ✨ Total surplus, and why the equilibrium maximizes it
- 🌊 What happens when the price is wrong
- ↔︎️ Shifts, the four step process and the four cases
- 🏦 Application: the credit market and the interest rate
- 🧾 Taxes: the wedge, incidence and deadweight loss
Two interactive widgets today. Bring a mouse. 🖱️
Part 1 · Equilibrium and Total Surplus
🤝 Bringing the Two Sides Together
Demand is the sorted list of values. Supply is the sorted list of marginal costs.
Both are functions of the same price. So ask the only question left: at which price do the two quantities agree?
Equilibrium: the price \(p^{*}\) at which \(Q_d(p^{*}) = Q_s(p^{*})\).
⚖️ Equilibrium
Consumer surplus above the price, producer surplus below it. Both already defined, now in one picture.
✨ Total Surplus
Total surplus (TS): consumer surplus plus producer surplus. What the market as a whole gains from trading.
\[\text{TS} = \text{CS} + \text{PS} = \sum_i (v_i - p) + \sum_j (p - c_j) = \sum (v_i - c_j)\]
The price cancels. It decides how the gains are split, not how big they are. 🎯
📖 Source: OpenStax Micro 3e, §3.5 “Demand, Supply, and Efficiency”, where CS + PS is called social surplus.
🎯 Why the Equilibrium Maximizes It
Trade unit by unit, highest value buyer against lowest cost seller. Each trade adds \(v - c\) to the total.
Below \(Q^{*}\): some buyer still has \(v > c\) of some idle seller. A gain is being left on the table.
Above \(Q^{*}\): the next trade would need \(v < c\). It destroys value.
So the total stops rising exactly where \(v = c\), which is \(Q^{*}\). A marginal condition again, just like session 1. 🎯
Say it carefully: the equilibrium maximizes the size of the pie. It says nothing about whether the split is fair. ⚠️
🌊 What if the Price Is Wrong?
Nobody in the market is told what \(p^{*}\) is. So suppose the price is simply wrong, and watch what happens.
There are only two ways to be wrong: too high, or too low. 👇
📈 Too High: Excess Supply
At 8 euros sellers offer 8 units and buyers want 2. Six units go unsold.
🔻 The Pressure That Follows
Unsold stock is expensive: storage, spoilage, capital tied up in a warehouse.
Any single seller who shaves the price sells first. So somebody always shaves it. 📉
The price falls, the orange gap narrows, and it keeps falling while any gap remains.
📖 Source: OpenStax Micro 3e, §3.1, “Equilibrium—Where Demand and Supply Intersect”, which calls this excess supply, or a surplus.
📉 Too Low: Excess Demand
At 2 euros buyers want 8 units and sellers offer 2. Six buyers go home empty-handed.
🔺 The Pressure That Follows
Some of those buyers value the good far above 2 euros. Offering a bit more beats going without.
And sellers notice they can raise the price and still sell everything they have. 📈
The price rises until the queue disappears.
📖 Source: OpenStax Micro 3e, §3.1, “Equilibrium—Where Demand and Supply Intersect”, which calls this excess demand, or a shortage.
🧲 The Equilibrium Is a Magnet
Write the adjustment as a rule: the price moves in the direction of the gap,
\[\dot p = \lambda \,\big[\,Q_d(p) - Q_s(p)\,\big], \qquad \lambda > 0\]
\(Q_d\) is decreasing in \(p\) and \(Q_s\) is increasing, so the bracket is strictly decreasing: positive below \(p^{*}\), negative above it.
That is precisely the condition for \(p^{*}\) to be a stable rest point. Nobody has to know \(p^{*}\); the sign of the gap points the way. 🧭
Note the assumption doing the work: demand slopes down, supply slopes up. Break either, and the magnet can turn into a repellent.
🧪 Try It: Set the Price Yourself
Drag the price. Watch the two quantities separate, and read off which way the pressure pushes. Only one price leaves no gap. 🖱️
Price: euros
Quantity demanded =
Quantity supplied =
📝 Review · Part 1
Two multiple choice questions and one exercise. ✍️
❓ Multiple Choice 1
At a price above the equilibrium:
A. There is excess demand and the price is pushed up.
B. The market is at rest, since sellers are happy.
C. The demand curve shifts to the left.
D. There is excess supply and sellers face pressure to cut the price.
✅ D. Above \(p^{*}\) the quantity supplied exceeds the quantity demanded. Unsold stock is what does the pushing.
❓ Multiple Choice 2
Total surplus at the equilibrium does not depend on:
A. The values buyers place on the good.
B. The price at which the market clears.
C. The marginal costs of the sellers.
D. The quantity traded.
✅ B. In \(\text{TS} = \sum (v_i - c_j)\) the price cancels: it moves surplus between buyers and sellers without changing the total. The other three all enter the sum directly.
🧮 Numerical Exercise
Demand \(P = 10 - Q\) and supply \(P = Q\), with \(P\) in euros. The industry also pays a fixed cost of 8 euros in total.
a) Find the equilibrium price and quantity.
b) Compute consumer surplus.
c) Compute producer surplus.
d) Compute total surplus.
e) Compute industry profit, and explain the difference from (c).
✅ Solution
a) \(10 - Q = Q \Rightarrow Q^{*} = 5\) and \(P^{*} = 5\) euros.
b) CS is the triangle between demand and the price: \(\tfrac{1}{2} \times 5 \times (10 - 5) = 12.5\) euros.
c) PS is the triangle between the price and supply: \(\tfrac{1}{2} \times 5 \times (5 - 0) = 12.5\) euros.
d) \(\text{TS} = 12.5 + 12.5 = 25\) euros. Check: \(\int_0^5 [(10 - q) - q]\,dq = 25\). ✅
e) Profit \(= \text{PS} - FC = 12.5 - 8 = 4.5\) euros. PS ignores the fixed cost; profit does not, exactly as session 4 said.
Part 2 · Comparative Statics
↔︎️ Moving Along vs Shifting
Now the curves themselves move. Watch out for the classic confusion:
A change in the price of the good moves us along the same curve. The price is an axis, so it cannot move the curve.
A change in anything else (income, tastes, costs, technology) shifts the whole curve.
Curves expand and contract. Saying “demand rises” is how you end up confusing the two. 🧠
🛒 What Shifts Demand
💶 Income: more income expands demand, for a normal good. For an inferior good it contracts. That is income elasticity from session 3, with a sign.
😋 Tastes and fashion: the product becomes popular and demand expands.
🔀 The price of other goods: a dearer substitute expands demand, a dearer complement contracts it. Cross-price elasticity, again with a sign.
👥 Number of buyers and 🔮 expected future prices.
📖 Source: OpenStax Micro 3e, §3.2, “What Factors Affect Demand?”, which runs through the same list one shifter at a time.
🏭 What Shifts Supply
🧰 Input costs: dearer raw materials, wages or energy contract supply.
⚙️ Technology: better technology expands supply, because marginal cost falls.
🌦️ External conditions: weather, regulation, the number of firms in the market.
Notice the pattern: anything that changes marginal cost shifts supply. That is the only channel, and session 4 told you why: supply is marginal cost. 🔑
📖 Source: OpenStax Micro 3e, §3.2, “How Production Costs Affect Supply” and “Other Factors That Affect Supply”.
🧭 The Four Step Process
A recipe for predicting any market change:
1. Draw supply and demand before the change, and mark the initial equilibrium.
2. Decide: does the change hit demand or supply?
3. Decide the direction: does that curve expand or contract?
4. Compare the new equilibrium with the old one.
📖 Source: OpenStax Micro 3e, §3.3 “Changes in Equilibrium Price and Quantity: The Four-Step Process”, where the recipe is applied to a summer of unusually good weather for salmon fishing.
🔎 Demand Expands
Supply is unchanged, so we slide up along it: the price rises and the quantity rises.
🔎 Supply Expands
Demand is unchanged, so we slide down along it: the price falls and the quantity rises.
🧭 The Four Cases
| Shock | \(P^{*}\) | \(Q^{*}\) |
|---|---|---|
| Demand expands | rises | rises |
| Demand contracts | falls | falls |
| Supply expands | falls | rises |
| Supply contracts | rises | falls |
Rule of thumb: with a demand shock, price and quantity move together. With a supply shock, they move opposite. 🎯
The reverse reading works too: price and quantity moving together in the data suggests demand moved. 🔍
🤝 And if Both Shift at Once?
If demand and supply move at the same time, one variable is determined and the other is ambiguous.
Both expand: the quantity rises for certain, the price can go either way. It depends on which shift is larger.
Demand expands while supply contracts: the price rises for certain, the quantity is ambiguous.
No arm-waving will settle the ambiguous one. Only the numbers will. 🧮
🧪 Try It: Move the Curves
Shift demand and supply, one at a time, and check the four cases. Then move both together and watch one of the two answers become ambiguous. 🖱️
Demand shift: 0
Supply shift: 0
P* = Q* =
☕ Example: A Good Coffee Harvest
An exceptional harvest at origin makes coffee cheaper to produce.
Step 2: it hits supply, because it changes the cost of producing.
Step 3: supply expands, the curve shifts right.
Step 4: the price falls and the quantity rises. Cheaper coffee, and more of it. ☕
📖 The mirror image is in OpenStax Micro 3e, §3.5: a terrible frost hits the Brazilian coffee crop, supply contracts and the price rises. Same market, opposite shock.
🍵 Example: A Change in Tastes
Consumers start to prefer tea over coffee. What happens in the coffee market?
Step 2: it hits demand, since it is a change in tastes.
Step 3: demand for coffee contracts, the curve shifts left.
Step 4: the price falls and the quantity falls. Both together, as the four cases predict. 📉
📖 Source: OpenStax Micro 3e, §3.2, “Changing Tastes or Preferences”.
⚡ Example: An Energy Price Shock
Energy prices in Europe jump. What happens in a market where energy is a large input?
Step 2: supply, through marginal cost. Step 3: supply contracts.
Step 4: the price rises and the quantity falls. Higher prices and a smaller market at the same time.
That combination has a name you will meet in Block 2: a supply shock, and it is why inflation and weak output can arrive together. 🔁
🏦 Application: The Interest Rate Is a Price
Here is the central application for anyone heading into finance:
The interest rate is the price of money over time: the price of using someone else’s funds today.
Like any price, it comes out of the meeting of a supply and a demand. Everything from Part 1 and Part 2 applies unchanged. 💶
💰 Who Supplies and Who Demands Funds
🟥 Supply of funds (saving): those with savings, lending them out. A higher rate makes saving more attractive, so the curve slopes up.
🟦 Demand for funds (credit): firms and households seeking financing. A higher rate kills the marginal project, so the curve slopes down.
Same reservation-price logic: a firm borrows while the return on the project exceeds the rate. The demand for funds is the sorted list of project returns. 🔑
📖 Source: OpenStax Micro 3e, §4.2 “Demand and Supply in Financial Markets”, which runs the argument through the market for credit card borrowing.
📈 Equilibrium in the Credit Market
The equilibrium rate \(r^{*}\) is what balances saving against borrowing.
↔︎️ Comparative Statics in Credit
Firms turn optimistic and want to invest more. Run the four steps.
Demand for credit expands, shifting right. Saving is unchanged.
So the rate rises and the quantity of funds traded rises. Price and quantity together: a demand shock, exactly as the table said. 📈
And a savings glut?
Supply expands, so the rate falls and the quantity still rises. Opposite directions: a supply shock. 🔁
📖 Source: OpenStax Micro 3e, §4.2, “Shifts in Demand and Supply in Financial Markets”.
💹 The Mirror Image: Bond Prices
A bond paying coupons \(C_t\) and maturing at \(T\) is worth, today,
\[P = \sum_{t=1}^{T} \frac{C_t}{(1 + r)^{t}}\]
Every single term falls when \(r\) rises, so \(\dfrac{dP}{dr} < 0\), always. The bond price and the interest rate move in opposite directions, mechanically.
So when the credit market pushes \(r^{*}\) up, every fixed income portfolio in Europe is repriced downward the same afternoon. 📉
The ECB moves these rates directly, which is why a room full of people watches a press conference. We return to it in monetary policy, session 11. 🏦
📝 Review · Part 2
Two multiple choice questions and one exercise. ✍️
❓ Multiple Choice 3
Demand and supply both expand at the same time. Then:
A. The quantity rises for certain, and the price can go either way.
B. The price rises for certain, and the quantity can go either way.
C. Both the price and the quantity rise for certain.
D. Nothing at all can be said.
✅ A. Both shifts push the quantity up, so that one is safe. On the price they pull in opposite directions, so the answer depends on which shift is larger. Note that D is too pessimistic: one of the two is always pinned down.
❓ Multiple Choice 4
Interest rates rise. Other things equal, the price of a bond already issued, with a fixed coupon:
A. Rises, because the bond now pays more.
B. Does not change, because the coupon is fixed.
C. Falls, because its fixed flows are discounted more heavily.
D. Rises or falls, depending on the issuer.
✅ C. The coupon being fixed is exactly the point: the flows do not move, but the discount factor does. Every term of the sum shrinks.
🧮 Numerical Exercise
Credit market: demand \(r = 12 - Q\) and supply \(r = 2 + Q\), with \(r\) in percent and \(Q\) in billion euros.
a) Find the equilibrium rate and quantity.
b) Firms turn optimistic and demand becomes \(r = 15 - Q\). Find the new equilibrium.
c) Which of the four cases is this, and does the answer match?
d) What happens to the price of bonds already issued?
✅ Solution
a) \(12 - Q = 2 + Q \Rightarrow 10 = 2Q\), so \(Q^{*} = 5\) billion euros and \(r^{*} = 7\) percent.
b) \(15 - Q = 2 + Q \Rightarrow 13 = 2Q\), so \(Q^{*} = 6.5\) billion euros and \(r^{*} = 8.5\) percent.
c) Demand expanded, so price and quantity should move together. Both rose. ✅
d) The rate rose, so existing bond prices fall: the same fixed flows, discounted at a higher rate. 📉
Part 3 · Taxes
🧾 The Incidence of a Tax
The state levies a per unit tax. Who actually pays it, the buyer or the seller?
Incidence does not depend on who hands the tax to the state, but on the elasticities.
The more inelastic side bears most of the tax. 🎯
📖 Source: OpenStax Micro 3e, §5.3, “Elasticity and Tax Incidence”, worked through with a tax on cigarettes.
🪓 The Tax Wedge
A per unit tax \(t\) drives a wedge between the two prices:
\[P_b = P_s + t\]
The buyer pays \(P_b\), the seller keeps \(P_s\), and the state takes the difference. The market clears at \(Q_d(P_b) = Q_s(P_s)\).
Nothing in these two equations mentions who hands the money over. That is why the legal side of the tax is irrelevant to who really pays. 🎯
📉 The Wedge, in a Figure
The tax shrinks the quantity traded. The shaded rectangle goes to the state; the grey triangle goes nowhere.
⚖️ The Incidence Rule
The share of the tax borne by consumers is
\[ \frac{\varepsilon_{supply}}{\varepsilon_{supply} + |\varepsilon_{demand}|}. \]
Read it as a tug of war: the side that can move escapes, and the side that cannot is left holding the tax. Elasticity is the ability to walk away.
Two limits to sanity check it. Perfectly inelastic demand (\(\varepsilon_d = 0\)): the share is \(1\), consumers pay all of it. Perfectly elastic demand: the share is \(0\), producers pay all of it.
Very inelastic demand (tobacco, fuel): almost the whole tax falls on the consumer. Which is exactly why those are the goods governments tax. 🚬
⚖️ Deadweight Loss
The rectangle is a transfer: buyers and sellers lose it, the state gains it. The triangle is not.
Deadweight loss (DWL): the value of the beneficial trades that no longer happen because a wedge stopped the price from clearing.
For a wedge \(t\) that cuts the quantity by \(\Delta Q\), the lost triangle is \[\text{DWL} \approx \tfrac{1}{2}\, t \, \Delta Q\]
Those are trades where \(v > c\) and which nobody now makes. Nobody collects that value: it is destroyed, not moved. 📐
📖 Source: OpenStax Micro 3e, §3.5, “Demand, Supply, and Efficiency”, which shows the same lost triangle without the algebra.
📈 Why the Loss Grows with the Square
\(\Delta Q\) is itself roughly proportional to \(t\), so \(\text{DWL} \approx \tfrac{1}{2} t \,\Delta Q\) grows with \(t^{2}\).
A tax twice as large costs about four times as much in lost surplus. That is the standard argument for many small taxes rather than one large one. 📐
And the loss grows with the elasticities, because \(\Delta Q\) does. Tax the side that cannot walk away and you raise revenue with less damage.
Which sets up an uncomfortable fact: the efficient tax and the fair tax are often not the same tax. 🧾
🎯 So Is Taxing Wrong?
No. Efficiency is one criterion, not the only one. Distribution, insurance and public goods are real objectives, and the market does not deliver any of them by itself.
What the analysis buys you is the price tag: this is what the objective costs in forgone surplus, and here is who pays it.
An economist’s contribution to that debate is the number, not the verdict. 🧾
📝 Review · Part 3
Two multiple choice questions and one exercise. ✍️
❓ Multiple Choice 5
A per unit tax is levied on sellers rather than buyers. Compared with levying it on buyers, the share actually borne by consumers:
A. Is smaller, since sellers hand over the money.
B. Is exactly the same.
C. Is larger, since sellers pass everything on.
D. Cannot be determined without knowing the tax rate.
✅ B. The wedge \(P_b = P_s + t\) and the clearing condition never mention who pays the state. Incidence is settled by the two elasticities alone.
❓ Multiple Choice 6
A government doubles a per unit tax. The deadweight loss, roughly:
A. Stays the same, since the market is the same.
B. Doubles.
C. Quadruples.
D. Halves, since the quantity traded falls.
✅ C. With \(\text{DWL} \approx \tfrac{1}{2} t\,\Delta Q\) and \(\Delta Q\) itself proportional to \(t\), the loss grows with \(t^{2}\).
🧮 Numerical Exercise
Demand \(Q_d = 20 - 2P\) and supply \(Q_s = 2P\), with \(P\) in euros. A per unit tax of \(t = 2\) euros is imposed.
a) Find the equilibrium before the tax.
b) Find \(P_b\), \(P_s\) and \(Q\) after the tax.
c) How much revenue does the state collect?
d) Compute the deadweight loss.
✅ Solution
a) \(20 - 2P = 2P \Rightarrow P^{*} = 5\) euros and \(Q^{*} = 10\).
b) Clearing needs \(20 - 2P_b = 2P_s\) with \(P_b = P_s + 2\). Substituting, \(20 - 2P_s - 4 = 2P_s\), so \(P_s = 4\), \(P_b = 6\) and \(Q = 8\).
c) \(t \times Q = 2 \times 8 = 16\) euros. Note the tax split evenly, one euro each way: here the two elasticities are equal in size.
d) \(\tfrac{1}{2} \times t \times \Delta Q = \tfrac{1}{2} \times 2 \times (10 - 8) = 2\) euros of surplus destroyed. ✅
Wrap-Up
🎯 What to Take From This Session
⚖️ Equilibrium is where the two quantities agree. Off it there is a gap, and the sign of the gap moves the price. Nobody needs to know \(p^{*}\).
✨ Total surplus is \(\sum (v_i - c_j)\): the price cancels out of it. The equilibrium maximizes the size of the pie and says nothing about the split.
↔︎️ Demand shocks move price and quantity together; supply shocks move them opposite. The four steps handle any shock, and both shifting leaves one answer ambiguous.
🏦 The interest rate is the price of funds, and bond prices are its mirror image.
🧾 A tax drives a wedge. Incidence falls on the more inelastic side whatever the law says, and the deadweight loss grows with \(t^{2}\).
👋 Next Session
Uncertainty, Risk and Information.
Every price so far was known with certainty. Next we drop that: expected utility, risk aversion, and what happens when one side of a trade knows more than the other. 🎲
See you next week. 🙌