Chapter 5: Market Clearing and General Equilibrium. The Encyclopedia of Economic Principals, interactive companion

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Which curve moved?

Quantity rose to 70 crates. Did demand or supply move, and what does the price say?

A demand shift moves price and quantity in the same direction; a supply shift moves them in opposite directions. The price change tells you which schedule moved.

Equation, written in LaTeX: Q_D=100-2p,

Equation, written in LaTeX: Q_S=20+2p,

Equation, written in LaTeX: 120-2p=20+2p,

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p is dollars per crate and Q is crates per day. A demand shift adds to the demand intercept 100; a supply shift adds to the supply intercept 20.

Predict first. If demand and supply both shift out by 20, what happens to the price?

Your prediction It stays at 20 It rises It falls

Demand shift

Supply shift

Figure: Which curve moved? Demand Q = 100 - 2p and supply Q = 20 + 2p cross at quantity 60 and price 20; the original equilibrium was 60 crates at 20. — Demand shift: 0, Supply shift: 0
Constructed example: the chapter's hypothetical crate market (demand 100 - 2p, supply 20 + 2p, shifts of 20); shifts of 40 are added for comparison.

Equilibrium price p*: 20
Equilibrium quantity Q*: 60
Price change from 20: 0
Quantity change from 60: 0
Excess demand at the old price 20: 0

Clearing needs 100 - 2p = 20 + 2p, so 4p = 80 and p* = 20; then Q* = 100 - 2 x 20 = 60. Compared with (60, 20), quantity changes by 0 and price is unchanged. At the old price of 20, excess demand is (100 - 40) - (20 + 40) = 0.

Worked steps

100 - 2p = 20 + 2p
4p = 100 - 20 = 80
p* = 80 / 4 = 20
Q* = 100 - 2 x 20 = 60
Excess demand at 20: 60 - 60 = 0

Use the idea

Before reading a rise in sales as stronger demand, check whether the price rose or fell with it.

Where the conclusion applies

Linear schedules, a single market and full adjustment to the new crossing. Shifts in other markets that feed back into this one are ignored.

Check your understanding: With a demand shift of +40 and a supply shift of +20, what are price and quantity?

140 - 2p = 40 + 2p gives 4p = 100, so p = 25 and Q = 140 - 50 = 90.

Chapter 5 source: section "Supply-and-demand market clearing".

Demonstration 2 of 4

Ceilings, shortages, and imports

How big is the shortage under a price ceiling, and what do imports change?

A binding ceiling raises the quantity demanded and lowers the quantity supplied. Trade falls to the supply side, and the gap is unmet planned demand, not a cleared market.

Equation, written in LaTeX: Q_D=120-2P,

Equation, written in LaTeX: S=90-50=40

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Demand for grain is 120 - 2P and supply is 20 + 2P in thousands of units. Imports add the same amount at every price. A ceiling binds only when it is below the clearing price.

Predict first. Does a ceiling of 20, without imports, cause a shortage?

Your prediction Yes, of 20 thousand units No, it does not bind

Price ceiling

Imports at every price

Figure: Ceilings, shortages, and imports. Grain market with supply 20 + 2P and a ceiling at 15; demand 90, supply 50, shortage 40. — Price ceiling: 15, Imports at every price: 0
Constructed example: the chapter's hypothetical grain market (ceiling 15, imports 20, illicit price 27 and penalty 8); ceilings of 20 and 10 are added for comparison.

Clearing price P*: 25
Legal price: 15
Quantity demanded: 90
Quantity supplied: 50
Shortage: 40
Ceiling binds: yes

Clearing solves 120 - 2P = 20 + 2P, so P* = 25. At the ceiling of 15, demand is 120 - 2 x 15 = 90 and supply is 20 + 2 x 15 = 50, so the shortage is 90 - 50 = 40 thousand units. An illicit seller paid 27 with penalty costs of 8 nets 27 - 8 = 19, above the legal price of 15.

Worked steps

P* = (120 - 20) / 4 = 25
Legal price = 15 (ceiling binds)
Demand = 120 - 2 x 15 = 90
Supply = 20 + 2 x 15 = 50
Shortage = 40
Illicit net = 27 - 8 = 19

Use the idea

Measure a control's effect by the gap between planned demand and legal supply at the ceiling, not by how much was sold.

Where the conclusion applies

Linear schedules, a ceiling that is enforced in the legal channel and imports that do not depend on price. Rationing, queues and quality changes are not modelled.

Check your understanding: With a ceiling of 10 and imports of 20, what is the shortage?

Demand 120 - 20 = 100; supply 20 + 20 + 20 = 60; shortage 40 thousand units.

Chapter 5 source: section "Price Controls and Black Markets".

Demonstration 3 of 4

Price adjustment that converges, cycles, or explodes

With the same equilibrium, how does the adjustment speed decide whether prices find it?

For this linear excess demand the gap to equilibrium is multiplied by 1 - 2 alpha each round. Below size 1 it shrinks, at -1 it cycles, beyond it grows.

Equation, written in LaTeX: z_1(p)=10-2p, z_2(p)=-p z_1(p).

$Equation, written in LaTeX: p^{t+1}-5=0.5(p^t-5),$

$Equation, written in LaTeX: p^{t+1}-5=-1.2(p^t-5),$

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p is the relative price of good 1 and z1 its excess demand. Each round the quote moves by alpha times excess demand: p(t+1) = p(t) + alpha z1(p(t)).

Predict first. Which adjustment speed reaches the equilibrium in one round?

Your prediction 0.5 0.25 1

Adjustment speed alpha

Starting price

Figure: Price adjustment that converges, cycles, or explodes. Price path over 8 rounds with adjustment speed 0.25 from 2: 3.500, 4.250, 4.625 and so on, against the equilibrium at 5. — Adjustment speed alpha: 0.25, Starting price: 2
Constructed example: the chapter's hypothetical two-good economy (start 2, speeds 0.25, 1 and 1.1); speed 0.5 and start 8 are added for comparison.

p1: 3.500
p2: 4.250
p3: 4.625
Factor 1 - 2 alpha: 0.5
Gap after 8 rounds: -0.012

From p0 = 2, excess demand is 10 - 2 x 2 = 6, so p1 = 2 + 0.25 x 6 = 3.500. Because z is linear, p(t+1) - 5 = (1 - 2 x 0.25)(p(t) - 5) = 0.5(p(t) - 5). The factor 0.5 has size below 1, so the gap to 5 shrinks every round.

Worked steps

p1 = 2.000 + 0.25 x (6.000) = 3.500
p2 = 3.500 + 0.25 x (3.000) = 4.250
p3 = 4.250 + 0.25 x (1.500) = 4.625
Factor = 1 - 2 x 0.25 = 0.5

Use the idea

An equilibrium existing does not mean a given price discovery rule will find it; check the step size against the slope of excess demand.

Where the conclusion applies

A fictitious auctioneer, no trade before clearing and a linear excess demand. Real markets trade at disequilibrium prices and use other rules.

Check your understanding: With alpha = 0.5 from p0 = 2, what is p1?

p1 = 2 + 0.5 x (10 - 4) = 5. The factor 1 - 2 x 0.5 = 0, so it lands on 5 at once.

Chapter 5 source: section "Walrasian tatonnement".

Demonstration 4 of 4

Three equilibria from one aggregate demand

Can a well-behaved economy have several equilibria, and where does price end up?

Aggregate restrictions such as Walras' law and homogeneity allow almost any excess demand shape, so several equilibria are possible. Where price ends depends on where it starts.

Equation, written in LaTeX: z_1(p)=-(p-1)(p-2)(p-3),

Equation, written in LaTeX: pz_1(p)+z_2(p)=0,

Equation, written in LaTeX: z_1(1.5)=-(0.5)(-0.5)(-1.5)=-0.375,

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p is the relative price of good 1, z1 its excess demand and z2 = -p z1 the excess demand for good 2, so Walras' law holds. Price moves in the direction of excess demand: dp/dt = z1(p).

Predict first. Starting at 3.5 with the cubic, where does price end?

Your prediction At 3 At 2 At 1

Starting price

Excess demand

z1 at the start: -0.375
Direction of price: falls
Price settles at: 1.00
Number of equilibria: 3

At p = 1.5, z1 = -(0.5) x (-0.5) x (-1.5) = -0.375, so under dp/dt = z1 the price falls and settles at 1.00. Markets clear at p = 1, 2 and 3; 1 and 3 are stable and 2 is unstable (open dot).

Worked steps

z1(1.5) = -(0.5) x (-0.5) x (-1.5) = -0.375
Sign negative, so price falls
Nearest stable equilibrium in that direction: 1.00

Use the idea

Do not assume a unique, stable equilibrium from individual rationality alone; uniqueness needs extra structure such as gross substitutes.

Where the conclusion applies

An illustrative polynomial, not an estimated demand, and the particular rule dp/dt = z1(p).

Check your understanding: What is z1(3.5) under the cubic, and which way does price move?

z1(3.5) = -(2.5)(1.5)(0.5) = -1.875, negative, so price falls toward 3.

Chapter 5 source: section "Sonnenschein-Mantel-Debreu theorem".

Which curve moved?

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Ceilings, shortages, and imports

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Price adjustment that converges, cycles, or explodes

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Three equilibria from one aggregate demand

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies