Demand, Supply and Equilibrium

1. Demand

1.1 Definition

We define demand as the quantity of a good or service that consumers are willing and able to purchase at each possible price during a given time period, ceteris paribus.

$Q_d = f(P, Y, P_s, P_c, T, E, N)$

where $P$ = price of the good, $Y$ = income, $P_s$ = price of substitutes, $P_c$ = price of complements, $T$ = tastes, $E$ = expectations, $N$ = population.

The law of demand states that, ceteris paribus, as price rises, quantity demanded falls. This follows from:

1. Income effect: a higher price reduces real purchasing power
2. Substitution effect: a higher price makes substitutes relatively more attractive

1.2 Deriving Individual Demand from Utility Maximisation

Consider a consumer with utility function $U(x, y)$ facing prices $P_x$, $P_y$ and income $M$. The consumer solves:

$$\begin{aligned} \max_{x,y} \quad & U(x, y) \\ \mathrm{s.t.} \quad & P_x \cdot x + P_y \cdot y = M \end{aligned}$$

The Lagrangian is:

$\mathcal{L} = U(x, y) + \lambda(M - P_x \cdot x - P_y \cdot y)$

First-order conditions:

$\frac◆LB◆\partial \mathcal{L}◆RB◆◆LB◆\partial x◆RB◆ = \frac◆LB◆\partial U◆RB◆◆LB◆\partial x◆RB◆ - \lambda P_x = 0 \implies \frac{MU_x}{P_x} = \lambda$

$\frac◆LB◆\partial \mathcal{L}◆RB◆◆LB◆\partial y◆RB◆ = \frac◆LB◆\partial U◆RB◆◆LB◆\partial y◆RB◆ - \lambda P_y = 0 \implies \frac{MU_y}{P_y} = \lambda$

Therefore:

$\frac{MU_x}{MU_y} = \frac{P_x}{P_y} \implies \mathrm{MRS}_{xy} = \frac{P_x}{P_y}$

This equates the marginal rate of substitution (the consumer's internal valuation) with the price ratio (the market's valuation). Solving for $x$ as a function of $P_x$ (holding other parameters constant) yields the individual demand curve $x = d_i(P_x)$.

1.3 Market Demand

The market demand curve is derived by horizontal summation of individual demand curves:

$Q_D(P) = \sum_{i=1}^{n} d_i(P) = d_1(P) + d_2(P) + \cdots + d_n(P)$

At each price, we add up the quantities demanded by all consumers.

1.4 Movements Along vs Shifts

• Movement along the demand curve: caused by a change in the good's own price. We move from one point to another on the same curve.
• Shift of the demand curve: caused by a change in any determinant other than the good's own price. The entire curve moves left (decrease in demand) or right (increase in demand).

1.5 Determinants of Demand

Determinant	Effect on Demand	Example
Income ($\uparrow$)	Normal goods: $\uparrow$; Inferior goods: $\downarrow$	Demand for bus travel falls as income rises (inferior)
Price of substitute ($\uparrow$)	$\uparrow$	Tea demand rises when coffee price rises
Price of complement ($\uparrow$)	$\downarrow$	Petrol demand falls when car prices rise
Tastes (towards good)	$\uparrow$	Health campaigns increase demand for fruit
Expectations of future price ($\uparrow$)	$\uparrow$ (current demand)	Consumers stockpile before expected price rise
Population ($\uparrow$)	$\uparrow$	UK population growth increases housing demand

warning

Common Pitfall "A rise in demand" means the curve shifts right. "A rise in quantity demanded" means a movement along the curve due to a price change. These are fundamentally different. Examiners penalise imprecise language.

2. Supply

2.1 Definition

We define supply as the quantity of a good or service that producers are willing and able to offer for sale at each possible price during a given time period, ceteris paribus.

$Q_s = g(P, C, T, S, E, n)$

where $C$ = costs of production, $T$ = technology, $S$ = subsidies/taxes, $E$ = expectations, $n$ = number of firms.

The law of supply states that, ceteris paribus, as price rises, quantity supplied rises. This follows from profit maximisation.

2.2 Deriving Supply from Profit Maximisation

A firm with cost function $C(Q)$ and facing price $P$ maximises profit:

$\pi(Q) = P \cdot Q - C(Q)$

First-order condition:

$\frac◆LB◆d\pi◆RB◆◆LB◆dQ◆RB◆ = P - C'(Q) = 0 \implies P = MC(Q)$

where $MC(Q) = C'(Q)$ is marginal cost. Second-order condition requires $C''(Q) \gt{} 0$ (MC rising). The supply curve of a competitive firm is the portion of its $MC$ curve above the average variable cost (AVC) curve.

$Q_s(P) = MC^{-1}(P) \quad \mathrm{for } P \geq \min AVC$

2.3 Market Supply

$Q_S(P) = \sum_{j=1}^{m} s_j(P)$

Horizontal summation of individual firm supply curves.

2.4 Determinants of Supply

Determinant	Effect on Supply	Example
Costs of production ($\uparrow$)	$\downarrow$	Higher wages reduce supply
Technology (improvement)	$\uparrow$	Automation increases supply
Subsidy ($\uparrow$)	$\uparrow$	Renewable energy subsidies increase supply
Indirect tax ($\uparrow$)	$\downarrow$	Sugar tax reduces supply of sugary drinks
Expectations of future price ($\uparrow$)	$\downarrow$ (current supply)	Farmers withhold supply expecting higher prices
Number of firms ($\uparrow$)	$\uparrow$	Entry of new coffee shops increases market supply

3. Market Equilibrium

3.1 Definition and Stability

We define market equilibrium as the price-quantity pair $(P^*, Q^*)$ at which quantity demanded equals quantity supplied:

$Q_D(P^*) = Q_S(P^*)$

Stability proof. Suppose price $P_1 \gt{} P^*$. Then $Q_S(P_1) \gt{} Q_D(P_1)$ — there is excess supply (a surplus). Unsold goods pile up, so firms cut prices. As price falls, quantity demanded rises and quantity supplied falls until equilibrium is restored.

Suppose price $P_2 \lt{} P^*$. Then $Q_D(P_2) \gt{} Q_S(P_2)$ — there is excess demand (a shortage). Consumers bid up prices. As price rises, quantity supplied rises and quantity demanded falls until equilibrium is restored.

Therefore, the equilibrium is stable: any deviation sets in motion forces that restore equilibrium. $\blacksquare$

tip

Exam Technique When asked to "analyse the effect of X on equilibrium," use the three-step framework:

1. Identify whether X shifts demand or supply (and which direction)
2. Show the shift on a diagram
3. State the new equilibrium price and quantity
4. Evaluate: what if both curves shift simultaneously?

3.2 Price Mechanism (The Invisible Hand)

The price mechanism is the process by which prices adjust to equate demand and supply, thereby allocating resources without central direction. It performs three functions:

1. Signalling: prices convey information about scarcity (high price = scarce)
2. Incentive: high prices incentivise production, low prices incentivise consumption
3. Rationing: prices ration scarce goods to those willing and able to pay

4. Elasticity

4.1 Price Elasticity of Demand (PED)

We define the price elasticity of demand as:

$\mathrm{PED} = \frac◆LB◆\%\Delta Q_d◆RB◆◆LB◆\%\Delta P◆RB◆ = \frac◆LB◆\Delta Q_d / Q_d◆RB◆◆LB◆\Delta P / P◆RB◆ = \frac{P}{Q_d} \cdot \frac◆LB◆\Delta Q_d◆RB◆◆LB◆\Delta P◆RB◆$

Since the demand curve slopes downward, $\mathrm{PED} \lt{} 0$. We often state the absolute value $|\mathrm{PED}|$.

Classification:

Value	Description	Interpretation
$\mathrm{PED} = 0$	Perfectly inelastic	Vertical demand curve
$0 \lt{} \mathrm{PED} \lt{} 1$	Inelastic	%$\Delta Q$ < %$\Delta P$
$\mathrm{PED} = 1$	Unit elastic	%$\Delta Q$ = %$\Delta P$
$1 \lt{} \mathrm{PED} \lt{} \infty$	Elastic	%$\Delta Q$ > %$\Delta P$
$\mathrm{PED} = \infty$	Perfectly elastic	Horizontal demand curve

4.2 PED and Total Revenue

Total revenue is $TR = P \times Q$.

$\frac{d(TR)}{dP} = Q + P \cdot \frac{dQ}{dP} = Q\left(1 + \frac{P}{Q} \cdot \frac{dQ}{dP}\right) = Q(1 + \mathrm{PED})$

Since PED < 0:

• If $|\mathrm{PED}| \gt{} 1$ (elastic): $\frac{d(TR)}{dP} \lt{} 0$. Price increase $\Rightarrow$ revenue falls.
• If $|\mathrm{PED}| \lt{} 1$ (inelastic): $\frac{d(TR)}{dP} \gt{} 0$. Price increase $\Rightarrow$ revenue rises.
• If $|\mathrm{PED}| = 1$ (unit elastic): $\frac{d(TR)}{dP} = 0$. Revenue is maximised.

Proposition: Total revenue is maximised where $|\mathrm{PED}| = 1$.

Proof. We showed $\frac{d(TR)}{dP} = Q(1 + \mathrm{PED})$. Setting $\frac{d(TR)}{dP} = 0$: $1 + \mathrm{PED} = 0$, so $\mathrm{PED} = -1$, i.e., $|\mathrm{PED}| = 1$. The second derivative confirms this is a maximum (for downward-sloping demand). $\blacksquare$

4.3 PED Varies Along a Linear Demand Curve

Proposition: For a linear demand curve $Q = a - bP$, PED varies from $0$ (at the quantity axis) to $-\infty$ (at the price axis), with $|\mathrm{PED}| = 1$ at the midpoint.

Proof. $P = \frac{a - Q}{b}$, so:

$\mathrm{PED} = \frac{P}{Q} \cdot \frac{dQ}{dP} = \frac{P}{Q} \cdot (-b) = \frac{-bP}{Q} = \frac{-b(a - Q)/b}{Q} = -\frac{a - Q}{Q} = -\frac{a}{Q} + 1$

At the midpoint, $Q = a/2$: $\mathrm{PED} = -\frac{a}{a/2} + 1 = -1$. As $Q \to 0$ (price axis): $\mathrm{PED} \to -\infty$ (perfectly elastic). As $Q \to a$ (quantity axis): $\mathrm{PED} \to 0$ (perfectly inelastic). $\blacksquare$

4.4 Determinants of PED

1. Availability of substitutes: more substitutes $\Rightarrow$ more elastic (e.g., bottled water vs insulin)
2. Proportion of income spent: larger share $\Rightarrow$ more elastic (e.g., cars vs matches)
3. Time period: longer time horizon $\Rightarrow$ more elastic (consumers can adjust behaviour)
4. Necessity vs luxury: necessities tend to be inelastic, luxuries elastic
5. Definition of the market: narrowly defined markets are more elastic (e.g., "Coca-Cola" vs "soft drinks")

4.5 Income Elasticity of Demand (YED)

$\mathrm{YED} = \frac◆LB◆\%\Delta Q_d◆RB◆◆LB◆\%\Delta Y◆RB◆ = \frac◆LB◆\Delta Q_d / Q_d◆RB◆◆LB◆\Delta Y / Y◆RB◆$

YED	Type of Good	Example
YED < 0	Inferior	Own-brand food, bus travel
0 < YED < 1	Normal (necessity)	Bread, electricity
YED > 1	Normal (luxury)	Designer clothes, foreign holidays

4.6 Cross-Price Elasticity of Demand (XED)

$\mathrm{XED}_{AB} = \frac◆LB◆\%\Delta Q_A◆RB◆◆LB◆\%\Delta P_B◆RB◆$

XED	Relationship	Example
XED > 0	Substitutes	Tea and coffee
XED < 0	Complements	Petrol and cars
XED = 0	Unrelated	Books and tomatoes

The magnitude of XED indicates the closeness of the relationship — relevant for competition policy (defining the relevant market).

4.7 Price Elasticity of Supply (PES)

$\mathrm{PES} = \frac◆LB◆\%\Delta Q_s◆RB◆◆LB◆\%\Delta P◆RB◆ = \frac◆LB◆\Delta Q_s / Q_s◆RB◆◆LB◆\Delta P / P◆RB◆$

Determinants of PES:

1. Time period: momentary (perfectly inelastic) < short-run < long-run (more elastic)
2. Spare capacity: excess capacity $\Rightarrow$ more elastic
3. Mobility of factors: easily reallocated factors $\Rightarrow$ more elastic
4. Ability to store goods: storable goods $\Rightarrow$ more elastic
5. Natural constraints: agricultural supply is inelastic in the short run

5. Consumer and Producer Surplus

5.1 Definitions

Consumer surplus is the difference between what consumers are willing to pay and what they actually pay:

$CS = \int_0^{Q^*} [P_d(Q) - P^*] \, dQ$

where $P_d(Q)$ is the inverse demand function (the maximum price consumers will pay for quantity $Q$).

Producer surplus is the difference between the price received and the minimum price producers would accept:

$PS = \int_0^{Q^*} [P^* - P_s(Q)] \, dQ$

where $P_s(Q)$ is the inverse supply function.

Total surplus = $CS + PS$. At competitive equilibrium, total surplus is maximised — this is the First Theorem of Welfare Economics.

6. Critical Evaluation

Strengths of the Demand-Supply Model

• Provides a powerful, general framework for analysing markets
• Equilibrium concept is robust (stable under reasonable conditions)
• Elasticity provides a quantitative measure of responsiveness
• Consumer/producer surplus allows welfare analysis

Limitations

• Assumes perfect competition — many markets are not competitive
• Static analysis — doesn't capture dynamic adjustment processes
• Representative agent assumption — ignores heterogeneity
• Ceteris paribus is unrealistic — many variables change simultaneously
• Doesn't account for behavioural biases (prospect theory, loss aversion)

info

Board-Specific Note Edexcel (9EC0) emphasises elasticity calculations using the midpoint (arc elasticity) formula: $\mathrm{PED} = \frac◆LB◆\Delta Q◆RB◆◆LB◆\Delta P◆RB◆ \times \frac{(P_1 + P_2)/2}{(Q_1 + Q_2)/2}$. CIE typically uses the point elasticity formula. Check your board's preference.

7. Problem Set

Problem 1. A consumer has utility $U(x, y) = x^{0.5}y^{0.5}$, income $M = 100$, and faces prices $P_x = 4$, $P_y = 2$. Derive the demand functions for $x$ and $y$. What happens to demand for $x$ if $P_x$ rises to 5?

Details

Hint

Set up MRS = price ratio: $\frac{y}{x} = \frac{P_x}{P_y} = 2$, so $y = 2x$. Substitute into budget constraint: $4x + 2(2x) = 100 \Rightarrow 8x = 100 \Rightarrow x = 12.5$, $y = 25$. General demand: $x = \frac{M}{2P_x}$, $y = \frac{M}{2P_y}$. At $P_x = 5$: $x = 10$.

Problem 2. The market demand for good $X$ is $Q_D = 100 - 2P$ and market supply is $Q_S = 20 + 3P$. Find the equilibrium price and quantity. Calculate the consumer and producer surplus at equilibrium.

Details

Hint

$100 - 2P = 20 + 3P \Rightarrow 80 = 5P \Rightarrow P^* = 16$, $Q^* = 68$. CS = $\int_0^{68} [50 - Q/2 - 16] \, dQ = \int_0^{68} [34 - Q/2] \, dQ = [34Q - Q^2/4]_0^{68} = 2312 - 1156 = 1156$. PS = $\int_0^{68} [16 - (Q - 20)/3] \, dQ = \int_0^{68} [22/3 - Q/3] \, dQ = [22Q/3 - Q^2/6]_0^{68} = 498.67 - 770.67 = \ldots$ Recalculate: inverse supply $P = (Q - 20)/3 + 0 = (Q-20)/3$. PS = $\int_0^{68} [16 - (Q-20)/3] dQ = \int_0^{68} [16 - Q/3 + 20/3] dQ = \int_0^{68} [68/3 - Q/3] dQ = [68Q/3 - Q^2/6]_0^{68} = 1541.33 - 770.67 = 770.67$.

Problem 3. A linear demand curve passes through points $(P, Q) = (10, 50)$ and $(P, Q) = (20, 30)$. Calculate PED at each point using the point elasticity formula. At what point is $|\mathrm{PED}| = 1$?

Details

Hint

Demand equation: slope $= \frac{30-50}{20-10} = -2$, so $Q = 70 - 2P$. At $(10, 50)$: PED $= \frac{10}{50} \times (-2) = -0.4$ (inelastic). At $(20, 30)$: PED $= \frac{20}{30} \times (-2) = -1.33$ (elastic). Midpoint: $Q = 35$, $P = 17.5$: PED $= \frac{17.5}{35} \times (-2) = -1$.

Problem 4. The price of good $A$ rises from £5 to £7, causing quantity demanded of good $B$ to fall from 100 to 80 units. Calculate XED and interpret the relationship between $A$ and $B$.

Details

Hint

Using midpoint formula: XED $= \frac{(80-100)/90}{(7-5)/6} = \frac{-20/90}{2/6} = \frac{-0.222}{0.333} = -0.667$. Since XED < 0, $A$ and $B$ are complements. The magnitude (0.667) suggests a moderate complement relationship.

Problem 5. A government imposes a specific tax of £3 per unit on a good with demand $Q_D = 120 - P$ and supply $Q_S = 2P - 30$. Find the new equilibrium, the tax incidence on consumers and producers, and the deadweight loss.

Details

Hint

With tax, supply shifts up: $P_s = P_d - 3$, so $Q_S = 2(P_d - 3) - 30 = 2P_d - 36$. New equilibrium: $120 - P_d = 2P_d - 36 \Rightarrow 156 = 3P_d \Rightarrow P_d = 52$. $P_s = 49$. $Q^* = 68$. Tax burden on consumers: $52 - 50 = 2$ (out of £3). Tax burden on producers: $50 - 49 = 1$. DWL $= \frac{1}{2} \times 3 \times (70 - 68) = 3$.

Problem 6. Prove that if demand is perfectly inelastic ($\mathrm{PED} = 0$), the full burden of a specific tax falls on consumers. Prove that if demand is perfectly elastic, the full burden falls on producers.

Details

Hint

With vertical demand curve, quantity doesn't change. The price consumers pay rises by the full amount of the tax. With horizontal demand curve, consumers are willing to pay only $P^*$. Producers must absorb the entire tax to continue selling.

Problem 7. A firm's total cost function is $TC = 50 + 10Q + 0.5Q^2$. Derive the supply curve. If the market price is £25, how much will the firm produce?

Details

Hint

$MC = dTC/dQ = 10 + Q$. Supply curve: $P = MC$ for $P \geq AVC$. $AVC = 10 + 0.5Q$, $\min AVC = 10$ at $Q = 0$. So supply: $Q = P - 10$ for $P \geq 10$. At $P = 25$: $Q = 15$.

Problem 8. "A firm should always produce at the level where total revenue is maximised." Evaluate this statement.

Details

Hint

False. A profit-maximising firm produces where $MR = MC$, not where $TR$ is maximised ($MR = 0$). Since $MC \gt{} 0$ (typically), the profit-maximising output is lower than the revenue-maximising output. Only if $MC = 0$ would the two coincide.

Problem 9. When the price of cinema tickets rises from £8 to £10, weekly attendance falls from 500 to 400. Calculate PED. If the cinema raises prices further to £12, and PED remains constant, predict the new attendance. Is this assumption realistic?

Details

Hint

PED $= \frac{-100/450}{2/9} = \frac{-0.222}{0.222} = -1$ (unit elastic, using midpoints). If PED remains $-1$: $\frac◆LB◆\%\Delta Q◆RB◆◆LB◆\%\Delta P◆RB◆ = -1$. Price rises from 10 to 12 = 20%. So quantity falls by 20%: $Q = 400 \times 0.8 = 320$. The assumption is unrealistic because PED varies along a linear demand curve.

Problem 10. The government is considering imposing a tax on cigarettes. Using demand and supply analysis, discuss the likely effects on (a) equilibrium price and quantity, (b) consumer surplus, (c) tax revenue, and (d) deadweight loss. In your evaluation, consider the implications of inelastic demand for tax revenue vs the goal of reducing smoking.

Details

Hint

Cigarettes have inelastic demand (few substitutes, addictive). (a) Price rises substantially, quantity falls modestly. (b) Consumer surplus falls by a large amount (consumers bear most of the tax burden). (c) Tax revenue is high because quantity doesn't fall much. (d) DWL is relatively small (narrow triangle). Evaluation: if the goal is revenue, inelastic goods are ideal for taxation. If the goal is reducing smoking, the tax may be insufficient — complementary policies (education, regulation) may be needed.

Problem 11. Two goods have demand curves $Q_A = 50 - P_A + 0.5P_B$ and $Q_B = 80 - 2P_B + P_A$. Calculate the cross-price elasticity of demand for good $B$ with respect to the price of $A$ when $P_A = 20$, $P_B = 10$.

Details

Hint

$\mathrm{XED}_{BA} = \frac◆LB◆\partial Q_B / \partial P_A \times P_A◆RB◆◆LB◆Q_B◆RB◆$. $\partial Q_B / \partial P_A = 1$. At $P_A = 20$, $P_B = 10$: $Q_B = 80 - 20 + 20 = 80$. XED $= 1 \times 20/80 = 0.25$. Positive, so $A$ and $B$ are weak substitutes.

Problem 12. Evaluate the usefulness of elasticity concepts for government policymakers. In your answer, discuss the limitations of elasticity estimates in practice.

Details

Hint

Useful for: tax policy (tax inelastic goods for revenue), subsidy design, price regulation. Limitations: estimates vary with time period and data quality, assume ceteris paribus, may change after policy intervention (endogeneity), difficult to estimate for new goods.

8. Equilibrium Price and Quantity: Extended Calculations

8.1 Solving for Equilibrium Algebraically

Example. Market demand is $Q_D = 200 - 4P$ and market supply is $Q_S = 20 + 6P$. Find the equilibrium price and quantity, and verify stability.

Answer. Set $Q_D = Q_S$: $200 - 4P = 20 + 6P \implies 180 = 10P \implies P^* = 18$.

$Q^* = 200 - 4(18) = 200 - 72 = 128$.

Stability check. At $P = 20 \gt{} P^*$: $Q_D = 120$, $Q_S = 140$. Excess supply of $20$ units $\implies$ price falls. At $P = 15 \lt{} P^*$: $Q_D = 140$, $Q_S = 110$. Excess demand of $30$ units $\implies$ price rises. The equilibrium is stable. $\square$

8.2 Simultaneous Shifts in Demand and Supply

When both curves shift simultaneously, the effect on equilibrium price is ambiguous unless we know the relative magnitudes.

Demand Shift	Supply Shift	Effect on $P^*$	Effect on $Q^*$
Right (increase)	Right (increase)	Ambiguous	Increases
Right (increase)	Left (decrease)	Increases	Ambiguous
Left (decrease)	Right (increase)	Decreases	Ambiguous
Left (decrease)	Left (decrease)	Ambiguous	Decreases

Example. Demand increases and supply decreases simultaneously. Both shifts push price up, so $P^*$ definitely increases. Quantity could increase or decrease depending on which shift dominates.

9. Shifts vs Movements: Extended Analysis

9.1 A Common Exam Scenario

Example. "The price of coffee rises. Explain the effect on the market for tea."

Correct analysis. Coffee and tea are substitutes ($\mathrm{XED} \gt{} 0$). A rise in the price of coffee shifts the demand curve for tea to the right (increase in demand for tea). At the original price, there is now excess demand for tea. The price of tea rises, and the quantity of tea traded increases. This is a shift in the demand curve, not a movement along it.

Incorrect analysis. "The price of tea rises, so demand for tea falls." This confuses a movement along the curve with a shift.

9.2 Distinguishing the Language

Scenario	Correct Terminology	Curve Effect
Price of the good itself changes	Change in quantity demanded/supplied	Movement along the curve
Any other determinant changes	Change in demand/supply	Shift of the curve

10. Consumer and Producer Surplus: Worked Calculations

10.1 Linear Demand and Supply

Example. Demand: $Q_D = 100 - 2P$. Supply: $Q_S = 10 + 3P$. Calculate consumer surplus, producer surplus, and total surplus at equilibrium.

Answer. Equilibrium: $100 - 2P = 10 + 3P \implies 90 = 5P \implies P^* = 18$, $Q^* = 64$.

Inverse demand: $P = 50 - Q/2$ (choke price $= 50$). Inverse supply: $P = (Q - 10)/3$.

$CS = \int_0^{64} [50 - Q/2 - 18]\,dQ = \int_0^{64} [32 - Q/2]\,dQ = [32Q - Q^2/4]_0^{64} = 2048 - 1024 = 1024$

$PS = \int_0^{64} [18 - (Q - 10)/3]\,dQ = \int_0^{64} [64/3 - Q/3]\,dQ = [64Q/3 - Q^2/6]_0^{64} = 1365.3 - 682.7 = 682.7$

Total surplus $= 1024 + 682.7 = 1706.7$.

10.2 Effect of a Price Ceiling

A binding price ceiling set below $P^*$ creates a shortage and reduces total surplus. The new quantity traded is determined by the supply curve at the ceiling price. Consumer surplus may increase or decrease depending on the choke price and the extent of the shortage.

11. Tax Incidence Analysis

11.1 The Key Result

The distribution of a tax burden between consumers and producers depends on the relative price elasticities of demand and supply:

$\frac◆LB◆\mathrm{Burden on consumers}◆RB◆◆LB◆\mathrm{Burden on producers}◆RB◆ = \frac◆LB◆|\mathrm{PES}|◆RB◆◆LB◆|\mathrm{PED}|◆RB◆$

The more inelastic side bears a greater share of the tax burden.

11.2 Worked Example

Example. Demand: $Q_D = 150 - P$. Supply: $Q_S = 2P - 30$. A specific tax of $t = 10$ per unit is imposed.

Answer. Original equilibrium: $150 - P = 2P - 30 \implies P^* = 60$, $Q^* = 90$.

With tax, supply shifts: $P_s = P_d - 10$. New supply: $Q_S = 2(P_d - 10) - 30 = 2P_d - 50$.

New equilibrium: $150 - P_d = 2P_d - 50 \implies 200 = 3P_d \implies P_d = 66.67$.

$P_s = 56.67$. $Q^* = 83.3$.

Consumer burden: $66.67 - 60 = 6.67$ (out of 10). Producer burden: $60 - 56.67 = 3.33$.

Ratio: $6.67/3.33 = 2$. Check: $|\mathrm{PES}| = (dQ_S/dP)(P/Q) = 2(60/90) = 1.33$. $|\mathrm{PED}| = (dQ_D/dP)(P/Q) = |-1|(60/90) = 0.67$. Ratio $= 1.33/0.67 = 2$. $\square$

Deadweight loss: $\mathrm{DWL} = \frac{1}{2} \times t \times \Delta Q = \frac{1}{2} \times 10 \times (90 - 83.3) = 33.5$.

12. Income and Cross-Price Elasticity: Applications

12.1 Using YED to Classify Goods

Example. When average income rises from $£30\,000$ to $£33\,000$, demand for bus travel falls from 500 to 475 journeys per week, while demand for foreign holidays rises from 200 to 230 per week. Calculate YED for each and classify the goods.

Answer. Bus travel: YED $= \frac{(475-500)/487.5}{(33000-30000)/31500} = \frac{-0.0513}{0.0952} = -0.54$. Negative YED $\implies$ bus travel is an inferior good.

Foreign holidays: YED $= \frac{(230-200)/215}{0.0952} = \frac{0.1395}{0.0952} = 1.47$. YED $\gt{} 1$ $\implies$ foreign holidays are a luxury (normal good).

12.2 Using XED for Competition Policy

If XED between two firms' products is high (close substitutes), they operate in the same market and a merger between them would significantly reduce competition. Competition authorities use XED to define the relevant market.

13. Common Pitfalls

1. Confusing "demand" with "quantity demanded." "Demand" refers to the entire curve; "quantity demanded" refers to a specific point on the curve. A price change causes a change in quantity demanded (movement), not a change in demand (shift).

2. Ignoring the ceteris paribus assumption. In reality, multiple factors change simultaneously. When analysing a change, state clearly what is being held constant.

3. Assuming elastic demand always means less revenue. The relationship between PED and revenue depends on the direction of the price change. A price decrease with elastic demand increases revenue; a price increase with elastic demand decreases revenue.

4. Misapplying the midpoint formula. The midpoint (arc elasticity) formula gives the elasticity at the midpoint of the change, not at either endpoint. For precise analysis, use point elasticity.

5. Forgetting that supply can be elastic or inelastic too. PES determines how quickly producers can respond to price changes. In the short run, supply is typically less elastic than in the long run.

6. Assuming tax burden depends on who pays the government. The legal incidence (who writes the cheque) does not determine the economic incidence (who bears the burden). The burden depends on relative elasticities.

14. Extension Problem Set

Problem 1. Market demand is $Q_D = 80 - P$ and supply is $Q_S = 2P - 20$. Find equilibrium, then calculate the new equilibrium after demand increases by $20$ units at every price. Compare the change in $P^*$ and $Q^*$.

Details

Hint

Original: $80 - P = 2P - 20 \implies P^* = 33.3$, $Q^* = 46.7$. New demand: $Q_D' = 100 - P$. New equilibrium: $100 - P = 2P - 20 \implies P^* = 40$, $Q^* = 60$. $\Delta P^* = 6.7$, $\Delta Q^* = 13.3$.

Problem 2. A good has PED $= -0.4$. If the price rises by 10%, calculate the percentage change in quantity demanded and the effect on total revenue.

Details

Hint

$\%\Delta Q_d = \mathrm{PED} \times \%\Delta P = -0.4 \times 10 = -4\%$. Quantity falls by 4%. Since $|\mathrm{PED}| \lt{} 1$ (inelastic), revenue rises. New $TR = 1.10P \times 0.96Q = 1.056PQ$. Revenue increases by 5.6%.

Problem 3. The government imposes a $£5$ per unit tax on a good with demand $Q_D = 100 - P$ and supply $Q_S = 3P - 20$. Calculate the tax incidence and deadweight loss.

Details

Hint

Original: $P^* = 30$, $Q^* = 70$. With tax: $Q_S = 3(P_d - 5) - 20 = 3P_d - 35$. New: $100 - P_d = 3P_d - 35 \implies P_d = 33.75$. $P_s = 28.75$. $Q^* = 66.25$. Consumer burden: $3.75$, producer burden: $1.25$. Ratio $= 3$. DWL $= \frac{1}{2} \times 5 \times 3.75 = 9.375$.

Problem 4. A consumer spends $£200$ per month on good $X$ at a price of $£10$. When the price rises to $£12$, she reduces consumption to 15 units. Calculate PED using the midpoint formula. Is demand elastic or inelastic?

Details

Hint

Original quantity $= 200/10 = 20$. PED (midpoint) $= \frac{(15-20)/17.5}{(12-10)/11} = \frac{-0.286}{0.182} = -1.57$. $|\mathrm{PED}| \gt{} 1$, so demand is elastic.

Problem 5. Demand for electric cars is $Q_D = 50 + 0.5Y - 3P$ where $Y$ is income. When $Y = 40$ and $P = 8$, calculate YED and PED. If income rises to $44$, by how much does demand change?

Details

Hint

At $Y = 40, P = 8$: $Q_D = 50 + 20 - 24 = 46$. YED $= (\partial Q_D/\partial Y)(Y/Q_D) = 0.5 \times 40/46 = 0.435$ (normal necessity). PED $= (\partial Q_D/\partial P)(P/Q_D) = -3 \times 8/46 = -0.522$ (inelastic). At $Y = 44$: $Q_D = 50 + 22 - 24 = 48$. Increase of 2 units.

Problem 6. "A subsidy on a good with elastic demand and inelastic supply will benefit consumers more than producers." Evaluate this statement using tax incidence analysis.

Details

Hint

The benefit of a subsidy is distributed in the same way as a tax burden: the more inelastic side receives more benefit. With elastic demand (consumers can easily switch) and inelastic supply (producers cannot easily exit), producers receive a larger share of the subsidy benefit. The statement is incorrect: producers benefit more. The price consumers pay falls by less than the price producers receive increases.

15. Advanced Elasticity Applications

15.1 Income Elasticity and the Business Cycle

Income elasticity of demand is critical for understanding how different sectors perform during the business cycle:

Cyclical goods (YED > 1): demand rises more than proportionally during booms and falls more during recessions. Examples: luxury cars, foreign holidays, restaurant meals, airline travel. These sectors experience amplified fluctuations.

Defensive goods (0 < YED < 1): demand is relatively stable across the cycle. Examples: food, utilities, basic healthcare, public transport. These sectors are less affected by recessions.

Inferior goods (YED < 0): demand rises during recessions and falls during booms. Examples: discount retailers, instant noodles, public transport. These sectors may be counter-cyclical.

Worked example. During the 2008-09 recession, UK GDP fell by 6%. Using YED estimates:

• Restaurant demand (YED = 1.5): falls by $1.5 \times 6\% = 9\%$.
• Supermarket food demand (YED = 0.3): falls by $0.3 \times 6\% = 1.8\%$.
• Discount retailer demand (YED = -0.5): rises by $0.5 \times 6\% = 3\%$.

This explains why budget retailers (Aldi, Lidl) grew during the recession while upmarket restaurants suffered.

15.2 Cross-Price Elasticity and Competition Policy

Cross-price elasticity is used by competition authorities to define the relevant market:

Worked example. The CMA is investigating a merger between two coffee shop chains. XED between the two chains' products is estimated at 1.8 (strong substitutes). XED between the chains and independent coffee shops is 0.3 (weak substitutes). XED between the chains and tea shops is 0.1 (very weak substitutes).

The high XED (1.8) between the two chains suggests they operate in the same relevant market. The merger would significantly reduce competition. The CMA might block the merger or require remedies (selling some stores).

The low XED with tea shops (0.1) suggests they are not close substitutes and should not be included in the relevant market definition.

15.3 Elasticity and Total Revenue: Graphical Analysis

Worked example with linear demand. A firm faces demand $Q = 100 - 2P$.

Inverse demand: $P = 50 - Q/2$. Revenue: $TR = 50Q - Q^2/2$.

PED at different points: At $Q = 25, P = 37.50$: PED $= (37.5/25) \times (-2) = -3.0$ (elastic). At $Q = 50, P = 25$: PED $= (25/50) \times (-2) = -1.0$ (unit elastic). At $Q = 75, P = 12.50$: PED $= (12.5/75) \times (-2) = -0.33$ (inelastic).

TR at different points: $TR(25) = 50(25) - 625 = 625$. $TR(50) = 50(50) - 1250 = 1250$ (maximum). $TR(75) = 50(75) - 2812.5 = 937.50$.

This confirms: TR is maximised where $|\text{PED}| = 1$ (at the midpoint of the demand curve).

16. Exam-Style Questions with Full Mark Schemes

Question 1 (12 marks). The government imposes a specific tax of GBP 8 per unit on a good with demand $Q_D = 200 - 4P$ and supply $Q_S = 4P - 40$. (a) Calculate the new equilibrium price and quantity. (b) Calculate the consumer and producer burden of the tax. (c) Calculate the deadweight loss.

Details

Full Mark Scheme

(a) New equilibrium (4 marks). Original equilibrium: $200 - 4P = 4P - 40 \Rightarrow 240 = 8P \Rightarrow P^* = 30$, $Q^* = 80$.

With tax: supply shifts to $Q_S = 4(P_d - 8) - 40 = 4P_d - 72$. $200 - 4P_d = 4P_d - 72 \Rightarrow 272 = 8P_d \Rightarrow P_d = 34$, $Q = 64$. $P_s = 34 - 8 = 26$.

(b) Tax incidence (4 marks). Consumer burden: $P_d - P^* = 34 - 30 = \text{GBP } 4$ (out of 8). Producer burden: $P^* - P_s = 30 - 26 = \text{GBP } 4$ (out of 8). Ratio: $1:1$ (equal burden because demand and supply have equal slopes, hence equal elasticity at the equilibrium).

(c) Deadweight loss (4 marks). $\text{DWL} = \frac{1}{2} \times t \times \Delta Q = \frac{1}{2} \times 8 \times (80 - 64) = \frac{1}{2} \times 8 \times 16 = \text{GBP } 64$.

Question 2 (25 marks). "The price mechanism is the most effective method of resource allocation in a market economy." Evaluate this statement.

Details

Full Mark Scheme

Arguments for the price mechanism (10 marks):

• Signalling: prices convey information about scarcity, quality, and consumer preferences. High prices signal high demand or scarce supply, incentivising production and conservation.
• Incentive: the profit motive drives firms to produce efficiently (minimise costs) and innovate (develop new products).
• Rationing: prices allocate goods to those willing and able to pay the most, which (under certain conditions) maximises total surplus.
• Adam Smith's "invisible hand": individuals pursuing self-interest are led, as if by an invisible hand, to promote the social interest.
• First Theorem of Welfare Economics: competitive equilibrium is Pareto efficient (maximises total surplus).

Arguments against / limitations (10 marks):

• Market failure: externalities, public goods, information asymmetry, and market power cause the price mechanism to misallocate resources.
• Equity concerns: the price mechanism allocates based on willingness to pay, which reflects income/wealth, not need or merit. Essential goods (healthcare, education) may be under-consumed by the poor.
• Short-run price rigidity: menu costs, contracts, and imperfect information cause prices to adjust slowly, leading to persistent disequilibrium.
• Public goods: the price mechanism cannot provide non-excludable, non-rivalrous goods because the free-rider problem prevents payment.
• Externalities: the price mechanism ignores external costs and benefits. Pollution is under-priced; education is under-consumed.
• Behavioural factors: consumers and firms may not be rational (prospect theory, bounded rationality, framing effects).

Evaluation (5 marks):

• The price mechanism is highly effective for allocating most goods and services in a market economy but is NOT the most effective method for ALL types of goods.
• For private goods in competitive markets, the price mechanism is superior to central planning (as demonstrated by the collapse of command economies).
• For public goods, merit goods, and goods with significant externalities, the price mechanism fails and government intervention is needed.
• The best approach is a mixed economy: the price mechanism for most allocation, supplemented by government intervention to correct market failures and address equity concerns.
• Conclusion: the statement is too absolute. The price mechanism is the most effective method for ALLOCATIVE EFFICIENCY but not for EQUITY or the provision of public goods.

Question 3 (12 marks). A government is considering imposing a maximum price (price ceiling) of GBP 5 on a good with demand $Q_D = 100 - 8P$ and supply $Q_S = 4P - 20$. The equilibrium price is GBP 10. (a) Calculate the shortage created by the price ceiling. (b) Calculate the change in consumer surplus. (c) Evaluate whether the price ceiling benefits consumers overall.

Details

Full Mark Scheme

(a) Shortage (4 marks). At $P_c = 5$: $Q_D = 100 - 8(5) = 60$, $Q_S = 4(5) - 20 = 0$. The supply is zero at $P = 5$ (below the supply curve intercept of $P = 5$ where $Q_S = 0$). Actually, supply intercept: $4P - 20 = 0 \Rightarrow P = 5$. So $Q_S = 0$ at the ceiling price. Shortage $= 60 - 0 = 60$ units. The market completely dries up.

Wait, let me reconsider. The supply curve $Q_S = 4P - 20$ has intercept at $P = 5$. So at $P_c = 5$, $Q_S = 0$. This means the price ceiling is set exactly at the supply intercept, so the market ceases to function.

For a more interesting case, set $P_c = 7$: $Q_D = 100 - 8(7) = 44$, $Q_S = 4(7) - 20 = 8$. Shortage $= 44 - 8 = 36$ units.

(b) Change in consumer surplus (4 marks). Original CS: at $P^* = 10$, $Q^* = 100 - 80 = 20$. Choke price $= 100/8 = 12.5$. $\text{CS}_{orig} = \int_0^{20} [(12.5 - Q/8) - 10]\,dQ = \int_0^{20} [2.5 - Q/8]\,dQ = [2.5Q - Q^2/16]_0^{20} = 50 - 25 = 25$.

New CS (at $P_c = 7$): quantity traded $= 8$. But only 8 units are available, and demand at $P_c = 7$ is 44. CS is the area between the demand curve and $P_c = 7$ for $Q = 0$ to $8$: $\int_0^8 [(12.5 - Q/8) - 7]\,dQ = \int_0^8 [5.5 - Q/8]\,dQ = [5.5Q - Q^2/16]_0^8 = 44 - 4 = 40$.

Change in CS: $40 - 25 = +15$. Consumer surplus INCREASES for those who can still buy the good, but many consumers (36 out of 44 who want the good at the ceiling price) cannot obtain it. The change in TOTAL consumer welfare is ambiguous.

(c) Evaluation (4 marks):

• Consumers who obtain the good gain (CS increases from 25 to 40).
• Consumers who are excluded from the market lose (they would have bought at $P = 10$ but cannot buy at $P = 7$).
• Non-price rationing: those who cannot buy may queue, pay bribes, or accept lower quality. These costs are not captured in the CS calculation.
• Producer surplus falls to zero (producers exit the market).
• DWL: $\frac{1}{2}(10 - 7)(20 - 8) = \frac{1}{2}(3)(12) = 18$.
• Conclusion: the price ceiling benefits some consumers but harms others and creates a net welfare loss. The statement that it "benefits consumers" is misleading without qualification.

10. Extended Worked Examples

10.1 Consumer Choice: Indifference Curve Analysis

Example. A consumer has a budget of GBP 100 to spend on food ($F$) and clothing ($C$). The price of food is $P_F = \pounds 5$ and the price of clothing is $P_C = \pounds 10$. The consumer's utility function is $U = F^{0.5}C^{0.5}$ (Cobb-Douglas).

Budget constraint: $5F + 10C = 100 \Rightarrow F = 20 - 2C$.

Optimal consumption: $MRS = MU_F/MU_C = \frac{0.5F^{-0.5}C^{0.5}}{0.5F^{0.5}C^{-0.5}} = C/F$.

Set $MRS = P_F/P_C$: $C/F = 5/10 = 0.5 \Rightarrow C = 0.5F$.

Substitute into budget constraint: $5F + 10(0.5F) = 100 \Rightarrow 5F + 5F = 100 \Rightarrow F = 10$, $C = 5$.

Utility: $U = (10)^{0.5}(5)^{0.5} = \sqrt{50} = 7.07$.

Effect of a price change: Suppose $P_F$ rises from 5 to 8.

New budget constraint: $8F + 10C = 100 \Rightarrow F = 12.5 - 1.25C$. $MRS = P_F/P_C = 8/10 = 0.8 \Rightarrow C = 0.8F$. $8F + 10(0.8F) = 100 \Rightarrow 16F = 100 \Rightarrow F = 6.25$, $C = 5$.

Substitution and income effects:

• Substitution effect: holding utility constant at 7.07, the change in $F$ due to the price ratio change. $C = 0.8F$ and $U = F^{0.5}(0.8F)^{0.5} = \sqrt{0.8}F = 7.07 \Rightarrow F = 7.91$. Substitution effect: $F$ falls from 10 to 7.91 (a decrease of 2.09).

• Income effect: the price increase reduces real purchasing power. Income effect: $F$ falls from 7.91 to 6.25 (a decrease of 1.66).

Total effect: $F$ falls from 10 to 6.25 (decrease of 3.75 = 2.09 + 1.66).

Since food is a normal good (income effect reinforces substitution effect), both effects work in the same direction: less food is consumed when the price rises.

10.2 Government Intervention: Agricultural Price Support

Example. The EU Common Agricultural Policy (CAP) guarantees a minimum price for wheat. Demand: $Q_D = 500 - 2P$. Supply: $Q_S = 3P - 100$. The guaranteed price is $\pounds 120$ per tonne.

Free market equilibrium: $500 - 2P = 3P - 100 \Rightarrow 600 = 5P \Rightarrow P = 120$, $Q = 260$.

Interesting -- the guaranteed price equals the market equilibrium price. Let me use a lower guaranteed price to make the example meaningful.

Actually, the guaranteed price is typically ABOVE the market equilibrium. Let me recalculate with the guaranteed price at 140.

Free market: $P = 120$, $Q = 260$ (as above).

With price floor at 140: $Q_D = 500 - 280 = 220$. $Q_S = 420 - 100 = 320$. Excess supply: $320 - 220 = 100$ tonnes. The government must buy 100 tonnes.

Cost to the government: $140 \times 100 = \pounds 14\,000$.

Welfare analysis: CS before: $\frac{1}{2}(250 - 120)(260) = 16\,900$. (Demand choke price: $Q = 0 \Rightarrow P = 250$.) CS after: $\frac{1}{2}(250 - 140)(220) = 12\,100$. Change: $-4\,800$.

PS before: $\frac{1}{2}(120 - 33.33)(260) = 11\,267$. (Supply choke: $Q = 0 \Rightarrow P = 100/3 = 33.33$.) PS after: $\frac{1}{2}(140 - 33.33)(320) = 17\,067$. Change: $+5\,800$.

Government cost: 14,000. Net welfare change: $-4\,800 + 5\,800 - 14\,000 = -13\,000$.

Deadweight loss: The price support creates DWL through two channels:

1. Overproduction: $320 - 260 = 60$ tonnes produced at a cost exceeding the value to consumers. DWL $= \frac{1}{2}(140 - 120)(60) = 600$.
2. Underconsumption: $260 - 220 = 40$ tonnes not consumed despite consumer willingness to pay exceeding the cost of production. DWL $= \frac{1}{2}(140 - 120)(40) = 400$. Total DWL $= 1000$.

The remaining 12,000 of welfare loss is a pure transfer from taxpayers to producers (via government purchases). This is not a DWL per se, but it represents a redistribution that may be considered undesirable on equity grounds.

10.3 Multi-Market Analysis: Indirect Taxes and Cross-Price Effects

Example. The government imposes a tax on petrol. Petrol demand: $Q_P = 1000 - 5P_P$. Petrol supply: $Q_P = 4P_P - 200$. The cross-price elasticity between petrol and electric vehicles (EVs) is $+0.3$. EV demand: $Q_E = 200 + 0.5P_P$ (EV demand increases when petrol price rises).

Initial petrol equilibrium: $1000 - 5P_P = 4P_P - 200 \Rightarrow 1200 = 9P_P \Rightarrow P_P = 133.33$, $Q_P = 333.33$.

Tax of GBP 20 per litre on petrol: $Q_P = 4(P_P - 20) - 200 = 4P_P - 280$. $1000 - 5P_P = 4P_P - 280 \Rightarrow 1280 = 9P_P \Rightarrow P_P = 142.22$. Consumers pay 142.22. Producers receive $142.22 - 20 = 122.22$. $Q_P = 1000 - 5(142.22) = 288.89$.

Effect on EVs: Petrol price rises from 133.33 to 142.22 ($\% \Delta P_P = (142.22 - 133.33)/133.33 = 6.67\%$). $\% \Delta Q_E = XED \times \% \Delta P_P = 0.3 \times 6.67 = 2.0\%$. EV demand increases by 2.0%.

Environmental benefit: If the average petrol car emits 120g/km of $\text{CO}_2$ and the average EV emits 0g/km (indirectly), and each car drives 12,000 km/year: $\text{CO}_2$ saved per switched car $= 120 \times 12000 / 10^6 = 1.44$ tonnes/year.

If 2% of the 333,333 car owners switch: $0.02 \times 333\,333 = 6\,667$ cars switch. Total $\text{CO}_2$ saved $= 6\,667 \times 1.44 = 9\,600$ tonnes/year.

Revenue and DWL: Tax revenue $= 20 \times 288.89 = 5\,778$. DWL $= \frac{1}{2} \times 20 \times (333.33 - 288.89) = \frac{1}{2} \times 20 \times 44.44 = 444.4$.

Net environmental benefit: If the social cost of carbon is GBP 50/tonne: $9\,600 \times 50 = 480\,000$. Net benefit $= 480\,000 - 444.4 = 479\,556$. The tax generates a large net social benefit through the EV switching channel, far exceeding the DWL.

10.4 Subsidy Analysis with Elasticity

Example. The government provides a production subsidy for solar panels. Demand: $P = 5000 - 0.5Q$. Supply: $P = 2000 + 0.3Q$. The subsidy is GBP 500 per panel.

Without subsidy: $5000 - 0.5Q = 2000 + 0.3Q \Rightarrow 3000 = 0.8Q \Rightarrow Q = 3750$, $P = 3125$.

With subsidy: Supply shifts down: $P = 1500 + 0.3Q$. $5000 - 0.5Q = 1500 + 0.3Q \Rightarrow 3500 = 0.8Q \Rightarrow Q = 4375$.

Consumer price: $P_c = 5000 - 0.5(4375) = 2812.5$. Consumers save $3125 - 2812.5 = 312.5$ per panel. Producer receives: $P_p = 2812.5 + 500 = 3312.5$. Producers gain $3312.5 - 3125 = 187.5$ per panel.

Incidence: Consumer share $= 312.5/500 = 62.5\%$. Producer share $= 187.5/500 = 37.5\%$. Consumers bear a larger share because demand is less elastic than supply: $PED = -0.5 \times 3125/3750 = -0.417$ (inelastic). $PES = (1/0.3) \times 3125/3750 = 2.78$ (elastic).

Government cost: $500 \times 4375 = 2\,187\,500$.

Positive externality correction: If the marginal external benefit per solar panel is GBP 600 (reduced pollution, energy security): Social optimum: $MSB = MPB + 500$. $5000 + 500 - 0.5Q = 2000 + 0.3Q \Rightarrow 3500 = 0.8Q \Rightarrow Q = 4375$.

The subsidy of GBP 500 achieves an output of 4375, which is close to (but not exactly) the social optimum of 4375. If the MEB were exactly 500, the subsidy would be perfectly set. In this case, the subsidy slightly under-corrects (optimal subsidy would be 600, not 500). The remaining DWL from under-correction: $\frac{1}{2}(600 - 500)(Q^*_{MEB=600} - 4375)$. Since the MEB is 600 but the subsidy is 500, there is still a small under-provision.

DWL of the subsidy (relative to the first-best): With MEB = 600 and subsidy = 500: the effective MPC becomes $2000 + 0.3Q - 500 = 1500 + 0.3Q$. MSB = MPB + MEB = $5000 - 0.5Q + 600 = 5600 - 0.5Q$. Social optimum: $5600 - 0.5Q = 2000 + 0.3Q \Rightarrow 3600 = 0.8Q \Rightarrow Q^* = 4500$.

The subsidy achieves $Q = 4375$ vs the social optimum of 4500. The remaining DWL: $DWL = \frac{1}{2}(MSB_{4375} - MSC_{4375})(4500 - 4375)$. $MSB_{4375} = 5600 - 0.5(4375) = 3412.5$. $MSC_{4375} = 2000 + 0.3(4375) = 3312.5$. $DWL = \frac{1}{2}(3412.5 - 3312.5)(125) = \frac{1}{2}(100)(125) = 6250$.

The subsidy is close to optimal but not perfect. A subsidy of GBP 600 would be first-best.

11. Extended Worked Examples

11.1 Market Equilibrium with Multiple Interventions

Example. The market for cigarettes has demand $Q_D = 200 - 4P$ and supply $Q_S = 6P - 80$. The government imposes: (1) a specific tax of GBP 5 per pack, (2) a price floor of GBP 30, and (3) a maximum production quota of 40 packs.

Step 1: Free market equilibrium. $200 - 4P = 6P - 80 \Rightarrow 280 = 10P \Rightarrow P = 28$, $Q = 88$.

Step 2: With tax only. $Q_S = 6(P - 5) - 80 = 6P - 110$. $200 - 4P = 6P - 110 \Rightarrow 310 = 10P \Rightarrow P = 31$ (consumer price). Producer receives 26. $Q = 200 - 124 = 76$.

Step 3: With tax AND price floor of 30. The price floor of 30 is BELOW the consumer price with tax (31), so the price floor is NOT binding. The equilibrium is the same as Step 2: $P = 31$, $Q = 76$.

If the price floor were 35: consumers pay 35. $Q_D = 200 - 140 = 60$. $Q_S$ at producer price $= 35 - 5 = 30$: $Q_S = 6(30) - 80 = 100$. Excess supply $= 100 - 60 = 40$. The government would need to buy 40 packs (costing $40 \times 30 = 1200$).

Step 4: With tax AND production quota of 40. The quota restricts supply to 40 units. At $Q = 40$: demand price $= P_D = (200 - 40)/4 = 40$. Supply price $= P_S = (40 + 80)/6 = 20$. With tax: producer receives $P_S = P_D - 5 = 35$. But $P_S$ at $Q = 40$ is 20, so producers are willing to supply at 20 but receive 35. The tax drives a wedge.

Actually, with the quota: supply is fixed at 40. The price is determined by demand: $P_D = 40$. Producer receives $40 - 5 = 35$. The quota rent is $35 - 20 = 15$ per unit (the difference between what producers receive and their minimum supply price). Total quota rent $= 15 \times 40 = 600$.

Combined effect: The quota (40) is more restrictive than the tax alone (76). The combined policy results in:

• Higher consumer price (40 vs 31 with tax alone).
• Higher producer revenue per unit (35 vs 26 with tax alone).
• Lower quantity (40 vs 76).
• Quota rent of 600 (captured by whoever holds the quota rights).
• Tax revenue $= 5 \times 40 = 200$ (vs $5 \times 76 = 380$ with tax alone).

Total government revenue: tax revenue (200) + quota auction revenue (if quotas are auctioned, 600) = 800.

Welfare comparison:

Policy	P consumer	Q	CS	PS	Govt revenue	DWL
None	28	88	968	361	0	0
Tax only	31	76	578	288	380	120
Tax + quota	40	40	100	400 + 600	200	560

The combined tax and quota is highly distortionary: DWL increases from 120 (tax only) to 560 (tax + quota). The quota is a blunt instrument that creates more DWL than the tax because it prevents the market from adjusting along the supply curve.

11.2 Agricultural Markets: Buffer Stock Scheme

Example. The government establishes a buffer stock scheme for wheat. The target price is GBP 200 per tonne. Demand: $Q_D = 1000 - 2P$. Supply: $Q_S = 3P - 200$.

Free market equilibrium: $1000 - 2P = 3P - 200 \Rightarrow 1200 = 5P \Rightarrow P = 240$, $Q = 520$.

Since the target price (200) is BELOW the market price (240), the buffer stock is irrelevant -- the market price is already above the target. The government does not need to intervene.

If the target price is 280 (above market price): At $P = 280$: $Q_D = 1000 - 560 = 440$. $Q_S = 840 - 200 = 640$. Excess supply $= 640 - 440 = 200$ tonnes. The government buys 200 tonnes and stores them.

Cost to government: $280 \times 200 = 56\,000$. Storage cost: GBP 10/tonne/year. Annual storage cost $= 2000$.

If a bad harvest reduces supply: New supply $Q_S = 2P - 100$. New equilibrium: $1000 - 2P = 2P - 100 \Rightarrow 1100 = 4P \Rightarrow P = 275$, $Q = 450$. The market price (275) is still below the target (280), so the government releases some stocks. Government releases: enough to push price to 280. At $P = 280$: $Q_D = 440$, $Q_S = 460$. Excess supply of 20. The government does NOT need to release stocks (supply exceeds demand even at the target price).

If supply falls further: $Q_S = P$. $1000 - 2P = P \Rightarrow P = 333.33$, $Q = 333.33$. Market price (333.33) is above target (280). The government sells stocks: $Q_D(280) = 440$, $Q_S(280) = 80$. Shortage $= 360$. The government sells 360 tonnes from its buffer stock.

Revenue from sales: $280 \times 360 = 100\,800$.

Buffer stock balance sheet:

• Bought: 200 tonnes at 280 = 56,000.
• Sold: 360 tonnes at 280 = 100,800.
• Storage costs: 200 tonnes x 10/tonne x N years.
• Net profit: $100\,800 - 56\,000 - 2000N = 44\,800 - 2000N$.

If stored for 5 years: net profit $= 44\,800 - 10\,000 = 34\,800$. If stored for 22.4 years: net profit $= 0$.

Problems with buffer stock schemes:

1. Expensive: storage costs erode the profit margin.
2. Market distortion: the target price sends the wrong signal to farmers (over-production).
3. Quality deterioration: stored commodities may deteriorate.
4. Political manipulation: governments may set the target price too high to win farmer votes.
5. International trade: buffer stocks may conflict with WTO rules (they are a form of domestic support).

EU Common Agricultural Policy (CAP): The EU operated a buffer stock scheme from the 1960s to the 1990s. By the 1980s, the "butter mountains" and "wine lakes" had become a political embarrassment. The scheme was reformed in 1992 (MacSharry reforms) and again in 2003, shifting from price support to direct income support for farmers.

11.3 Black Markets: Price Controls with Enforcement

Example. The government imposes a maximum rent of GBP 1,000/month on apartments. The free market rent is GBP 1,500/month. Demand: $Q_D = 2000 - P$ (where P is monthly rent). Supply: $Q_S = P - 500$.

Free market: $2000 - P = P - 500 \Rightarrow 2500 = 2P \Rightarrow P = 1250$, $Q = 750$.

Wait, the free market rent is given as 1500 but the equilibrium gives 1250. Let me adjust the functions.

Demand: $Q_D = 3000 - P$. Supply: $Q_S = P - 1500$. $3000 - P = P - 1500 \Rightarrow 4500 = 2P \Rightarrow P = 2250$, $Q = 750$.

Hmm, still not matching. Let me just use the given free market equilibrium:

At free market: $P = 1500$, $Q = 750$. Demand: $Q_D = a - bP$. $750 = a - 1500b$. Supply: $Q_S = cP - d$. $750 = 1500c - d$.

Let me use: $Q_D = 2000 - 0.833P$, $Q_S = P - 750$. At $P = 1500$: $Q_D = 750.5 \approx 750$, $Q_S = 750$. Good enough.

With rent control at P = 1000: $Q_D = 2000 - 833 = 1167$. $Q_S = 1000 - 750 = 250$. Shortage $= 1167 - 250 = 917$ apartments.

Black market: Landlords and tenants may illegally trade at above the controlled price. Suppose enforcement is imperfect: probability of detection for illegal renting is 20%, and the fine is GBP 5,000.

Landlord's expected revenue from black market: If the black market price is $P_b$: Expected revenue $= P_b \times 0.80 - 5000 \times 0.20 = 0.8P_b - 1000$.

Landlords will participate in the black market if expected revenue exceeds the controlled price: $0.8P_b - 1000 > 1000 \Rightarrow P_b > 2500$.

At $P_b = 2500$: expected revenue $= 1000$ (same as legal rent). The black market price must exceed 2500 for landlords to participate. This is well above the free market price of 1500, suggesting that the black market is not profitable for landlords.

But this assumes the fine is 5000 and detection probability is 20%. If the fine is lower (1000) or detection is less likely (5%):

Expected revenue $= P_b \times 0.95 - 1000 \times 0.05 = 0.95P_b - 50$. $0.95P_b - 50 > 1000 \Rightarrow P_b > 1105$.

Now the black market price only needs to exceed 1105 for landlords to participate. The black market would operate at a price between 1105 and 1500 (the free market price).

With weak enforcement: approximately $Q_S = 750$ apartments are available (same as free market), but they trade at $P_b \approx 1400$ (slightly below free market due to the risk premium). Consumers pay 1400 instead of 1000 (the controlled price). The rent control is entirely ineffective -- the black market undermines it.

Key insight: price controls create black markets unless enforcement is extremely rigorous. The resources devoted to enforcement (inspections, legal proceedings) are a deadweight loss. In practice, most economists recommend income support (housing benefit) rather than rent control as a way to help low-income renters without distorting the market.