Demand, Supply, and Equilibrium — Diagnostic Tests

Unit Tests

UT-1: Price Elasticity of Demand Calculation

Question: When the price of coffee rises from $\pounds 2.50$ to $\pounds 3.00$ per cup, the quantity demanded falls from 800 cups per day to 600 cups per day. Calculate the PED using the midpoint (arc elasticity) method. Is demand elastic or inelastic? What happens to total revenue?

Solution:

Midpoint method: $\text{PED} = \frac◆LB◆\% \Delta Q_d◆RB◆◆LB◆\% \Delta P◆RB◆ = \frac◆LB◆\Delta Q / \bar{Q}◆RB◆◆LB◆\Delta P / \bar{P}◆RB◆$

$\Delta Q = 600 - 800 = -200$, $\bar{Q} = (800 + 600)/2 = 700$

$\Delta P = 3.00 - 2.50 = 0.50$, $\bar{P} = (2.50 + 3.00)/2 = 2.75$

$\text{PED} = \frac{-200/700}{0.50/2.75} = \frac{-0.2857}{0.1818} = -1.57$

$|\text{PED}| = 1.57 \gt 1$: Demand is elastic.

Total revenue at $\pounds 2.50$: $\text{TR}_1 = 2.50 \times 800 = \pounds 2000$ Total revenue at $\pounds 3.00$: $\text{TR}_2 = 3.00 \times 600 = \pounds 1800$

Total revenue decreased from $\pounds 2000$ to $\pounds 1800$. This is consistent with elastic demand: when price rises and demand is elastic, the percentage decrease in quantity demanded exceeds the percentage increase in price, so revenue falls.

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UT-2: Price Elasticity of Supply

Question: The supply of hand-sanitiser is given by $Q_s = -100 + 50P$. Calculate the PES when price rises from $\pounds 3$ to $\pounds 5$. Classify the elasticity. Explain why the supply of hand-sanitiser might be less elastic in the very short run than in the long run.

Solution:

At $P = 3$: $Q_s = -100 + 50(3) = 50$ At $P = 5$: $Q_s = -100 + 50(5) = 150$

Midpoint method: $\text{PES} = \frac◆LB◆\% \Delta Q_s◆RB◆◆LB◆\% \Delta P◆RB◆ = \frac◆LB◆\Delta Q / \bar{Q}◆RB◆◆LB◆\Delta P / \bar{P}◆RB◆$

$\Delta Q = 150 - 50 = 100$, $\bar{Q} = (50 + 150)/2 = 100$

$\Delta P = 5 - 3 = 2$, $\bar{P} = (3 + 5)/2 = 4$

$\text{PES} = \frac{100/100}{2/4} = \frac{1}{0.5} = 2.0$

$\text{PES} = 2.0 \gt 1$: Supply is elastic (responsive to price changes).

In the very short run, supply is less elastic because: (1) firms have limited stockpiles and cannot instantly increase production; (2) hiring and training new workers takes time; (3) acquiring additional raw materials and manufacturing capacity requires investment with a time lag; (4) factories operate at or near capacity during a sudden demand surge. In the long run, firms can expand production capacity, new firms can enter the market, and supply chains can adjust, making supply more elastic.

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UT-3: Consumer and Producer Surplus

Question: The demand curve is given by $P = 20 - 0.5Q$ and the supply curve by $P = 2 + 0.5Q$. Calculate: (a) the equilibrium price and quantity, (b) consumer surplus, (c) producer surplus, (d) total surplus. Illustrate the deadweight loss that would result from a price ceiling at $\pounds 8$.

Solution:

(a) Equilibrium: set $D = S$.

$20 - 0.5Q = 2 + 0.5Q$

$18 = Q$, so $Q^* = 18$.

$P^* = 20 - 0.5(18) = 20 - 9 = \pounds 11$.

(b) Consumer surplus (CS) $= \frac{1}{2} \times Q^* \times (P_{\text{max}} - P^*)$

$P_{\text{max}}$ (where $Q = 0$ on demand): $P = 20$.

$\text{CS} = \frac{1}{2} \times 18 \times (20 - 11) = \frac{1}{2} \times 18 \times 9 = \pounds 81$.

(c) Producer surplus (PS) $= \frac{1}{2} \times Q^* \times (P^* - P_{\text{min}})$

$P_{\text{min}}$ (where $Q = 0$ on supply): $P = 2$.

$\text{PS} = \frac{1}{2} \times 18 \times (11 - 2) = \frac{1}{2} \times 18 \times 9 = \pounds 81$.

(d) Total surplus $= 81 + 81 = \pounds 162$.

Price ceiling at $\pounds 8$:

Quantity demanded at $\pounds 8$: $8 = 20 - 0.5Q_d$, $Q_d = 24$. Quantity supplied at $\pounds 8$: $8 = 2 + 0.5Q_s$, $Q_s = 12$.

Since $Q_s \lt Q_d$, the binding price ceiling creates a shortage of $24 - 12 = 12$ units. The quantity traded is $Q = 12$.

New CS: area below the demand curve and above $\pounds 8$, from $Q = 0$ to $Q = 12$.

Demand price at $Q = 0$: $\pounds 20$. Demand price at $Q = 12$: $20 - 6 = \pounds 14$.

$\text{CS}_{\text{new}}$ is a trapezoid: $\frac{1}{2}(20 + 14 - 2 \times 8) \times 12 = \frac{1}{2}(34 - 16) \times 12 = \frac{1}{2} \times 18 \times 12 = \pounds 108$.

New PS: area above supply and below $\pounds 8$, from $Q = 0$ to $Q = 12$:

Supply price at $Q = 0$: $\pounds 2$. Supply price at $Q = 12$: $2 + 6 = \pounds 8$.

$\text{PS}_{\text{new}} = \frac{1}{2}(8 - 2) \times 12 = \frac{1}{2} \times 6 \times 12 = \pounds 36$.

New total surplus: $108 + 36 = \pounds 144$.

Deadweight loss $= 162 - 144 = \pounds 18$.

Alternatively, DWL as a triangle: base $= 18 - 12 = 6$, height $= 14 - 8 = 6$.

$\text{DWL} = \frac{1}{2} \times 6 \times 6 = \pounds 18$.

Integration Tests

IT-1: Taxation and Welfare Analysis (with Market Failure)

Question: The government imposes a specific tax of $\pounds 4$ per unit on a good with demand $P = 50 - Q$ and supply $P = 10 + Q$. Calculate: (a) the pre-tax and post-tax equilibrium, (b) the tax incidence on consumers and producers, (c) the deadweight loss, (d) the tax revenue. Explain why the deadweight loss occurs.

Solution:

(a) Pre-tax equilibrium: $50 - Q = 10 + Q$, so $40 = 2Q$, $Q^* = 20$, $P^* = \pounds 30$.

Post-tax: supply shifts up by $\pounds 4$. New supply: $P = 14 + Q$.

$50 - Q = 14 + Q$, so $36 = 2Q$, $Q_t = 18$, $P_{\text{buyer}} = 50 - 18 = \pounds 32$.

$P_{\text{seller}} = P_{\text{buyer}} - \text{tax} = 32 - 4 = \pounds 28$.

(b) Tax incidence:

• Consumers pay $\pounds 32$ instead of $\pounds 30$: burden $= \pounds 2$ out of $\pounds 4$ (50%).
• Producers receive $\pounds 28$ instead of $\pounds 30$: burden $= \pounds 2$ out of $\pounds 4$ (50%).

The equal incidence arises because demand and supply have the same slope magnitude (both $|\text{slope}| = 1$), so they have equal elasticities at equilibrium.

(c) Deadweight loss: $\text{DWL} = \frac{1}{2} \times \text{tax} \times \Delta Q = \frac{1}{2} \times 4 \times (20 - 18) = \frac{1}{2} \times 4 \times 2 = \pounds 4$.

(d) Tax revenue $= \text{tax} \times Q_t = 4 \times 18 = \pounds 72$.

The deadweight loss occurs because the tax discourages 2 units of mutually beneficial trade. For these 2 units, the marginal benefit to consumers (given by the demand curve) exceeds the marginal cost to producers (given by the supply curve), but the tax makes these trades unprofitable. The surplus that would have been gained from these trades is lost to society -- it is not transferred to the government as revenue, but simply destroyed.

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IT-2: Subsidy Analysis (with Theory of the Firm)

Question: The government provides a production subsidy of $\pounds 6$ per unit to wheat farmers. Demand: $P = 40 - 0.5Q$, Supply: $P = 4 + 0.5Q$. Calculate the change in consumer surplus, producer surplus, government expenditure, and deadweight loss. Discuss why subsidies can lead to overproduction.

Solution:

Pre-subsidy equilibrium: $40 - 0.5Q = 4 + 0.5Q$, $36 = Q$, $Q^* = 36$, $P^* = \pounds 22$.

With subsidy: effective supply shifts down by $\pounds 6$. New supply (from seller's perspective, the price they receive): $P_{\text{seller}} = 4 + 0.5Q$, but they receive $P_{\text{buyer}} + 6$.

Equilibrium: $40 - 0.5Q = 4 + 0.5Q - 6 = -2 + 0.5Q$. So $42 = Q$, $Q_s = 42$.

$P_{\text{buyer}} = 40 - 0.5(42) = \pounds 19$. $P_{\text{seller}} = 4 + 0.5(42) = \pounds 25$ (which is $P_{\text{buyer}} + 6 = 19 + 6 = 25$. Correct.)

Change in CS: CS was $\frac{1}{2}(40 - 22)(36) = \pounds 324$. New CS $= \frac{1}{2}(40 - 19)(42) = \frac{1}{2}(21)(42) = \pounds 441$. Change $= +\pounds 117$.

Change in PS: PS was $\frac{1}{2}(22 - 4)(36) = \pounds 324$. New PS $= \frac{1}{2}(25 - 4)(42) = \frac{1}{2}(21)(42) = \pounds 441$. Change $= +\pounds 117$.

Government expenditure $= 6 \times 42 = \pounds 252$.

Net welfare change $= +117 + 117 - 252 = -\pounds 18$.

Deadweight loss $= \pounds 18$.

Alternatively: $\text{DWL} = \frac{1}{2} \times \text{subsidy} \times \Delta Q = \frac{1}{2} \times 6 \times (42 - 36) = \pounds 18$.

Subsidies cause overproduction because they lower the effective cost of production, encouraging farmers to supply more than the socially optimal quantity. The additional 6 units produced (36 to 42) have a marginal cost to society (given by the supply curve) that exceeds the marginal benefit to consumers (given by the demand curve). The resources used to produce these extra units could have been allocated more efficiently elsewhere in the economy.

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IT-3: Multiple Market Equilibrium (with The Economic Problem)

Question: In a small island economy, the demand for fish is $P = 60 - Q$ and the supply is $P = 10 + Q$. The government simultaneously introduces a price floor at $\pounds 40$ for fish and a per-unit subsidy of $\pounds 5$ for fishermen. Calculate the resulting quantity traded, consumer surplus, and producer surplus. Explain whether these two policies are contradictory.

Solution:

Supply with subsidy: $P = 10 + Q - 5 = 5 + Q$ (fishermen receive $\pounds 5$ extra per unit).

At the price floor of $\pounds 40$:

Quantity demanded: $40 = 60 - Q_d$, $Q_d = 20$. Quantity supplied (with subsidy): fishermen receive $40 + 5 = \pounds 45$ per unit. So $45 = 10 + Q_s$, $Q_s = 35$.

The subsidy means the effective price received by fishermen is $\pounds 40 + \pounds 5 = \pounds 45$ per unit. The quantity supplied at this effective price: $45 = 10 + Q_s$, $Q_s = 35$.

Since $Q_s = 35 \gt Q_d = 20$, there is excess supply (surplus) of $35 - 20 = 15$ units. The quantity traded is determined by demand: $Q = 20$.

Consumer surplus at $P = 40$, $Q = 20$:

$P_{\text{demand}}$ at $Q = 0$ is $\pounds 60$, at $Q = 20$ is $\pounds 40$.

$\text{CS} = \frac{1}{2}(60 - 40)(20) = \frac{1}{2}(20)(20) = \pounds 200$.

Producer surplus: producers receive $\pounds 45$ per unit. The supply price at $Q = 0$ is $\pounds 10$, at $Q = 20$ is $10 + 20 = \pounds 30$.

$\text{PS}$ is the area above the supply curve and below the effective price received ($\pounds 45$), from $Q = 0$ to $Q = 20$:

Supply price at $Q = 0$: $\pounds 10$. Supply price at $Q = 20$: $10 + 20 = \pounds 30$.

$\text{PS} = \frac{1}{2}(45 - 10 + 45 - 30)(20) = \frac{1}{2}(35 + 15)(20) = \frac{1}{2}(50)(20) = \pounds 500$.

Government subsidy cost $= 5 \times 20 = \pounds 100$.

Are the policies contradictory? Yes, partially. The price floor is designed to help producers by keeping prices high (and typically reduces quantity traded). The subsidy also helps producers but encourages more production. The combination means consumers pay a high price ($\pounds 40$) while producers receive an even higher effective price ($\pounds 45$). The government spends on the subsidy while the price floor creates unsold surplus (15 units). This is inefficient: the price floor reduces quantity traded below equilibrium, while the subsidy pushes supply beyond what the market demands at that price, creating waste. A simpler approach would be to use either the price floor or the subsidy, not both.

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Section 3: Extended Quantitative Practice

UT-4 (Extension). A government imposes a per-unit subsidy of $\pounds 8$ on good $X$. Demand: $Q_D = 120 - P$. Supply: $Q_S = 2P - 40$. Calculate the pre- and post-subsidy equilibrium, the change in consumer and producer surplus, government expenditure, and deadweight loss.

Solution:

Pre-subsidy: $120 - P = 2P - 40 \Rightarrow 160 = 3P \Rightarrow P = 53.33$, $Q = 66.67$.

Post-subsidy: consumers pay $P_c$, producers receive $P_c + 8$. Supply: $Q_S = 2(P_c + 8) - 40 = 2P_c - 24$. $120 - P_c = 2P_c - 24 \Rightarrow 144 = 3P_c \Rightarrow P_c = 48$. $Q = 72$. Producers receive $48 + 8 = 56$.

Consumer surplus before: $\frac{1}{2}(120 - 53.33)(66.67) = \frac{1}{2}(66.67)(66.67) = 2222.2$. Consumer surplus after: $\frac{1}{2}(120 - 48)(72) = \frac{1}{2}(72)(72) = 2592$. Change in CS: $+369.8$.

Producer surplus before: $\frac{1}{2}(53.33 - 20)(66.67) = \frac{1}{2}(33.33)(66.67) = 1111.1$. (Supply intercept: $Q_S = 0 \Rightarrow P = 20$.) Producer surplus after: $\frac{1}{2}(56 - 20)(72) = \frac{1}{2}(36)(72) = 1296$. Change in PS: $+184.9$.

Government expenditure: $8 \times 72 = 576$.

Total welfare change: $+369.8 + 184.9 - 576 = -21.3$.

This is the deadweight loss of the subsidy: $\text{DWL} = \frac{1}{2} \times 8 \times (72 - 66.67) = \frac{1}{2} \times 8 \times 5.33 = 21.3$. Correct.

The DWL arises because the subsidy encourages overproduction (72 units vs the socially optimal 66.67). The marginal cost of the last 5.33 units exceeds the marginal benefit.

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UT-5 (Extension). The price elasticity of demand for bus travel in a city is $-0.4$ and the cross-price elasticity of demand between bus travel and taxi travel is $+0.6$. The city council reduces bus fares by 25%. Calculate: (a) the percentage change in bus passenger numbers, (b) the percentage change in bus revenue, (c) the percentage change in taxi demand.

Solution:

(a) $\% \Delta Q_{bus} = PED \times \% \Delta P = -0.4 \times (-25) = +10\%$. Bus passenger numbers rise by 10%.

(b) Revenue change: $\% \Delta R = \% \Delta P + \% \Delta Q + (\% \Delta P \times \% \Delta Q) / 100 = -25 + 10 + (-25 \times 10)/100 = -25 + 10 - 2.5 = -17.5\%$. Bus revenue falls by 17.5% because demand is inelastic (PED = -0.4). The council needs to subsidise bus services to maintain them.

(c) $\% \Delta Q_{taxi} = XED \times \% \Delta P_{bus} = 0.6 \times (-25) = -15\%$. Taxi demand falls by 15% (bus and taxi are substitutes). The bus fare reduction reduces congestion and pollution by shifting commuters from taxis to buses.

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UT-6 (Extension). A market has demand $Q_D = 200 - 4P$ and supply $Q_S = 6P - 80$. The government imposes a price ceiling at $\pounds 20$. Calculate: (a) the equilibrium before the ceiling, (b) the quantity traded after the ceiling, (c) the shortage created, (d) the change in consumer and producer surplus, (e) the deadweight loss.

Solution:

(a) $200 - 4P = 6P - 80 \Rightarrow 280 = 10P \Rightarrow P = 28$, $Q = 88$.

(b) At $P = 20$: $Q_D = 200 - 80 = 120$. $Q_S = 120 - 80 = 40$. Quantity traded $= \min(120, 40) = 40$ (supply is the binding constraint).

(c) Shortage $= 120 - 40 = 80$ units.

(d) CS before: $\frac{1}{2}(50 - 28)(88) = \frac{1}{2}(22)(88) = 968$. (Demand intercept: $Q = 0 \Rightarrow P = 50$.) CS after: $\frac{1}{2}(50 - 20)(40) = 600$. Change: $-368$. Note: some consumers gain from the lower price (those who can still buy), but the quantity restriction means many consumers lose access entirely.

PS before: $\frac{1}{2}(28 - 13.33)(88) = \frac{1}{2}(14.67)(88) = 645.5$. (Supply intercept: $Q = 0 \Rightarrow P = 80/6 = 13.33$.) PS after: $\frac{1}{2}(20 - 13.33)(40) = 133.4$. Change: $-512.1$.

(e) DWL $= \frac{1}{2}(28 - 20)(88 - 40) = \frac{1}{2}(8)(48) = 192$. The price ceiling creates a DWL of 192 because 48 units that were valued above cost (between $Q = 40$ and $Q = 88$) are no longer produced.

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IT-4 (Extension): Subsidy and Market Failure. Agricultural production creates negative externalities (pollution) with a marginal external cost of $MEC = 0.1Q$. The demand is $P = 100 - Q$ and private supply is $P = 20 + Q$. (a) Calculate the market equilibrium and the socially optimal output. (b) Calculate the deadweight loss of the market failure. (c) If the government provides a production subsidy of $\pounds 5$ per unit, calculate the new equilibrium. (d) Explain why a production subsidy makes the externality worse.

Solution:

(a) Market equilibrium: $100 - Q = 20 + Q \Rightarrow 80 = 2Q \Rightarrow Q = 40$, $P = 60$. Social optimum: $MSC = MPC + MEC = (20 + Q) + 0.1Q = 20 + 1.1Q$. Set $MSB = MSC$: $100 - Q = 20 + 1.1Q \Rightarrow 80 = 2.1Q \Rightarrow Q = 38.10$. Socially optimal output is 38.10, below the market output of 40.

(b) DWL $= \frac{1}{2}(MSC_{Q=40} - MSB_{Q=40})(40 - 38.10)$. $MSC_{40} = 20 + 1.1(40) = 64$. $MSB_{40} = 60$. DWL $= \frac{1}{2}(64 - 60)(1.90) = \frac{1}{2}(4)(1.90) = 3.81$.

(c) With subsidy of $\pounds 5$: supply shifts down to $P = 15 + Q$. $100 - Q = 15 + Q \Rightarrow 85 = 2Q \Rightarrow Q = 42.5$, $P = 57.5$. Output INCREASES from 40 to 42.5, making the externality worse.

(d) A production subsidy encourages MORE production, which increases the negative externality (pollution). The DWL increases because the gap between market output and socially optimal output widens. The correct policy for a negative externality is a tax (Pigouvian tax), not a subsidy. A subsidy is appropriate for positive externalities.

New DWL with subsidy: $MSC_{42.5} = 20 + 1.1(42.5) = 66.75$. $MSB_{42.5} = 57.5$. DWL $= \frac{1}{2}(66.75 - 57.5)(42.5 - 38.10) = \frac{1}{2}(9.25)(4.40) = 20.35$. The DWL has increased from 3.81 to 20.35 -- a more than five-fold increase. The subsidy has significantly worsened the market failure.