Difficulty: Introductory to Intermediate | Prerequisites: Basic algebra, supply and demand fundamentals.
Market equilibrium analysis is where supply and demand stop being abstract curves and start producing testable predictions. This topic teaches you to find the price and quantity where a market clears, then to trace what happens when the government intervenes with a tax. You should already be comfortable reading inverse demand and supply functions and setting them equal. If you are not, revisit the introductory supply-and-demand material first.
These notes walk through competitive equilibrium, the mechanics of a per-unit (specific) tax, how the tax burden splits between buyers and sellers, and how to calculate the deadweight loss triangle. This is standard exam fare for ECO 300 and most intermediate micro courses.
Set demand equal to supply to find the equilibrium price and quantity. A per-unit tax drives a wedge between what buyers pay and sellers receive, reducing the quantity traded and creating deadweight loss. The side of the market with less elastic behaviour bears more of the tax burden, regardless of who physically remits the tax.
Competitive equilibrium (market-clearing price and quantity)
The price and quantity at which quantity demanded equals quantity supplied. No shortage, no surplus. Found by setting the demand function equal to the supply function.
In simple terms, the point where the two curves cross.
Inverse demand function
Price expressed as a function of quantity demanded: P = f(Q_D). The exam's P = 100 - 4Q_D is in this form.
Think of it as "what price would make consumers want exactly this quantity?"
Inverse supply function
Price expressed as a function of quantity supplied: P = g(Q_S). The exam's P = 10 + 2Q_S is in this form.
Per-unit (specific) tax
A fixed dollar amount collected on each unit sold. It shifts the supply curve vertically upward by the tax amount (when levied on sellers) or the demand curve vertically downward by the tax amount (when levied on buyers). The economic outcome is the same either way.
Tax incidence (tax burden)
How the economic burden of a tax is split between buyers and sellers. This depends on the relative elasticities of demand and supply, not on who is legally responsible for paying the tax.
In simple terms, the less responsive side of the market gets stuck with more of the bill.
Deadweight loss (DWL)
The loss in total surplus (consumer surplus + producer surplus + tax revenue) caused by the tax. It arises because the tax prevents some mutually beneficial trades from occurring.
Think of it as the triangle of wasted value between the old and new equilibrium quantities.
Given:
Inverse demand: P = 100 - 4Q_D
Inverse supply: P = 10 + 2Q_S
At equilibrium, Q_D = Q_S = Q* and the prices are equal:
100 - 4Q = 10 + 2Q
90 = 6Q
Q* = 15
P* = 100 - 4(15) = 40
The competitive equilibrium is P* = $40, Q* = 15 units.
To verify: P = 10 + 2(15) = 40. Both sides agree.
The government places an $18 per-unit tax on sellers. This means sellers now face a cost of $18 per unit on top of their supply curve, so the new supply curve shifts up by $18:
New inverse supply: P = (10 + 2Q_S) + 18 = 28 + 2Q_S
Set the new supply equal to demand:
100 - 4Q = 28 + 2Q
72 = 6Q
Q_tax = 12
The price buyers pay: P_buyers = 100 - 4(12) = $52
The price sellers receive (after paying tax): P_sellers = 52 - 18 = $34
Alternatively, P_sellers = 10 + 2(12) = $34. Both routes confirm the same answer.
The tax wedge: buyers pay $52 (up from $40), sellers receive $34 (down from $40). The $18 tax sits in between.
Buyers' burden: they now pay $52 instead of $40, so they bear $12 of the $18 tax per unit.
Sellers' burden: they now receive $34 instead of $40, so they bear $6 of the $18 tax per unit.
Buyers bear the larger share ($12 vs $6).
Why? It comes down to relative elasticities. The demand curve has a slope of -4 (in inverse form) and the supply curve has a slope of +2. Demand is steeper (less elastic) than supply, so buyers bear a larger fraction of the tax. The general rule: the more inelastic side bears more of the burden. Here, the burden ratio is 2:1 (buyers:sellers), matching the slope ratio |slope of demand| / |slope of supply| = 4/2 = 2.
The deadweight loss is the area of the triangle between the demand and supply curves, from the new quantity (Q_tax = 12) to the old quantity (Q* = 15).
The height of the triangle is the tax: $18
The base of the triangle is the reduction in quantity: 15 - 12 = 3
DWL = 0.5 x base x height = 0.5 x 3 x 18 = $27
This $27 represents surplus that existed before the tax but is now lost entirely. It is not collected as revenue; it simply vanishes because three units that were worth trading no longer get traded.
For context, total tax revenue is $18 x 12 = $216. The deadweight loss is roughly 12.5% of tax revenue, a measure of the efficiency cost of raising this particular tax.
\text{Equilibrium: } P_D(Q) = P_S(Q) \Rightarrow 100 - 4Q = 10 + 2QQ^* = 15, \quad P^* = 40\text{With tax } t = 18: \quad 100 - 4Q = (10 + 2Q) + 18Q_{\text{tax}} = 12, \quad P_{\text{buyers}} = 52, \quad P_{\text{sellers}} = 34\text{DWL} = \tfrac{1}{2} \times (Q^* - Q_{\text{tax}}) \times t = \tfrac{1}{2} \times 3 \times 18 = 27Tax incidence analysis is how economists evaluate proposals like carbon taxes, sugar taxes, or payroll taxes. The key policy insight is that the statutory incidence (who writes the cheque) differs from the economic incidence (who bears the cost). Payroll taxes in the US are split 50/50 between employer and employee by law, but most economists conclude that employees bear the bulk of the economic burden because labour supply is relatively inelastic. Deadweight loss calculations inform debates about which taxes are least distortionary.
Students often believe that whoever legally pays the tax bears the full burden. This is wrong. Economic incidence depends on elasticities, not legal responsibility.
A common error is forgetting that the supply curve shifts up by the full tax amount when the tax is levied on sellers. Some students shift it by half the tax or treat it as a rotation.
Students sometimes calculate deadweight loss using the wrong triangle. The DWL triangle sits between the demand and supply curves (not the new supply curve), from Q_tax to Q*.
Another mistake is confusing a specific (per-unit) tax with an ad valorem (percentage) tax. The exam specifies per-unit, so the supply curve shifts by a constant vertical distance.
⚠️ You will almost certainly be asked to find equilibrium, then re-solve with a tax. Practise until the algebra is automatic.
⚠️ Tax incidence questions require you to state which side bears more and explain why (elasticities). A number alone is not enough.
⚠️ Deadweight loss is the area of a triangle, not a rectangle. Use 0.5 x base x height. The base is the change in quantity; the height is the tax.
⚠️ Know that it does not matter whether the tax is levied on buyers or sellers. The equilibrium quantities and the economic burden are the same either way.
True or false: If a per-unit tax is placed on sellers, only sellers bear the economic burden.
False. Buyers bear part of it too, through a higher price.
Fill in the blank: Deadweight loss is calculated as 0.5 x ________ x ________.
(Q* - Q_tax) x tax amount.
True or false: A larger tax always creates a proportionally larger deadweight loss.
False. DWL grows with the square of the tax (quadratically), so doubling the tax quadruples the DWL.
Fill in the blank: The side of the market with the more ________ curve bears a larger share of the tax.
Inelastic.
Q: Given inverse demand P = 100 - 4Q and inverse supply P = 10 + 2Q, what is the competitive equilibrium?
A: Q* = 15, P* = $40.
Q: An $18 per-unit tax is levied on sellers. What are the new equilibrium quantity, buyer price, and seller price?
A: Q_tax = 12. Buyers pay $52. Sellers receive $34.
Q: Who bears the larger tax burden, and why?
A: Buyers bear $12 of the $18 tax (sellers bear $6). Demand is less elastic than supply (steeper curve), so buyers absorb a larger share.
Q: What is the deadweight loss from this tax?
A: DWL = 0.5 x (15 - 12) x 18 = $27.
This connects to welfare economics (consumer and producer surplus), where the same graphical tools are used to evaluate any market intervention. It also links to elasticity of demand (covered in a separate set of notes from this exam), since tax incidence depends directly on relative elasticities. In more advanced courses, this analysis extends to general equilibrium and optimal taxation theory.
Market equilibrium, competitive equilibrium, supply and demand, inverse demand function, inverse supply function, per-unit tax, specific tax, ad valorem tax, tax incidence, tax burden, deadweight loss, welfare loss, efficiency cost of taxation, consumer surplus, producer surplus, tax revenue, elasticity and tax incidence, ECO 300, intermediate microeconomics