Market Equilibrium and Taxation -- ECON 300, Midterm Review
Source: ECON 300 Midterm Examination, Autumn 2016 (University of Washington, Prof. Greg Ellis)
Tags: market equilibrium, supply and demand, per-unit tax, specific tax, tax incidence, tax burden, deadweight loss, welfare loss, consumer surplus, producer surplus, inverse demand, inverse supply
Difficulty: Intermediate. Prerequisites: basic algebra, understanding of supply and demand curves, consumer and producer surplus.
Big Picture
Market equilibrium is where supply meets demand, and it is the starting point for almost every policy analysis in microeconomics. This topic takes that foundation and asks what happens when the government intervenes with a tax. Who really pays? (Spoiler: it depends on the relative slopes of supply and demand, not on who writes the cheque.) And how much value does society lose in the process? These questions connect directly to welfare economics and public finance, and they appear on virtually every intermediate micro exam.
Set quantity demanded equal to quantity supplied to find equilibrium price and quantity. A per-unit tax drives a wedge between the price buyers pay and the price sellers receive, reducing the equilibrium quantity. The side of the market with the less elastic curve bears the larger share of the tax burden. The deadweight loss is the triangle of surplus that neither buyers, sellers, nor the government captures.
Inverse demand function
Demand written as price as a function of quantity: P = f(Qd). In this problem, P = 100 - 4Qd. In simple terms, it tells you what price corresponds to each quantity on the demand curve.
Inverse supply function
Supply written as price as a function of quantity: P = g(Qs). In this problem, P = 10 + 2Qs. In simple terms, it tells you the minimum price sellers need to supply each quantity.
Market equilibrium
The price and quantity at which quantity demanded equals quantity supplied. Graphically, it is where the supply and demand curves cross. No buyer is left wanting and no seller is left with unsold stock.
Per-unit (specific) tax
A fixed dollar amount charged on each unit sold. In this problem, $18 per unit. Think of it as a wedge driven between the demand price (what buyers pay) and the supply price (what sellers keep).
Tax incidence
The division of the tax burden between buyers and sellers. It depends on the relative elasticities of supply and demand, not on which side is legally responsible for remitting the tax.
Deadweight loss (DWL)
The net loss of total surplus (consumer surplus + producer surplus + tax revenue) caused by the tax. It is the value of the transactions that no longer happen because the tax has pushed the price above what some buyers are willing to pay. In simple terms, it is the cost of the tax to society over and above what the government collects.
Consumer surplus
The difference between what consumers are willing to pay and what they actually pay, summed over all units purchased. Graphically, it is the area below the demand curve and above the market price.
Producer surplus
The difference between the price sellers receive and the minimum they would accept, summed over all units sold. Graphically, it is the area above the supply curve and below the price sellers receive.
Inverse demand: P = 100 - 4Qd
Inverse supply: P = 10 + 2Qs
Set demand price equal to supply price: 100 - 4Q = 10 + 2Q.
Solve: 90 = 6Q, so Q* = 15.
Substitute back: P* = 100 - 4(15) = 40. (Check: P* = 10 + 2(15) = 40.)
The competitive equilibrium is P* = $40, Q* = 15 units.
The government levies an $18 per-unit tax on sellers. Sellers must remit $18 for each unit sold, so their effective supply curve shifts up by $18.
New inverse supply: P = (10 + 2Qs) + 18 = 28 + 2Qs. This is the price buyers must pay.
Demand is unchanged: P = 100 - 4Qd.
Set the new supply equal to demand: 100 - 4Q = 28 + 2Q.
Solve: 72 = 6Q, so Qt = 12.
Price buyers pay: Pb = 100 - 4(12) = $52.
Price sellers receive (net of tax): Ps = 52 - 18 = $34. (Check: Ps = 10 + 2(12) = 34.)
Before tax, the price was $40.
After tax, buyers pay $52, an increase of $12.
Sellers receive $34, a decrease of $6.
Buyers bear $12 of the $18 tax. Sellers bear $6.
Buyers bear the larger share because demand is less elastic than supply at this point. The demand slope (in absolute value) is 4, while the supply slope is 2. The steeper (less elastic) side absorbs more of the tax.
The statutory incidence (sellers collect it) differs from the economic incidence (buyers pay two-thirds). This result is general: switching the legal responsibility to buyers would shift the demand curve down by $18 instead of the supply curve up, and the equilibrium prices paid and received would be identical.
The deadweight loss is the area of the triangle formed between the supply curve, the demand curve, and the vertical lines at the old and new equilibrium quantities.
Base of the triangle = tax per unit = $18.
Height of the triangle = change in quantity = Q* - Qt = 15 - 12 = 3 units.
DWL = 0.5 x base x height = 0.5 x 18 x 3 = $27.
Those 3 units (from Q = 12 to Q = 15) would have been traded in the absence of the tax, generating surplus for both buyers and sellers. The tax makes those trades unprofitable, and that surplus vanishes. The government does not collect it either, because those units are never sold.
Inverse demand: P = 100 - 4Q
Inverse supply: P = 10 + 2Q
Equilibrium condition: 100 - 4Q = 10 + 2Q
Pre-tax equilibrium: P* = $40, Q* = 15
Post-tax supply (with $18 tax on sellers): P = 28 + 2Q
Post-tax equilibrium: Pb = $52 (buyers pay), Ps = $34 (sellers receive), Qt = 12
Tax burden on buyers: Pb - P* = $12
Tax burden on sellers: P* - Ps = $6
DWL = 0.5 x t x (Q* - Qt) = 0.5 x 18 x 3 = $27
Tax revenue = t x Qt = 18 x 12 = $216
Tax incidence analysis is how economists evaluate proposals like carbon taxes, sugar taxes, or tariffs. The lesson from this topic, that the economic burden falls on the less elastic side regardless of who sends the cheque to the government, is one of the most policy-relevant results in all of economics. When governments tax cigarettes, for instance, demand is relatively inelastic (smokers keep buying), so consumers absorb most of the tax.
Students often think the side legally responsible for paying the tax bears more of the burden. This is wrong. Economic incidence depends on relative elasticities, not statutory incidence.
Students sometimes calculate DWL as a rectangle (tax times the reduction in quantity) rather than a triangle. The triangle is correct because the lost surplus tapers: the marginal units closest to the new equilibrium were only barely profitable.
Students occasionally forget to shift the correct curve. If the tax is on sellers, the supply curve shifts up. If the tax were on buyers, the demand curve would shift down. The end result is the same either way.
Students sometimes compute the tax revenue and call it the deadweight loss. Tax revenue (t x Qt = $216) is a transfer from buyers and sellers to the government. Deadweight loss ($27) is the surplus that disappears entirely.
⚠️ Finding the pre-tax equilibrium is straightforward algebra, but it is the foundation for every sub-part. Get this right first.
⚠️ When applying a per-unit tax on sellers, shift the supply curve up by the tax amount. Write out the new supply equation explicitly.
⚠️ To determine tax incidence, compare the price change for buyers (Pb - P*) against the price change for sellers (P* - Ps). State which is larger and why.
⚠️ DWL is a triangle, not a rectangle. The formula is 0.5 x t x ΔQ. Show the calculation clearly.
True or False: In a competitive market, equilibrium occurs where the demand curve intersects the supply curve.
Fill in the blank: A per-unit tax of $18 on sellers shifts the supply curve ______ by $______.
True or False: If demand is more inelastic than supply, sellers bear the larger share of the tax.
Fill in the blank: Deadweight loss is calculated as 0.5 x ______ x ______.
True or False: Tax revenue is part of deadweight loss.
Answers: 1. True. 2. Up, $18. 3. False (buyers bear more). 4. Tax per unit, change in quantity. 5. False (it is a transfer, not a loss).
Q: Given inverse demand P = 100 - 4Q and inverse supply P = 10 + 2Q, find the competitive equilibrium price and quantity.
A: Set 100 - 4Q = 10 + 2Q. Solve: 6Q = 90, Q* = 15. P* = 100 - 4(15) = $40.
Q: An $18 per-unit tax is placed on sellers. What is the new equilibrium quantity and the price buyers pay?
A: New supply: P = 28 + 2Q. Set 100 - 4Q = 28 + 2Q. Solve: 6Q = 72, Qt = 12. Pb = 100 - 4(12) = $52.
Q: What price do sellers receive after the tax? Who bears more of the tax burden?
A: Ps = 52 - 18 = $34. Buyers pay $12 more than before ($52 vs $40). Sellers receive $6 less ($34 vs $40). Buyers bear the larger share.
Q: Calculate the deadweight loss from the tax.
A: DWL = 0.5 x 18 x (15 - 12) = 0.5 x 18 x 3 = $27.
Q: How much tax revenue does the government collect?
A: Revenue = $18 x 12 = $216.
Market equilibrium connects back to consumer theory (the demand curve is the horizontal sum of individual Marshallian demands) and forward to welfare economics (surplus analysis). Tax incidence connects to elasticity of demand (Doc 3 in these notes): the more inelastic side bears more of the burden. Deadweight loss analysis is the foundation for evaluating any market intervention, including price floors, price ceilings, and subsidies.
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