Deadweight Loss Welfare Calculation Quiz

Deadweight loss from a per-unit tax is the triangular area between supply and demand curves over units no longer traded. For a linear model this equals one half times the per-unit tax times the change in quantity. The per-unit tax is the vertical height representing the gap between what buyers pay and what sellers receive, and the quantity reduction is the horizontal base of the triangle.

Deadweight loss equals one half times the per-unit tax times the reduction in quantity. The per-unit tax is six dollars and quantity falls by twenty units. One half times six times twenty equals sixty dollars. This triangle over the twenty units no longer traded represents the surplus that buyers and sellers would have shared on those transactions but is permanently lost due to the tax distortion.

The deadweight loss triangle has two defining dimensions. The height is the per-unit tax, equal to the vertical gap between the buyer price and the seller price after the tax. The base is the reduction in quantity caused by the tax. The area of this right triangle is one half times height times base, giving the standard formula used to calculate the welfare cost of any per-unit tax in a linear supply and demand model.

Deadweight loss from a price ceiling equals one half times the price gap times the reduction in quantity. The price gap is twelve minus eight equals four dollars. The quantity reduction is one hundred minus seventy equals thirty units. One half times four times thirty equals sixty dollars. This triangle between the supply and demand curves over the thirty untraded units measures the full welfare cost of the price ceiling.

The one-half factor reflects the geometric shape of the deadweight loss area. With linear supply and demand curves the region of prevented trades forms a right triangle. The demand curve slopes downward and the supply curve slopes upward, converging at equilibrium. The area of this triangle is one half times base times height, which is why the deadweight loss formula always includes the one-half factor when supply and demand are both linear.

The combined decline in consumer and producer surplus equals tax revenue plus deadweight loss. Tax revenue flows to the government as a transfer. Deadweight loss is the remaining portion permanently destroyed because the tax prevents beneficial trades. No party receives this lost amount. Together these two components account for the entire welfare reduction in the market, showing why tax revenue is a redistribution while deadweight loss is a net social cost.

With perfectly inelastic demand consumers purchase the same quantity regardless of price. A tax raises the price buyers pay but does not reduce quantity traded. Because no mutually beneficial trades are prevented, the base of the deadweight loss triangle is zero and the area is zero. The tax collects revenue without causing allocative inefficiency, making perfectly inelastic demand the special case where a per-unit tax imposes no welfare cost beyond the revenue transfer.

Without the tax, equating supply and demand: price minus two equals ten minus price gives equilibrium price of six and quantity of four. With a two-dollar tax the buyer pays seven and the seller receives five. New quantity equals five minus two equals three. Quantity falls from four to three, a reduction of one unit. Deadweight loss equals one half times two times one equals one dollar.

The base of the deadweight loss triangle equals the reduction in quantity traded. A larger quantity reduction means a wider base, which increases the total area of the triangle and therefore increases the deadweight loss. Each additional unit in the quantity reduction represents another beneficial trade that is prevented. The formula one half times tax times quantity change confirms that any increase in the quantity reduction directly raises the welfare cost of the distortion.

As the tax rate rises the height of the deadweight loss triangle grows because the price wedge is larger. At the same time the base grows because the larger wedge causes a greater quantity reduction. Since both height and base increase proportionally with the tax rate, the area grows with the square of the rate. Doubling a tax therefore more than doubles the welfare cost, making high marginal tax rates disproportionately damaging to allocative efficiency.

Calculating deadweight loss requires the per-unit tax as the triangle height, the quantity reduction as the triangle base, and the one-half factor because the deadweight loss is triangular. Government tax revenue is not part of the deadweight loss formula. Revenue is a separate transfer from buyers and sellers to the government and is excluded from the triangular area calculation that measures the net welfare cost of the tax.

Deadweight loss equals one half times five dollars times ten units equals twenty-five dollars. The combined decline in consumer and producer surplus is three hundred fifty dollars. Tax revenue equals three hundred fifty minus twenty-five equals three hundred twenty-five dollars. The government receives three hundred twenty-five dollars as a transfer while the twenty-five dollar deadweight loss is permanently destroyed with no party receiving it.

The ratio of deadweight loss to tax revenue measures how much welfare is destroyed for each dollar the government collects. A low ratio indicates a tax that raises substantial revenue with little allocative inefficiency. This metric guides tax design: taxes on goods with inelastic demand have low ratios and are preferred for raising revenue efficiently, while taxes on elastic goods carry high ratios meaning each dollar of revenue comes at a large welfare cost.

When the tax doubles in a linear market the height of the deadweight loss triangle doubles because the price wedge is twice as large. Because supply and demand are linear the quantity reduction also doubles, so the base doubles too. The area of the triangle equals one half times base times height. With both dimensions doubling the area increases by a factor of four, confirming that deadweight loss grows with the square of the tax rate.

Without the tax, equating supply and demand: two times price minus four equals twenty minus price gives three times price equals twenty-four, so equilibrium price equals eight and quantity equals twelve. With a two-dollar tax the buyer pays nine and the seller receives seven. New quantity supplied equals two times seven minus four equals ten. Quantity falls from twelve to ten, a reduction of two units. Deadweight loss equals one half times two times two equals two dollars.