Taxes and Subsidies: Tax Incidence, Revenue, Deadweight Loss

Your town puts a $3 tax on every large pizza, and the law says the pizza shop pays it, not you. So why does your pizza cost $2 more? Who writes the check to the government doesn't decide who really pays. This is Module 2.8A, Government Intervention: Taxes, Subsidies, and Market Efficiency.

Governments can step into markets with price controls, quantity limits, taxes, or subsidies. Each changes what buyers and sellers do. Here: taxes and subsidies.

A per-unit tax, or excise tax, is a set dollar amount on each unit sold. Ours is $3 per pizza. Before the tax, pizzas sell for $12, and 400 sell a day. Now the shop owes $3 on every pizza, so at any quantity it needs $3 more per pizza. So the supply curve shifts up by exactly $3. Up means left: a decrease in supply.

The new curve crosses demand at 300 pizzas. Buyers now pay $14. The shop sends $3 to the government and keeps $11. Exam trap: the tax is the vertical distance between the curves, $14 minus $11, or $3. It's not the $2 change in the buyers' price.

So what just happened with the tax? One: the $3 tax shifted supply up by $3, which is to the left. Two: quantity fell from 400 to 300 pizzas. Three: buyers now pay $14, sellers keep $11, and that $3 gap is the tax.

Back to the hook: buyers pay $2 more, and sellers keep $1 less. So buyers bear $2 of the $3 tax. Who really bears a tax is called tax incidence.

Quick refresher: elasticity is the percent change in quantity over the percent change in price, in absolute value, so drop any minus sign. The smaller that number, the less elastic the side. Drawn through the same point, the less elastic curve is the steeper one.

Why do buyers bear more here? Where the curves cross, demand is steeper than supply, so buyers are the less elastic side. It takes a bigger price change to make them cut back, so their price moves more.

Check it with numbers. Percent change is the change over the starting value. Both sides cut back 100 from 400 pizzas: 25%. Buyers' price rose $2 from $12: about 17%. And 25 over 17 is about 1.5.

Sellers' price fell $1 from $12: about 8%. And 25 over 8 is about 3. Both are above 1, but the buyers' 1.5 is smaller, so buyers are the less elastic side.

The rule: the less elastic side bears more of the tax, whoever legally sends the money. Test it: tax the buyers $3 instead, and demand shifts down by $3. Same 300 pizzas, same $14 paid, same $11 kept.

So what just happened to the $3? One: buyers bear $2 of it, and sellers bear $1. Two: buyers bear more because they are the less elastic side here. Three: taxing buyers instead gives the same split.

Tax revenue is the tax times the new quantity: $3 times 300 is $900 a day. On the graph it's a rectangle, $3 tall and 300 wide.

Quick refresher: a rectangle's area is height times width, with no one half. Here the height is the $3 from $11 to $14, and the width is 300 pizzas.

Buyers fund $2 times 300, or $600. Sellers fund $1 times 300, or $300. But the 100 pizzas from 300 to 400 no longer get made, though buyers valued each one more than it cost. That lost surplus is deadweight loss.

Quick refresher: a triangle's area is one half times base times height. Read the base along the quantity axis and the height up the price axis.

Deadweight loss is that triangle. Its height is the $3 tax, and its base is the 100 lost pizzas. One half times 3 times 100 is $150 a day. Exam trap: deadweight loss is the triangle, not the revenue rectangle. Consumer and producer surplus both shrink, but the rectangle isn't lost; it moves to the government.

So what just happened in the numbers? One: revenue is $3 times 300, a $900 rectangle that moves to the government. Two: deadweight loss is one half times 3 times 100, a $150 triangle that no one gets.

The rule: if a competitive market already makes the efficient quantity, any tax, subsidy, or price or quantity control can only lower efficiency. Not every tax shifts supply. A lump-sum tax is one flat fee, however many pizzas sell. The cost of one more pizza doesn't change, so in the short run price and quantity stay put.

A per-unit subsidy is the reverse: the government pays sellers a set amount for each unit. Say $3 a pizza. Sellers now need $3 less from buyers, so supply shifts down by $3, which means right, meeting demand at 500 pizzas. Buyers pay $10; sellers get $10 plus $3, or $13. Exam trap: buyers pay $10, not the old $12 minus $3. The $3 splits the same way as the tax: buyers gain $2 per pizza, and sellers gain $1. Paying buyers instead shifts demand up by $3, with the same result.

So what just happened with the subsidy? One: supply shifted down by $3, which is to the right. Two: quantity rose from 400 to 500. Three: buyers pay $10, not $12 minus $3, and sellers get $10 plus the $3, or $13.

The government pays $3 on each of the new 500 pizzas, not the old 400: $3 times 500 is $1,500 a day. On the graph, that cost is the whole rectangle from $10 up to $13, out to 500. It crosses the old $12 line, so shade all of it. Exam trap: a subsidy creates deadweight loss too, even though the new curves cross. The true cost of each pizza is still the original supply curve. The 500th pizza costs $13 to make, but buyers value it at only $10. Those extra 100 pizzas leave a triangle: one half times 3 times 100, or $150 a day.

Buyers and sellers both gain surplus, but the government spends $150 more than they gain.

So what just happened to the budget? One: the government pays $3 on each of the new 500 pizzas, $1,500, the whole rectangle from $10 to $13. Two: the extra 100 pizzas past the efficient 400 cost more than they're worth: a $150 deadweight loss.

Quick check. Before a per-unit tax, a good sold for $8. After it, buyers pay $9, sellers keep $6, and 200 units sell. What are the tax and the tax revenue? A) $1 and $200. B) $3 and $600. C) $3 and $1,800. D) $2 and $400.

B. The tax is the gap between what buyers pay and sellers keep: $9 minus $6 is $3. Revenue is $3 times 200, or $600. The trap is A: $1 is only how much the buyers' price rose. And sellers lost $2 to buyers' $1, so here supply is the less elastic side.

The government pays sellers $2 for every unit sold in a competitive market that was producing the efficient quantity. Which is true? A) Quantity falls, and there's deadweight loss. B) Quantity rises, and total surplus rises. C) Quantity rises, and there's deadweight loss. D) Quantity rises, and there's no deadweight loss at the new equilibrium.

C. The subsidy pushes quantity past the efficient amount, so some units cost more to make than they're worth to buyers. The trap is D: the new curves cross, but the true cost is still the original supply curve. B misses that the government pays more than buyers and sellers gain.

Recap: a per-unit tax is the vertical gap between the curves, and the less elastic side bears more. Revenue is the rectangle; deadweight loss, the triangle. A subsidy costs its amount times the new quantity, and leaves a triangle too. Next: Module 2.8B, Government Intervention: Price and Quantity Controls.