Who Really Bears a Tax? Tax Incidence, Elasticity, and Deadweight Loss

A tax law may name a buyer, a seller, or an employer as the person who sends money to the government. But that legal instruction does not yet tell us whose economic position becomes worse. To find that split, we have to watch the market adjust. Here is a numerical market. Demand is one hundred minus two Q, and supply is twenty plus Q. Demand records the highest price buyers will pay for each quantity. Supply records the lowest price sellers will accept. Before the tax, the two curves meet where quantity is about twenty-six point seven and price is about forty-six point seven. At that point, the price paid by a buyer is exactly the price received by a seller. Now impose a tax of fifteen dollars on every unit traded. A trade can occur only if the buyer is willing to pay fifteen dollars more than the seller needs to receive. The tax therefore drives a vertical gap between the two prices. Algebraically, take the price on demand and subtract the tax before comparing it with supply. Solving gives a new quantity of about twenty-one point seven. Five units that used to trade no longer do. At the new quantity, buyers pay about fifty-six point seven dollars. Sellers receive about forty-one point seven after the tax is accounted for. Those prices differ by exactly fifteen dollars. Compare each new price with the old price of forty-six point seven. Buyers now pay ten dollars more, while sellers receive five dollars less. Together, ten plus five is the full fifteen-dollar tax. So this tax is not automatically split fifty-fifty. In this market, buyers bear ten dollars per unit and sellers bear five. What determines that split is how strongly each side changes its behavior when its price changes. Now separate the economics from the paperwork. Suppose the law tells the seller to collect fifteen dollars and remit it. Demand must then cover the seller's required receipt plus the tax. Alternatively, suppose the law tells the buyer to send fifteen dollars. Then the buyer's willingness to pay, net of that payment, must cover what the seller requires. Rearranged, this is the same equation. Either legal rule produces the same wedge: the buyer price minus the seller price equals fifteen. The same quantity trades, buyers pay the same amount, and sellers receive the same amount. The person named on the tax form has statutory incidence. The people whose real prices change have economic incidence. Supply and demand determine the second one, even when legislation determines the first.

Responsiveness asks how much buying or selling changes when price changes. Economists summarize that responsiveness with elasticity. For incidence, the intuition is simple: the side that can most easily change quantity can escape more of the burden. Begin with buyers who respond only weakly to price. The green supply curve is twenty plus Q. The steep blue demand curve is one hundred minus three Q, and the original equilibrium is quantity twenty at price forty. Impose a twelve-dollar tax. Quantity falls only from twenty to seventeen. Buyers pay forty-nine, nine dollars above the old price. Sellers receive thirty-seven, only three dollars below it. The two colored segments make the full twelve-dollar wedge. Nine dollars sits above the old market price and only three sits below it. Buyers bear three quarters of the tax because their quantity barely responds. Now keep supply and the twelve-dollar tax exactly the same, but make buyers much more responsive. The flatter demand curve still crosses supply at quantity twenty and price forty, so the comparison starts from the same place. With responsive buyers, even a modest increase in their price causes a large reduction in purchases. Taxed quantity falls from twenty to twelve, a loss of eight units rather than three. Buyers now pay forty-four, only four dollars above the old price. Sellers receive thirty-two, eight dollars below it. The same twelve-dollar tax has shifted most of its burden to sellers. Why can responsive buyers avoid more of the tax? They reduce purchases sharply when their price rises. To keep enough buyers in the market, sellers must absorb more of the wedge through a lower net price. The table puts the contrast side by side. Weak buyer response produced a small quantity change and a nine-dollar buyer burden. Strong response produced a larger quantity change and only a four-dollar buyer burden. The lesson is not that buyers always bear more or sellers always bear more. The lesson is conditional: other things equal, the side less able to change quantity, find an alternative, or walk away bears more of the tax.

The same logic applies to sellers. A seller may be unable to reduce production quickly because equipment, land, training, or contracts are already committed. Another seller may be able to redirect resources almost immediately. Start with weakly responsive sellers. The blue demand curve is eighty minus Q. The steep green supply curve is three Q, and they meet at quantity twenty and price sixty. Apply the same twelve-dollar tax. Quantity slips only to seventeen. Buyers pay sixty-three, three dollars above the old price, while sellers receive fifty-one, nine dollars below it. Sellers bear three quarters of this tax. Their quantity changes very little when their net price falls, so a large price reduction is needed before they withdraw even a few units from the market. Now keep demand, the original equilibrium, and the twelve-dollar tax the same. Replace steep supply with a flatter curve. These sellers change quantity much more strongly when the price they receive changes. Taxed quantity now falls to twelve. Buyers pay sixty-eight, eight dollars above the old price. Sellers receive fifty-six, only four dollars below it. Responsive sellers can withdraw more output rather than accept a much lower net price. Buyers must offer a larger price increase to keep those units in the market, so most of the tax moves to the buyer side. Notice the symmetry. When buyers were relatively stuck, buyers bore more. When sellers are relatively stuck, sellers bear more. Incidence follows relative ability to adjust, not the name printed on the tax bill. The table records the reversal. Weak seller response leaves nine dollars of burden with sellers. Strong seller response cuts that share to four dollars and shifts eight dollars toward buyers. Elasticity does not mean that one side feels no pain. Both sides can lose. It tells us how the total wedge is divided and how far quantity contracts while the market searches for a new equilibrium.

A tax does more than transfer purchasing power from private traders to the government. It also prevents some trades. To see those missing trades, return to demand and supply before any tax is imposed. At the original equilibrium, every unit up to quantity twenty-six point seven has a buyer whose willingness to pay is at least the seller's cost. The last traded unit is where those two values are equal. Now begin with a six-dollar tax. The yellow segment is the tax wedge and the blue rectangle is tax revenue: six dollars collected on every unit that still trades. The red triangle is different. It lies between the taxed quantity and the original quantity. In that interval, buyers value each unit more than it costs sellers to provide, but the tax prevents the trade. Those are mutually beneficial trades that no longer occur. Their lost gains are called deadweight loss. This is a loss of total surplus, not a payment received by someone else. The deadweight-loss region is a triangle. Its vertical height is the tax t, and its horizontal base is the quantity reduction delta Q. Triangle area is one half times height times base. For these particular straight curves, demand minus supply falls by three dollars whenever quantity rises by one unit. A tax of t therefore reduces quantity by t divided by three. Substitute that quantity change into the triangle formula. Deadweight loss is one half times t times t over three, which is t squared over six. Now double the tax from six to twelve. The triangle becomes twice as high, and the quantity loss becomes twice as wide. Its area rises from six to twenty-four, which is four times as large. Set the tax at eighteen, three times the original six. Height is now three times the original height and the quantity loss is three times the original width. The area becomes fifty-four, nine times the original. That is why deadweight loss does not grow in direct proportion to a tax. One factor of t comes from the wedge itself. A second factor comes from the growing number of trades the tax eliminates. Doubling a small tax roughly quadruples its deadweight loss in this linear setting. Tripling it multiplies the loss by nine. The square law comes directly from the geometry of a triangle whose height and base both grow with the tax.

Payroll taxes are a useful test of the difference between law and economics. A common payroll-tax arrangement names the employer as the party that must remit the payment. But wages and employment can adjust before the final burden settles. Treat labor hours like the quantity in an ordinary market. Employers demand labor, shown by the blue curve. Workers supply labor, shown by the green curve. The vertical axis is the wage per hour. Employer willingness to pay is forty minus one half L. Worker willingness to supply is ten plus one quarter L. Before tax, the curves meet at forty units of labor and a twenty-dollar wage. Now suppose the law requires employers to send six dollars per hour to the government. Employers care about their total labor cost. Workers care about the wage that reaches them. Call the employer's total cost w E and the worker's received wage w W. The tax says their difference must be six. That difference is the yellow wedge in the labor market. Substitute the labor-demand and labor-supply equations. Combining like terms gives thirty minus zero point seven five L equals six. Solving gives employment of thirty-two units, an employer cost of twenty-four, and a worker wage of eighteen. Compare those numbers with the old wage of twenty. Employers now face a cost four dollars higher. Workers receive two dollars less. The six-dollar legal payment has become a four-dollar employer burden and a two-dollar worker burden. Employment also falls from forty to thirty-two. The missing employment relationships are the labor-market version of the mutually beneficial trades lost in any taxed market. Legal incidence asks who sends the payment. Under the stated rule, the employer writes the check for six dollars. That is an administrative fact about collection. Economic incidence asks whose real price changes after the labor market adjusts. Here the employer loses four dollars per hour and the worker loses two through a lower wage. The table compares two possible statutes. In the first, the employer remits the tax. In the second, the worker is legally instructed to remit it from the wage payment. The bookkeeping differs, but the market condition does not. Employers must still spend six dollars more than workers keep. The same demand and supply curves therefore produce the same employment, the same employer cost, and the same worker wage. In actual labor markets, the split depends on how readily firms change hiring and how readily workers change hours, jobs, locations, or labor-force participation. Those are the relevant demand and supply elasticities. So the concrete answer is no. An employer can be the legally designated payer of a six-dollar payroll tax without bearing all six dollars economically. In this example the employer sends six, bears four, and workers bear two through lower pay.