Every tax has a name on the cheque. Sales tax: the shopper pays. Payroll tax: half the bill goes to the employer. A tax on fuel suppliers: the supplier pays. Politicians love to say "we will make the companies pay", and the law really can say that.
But the law only decides who hands over the money. Who actually ends up poorer is a different question, and the answer is settled by something the law never mentions: who has an easier way out of the deal.
Picture a market with a price where buyers and sellers happily meet. Now add a tax of, say, 20 cents per item. Afterwards there are two prices instead of one: what the buyer pays, and what the seller keeps. The gap between them is the tax, and the government takes it.
The buyer's price goes up. The seller's take-home goes down. The question is how the 20 cents splits between them. It does not have to be 10 and 10.
The split is a tug of war. Each side can walk away, and the side that can walk away more easily gets to avoid more of the tax. A buyer who must have the thing (a diabetic and insulin) cannot walk away, so the price they pay climbs a lot. A buyer with plenty of substitutes simply buys something else, and the seller has to cut the price to keep the sale.
Start with the sliders in the middle. Drag the tax up and the cyan and amber boxes grow at the same rate: the price you pay goes up by a bit, and the price the seller keeps goes down by a bit. Now flip the switch from "Sellers" to "Buyers". The dashed line jumps (a seller-side tax lifts the supply curve; a buyer-side tax pushes the demand curve down) but the two boxes do not change at all. It made no difference who the law pointed at.
Now take away the buyers' escape routes. The demand line becomes steep, and the cyan box swallows almost the entire tax. Give the buyers lots of options and it flips: the amber box takes over and the sellers carry the bill. Economists call this willingness to walk away elasticity, and the rule is short: the side that is less elastic pays more of the tax.
In this toy, the lines meet at a fixed point, and the demand slope is b and the supply slope is d (steeper means fewer escape routes). With a tax of t per item, the number of trades falls by t / (b + d). The buyer's price rises by b × t / (b + d) and the seller's take falls by d × t / (b + d). Those two pieces add up to t, which is why the wedge always closes. The buyer's share is simply b / (b + d): if the demand curve is three times as steep as the supply curve, buyers pay three quarters of the tax.
There is a second, quieter cost. The hatched triangle is the set of trades that used to happen and no longer do: a buyer who would pay 45 and a seller who would sell at 44 now cannot meet, because the tax has made the deal cost 46. Nobody gets that money. The government never sees it, and it is not a transfer from one person to another. It is just gone. Economists call it the deadweight loss.
The triangle has a very particular shape: its area grows with the square of the tax. Double the tax and the lost trades do not double, they quadruple. Try it: set the tax to 6, note the hatched area, then set it to 12. This is the main reason many economists prefer broad, low taxes to narrow, high ones.
In the United States, the Social Security payroll tax is 6.2% on the employer and 6.2% on the employee, split evenly on paper. The even split is a legal fiction in exactly the sense above. Most economists who study it find that workers carry most of the employer's half as well, through lower wages than they would otherwise have got, because workers' supply of labour is fairly unresponsive compared with how easily firms can adjust.
In 1990, the US Congress added a luxury excise tax on yachts, charging 10% on the part of the price above $100,000. The idea was that rich buyers would pay. But rich buyers have an easy way out: do not buy the boat, or buy a smaller one. Boat sales slumped and boat builders' workers, who had no easy way out, were hurt. The tax was repealed in 1993. A tax can be aimed at the wealthy buyer and still land on the worker, depending on who has fewer options.
The takeaway. A tax is a wedge between two prices, and the legal label on it is decoration. The burden settles on whichever side finds it harder to walk away, and the more you tax, the more good trades disappear, at a rate that grows with the square of the tax.