Separate percentage responses, legal remittance, and economic burden.
By the end: Explain why changing who sends a tax payment does not necessarily change who bears its cost.
Price elasticity of demand compares the percentage change in quantity demanded with the percentage change in price. Its sign is usually negative; economists often discuss its absolute magnitude. A 2% quantity decline after a 10% price increase gives an approximate magnitude of 0.2. A 20% quantity decline gives about 2. The calculation must specify the comparison point or midpoint convention for larger changes.
Elasticity differs from a graph's slope because it accounts for units and starting levels. Changing kilograms to grams alters a slope without changing the underlying responsiveness. Time horizon matters too: households may have few immediate alternatives to a fuel but more alternatives after changing vehicles, homes, or work arrangements. Supply can also become more responsive as capacity adjusts.
A change of ten units is large for a shop selling twenty units and small for an industry selling ten thousand. Percentage changes divide the change by a reference quantity before comparing responsiveness. Suppose a paper market has a price of ten currency units per item and demand of two hundred items per week. Price rises to eleven and desired purchases fall to 190, with other demand conditions fixed. The price increase is one divided by ten, or ten percent. The quantity reduction is ten divided by two hundred, or five percent. Dividing negative five percent by ten percent gives an initial-base elasticity of negative one half.
Using final values as denominators changes the reported percentages. The same price change is one divided by eleven, about 9.09 percent; the quantity change is negative ten divided by 190, about negative 5.26 percent. Their ratio is about negative 0.579. A midpoint convention instead divides by the average price of 10.5 and average quantity of 195. Price changes about 9.52 percent and quantity about negative 5.13 percent, giving elasticity about negative 0.538. None changes the transactions being compared. They answer slightly different finite-change questions. Naming the convention prevents a disagreement about denominators from masquerading as different evidence about behavior.
A slope divides a change in quantity by a change in price, or the reverse for an inverse schedule. Its units matter. If the two hundred weekly items are measured as twenty boxes containing ten items each, the quantity reduction is one box rather than ten items. The numerical quantity slope changes by a factor of ten. The percentage reduction remains five percent because one box is five percent of twenty boxes. Elasticity therefore permits comparisons across measurement units that raw slopes do not. It still depends on the comparison point and on which other conditions are held constant.
Responsiveness is about alternatives under particular constraints. A household might reduce a purchase by switching products, using less, postponing replacement, or leaving the market entirely. A supplier might reduce output, redirect capacity, or change technology. A weekly response with installed equipment can differ from a response after leases expire and new equipment becomes available. These are mechanisms that can change elasticity, not numerical estimates of it. The lab supplies no time-horizon control, substitute-good slider, or observations from which to estimate a household response. Its preset slopes are assumed characteristics of constructed markets.
For an empirical elasticity, the demand condition matters as much as the division. Price eleven and sales 190 could result partly from an income decline that shifts demand. Dividing those observed changes would still produce a number, but it would not isolate the response to price along a fixed schedule. A credible estimate needs a reason that other demand determinants did not cause the relevant variation, or a research design that separates them. The accounting of percentage changes is exact once a denominator is chosen. Interpreting their ratio as a causal behavioral response requires additional assumptions and evidence.
Interpreting price 11 and sales 190
| Comparison | Mechanism | Evidence needed |
|---|---|---|
| Fixed demand | Price changes purchases | Other demand held fixed |
| Income also falls | Demand schedule shifts | Separate income effect |
| Longer horizon | Replacement now feasible | Adjustment time matched |
With a tax of 20 per unit, buyers pay a price 20 higher than sellers receive net of the tax. The quantity must satisfy demand at the buyer price and supply at the seller price. With the baseline schedules, quantity falls from 50 to 40, buyers pay 80 rather than 70, and sellers receive 60 rather than 70. The per-unit burden is split equally in this particular case.
If the tax is legally remitted by buyers rather than sellers, the same wedge conditions apply under identical enforcement and market assumptions. Actual administrative costs, evasion, salience, or bargaining can break that equivalence. Legal responsibility for paying the authority and the economic loss relative to the no-tax counterfactual are different concepts.
A per-unit tax names a currency amount for every item sold, unlike a percentage tax calculated as a fraction of a tax base. In the baseline lab, the tax is twenty currency units per item. Buyers and sellers now face different prices for the same transaction. Buyers decide purchases using what they pay, while sellers decide offers using what they keep after remittance. Clearing requires that these choices yield the same quantity. Subtracting the seller's net price from the buyer's price must also leave twenty. Those conditions jointly determine quantity and both prices under the specified schedules.
Derive the displayed baseline rather than adding twenty to the old price. Before the tax, the difference between the demand and supply intercepts is one hundred. Each additional unit reduces that gap by two because demand price falls by one and supply price rises by one. Reserving twenty of the gap for tax leaves eighty to close through quantity. Eighty divided by two gives forty units per period. Demand at forty gives buyer price eighty, and supply gives seller net price sixty. Relative to the no-tax seventy, buyers pay ten more and sellers receive ten less on each continuing trade.
Suppose sellers legally remit the tax. For each item they collect eighty from the buyer, send twenty to the authority, and retain sixty. Across forty items, buyers pay 3,200, sellers remit eight hundred, and sellers retain 2,400 before their production costs. The three amounts reconcile because 3,200 equals eight hundred plus 2,400. Remittance is the transfer to the authority. The economic price burden is the change relative to seventy, not the fact that the seller's bank account processed a payment of eight hundred. Production costs and lost trades enter a broader welfare comparison separately.
Now let buyers remit while leaving schedules and enforcement unchanged. The seller receives sixty directly. The buyer pays that sixty plus twenty to the authority, again spending eighty per item. Forty sales give sellers 2,400 and the authority eight hundred, with the same combined buyer expenditure of 3,200. The recipient of the initial payment changes, but the prices relevant to purchase and supply choices do not. This equivalence relies on identical effective enforcement, information, and costs. It is a conditional equilibrium result, not a statement that every actual tax collection arrangement has identical consequences.
Use the Seller sends tax and Buyer sends tax buttons to isolate that distinction. They alter the displayed remitter but do not change the quantity, burden bars, or prices. Selecting a market preset or moving Tax per unit changes economic assumptions instead. These controls update the comparison immediately; there is no sequence of past tax payments, budget balance, or calendar-time response. A statutory tax on a company also need not be borne entirely by its shareholders. Workers, customers, suppliers, and owners may face adjustments in a richer model. This experiment separates only the buyer and seller sides of one competitive market.
Use matching quantity and price changes, with other demand conditions held fixed. For large changes, state the percentage base or use a midpoint convention.
Current demand slope b: 1; supply slope d: 1.
| Measure | No tax | Selected tax |
|---|---|---|
| Buyer price | 70 | 80 |
| Seller net price | 70 | 60 |
| Quantity | 50 | 40 |
| Tax revenue | 0 | 800 |
Buyer price minus seller price: 20. Seller remits the tax; the economic outcome is unchanged by this choice under the assumptions.
| Baseline market | No tax | Tax 20 |
|---|---|---|
| Buyer price | 70 | 80 |
| Seller net price | 70 | 60 |
| Quantity | 50 | 40 |
| Tax revenue | 0 | 800 |
Select Buyers adjust less and inspect the no-tax column before interpreting the tax result. Inverse demand now falls three price units per additional unit of quantity, while supply still rises by one. The original intercept gap remains one hundred, but quantity closes it at four price units per item. Without tax, twenty-five units clear at price forty-five. With tax twenty, the remaining gap of eighty supports twenty units. Buyers then pay sixty, while sellers keep forty. Compared with this market's no-tax forty-five, buyers bear fifteen per continuing item and sellers five. Tax revenue is twenty times twenty, or four hundred per period.
The steeper demand line means a small quantity reduction can accommodate a relatively large buyer-price increase. Buyers cannot reduce quantity much without moving far up their willingness-to-pay schedule. That is the stipulated sense in which they adjust less. The burden split is fifteen twentieths to buyers and five twentieths to sellers, or seventy-five and twenty-five percent of the per-unit tax. These percentages refer to price changes on continuing trades. They are not automatically shares of each side's total welfare loss, since buyers and sellers also lose gains on the five transactions that disappear.
Next select Sellers adjust less. Inverse demand falls by one per extra unit, while supply rises by three. No-tax quantity is again twenty-five, but no-tax price is ninety-five. At tax twenty, quantity is twenty, buyer price is one hundred, and seller net price is eighty. Buyers pay five more than ninety-five; sellers retain fifteen less. Revenue is again four hundred. The mirrored quantities and reversed burden shares follow from exchanging the two slopes while keeping their sum four. The different no-tax prices follow from the distinct intercepts. Comparing one hundred with baseline seventy would confuse a new market with a tax effect.
For a controlled exploration, return to Equal responses and move Tax per unit from twenty to forty. The slider permits values between zero and forty and retains the selected slopes. At forty, quantity is thirty, buyer price ninety, seller net price fifty, and revenue 1,200. Moving to zero restores quantity fifty and common price seventy for these slopes. Choosing a named preset replaces the tax and both slopes with that preset's settings. It does not necessarily reset the remitter selection, which is separate state. There is no Reset playback button because the comparison has no playback or accumulating history.
A useful numerical check works for every displayed choice. Add the increase in buyer price to the reduction in seller net price, each measured against that choice's no-tax price. The sum must equal the selected per-unit tax. Then multiply the tax by the selected quantity to verify revenue. Neither calculation proves that the market assumptions describe an actual industry. It checks consistency within the model. In particular, the presets' slopes are not estimated elasticities, and changing them changes the no-tax benchmark as well as the burden split. A real incidence study would need evidence on substitution, supply adjustment, enforcement, and the relevant horizon.
The side that can adjust less tends to bear more of a tax in a competitive partial-equilibrium model. In the lab, steeper inverse demand gives buyers less quantity response per unit price change. Steeper inverse supply does the analogous thing for sellers. Each preset is a different market, so its no-tax price and quantity are shown separately rather than silently held fixed.
Tax revenue is the tax per unit multiplied by the quantity actually traded. It is a transfer to government, not automatically a destroyed social resource. When the original market has no externalities or other distortions, lost mutually beneficial trades can create deadweight loss. A corrective tax can instead improve outcomes when private transactions impose unpriced costs. Wider effects on wages, profits, and spending require more than this one-market model.
The distinction between transfers and forgone gains becomes concrete with a paper welfare calculation using the continuous baseline schedules. Before tax, buyer surplus is a triangle with width fifty and height fifty, so it equals 1,250 currency units per period. Seller surplus has the same dimensions and value. Total private surplus is 2,500. After tax twenty, forty units trade. Buyers pay eighty against a highest demand valuation of 120, leaving a triangle with width forty and height forty. Their surplus is eight hundred. Sellers receive sixty against a supply intercept of twenty, giving the same eight hundred of seller surplus.
Buyer surplus falls by 450 and seller surplus falls by 450. Together they lose nine hundred. The authority receives eight hundred, which remains purchasing power rather than vanishing as a resource cost. Adding post-tax private surplus of 1,600 to revenue of eight hundred gives 2,400. Compared with the original 2,500, the difference is one hundred. This is the modeled deadweight loss. The calculation treats the revenue as a transfer valued at its currency amount and assumes no administrative costs or unpriced effects. Judgments about how the authority uses the revenue require an additional analysis.
The missing one hundred also follows directly from lost trades. Quantity falls by ten. At forty units, buyers' marginal valuation exceeds sellers' marginal opportunity cost by twenty; at fifty that gap is zero. The lost gains form a triangle with width ten and height twenty, whose area is one hundred. It is incorrect to multiply the ten lost units by the full buyer price eighty. Most of that price would have paid for production resources or transferred income. Deadweight loss counts the net gain above opportunity cost that disappears, not gross receipts that would have changed hands.
Consider a contrasting paper case with the same private schedules but external harm of twenty for each item produced. Assume constant harm, complete enforcement, and no other effects. At fifty units, total harm is one thousand, so private surplus 2,500 minus harm gives a social total of 1,500 under this valuation. At forty units, private surplus plus revenue is 2,400, while harm is eight hundred. The social total is therefore 1,600. The reduction in harm of two hundred exceeds the private deadweight loss of one hundred. Relative to this specified benchmark, the tax raises the modeled social total by one hundred.
That contrasting calculation is not an externality switch in the widget and does not establish a measured environmental benefit. It demonstrates why a welfare conclusion depends on which costs were omitted from the initial private market. Distribution remains another question. An equal per-unit burden can consume different fractions of income for rich and poor buyers, and the spending of revenue can shift net benefits further. An incidence claim should therefore name the tax base, counterfactual prices, affected people, period, and omitted adjustments. The price split, welfare reconciliation, empirical effect, and judgment about acceptable burdens answer different questions even when each uses the same tax amount.
A tax of 12 per unit reduces sales from 100 to 80 units. What is tax revenue? Why is 1,200 wrong?
Use the post-tax quantity.
| Calculation | Result |
|---|---|
| 12 × 80 actual units | 960 |
| 12 × 100 old units | 1,200 |
| Overstatement | 240 |
Revenue is 960. Multiplying 12 by the old 100 assumes no quantity response and overstates revenue by 240. This calculation says nothing yet about administrative costs or the total welfare effect.
In a market with few short-run substitutes for buyers but easy redeployment for suppliers, which side is more likely to bear most of a small per-unit tax under competitive assumptions?
Compare each side's ability to avoid the taxed transaction.
| Side | Ability to adjust | Likely relative burden |
|---|---|---|
| Buyers | Low | Higher |
| Suppliers | High | Lower |
Buyers are more likely to bear most through a higher purchase price. Suppliers can withdraw more readily rather than accepting a large reduction in net price. This is conditional on the model and does not follow merely from who files the tax return.
The lessons and answers stand alone. These optional references identify underlying models, evidence, and perspectives. All numerical examples here are invented unless explicitly attributed; none is a forecast or a personal investment recommendation.
Competitive schedules, surplus, taxation, and limits of equilibrium reasoning. All numerical curves in these lessons are separately constructed teaching examples.