Elasticity

This is Unit Four of Principles of Microeconomics, part two of two: elasticity over time, the two elasticities read by their sign, and a tax, where elasticity decides who pays and how much is lost. Carry the same idea as part one: the size of a response decides the outcome. Here it decides who bears a tax, how much the government collects, and how much surplus a tax destroys; and it does so whichever side the law names. By the end you will be able to say why supply is more elastic over time. To compute income and cross-price elasticities and classify a good by the sign. To split a tax's burden between buyers and sellers by their elasticities. To compute a tax's revenue and deadweight loss, and show that a buyer's tax and a seller's tax give the same result. And to work one excise tax fully, with the elasticities reversed.

Four entries on time and the other elasticities: supply over time, income elasticity, cross-price elasticity, and classifying by sign.

Why time matters most here.

MIC 0318 Time and the Elasticity of Supply

Supply is more elastic the longer the period. In months, sellers can only work existing capacity harder; in years they can build, convert and enter. A rent rise adds a few attic flats this year and whole buildings in five. A demand shift that meets short-run supply moves the price sharply; the same shift against long-run supply moves quantity instead.

Responsiveness to income.

MIC 0319 Income Elasticity of Demand

Income elasticity of demand is the percentage change in quantity demanded at a given price divided by the percentage change in income. Positive means a normal good; negative, inferior. A ten percent income rise raising restaurant meals fifteen percent gives one point five. It measures a shift of the demand curve, not a movement along it, since the price is held fixed.

Responsiveness to another good's price.

MIC 0320 Cross-Price Elasticity of Demand

Cross-price elasticity is the percentage change in the quantity of one good demanded divided by the percentage change in the price of another. Positive means substitutes, negative complements, and near zero unrelated. Dearer tea raising coffee demand gives a positive number. The sign carries the classification and is never dropped here; elastic and inelastic are not used for these.

Classifying from the numbers.

MIC 0321 Classifying Goods by the Sign

Two elasticities are read by sign, not size. Income elasticity, quantity's percentage change over income's: positive is normal, negative inferior. Cross-price elasticity, one good's quantity change over the other good's price change: positive means substitutes, negative complements, zero unrelated. Incomes rise ten percent and bus rides fall three percent: minus zero point three, inferior. Bagel prices rise ten percent and cream cheese demand falls four percent: minus zero point four, complements. One streaming service's price rises ten percent and a rival's subscriptions rise six percent: plus zero point six, substitutes. The error is to drop the sign out of habit, which turns an inferior good into a normal one, or to divide by the good's own price instead of the other good's.

Three entries on a tax: the wedge it drives, who bears the burden, and splitting it with numbers.

What a tax does to a market.

MIC 0322 A Tax Drives a Wedge

A per-unit tax puts a wedge equal to the tax between what buyers pay and what sellers keep, and the market settles where buyers' quantity at the higher price equals sellers' quantity at the lower. The wedge is a vertical gap of the tax's height between the curves. Neither curve is redrawn, and the wedge works whichever side the law names as payer.

Who actually pays.

MIC 0323 Tax Incidence: Who Bears the Burden

Tax incidence is how the burden divides between buyers and sellers, and the more inelastic side bears more: inelastic demand puts most on buyers, inelastic supply puts most on sellers. A cigarette tax lands mostly on smokers; a beachfront hotel tax lands mostly on the hotels. Who writes the check is irrelevant; only the two elasticities decide the split.

Splitting the burden with numbers.

MIC 0324 Splitting the Burden

Find three prices: before the tax, what buyers pay after, and what sellers receive after. Buyers' share of each unit's tax is the after-tax buyer price minus the old price; sellers' share is the old price minus the after-tax seller price; the two add to the tax. The larger share falls on the side less able to change its quantity. Bottled juice clears at three dollars before a one-dollar tax; after it, buyers pay three seventy and sellers keep two seventy, so buyers bear seventy cents and sellers thirty, and demand is the more inelastic side. Hotel rooms: a twenty-dollar tax moves the buyer price from one hundred to one hundred five and the seller price to eighty-five, so buyers bear five and sellers fifteen, and supply is the inelastic side. The shares must add to the tax, and each is per unit, not a total.

Four entries on what a tax collects and destroys: revenue, the deadweight loss, its computation, and why the named payer does not matter.

What the government collects.

MIC 0325 Tax Revenue and Elasticity

Tax revenue is the tax per unit times the after-tax quantity: a rectangle the wedge high and the new quantity wide. More elastic curves mean a smaller quantity and less revenue, so the same tax raises far more on an inelastic good than on an elastic one. A revenue-raising tax and a consumption-reducing tax want opposite elasticities.

What the tax destroys.

MIC 0326 The Deadweight Loss of a Tax

The units the wedge stops were worth more to buyers than they cost sellers. That lost surplus is the tax's deadweight loss: the triangle between the curves over the lost quantity, beside the revenue rectangle. The loss grows with elasticity, as more units drop out, and faster than the tax: doubling the tax roughly quadruples the loss.

Measuring it.

MIC 0327 Computing a Tax's Deadweight Loss

With straight-line curves the loss is a triangle: height the tax, base the lost quantity. Find the quantity before the tax and after, subtract for the lost quantity, and take one half of the tax times that. Revenue sits beside it, the tax times the after-tax quantity. Bottled juice traded twelve thousand bottles a week at three dollars; a one-dollar tax moves buyers to three seventy and sellers to two seventy, and the quantity falls to nine thousand. The lost quantity is three thousand, so the loss is one half times one dollar times three thousand, fifteen hundred dollars a week; revenue is nine thousand dollars, and the two sides lost ten thousand five hundred of surplus in all. The base is the change in quantity, not the after-tax quantity, and the loss is not the revenue: revenue still exists, the triangle does not.

Why the named payer does not matter.

MIC 0328 Buyer's Tax or Seller's Tax: Same Result

Work a tax two ways. Levied on sellers, it lifts the supply curve by the tax. Levied on buyers, it lowers the demand curve by the tax. Either way the curves meet at the same quantity, buyers end up paying the same price, and sellers end up keeping the same price: one wedge from either side. In the juice market, a one-dollar tax on sellers clears at nine thousand bottles with buyers paying three seventy at the register; the same tax charged to buyers clears at nine thousand with a register price of two seventy, and buyers add the dollar to reach three seventy. Sellers keep two seventy in both cases. The posted price differs, which makes the error tempting; the burden does not move with the posting, so compare what buyers pay in total and what sellers keep.

Two entries to close: elasticity in factor markets, and one excise tax worked with the elasticities reversed.

Elasticity in factor markets.

MIC 0329 Elasticity of Labor Supply and of Saving

Labor supply elasticity is the percentage change in hours offered over the percentage change in the wage: high for teenagers, low and sometimes negative for the highest paid. Saving's response to the interest rate is measured the same way and is small. A negative value is unit three's backward bend as a number; both estimates are attributed and vary by study.

One tax fully worked.

MIC 0330 One Excise Tax, Fully Worked

A single excise tax exercises every idea in this unit, and working it twice with the elasticities reversed shows what elasticity decides. Start with the market before the tax. Insert the wedge; quantity falls until the two prices differ by the tax. Read four things. Incidence: buyer price minus old price is buyers' share, old price minus seller price is sellers', and the larger share falls on the more inelastic side. Revenue: the tax times the new quantity, a rectangle that was surplus and is now the government's. Deadweight loss: half the tax times the lost quantity, a triangle that was surplus and is now nobody's. Then reverse the elasticities, and the same tax shifts most of the burden to the other side, cuts quantity by more, raises less revenue and destroys more surplus, because the more elastic side escapes. Juice, inelastic demand: twelve thousand bottles at three dollars; after a one-dollar tax, nine thousand, buyers pay three seventy and sellers keep two seventy; buyers bear seventy cents, sellers thirty; revenue nine thousand, loss fifteen hundred. Elastic demand and inelastic supply, same start: seven thousand bottles, buyers pay three twenty-five and sellers keep two twenty-five; buyers bear twenty-five cents, sellers seventy-five; revenue seven thousand, loss two thousand five hundred. Same tax, two thousand dollars less revenue and a thousand more destroyed. Incidence is about elasticity, never about who remits; revenue and loss move opposite ways as elasticity rises; and the triangle's base is the lost quantity, not the quantity that remains. Unit eleven returns to the wedge as a remedy.

Look at what you can now do. You can say why supply is more elastic over time: in months sellers work capacity harder, in years they build and enter. You can compute income and cross-price elasticities and read the sign first: positive normal or substitute, negative inferior or complement. You can split a tax's burden from three prices, and say that the inelastic side pays more whoever writes the check. You can compute revenue as the tax times the after-tax quantity and the loss as half the tax times the lost quantity, and show that a buyer's tax and a seller's tax are one wedge. And you can work an excise tax twice and watch elasticity move the burden, the revenue and the loss. Unit five turns from the market to the buyer, and asks how a consumer chooses.