Market Prices and Elasticity

Prerequisites: Economic Choice and Analysis · Specialization, Comparative Advantage, and Exchange

Will a breakfast shop increase its revenue by raising the price by one yuan? Don’t rush to answer. The owner might sell fewer units, or they might coincidentally encounter the opening of a new office building, bringing in a batch of new customers. Price and sales figures alone do not distinguish these underlying changes. This article first explains who is willing to trade and how prices coordinate plans, then asks how large the quantity response actually is.

The following are fictional teaching markets. We temporarily fix product quality, income, and trading rules, excluding taxes and third-party effects; these conditions will be reintroduced one by one later. Distinguishing between “changing only the price” and “other conditions also changing” is a common prerequisite for understanding supply and demand diagrams and elasticity.

How Four Buyers and Four Sellers Choose

Each buyer wants to buy at most one breakfast, with maximum willingness-to-pay values of 12, 10, 8, and 6, respectively. Four independent sellers can each produce one unit, with opportunity costs of 2, 4, 6, and 8. All amounts are in yuan, and quantities are in units. Let’s try a price of 5: all four buyers want to buy, but only the two sellers with costs of 2 and 4 are willing to sell. Some buyers have money but cannot buy.

Here, the buyer’s valuation must be strictly greater than the price, and the seller’s cost must be strictly less than the price; those exactly equal are indifferent, so the table below avoids this boundary.

Assume the goods are identical, information is available, and prices can vary. At a price of 5, buyers who couldn’t buy might be willing to bid higher, and sellers who weren’t making breakfast might join; at a price of 9, sellers with unsold goods might lower prices to win orders. At 7, three buyers with valuations above 7 and three sellers with costs below 7 can transact; one more unit would mean the cost of 8 exceeds the remaining buyer’s valuation of 6.

This explains the pressure for price adjustment, but does not prove that reality always moves along this path. If breakfast is already prepared and the shop cannot change prices at any time, the short-term result might be queues or leftover goods. Equilibrium in the model means that buying and selling plans are compatible, not that “everyone is satisfied” or that “distribution is fair.” People with less money, even if they need breakfast very much, may not have a high monetary willingness to pay.

From Individual Choices to a Demand Relationship

Now, let’s test prices from low to high in sequence, recording the number of buyers at each price level. When the price crosses a person’s maximum willingness-to-pay, they drop out; thus, a few participants form steps rather than a naturally smooth line. The same applies to supply: when the price exceeds a seller’s opportunity cost, they join.

When there are enough participants and differences are fine enough, continuous relationships can approximate these steps. Below, we set up a larger breakfast market, using simplified linear relationships in the relevant price range:

Quantity demanded Qd = 120 − 10P
Quantity supplied Qs = 20 + 10P

This is not an equation derived precisely from the eight people above, nor is it a statistical estimate. It is a teaching model used jointly for the following analysis; for every 1 yuan increase in P, holding other demand conditions constant, 10 fewer units are bought, while sellers are willing to supply 10 more units. Do not arbitrarily extrapolate this local approximation to negative prices or impossible quantities.

Setting the two sides equal, 120−10P = 20+10P, we get P=5, Q=70. Let’s check: demand is 120−50=70, supply is 20+50=70. If the price is temporarily only 4, demand is 80, supply is 60; the difference of 20 is unmet purchase plans, not the quantity already sold. At a price of 6, there is an excess supply of 20 units.

Why Price and Quantity Can Rise Simultaneously

Suppose an office building opens, adding 20 units of demand at every price. Now it is Qd=140−10P, with the original supply relationship unchanged. Solving for the new equilibrium gives P=6, Q=80. The price is higher, and more units are traded, without overturning “when other conditions are fixed, raising the price reduces the quantity demanded.” We are comparing points on two different demand relationships.

Try to distinguish in one sentence: if the owner only changes the marked price from 5 to 6, with no other changes to customers’ conditions, it is moving along the original demand relationship from 70 to 60; new customers joining means demand changes at every price, which is a shift of the entire relationship. Do not automatically assume that only the former change occurs just because the price changed in reality.

First predict whether “new customers” and “supply reduction” will both push prices up, then switch. When supply decreases by 20, the new equilibrium is 6, 60; the direction of price change is the same for both changes, but the direction of quantity change is opposite. If they occur simultaneously, the quantity depends on the magnitude, and one cannot judge solely based on “costs rose and demand is also strong.”

Price ceilings can also be tracked using the same model: when the ceiling is an effective constraint at 4, 80 are willing to buy, 60 are willing to sell. Even if all 60 units are matched, some people still cannot buy. Policy evaluation must also ask who gets the goods, whether queuing is time-consuming, and how supply is supplemented, rather than treating the 80 units of demand as actual consumption or directly equating shortage with total evaluation.

Why Revenue Can Increase Even When Sales Decrease

The direction of price is clear, but how much “sales decrease” determines the operating revenue after a price hike. Now focus on a single store’s pricing comparison, explicitly switching to another set of store-level demand settings: original price 10, selling 100 units, operating revenue 1000; new price 12. The store’s demand is different from the overall market demand; whether customers can switch to other stores is a key condition.

First guess: if 90 units are still sold, is the revenue less than before? Break the change into two parts, so there is no need to guess the total based on intuition.

1. For the remaining 90 units, 2 yuan more is collected per unit, collecting an extra 180.
2. For the lost 10 units, the original payment of 10 yuan per unit is lost, losing 100.
3. Netting the two, revenue increases by 80; directly verify 12×90=1080.

If only 70 units remain, 140 is collected extra, but 300 is lost, resulting in a net decrease of 160, with revenue becoming 840. Note that the unsold portion is valued at the original price; if both parts are valued at the new price, the change will be double-counted.

The green area is the revenue increase from the price hike on retained sales, and the orange area is the original revenue lost due to vanished sales. Change the retained quantity from 90 to 70, and observe which of the two areas is larger. The threshold for unchanged revenue is 12Q=1000, meaning retaining about 83.33 units; continuous quantity is only an analytical approximation, and when selling whole units, it corresponds to the sides of adjacent integers.

Elasticity Places Reactions of Different Scales on the Same Scale

Saying “sales decreased by 10 units” has different meanings for a store selling 20 units a day and one selling 1000 units a day. Price elasticity of demand compares the percentage change in quantity to the percentage change in price. When comparing two endpoints, we use the midpoint method, so that calculating forward and backward uses the same baseline:

Quantity percentage change = (Q1−Q0) / ((Q1+Q0)/2)
Price percentage change = (P1−P0) / ((P1+P0)/2)
Absolute price elasticity of demand = |Quantity percentage change / Price percentage change|

From 10 to 12, the price change rate is 2/11≈18.18%. When 90 units remain, the quantity change rate is −10/95≈−10.53%, and the absolute value of elasticity is about 0.58; when 70 units remain, it is |(-30/85)/(2/11)|≈1.94. The former is less than 1, meaning the quantity percentage response is weak, called inelastic; the latter is greater than 1, called elastic.

Unit elasticity is the threshold where revenue increase and decrease exactly offset each other. It applies to this price change, this market, and this time period, not as a permanent label for the product. Changing “units” to “thousands of units” changes the numerical slope of the curve but does not change the percentage elasticity; on the same straight line, different base points yield different percentage responses.

Operating revenue is still not profit. Even if lowering the price increases sales and revenue, making more breakfasts consumes raw materials and labor. The next article will first look at the gains and losses of both parties in trade, and the business section will add costs to pricing; do not treat this revenue chart as an optimal pricing device.

Reintroducing a Real Condition: Substitutes and Time

If a similar store opens nearby, customers of this store are more likely to leave; studying the reaction of “this store’s breakfast” is not the same as studying the reaction of “breakfast in the whole city.” Short-term commuters may still take the original route, but after a few months, they can change their commuting and dining habits, so elasticity also has a time scope. On the supply side, equipment is fixed in the short term but can be expanded in the long term, so the reaction period must also be distinguished.

Income elasticity asks how much is bought when income changes, and cross-price elasticity asks how much is bought when the price of another good changes. They change the conditions of this good’s demand relationship, so do not confuse them with elasticity calculated under “this good’s price changes, other conditions remain constant.” The ratio of price to sales volume over two years in reality, if it simultaneously includes changes in income, population, and quality, has not yet identified the effect of price itself.

Try a Different Scenario

Ticket prices drop from 20 yuan to 18 yuan, and sales volume increases from 100 to 110. Does selling ten more tickets mean revenue increases? First, split the payment changes, then calculate the midpoint elasticity.

In a certain city, rent increased by 10% and the number of rentals increased by 5%. Someone calculated an elasticity of +0.5 and said the demand curve is upward sloping. What evidence do you need to supplement?

Sources and Further Reading

These references support concepts and statistical definitions; the numerical cases and diagrams are original synthetic teaching examples.

CORE · Firms and Markets for Goods and Services · OpenStax · Price Elasticity