Class 10: Market Demand
Optional textbook reading: Varian, Chapter 15: “Market Demand.”
Key Ideas
For eight classes the unit of analysis has been one consumer: a budget, a preference, a favorite affordable bundle, and the demand function that records how that choice responds to prices and income. Firms, governments, and the rest of this course care about a different object: how much the whole market buys at each price. Market demand is the sum of individual demands, taken price by price. Everything Class 7 said about one consumer’s demand carries over to the aggregate. A rise in income shifts market demand out if the good is normal for most buyers, and a change in another good’s price shifts it according to whether the two goods are substitutes or complements on balance. The first problem today shows where the summation is less innocent than it looks.
The second idea is a new way to describe the shape of a demand curve. The slope \(dq/dp\) depends on units: it changes when quantity is measured in cups rather than liters, or price in cents rather than dollars, so a slope of \(-2\) says nothing by itself about how sensitive buyers are. The elasticity of demand, \[ \varepsilon = \frac{dq}{dp} \cdot \frac{p}{q}, \] is a ratio of percentage changes and is unit free. Demand is elastic where \(|\varepsilon| > 1\), inelastic where \(|\varepsilon| < 1\), and unit elastic where \(|\varepsilon| = 1\). Elasticity is the one number a seller needs in order to know whether a price rise will raise or lower revenue; the second and third problems work out why, and why the answer can differ for a market and for a single seller within it.
Problems of the Day
Q31: The Intern’s Formula
Rocco’s juice cart has exactly two customers. Ana’s demand is \(q_A = 10 - p\) and Ben’s is \(q_B = 6 - 2p\). Rocco’s intern adds the two equations and proudly reports the market demand: \(Q = 16 - 3p\). Rocco tests it. At $2 the formula predicts sales perfectly. At $5 it predicts one juice, yet Rocco sells five. “Your algebra is broken,” says Rocco. “Algebra is never broken,” replies the intern.
Who is right? Work out what each customer buys at $2 and at $5, then find and plot the correct market demand at every price. What happens at $3, and why? Write market demand as a piecewise formula and sketch the curve.
Extension: A third customer arrives with demand \(q_C = 4 - 4p\). How many kinks does market demand have now? State the general rule, and say why market demand curves for goods with thousands of buyers look smooth.
Q32: Elasticity Is Escape
The Starlight Theatre charges $20 a ticket and fills 80 seats a night; its demand curve is \(q = 120 - 2p\). The manager is weighing a price rise and wants one number: how many customers would “escape” to alternatives?
Economists measure escape with the elasticity of demand, \[ \varepsilon = \frac{\%\Delta q}{\%\Delta p} = \frac{dq}{dp} \cdot \frac{p}{q}, \] the percentage of customers who escape when the price rises by one percent.
Compute \(\varepsilon\) at \(p = 10, 20, 30, 40, 50\). The slope \(dq/dp\) is the same at all five prices. Why does the elasticity change, and in which direction?
Total revenue is \(TR = p \cdot q(p)\). Use the product rule to write \(dTR/dp\) in terms of \(q\) and \(\varepsilon\).
Show that when \(\varepsilon = -1\), a small price change leaves total revenue unchanged. Then say what a small price rise does to revenue when \(\varepsilon = -1.5\), and when \(\varepsilon = -0.5\). Where on the Starlight’s demand curve is each case? What should the manager charge if she cares only about revenue?
Q33: The Curse of Good Weather
Rain fell perfectly on Millbrook this year, and its hundred wheat farmers hauled in a harvest 10% larger than usual. At the market, old Tomasz groans: “Ruin. I have seen it before.” The market elasticity of demand for wheat is \(-0.5\).
Explain Tomasz’s groan with numbers, working in percentages: roughly what happens to the price of wheat, and to the farmers’ total revenue? Then resolve the paradox that follows: if bigger harvests bring less revenue, why does each individual farmer still fertilize, weed, and pray for her own biggest possible crop? Describe the demand curve facing one small farmer, and compare it with the market demand curve.
Extension: Tomasz proposes a pact: every farmer burns a tenth of their harvest. Would total revenue rise? Why is his pact almost impossible to keep?
Answers
Q31: The Intern’s Formula. Both are right, in a sense, and the intern’s formula is still wrong. At $2 Ana buys \(10 - 2 = 8\) juices and Ben buys \(6 - 4 = 2\), so the market buys 10, and the formula gives \(16 - 6 = 10\). At $5 Ana buys 5 and Ben buys \(6 - 10 = -4\), which is not a quantity of juice. Ben’s demand equation describes him only up to his choke price of $3; above that he buys zero. The intern’s sum \(16 - 3p\) silently lets Ben buy negative juice, and at $5 that cancels four of Ana’s five. The algebra is fine; the mistake was adding a line beyond the range where it means anything.
Market demand is the horizontal sum of the two individual curves: at each price, add the quantities, with each quantity floored at zero. \[ Q(p) = \begin{cases} 16 - 3p & 0 \le p \le 3 \\ 10 - p & 3 \le p \le 10 \\ 0 & p \ge 10. \end{cases} \] At $3 both pieces give \(Q = 7\), so the curve is continuous, but its slope jumps from \(-3\) to \(-1\). That is the kink. Below $3 a one dollar price cut attracts three extra juices, two from Ben and one from Ana; above $3 Ben is out of the market and only Ana responds. In the figure, with price on the vertical axis, market demand runs along Ana’s curve from $10 down to $3 and then swings outward to a flatter segment.
Extension. Carla’s choke price is $1, so there are now two kinks, at $1 and at $3: \[ Q(p) = \begin{cases} 20 - 7p & 0 \le p \le 1 \\ 16 - 3p & 1 \le p \le 3 \\ 10 - p & 3 \le p \le 10. \end{cases} \] The general rule: market demand kinks at every price at which some buyer enters or leaves the market. With \(n\) buyers who have distinct choke prices there are \(n - 1\) interior kinks, one at each choke price except the highest, where demand reaches zero. With thousands of buyers there are thousands of kinks, each one a tiny change of slope spaced a few cents from the next. At any scale you could draw, the curve looks smooth, and treating it as smooth is harmless.
Q32: Elasticity Is Escape. The two expressions agree because a percentage change is a change divided by a level: \[ \frac{\%\Delta q}{\%\Delta p} = \frac{\Delta q / q}{\Delta p / p} = \frac{\Delta q}{\Delta p} \cdot \frac{p}{q}, \] and for a small price change \(\Delta q / \Delta p\) is the slope \(dq/dp\). Elasticity is escape: raise the price by 1% and \(|\varepsilon|\) percent of the customers walk out.
(a) The slope is \(-2\) everywhere, so \(\varepsilon = -2p/q = -2p/(120 - 2p)\). At $10 the theatre sells 100 seats and \(\varepsilon = -0.2\); at $20, 80 seats and \(-0.5\); at $30, 60 seats and \(-1\); at $40, 40 seats and \(-2\); at $50, 20 seats and \(-5\). The slope says that each extra dollar costs two seats, at every price. Elasticity changes because the same two seats mean different things in different places. At $10 a dollar is a 10% price rise and two seats are 2% of a full house; at $50 a dollar is only a 2% rise and two seats are 10% of a thin crowd. Moving up the line, price rises and quantity falls, so \(p/q\) grows without bound: demand is nearly perfectly inelastic where the line meets the quantity axis and becomes infinitely elastic as it approaches the choke price of $60. It is unit elastic at the midpoint, $30.
(b) By the product rule, \[ \frac{dTR}{dp} = q + p \frac{dq}{dp} = q \left( 1 + \frac{dq}{dp} \cdot \frac{p}{q} \right) = q \, (1 + \varepsilon). \]
(c) When \(\varepsilon = -1\) the bracket is zero, so \(dTR/dp = 0\): a small price change leaves revenue where it was. The two effects cancel exactly. Each remaining customer pays 1% more, and 1% of the customers have escaped. For the Starlight that is $30.
When \(\varepsilon = -1.5\), \(dTR/dp = -0.5q < 0\). A 1% price rise collects 1% more from each customer but loses 1.5% of them, and revenue falls. This is the elastic region above $30, for instance $40, where \(\varepsilon = -2\). When \(\varepsilon = -0.5\), \(dTR/dp = +0.5q > 0\). Only half a percent of customers escape for each percent of price, and revenue rises. This is the inelastic region below $30, which is where the Starlight sits now at $20.
The manager should raise the price to $30. Revenue is \(p(120 - 2p)\), which is $1600 at $20 and $1800 at $30, and it falls on either side: at $40 it is back to $1600. Revenue peaks exactly where elasticity passes through \(-1\), which is what (b) says it must.
Q33: The Curse of Good Weather. Elasticity of \(-0.5\) means that each 1% change in quantity requires a 2% change in price in the opposite direction. A 10% larger harvest therefore pushes the price down by about 20%. Revenue is price times quantity, so it changes by roughly \(+10\% - 20\% = -10\%\); the exact figure is \(1.10 \times 0.80 = 0.88\), a fall of about 12%. The farmers of Millbrook worked harder, grew more, and earned less. That is Tomasz’s groan: when demand is inelastic, more of the good means less money for the people who sell it.
The paradox dissolves once we look at one farmer instead of all of them. Each farmer is one hundredth of the market. If she alone grows 10% more, market quantity rises by 0.1% and the price falls by about 0.2%, which she will never notice. Her revenue rises by roughly 9.8%. For her the price is effectively fixed, and the demand curve she faces is nearly horizontal at the market price. Put as an elasticity, her own demand is about \(-0.5 / 0.01 = -50\), extremely elastic, while the market curve is steep at \(-0.5\). The market curve is inelastic because wheat has few substitutes; the curve facing one farmer is elastic because her wheat has a hundred perfect substitutes, namely everyone else’s. Each farmer fertilizes and prays for rain because, taking the price as given, more wheat is more money. The curse falls only when everyone’s prayers are answered at once.
Extension. If every farmer burns a tenth of the crop, market quantity falls 10%, price rises about 20%, and total revenue rises by about \(0.9 \times 1.2 = 1.08\), 8%. The pact works for the group. It fails for the individual. Once the other ninety-nine have burned their tenth and pushed the price up, any single farmer does best by quietly selling her whole crop at the new high price; her extra wheat barely moves the price, and the 10% she was meant to burn is pure revenue to her. Every farmer faces the same temptation, so every farmer cheats, the full harvest reaches the market, and the price falls back. Each farmer’s demand curve is flat, so each one acts as if restricting output is pointless, even though the group’s demand curve is steep and restricting output is exactly what would help. Holding the pact together would require someone to monitor a hundred fields and punish the cheaters, which is why cartels of many small sellers almost never last.