Source: Demand analysis and Optimal Pricing lecture, Texas A&M
Tags: demand function, demand curve, demand shifts, inverse demand, normal goods, inferior goods, substitutes, complements, information goods, network externalities, Qd, graphing demand
The demand function expresses quantity demanded as a function of price, income, competitor prices, and other factors. When you graph it, only price lives on the axes; everything else is baked into the intercept, so a change in income or competitor price shifts the entire curve. A change in the good's own price is just movement along the existing curve.
Demand function
The full relationship Qd = f(P, P0, Y, ...) showing how quantity demanded depends on own price, competitor prices, income, and other variables.
Inverse demand
The demand function rearranged to put price on the left-hand side, e.g. P = 212.5 – Qd/2. This is the form you plot, because economists place P on the vertical axis.
Normal good
A good whose quantity demanded rises when income rises (and falls when income falls). Examples: clothing, restaurant meals.
Inferior good
A good whose quantity demanded falls when income rises. Examples: ground chuck, corned beef, instant noodles.
Substitutes
Two goods where a rise in the price of one increases quantity demanded of the other. Example: rail travel and air travel.
Complements
Two goods where a rise in the price of one decreases quantity demanded of the other. Example: coffee and milk.
Information goods
Products with near-zero marginal cost of serving an additional user once developed: software, online subscriptions, music, online lectures.
Network externalities
The phenomenon where a product's value to each user increases as total users increase. Creates lock-in and high switching costs.
Qd depends on many variables, but economists focus on own price (P) as the main component.
General form: Qd = f(P, P0, Y, etc.)
Textbook example: Qd = 25 + 3Y + P0 – 2P
Y = index of income
P0 = price of a competing firm's good
P = own price
To graph, plug in values for every variable except P.
If Y = 100 and P0 = $100: Qd = 25 + 3(100) + 100 – 2P = 425 – 2P
Economists put P on the y-axis, so rearrange: P = 212.5 – Qd/2. This is the inverse demand.
Plot with the y-intercept at 212.5 and the x-intercept at 425.
A change in the good's own price (P) is movement along the curve. Nothing shifts; you simply read off a different point.
A change in any other variable (Y, P0, tastes, population) shifts the entire curve.
Example: if Y rises from 100 to 120, the new demand is Qd = 485 – 2P, and the inverse demand becomes P = 242.5 – Qd/2. The curve shifts right (outward), with a higher y-intercept.
Use the coefficients directly.
Exercise: Y rises by $3,000 (and Y is in thousands, so ΔY = 3) and the competitor (Subway) drops its price by $2 (so ΔP0 = +2 from the buyer's perspective, since the coefficient on P0 is positive).
ΔQd = 3(3) + 1(2) = 11 units increase in quantity demanded.
Income
Normal goods: Qd moves in the same direction as income.
Inferior goods: Qd moves in the opposite direction to income.
Prices of other goods
Substitutes: Qd moves in the same direction as the other good's price.
Complements: Qd moves in the opposite direction to the other good's price.
Other factors
Population size, demographics, tastes, expectations, anything that is not the good's own price.
Near-zero marginal cost once developed, so suppliers behave as "pure sellers" (revenue maximisers).
Common pricing strategies:
User fees (subscriptions)
Ad-supported free usage (Google, YouTube)
Versioning: stripped-down version free, premium version paid
Network externalities mean the best product may not win. Lock-in occurs because switching costs are high (everyone already uses the incumbent, e.g. MS Word).
Demand function (general): Qd = f(P, P0, Y, ...)
Example demand: Qd = 25 + 3Y + P0 – 2P
Inverse demand (for graphing): Rearrange so P is on the left. For the example with Y = 100, P0 = 100: P = 212.5 – Qd/2
Shift magnitude: ΔQd = (coefficient on Y)(ΔY) + (coefficient on P0)(ΔP0)
⚠️ A price change is movement along the curve, not a shift. This distinction is tested constantly.
⚠️ When real-world data shows price up and quantity up simultaneously, that does not violate the law of demand. It means the demand curve itself shifted (e.g. population growth, changed preferences). You cannot estimate elasticity from such data because the curve was not stable.
⚠️ When converting from the demand function to inverse demand for graphing, the slope flips. The slope of the demand curve (ΔQd/ΔP) is –2, but the slope of the inverse demand (ΔP/ΔQd) is –1/2.
⚠️ The coefficient on P0 tells you the relationship: positive means substitutes, negative means complements.
Q: If Qd = 25 + 3Y + P0 – 2P, and income rises by 10 units while everything else stays constant, what happens to the demand curve?
A: Qd increases by 3(10) = 30 at every price. The entire demand curve shifts right by 30 units. The inverse demand's y-intercept rises by 30/2 = 15.
Q: Why can we not use the Texas A&M tuition/enrolment data (price rose from $11,361 to $11,817, enrolment rose from 64,183 to 65,684) to calculate price elasticity of demand?
A: Because the demand curve shifted during the same period (likely due to COVID effects, population growth, and changing preferences). The law of demand says Qd falls when P rises along a stable demand curve. Here both P and Q rose, which signals a rightward demand shift, not a movement along the curve.
Q: A product exhibits network externalities. Explain why a superior competitor might still fail to displace it.
A: Users value compatibility with existing users (file sharing, common knowledge of features). Switching costs are high because the installed base is large. Even if a new product is objectively better, the network effect of the incumbent keeps users locked in.
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