Difficulty: Intermediate | Prerequisites: Basic calculus (partial derivatives), introductory supply and demand.
This material sits at the core of intermediate microeconomics. You are moving beyond the introductory "supply goes up, demand goes down" intuition and learning to work with multi-variable demand functions, where price, the prices of related goods, and income all appear as separate terms. If you missed introductory micro or are shaky on partial derivatives, revisit those first. Everything here feeds directly into elasticity, welfare analysis, and market intervention topics later in the course.
A demand function expresses quantity demanded as a function of own price, prices of related goods, and income. You classify goods as substitutes or complements by taking partial derivatives with respect to their prices. The inverse demand function simply rearranges the demand function to express price as a function of quantity, which is what you graph with price on the vertical axis.
Demand function
An equation that expresses quantity demanded as a function of own price, prices of related goods, and consumer income. Think of it as the full recipe for how much of a good people want to buy, given everything that matters.
Partial derivative (∂Q/∂x)
The rate of change of quantity demanded with respect to one variable, holding all others constant. In simple terms, this tells you how quantity responds when you nudge just one factor (like the price of beef) while keeping everything else fixed.
Substitute goods
Two goods where an increase in the price of one raises demand for the other. Formally, ∂Q/∂p_other > 0. Think of it as: when beef gets expensive, people buy more pork instead.
Complement goods
Two goods where an increase in the price of one lowers demand for the other. Formally, ∂Q/∂p_other < 0. Think of it as: when the price of printers rises, demand for ink cartridges falls.
Normal good
A good for which demand increases as income rises. Formally, ∂Q/∂Y > 0. In simple terms, when people earn more, they buy more of it.
Inferior good
A good for which demand decreases as income rises. Formally, ∂Q/∂Y < 0. When people earn more, they switch away from it.
Inverse demand function
The demand function rearranged to express price as a function of quantity: p = f(Q). This is the form you graph, because economics convention puts price on the vertical axis.
Equilibrium
The price and quantity where quantity demanded equals quantity supplied. The market clears, with no pressure to move.
Comparative statics
The method of comparing two equilibria to see how a change in an exogenous variable (like temperature or the price of a related good) shifts the outcome. You are not tracking the path between them, just the before and after.
A typical multi-variable demand function looks like this:
Q = 171 - 20p + 20p_b + 3p_c + 2Y
where Q is quantity of pork (millions of kg), p is own price, p_b is the price of beef, p_c is the price of chicken, and Y is average income (thousands of dollars).
Each coefficient tells you the direction and magnitude of the relationship between that variable and quantity demanded, holding everything else constant.
Classifying the relationship with beef
Take the partial derivative: ∂Q/∂p_b = +20
The sign is positive, so when beef prices rise, pork demand rises.
Beef and pork are therefore substitutes.
The intuition: as beef becomes more expensive, consumers switch to pork.
Classifying the relationship with income
Take the partial derivative: ∂Q/∂Y = +2
The sign is positive, so when income rises, pork demand rises.
Pork is therefore a normal good.
If the coefficient had been negative, it would be an inferior good.
Simplifying the demand function
When you hold related prices and income at their typical values, you collapse the multi-variable function into a two-variable demand function (Q as a function of own price only).
With p_b = 4, p_c = 3.33, Y = 12.5:
Q = 171 - 20p + 20(4) + 3(3.33) + 2(12.5)
Q = 171 - 20p + 80 + 9.99 + 25
Q = 285.99 - 20p
This is the demand curve you would graph in price-quantity space.
Shifting the demand curve
If beef price rises from p_b = 4 to p_b = 5, the new simplified demand function becomes:
Q = 171 - 20p + 20(5) + 9.99 + 25 = 305.99 - 20p
The entire demand curve has shifted right (outward). At every price, more pork is demanded. The slope has not changed, only the intercept.
The inverse demand function rearranges Q = f(p) into p = g(Q). You need this because the convention in economics is to put price on the vertical axis.
Working through the algebra
Starting from Q = 285.99 - 20p:
20p = 285.99 - Q
p = 14.30 - 0.05Q
This is a straight line with a vertical intercept of 14.30 (the "choke price," above which nobody buys) and a slope of -0.05.
What to label on the graph
Vertical axis: price (p)
Horizontal axis: quantity (Q)
Vertical intercept: p = 14.30 (set Q = 0)
Horizontal intercept: Q = 285.99 (set p = 0)
Slope of the inverse demand curve: -0.05 (= -1/20)
After a shift
When beef price rises to p_b = 5, the new inverse demand is:
p = 15.30 - 0.05Q
The curve shifts up (equivalently, right). The slope is unchanged. The vertical intercept moves from 14.30 to 15.30. The horizontal intercept moves from 285.99 to 305.99. On your graph, draw the new line parallel to the original, shifted outward.
Comparative statics asks: how does the equilibrium change when an exogenous variable changes?
Roses example: temperature and equilibrium
Demand: Q = a - bp (temperature does not affect demand here)
Supply: Q = c + ep + ft (temperature t shifts supply)
Finding equilibrium
Set demand equal to supply:
a - bp = c + ep + ft
Solve for the equilibrium price:
a - c - ft = bp + ep = (b + e)p
p* = (a - c - ft) / (b + e)
Substitute back into demand to get the equilibrium quantity:
Q* = a - b × [(a - c - ft) / (b + e)]
Q* = (ae + bc + bft) / (b + e)
How equilibrium varies with temperature
dp*/dt = -f / (b + e)
Since f, b, and e are all positive, dp*/dt < 0.
As temperature rises, the equilibrium price falls.
dQ*/dt = bf / (b + e)
Since b and f are positive, dQ*/dt > 0.
As temperature rises, the equilibrium quantity rises.
The intuition: warmer weather shifts the supply curve right (roses grow more easily). More supply at every price pushes the equilibrium price down and the equilibrium quantity up. This is a movement along the demand curve caused by a shift in supply.
General linear demand function
Q = α₀ + α₁p + α₂p_related + α₃Y
where α₁ < 0 (law of demand), and the signs of α₂ and α₃ classify the related good and the income relationship.
Inverse demand (linear case)
Rearrange Q = a - bp to get:
p = (a/b) - (1/b)Q
Vertical intercept = a/b (the choke price). Slope = -1/b.
Equilibrium condition
Q_d = Q_s
Solve the resulting equation for p*, then substitute back for Q*.
Comparative statics derivative
To find how equilibrium price or quantity responds to a parameter (like temperature t), differentiate p* or Q* with respect to that parameter. The sign tells you the direction.
The pork demand function is estimated from real Canadian market data (Moschini and Meilke, 1992). Governments and agricultural boards use exactly these kinds of estimated demand functions to predict how changes in cattle prices, trade policy, or income growth will affect pork consumption, which feeds into subsidy design and trade negotiations.
The roses-and-temperature example mirrors how agricultural economists model seasonal supply shifts. Flower growers, for instance, know that a warm February shifts Valentine's Day supply outward, which pushes wholesale rose prices down.
Students often think a positive coefficient on p_b means beef and pork are complements. It does not. A positive ∂Q_pork/∂p_beef means that when beef gets more expensive, people buy more pork. That is the definition of substitutes.
Students confuse a movement along the demand curve with a shift of the demand curve. A change in own price causes movement along the curve. A change in income, the price of a related good, or tastes shifts the entire curve.
Students sometimes graph the demand function Q = f(p) directly, putting Q on the vertical axis. In economics, the convention is to invert it and put price on the vertical axis. Always convert to the inverse demand function before graphing.
Students forget that the slope of the inverse demand curve is the reciprocal of the coefficient on p in the original demand function (with a sign flip if needed). If Q = 286 - 20p, the inverse demand slope is -1/20, not -20.
⚠️ You will almost certainly be asked to take a partial derivative to classify a good as a substitute, complement, normal, or inferior. Know the sign conventions cold.
⚠️ Converting between demand and inverse demand is a bread-and-butter exam skill. Expect to be asked to graph the inverse demand, label intercepts, and then show how the curve shifts when a parameter changes.
⚠️ Comparative statics problems (like the roses example) test whether you can go from "set Q_d = Q_s" through to "differentiate the equilibrium with respect to a parameter" and interpret the sign. This is a common exam structure.
⚠️ When a demand curve shifts, the slope of the inverse demand stays the same for linear functions. Only the intercept changes. Exams love testing whether you understand this.
True or false: If ∂Q/∂p_b > 0, then good b is a complement to the good in question.
False. A positive partial derivative means an increase in the price of b raises demand for the good. That makes them substitutes.
Fill in the blank: To graph the demand curve, you must convert the demand function into the ______ demand function.
Inverse.
True or false: When the price of a related good changes, the demand curve for the original good shifts, and its slope changes.
False (for linear functions). The slope stays the same; only the intercept changes.
Fill in the blank: When supply shifts right and demand is unchanged, equilibrium price ______ and equilibrium quantity ______.
Falls; rises.
True or false: If ∂Q/∂Y < 0, the good is a normal good.
False. A negative income derivative means the good is inferior.
Q: Given Q = 171 - 20p + 20p_b + 3p_c + 2Y, what type of good is chicken with respect to pork, and how do you know?
A: ∂Q/∂p_c = +3, which is positive. When chicken prices rise, pork demand rises. Chicken is a substitute for pork.
Q: Using Q = 285.99 - 20p, what is the inverse demand function, and what are the vertical and horizontal intercepts?
A: p = 14.30 - 0.05Q. Vertical intercept (choke price): p = 14.30 when Q = 0. Horizontal intercept: Q = 285.99 when p = 0.
Q: In the roses market, if temperature rises, what happens to the supply curve and why?
A: The supply curve shifts right (outward). The supply function Q_s = c + ep + ft has a positive coefficient on t, so higher temperature increases quantity supplied at every price.
Q: After solving for equilibrium price p in the roses market, you find dp/dt = -f/(b+e). Interpret this result.**
A: Since f, b, and e are all positive, dp*/dt is negative. Higher temperature lowers the equilibrium price. The intuition is that more supply pushes price down.
Q: If p_b increases from 4 to 5 in the pork demand function, does the demand curve shift left or right? Does the slope of the inverse demand change?
A: The demand curve shifts right (more pork demanded at every price, because the substitute has become more expensive). The slope of the inverse demand does not change; it remains -1/20.
This material connects directly to elasticity. Once you know the demand function and can take partial derivatives, elasticity is just one more step: multiply the partial derivative by the ratio of the variable to quantity. That is the next major topic.
Comparative statics here is a simpler version of what you will do later with taxes, price ceilings, and welfare analysis, where you compare the pre-intervention equilibrium to the post-intervention one.
The classification of goods as substitutes, complements, normal, and inferior reappears in consumer theory when you decompose the effect of a price change into substitution and income effects.
Related Terms / Search Tags
demand function, inverse demand, partial derivative, substitutes, complements, normal good, inferior good, comparative statics, equilibrium, demand shift, choke price, multi-variable demand, cross-price relationship, Moschini and Meilke, Canadian pork demand, roses supply and demand, temperature and equilibrium, ECON 300A, intermediate microeconomics, University of Washington