Source: Microeconomic Theory, Texas A&M University, Chapter 4
Tags: individual demand, price-consumption curve, income-consumption curve, Engel curve, normal good, inferior good, substitutes, complements, budget constraint, utility maximisation, MRS, demand function
A consumer's demand curve comes from their utility-maximising choices under a budget constraint. When the price of a good changes, we trace a price-consumption curve; when income changes, we trace an income-consumption curve (and from it, an Engel curve). Whether a good is normal or inferior, and whether two goods are substitutes or complements, can be read off these curves or derived analytically from the demand functions Q_X(P_X, P_Y, I) and Q_Y(P_X, P_Y, I).
Price-consumption curve
The curve that traces utility-maximising market baskets as the price of one good changes, holding income and the other good's price fixed.
Individual demand curve
A curve relating the quantity of a good that a single consumer will buy to its price.
Income-consumption curve
The curve tracing utility-maximising market baskets as the consumer's income changes, holding prices fixed.
Engel curve
A curve that relates the quantity of a good consumed to the consumer's income level (income on the vertical axis, quantity on the horizontal axis).
Normal good
A good for which an increase in income leads to more consumption. The Engel curve slopes upward and the income-consumption curve slopes in the direction of increasing quantity.
Inferior good
A good for which an increase in income leads to less consumption. The Engel curve bends backward and the income-consumption curve bends away from that good's axis.
Substitutes
Two goods are substitutes if an increase in the price of one leads to an increase in the quantity demanded of the other.
Complements
Two goods are complements if an increase in the price of one leads to a decrease in the quantity demanded of the other.
Independent goods
Two goods are independent if a change in the price of one has no effect on the quantity demanded of the other.
Marginal rate of substitution (MRS)
The rate at which a consumer is willing to trade one good for another while staying on the same indifference curve. At the optimum, MRS = P_X / P_Y.
The demand curve follows from the consumption choices a person makes when facing a budget constraint.
Fix income I = $20 and the price of clothing P_C = $2. Let the price of food vary: P_F = $2, $1, $0.50.
At each price, the consumer picks the bundle where the budget line is tangent to the highest attainable indifference curve.
P_F = $2: optimal bundle A, food = 4 units
P_F = $1: optimal bundle B, food = 12 units
P_F = $0.50: optimal bundle D, food = 20 units
Connecting these optimal bundles in goods-space gives the price-consumption curve.
Plotting each (quantity, price) pair gives the individual demand curve.
Three key remarks about the demand curve:
The level of utility attained changes as you move along the demand curve (you reach higher indifference curves at lower prices).
At every point, the consumer maximises utility by setting MRS = P_F / P_C.
The demand curve reflects willingness to pay. At bundle A (F = 4), MRS = 1, meaning the consumer would give up 1 unit of clothing for 1 more unit of food. Since P_C = $2, the willingness to pay for food at that point is $2.
Fix both prices (P_F = 1, P_C = 2) and vary income: I = $10, $20, $30.
Optimal bundles shift outward as income rises, tracing the income-consumption curve.
I = $10: food = 4
I = $20: food = 10
I = $30: food = 16
When income increases, the demand curve for food shifts to the right.
To build the Engel curve, plot these (food, income) pairs with income on the vertical axis and food on the horizontal axis.
For perfect complements (indifference curve kinks on the 45-degree line), both the price-consumption curve and the income-consumption curve are increasing straight lines.
Two methods:
Method 1: Look at the slope of the income-consumption curve. If it moves toward more of the good as income rises, the good is normal. If it bends backward (less of the good), the good is inferior over that income range.
Method 2: Look at the slope of the Engel curve. Upward-sloping means normal; backward-bending means inferior.
A good can switch between normal and inferior at different income levels. The classic textbook example: hamburger is a normal good for income below $20/month but an inferior good for income above $20/month (the income-consumption curve bends backward between points B and C).
U(x, y) = x^0.5 · y^0.5
MU_x = 0.5 · x^(−0.5) · y^(0.5), MU_y = 0.5 · x^(0.5) · y^(−0.5)
Optimality requires MRS = MU_x / MU_y = y/x = P_X / P_Y, together with the budget constraint P_X · x + P_Y · y = I.
Solving gives:
x* = 0.5I / P_X
y* = 0.5I / P_Y
Demand curve of X (fix P_Y = 10, I = 100): Q_X(P_X) = 50 / P_X
Engel curve of X (fix P_X = 20, P_Y = 10): Q_X(I) = 0.025I
With Cobb-Douglas preferences, the demand for each good depends only on its own price and income, not on the other good's price. The two goods are therefore independent.
Kinks of the indifference curve fall on the line y = 2x. The consumer always buys in the ratio y = 2x, so we substitute into the budget constraint:
P_X · x + P_Y · (2x) = I
Solving:
Q_X(P_X, P_Y, I) = I / (P_X + 2P_Y)
Q_Y(P_X, P_Y, I) = 2I / (P_X + 2P_Y)
Demand curve of X (P_Y = 10, I = 100): Q_X(P_X) = 100 / (P_X + 20)
Engel curve of X (P_X = 20, P_Y = 10): Q_X(I) = I / 40. X is a normal good (consumption rises with income).
Substitute or complement? When P_X increases, Q_Y decreases; when P_Y increases, Q_X decreases. The two goods are complements.
When the price of food falls and the budget line rotates outward:
From A to B, clothing consumption decreases, so food and clothing act as substitutes over that range.
From B to D, clothing consumption increases, so food and clothing act as complements over that range.
Two goods can therefore be substitutes at some prices and complements at others.
The demand functions Q_X(P_X, P_Y, I) and Q_Y(P_X, P_Y, I) are solved from the utility-maximisation problem. From these functions you can extract everything:
Demand curve of X: fix P_Y and I, express Q_X as a function of P_X alone.
Demand curve of Y: fix P_X and I, express Q_Y as a function of P_Y alone.
Engel curve of X: fix P_X and P_Y, express Q_X as a function of I alone.
Engel curve of Y: fix P_X and P_Y, express Q_Y as a function of I alone.
Substitutes or complements? Check whether Q_X is increasing in P_Y (substitutes) or decreasing in P_Y (complements). Equivalently, check whether Q_Y is increasing in P_X.
Suppose Q_X(P_X, P_Y, I) = (I − P_X + P_Y) / (2P_X).
Fix P_Y = 10, I = 100: demand curve Q_X(P_X) = (110 − P_X) / (2P_X)
Fix P_X = 20, P_Y = 10: Engel curve Q_X(I) = (I − 10) / 40
Q_X is increasing in P_Y, so the two goods are substitutes.
If Q_X(P_X, P_Y, I) = 1 / (2P_X), the function does not contain P_Y at all. X and Y are independent.
If Q_X(P_X, P_Y, I) = I / (2P_X + P_Y), Q_X is decreasing in P_Y. X and Y are complements.
Utility maximisation condition (interior solution): MRS = MU_X / MU_Y = P_X / P_Y, subject to P_X · x + P_Y · y = I
Cobb-Douglas demand (U = x^0.5 · y^0.5): x* = 0.5I / P_X, y* = 0.5I / P_Y
Perfect complements demand (y = 2x): x* = I / (P_X + 2P_Y), y* = 2I / (P_X + 2P_Y)
Price elasticity of demand: E_P = (ΔQ/Q) / (ΔP/P) = Q′(P) · P / Q
⚠️ The demand curve, the Engel curve, and the substitute/complement classification all come from the same demand function Q_X(P_X, P_Y, I). Learn to extract each one by fixing the right variables.
⚠️ A good can be normal over one income range and inferior over another. Do not assume a good is always one or the other.
⚠️ With Cobb-Douglas utility, the two goods are always independent (demand for each depends only on its own price and income). This is a common exam trick.
⚠️ For perfect complements, always substitute the fixed-proportion condition (e.g. y = 2x) into the budget constraint. Do not set MRS = price ratio (MRS is undefined at the kink).
⚠️ The price-consumption curve lives in goods-space (x vs. y). The demand curve lives in price-quantity space. Do not confuse the two graphs.
Q: What is the difference between the price-consumption curve and the individual demand curve?
A: The price-consumption curve traces optimal bundles in goods-space (both goods on the axes) as one good's price changes. The individual demand curve plots the same information in price-quantity space (price on the vertical axis, quantity of that good on the horizontal axis).
Q: Given U(x,y) = x^0.5 · y^0.5, P_Y = 10, I = 100, derive the demand curve for X.
A: From the optimality condition, x* = 0.5I / P_X. Substituting I = 100 gives Q_X(P_X) = 50 / P_X.
Q: How do you determine whether two goods are substitutes, complements, or independent using the demand function?
A: Check whether Q_X is increasing, decreasing, or constant with respect to P_Y. Increasing means substitutes; decreasing means complements; constant means independent. You can equivalently check Q_Y with respect to P_X.
Q: For perfect complements with y = 2x, P_X = 20, P_Y = 10, and I = 100, what is Q_X? Is X normal or inferior?
A: Q_X = 100 / (20 + 20) = 2.5. The Engel curve is Q_X(I) = I/40, which is increasing in I, so X is a normal good.
Q: Can a good be a normal good at one income level and an inferior good at another?
A: Yes. The textbook example is hamburger: normal for income below $20/month and inferior for income above $20/month, where the income-consumption curve bends backward.
individual demand, market basket, budget constraint, budget line, indifference curve, utility maximisation, marginal rate of substitution, MRS, price-consumption curve, income-consumption curve, Engel curve, normal good, inferior good, substitutes, complements, independent goods, Cobb-Douglas utility, perfect complements, demand function, willingness to pay, ECON 323, microeconomic theory, Texas A&M, chapter 4