Difficulty: Intermediate | Prerequisites: Consumer Theory Foundations (Doc 1), comfort with Cobb-Douglas optimisation and graphing budget constraints.
Once you can find the consumer's optimal bundle (Doc 1), the next step is to see how that bundle responds to changes in prices and income. This is where demand curves and Engel curves come from. You are not learning new optimisation techniques here; you are applying the same tangency or kink conditions at different parameter values and tracing the results. This material is tested directly in two of the four long exam problems.
Vary one price while holding income and the other price fixed, re-solve the optimum each time, and you trace out the price consumption curve in (x₁, x₂) space and the demand curve in (p₁, x₁) space. Vary income instead and you get the income consumption curve and the Engel curve. Whether a good is normal, inferior, or Giffen depends on how quantity demanded responds to these changes.
Price consumption curve (PCC)
The locus of optimal bundles in (x₁, x₂) space as the price of one good varies, holding income and the other price constant. Think of it as connecting the dots of all optimal bundles as you pivot the budget line.
Demand curve
The relationship between the price of a good and the quantity demanded, holding income and other prices fixed. It is derived directly from the PCC by plotting (p₁, x₁*) pairs. For standard preferences it slopes downward.
Income consumption curve (ICC)
The locus of optimal bundles as income varies, holding all prices constant. On the graph it connects optimal bundles as the budget line shifts outward in parallel.
Engel curve
The relationship between income and the quantity demanded of a good, holding prices constant. Plotted in (m, x₁*) space. For a normal good it slopes upward; for an inferior good it slopes downward (at least over some range).
Normal good
A good whose quantity demanded increases when income rises. The Engel curve slopes upward.
Inferior good
A good whose quantity demanded falls when income rises. The Engel curve slopes downward over that income range.
Giffen good
A special (and rare) case of an inferior good where the income effect is so strong that it overwhelms the substitution effect, causing the demand curve to slope upward. Quantity demanded rises when the good's own price rises.
Necessary and sufficient conditions
A necessary condition must hold for the optimum to exist but does not guarantee it alone (e.g., MRS = MRT is necessary for an interior optimum). A sufficient condition guarantees the result (e.g., MRS = MRT plus strict convexity of preferences is sufficient for a unique interior maximum).
Cobb-Douglas: U = x₁^a · x₂^b
Tangency condition: MRS = (a·x₂)/(b·x₁) = p₁/p₂. Combine with the budget constraint p₁x₁ + p₂x₂ = m.
Demand functions: x₁* = [a/(a+b)] · m/p₁ and x₂* = [b/(a+b)] · m/p₂. The consumer always spends the fraction a/(a+b) of income on good 1. Both goods are always normal.
Perfect complements: U = min(ax₁, bx₂)
The consumer chooses the kink of the L-shaped indifference curve, where ax₁ = bx₂. Substitute into the budget constraint to get x₁* = bm/(bp₁ + ap₂) and x₂* = am/(bp₁ + ap₂). There is no substitution between the goods, so price changes create only income effects.
Perfect substitutes: U = ax₁ + bx₂
Compare MRS = a/b to MRT = p₁/p₂.
If a/b > p₁/p₂: spend everything on good 1. x₁* = m/p₁, x₂* = 0.
If a/b < p₁/p₂: spend everything on good 2. x₁* = 0, x₂* = m/p₂.
If a/b = p₁/p₂: any bundle on the budget line is optimal.
To build these, hold m and p₂ fixed and let p₁ take several different values. At each p₁, solve for the optimal bundle (x₁*, x₂*).
Price consumption curve (PCC): plot each optimal bundle in (x₁, x₂) space and connect them. The PCC shows how the mix of goods changes as good 1 becomes cheaper or more expensive.
Demand curve: take the same data and plot (p₁, x₁*) in a separate diagram. This is the individual demand curve for good 1.
For Cobb-Douglas utility, the demand function x₁* = [a/(a+b)] · m/p₁ is a rectangular hyperbola (downward-sloping, convex). The PCC for Cobb-Douglas is a horizontal line at x₂* = [b/(a+b)] · m/p₂, because spending on good 2 does not change when p₁ changes.
For perfect complements, both quantities change when p₁ changes (because the consumer must maintain the fixed ratio). The demand curve still slopes downward.
For perfect substitutes, demand jumps discretely: the consumer buys all of good 1 when p₁ is low enough, none when it is high, and is indifferent at the threshold price.
Now hold both prices fixed and vary income m.
Income consumption curve (ICC): plot the optimal bundle at each income level in (x₁, x₂) space. For Cobb-Douglas, the ICC is a straight line through the origin (both goods grow proportionally with income). For other utility functions it may curve.
Engel curve: plot (m, x₁*) in a separate diagram.
For a normal good, the Engel curve slopes upward.
For an inferior good, the Engel curve bends backward (slopes downward) beyond some income level.
For Cobb-Douglas, the Engel curve is a straight line through the origin: x₁* = [a/(a+b)] · m/p₁, which is linear in m.
Normal good: quantity demanded rises with income. Both goods in a Cobb-Douglas utility function are always normal. Engel curve slopes upward.
Inferior good: quantity demanded falls as income rises, at least over some range. Cheap staples (instant noodles, bus tickets) are textbook examples. The income consumption curve bends back toward the axis of the inferior good.
Giffen good: a special inferior good where the (positive) income effect of a price increase is so large that it outweighs the (negative) substitution effect. The result is that quantity demanded rises when the good's own price rises, producing an upward-sloping demand curve. Giffen goods are theoretically possible but extremely rare in practice.
Key logical chain:
All Giffen goods are inferior, but not all inferior goods are Giffen.
A normal good can never be Giffen.
Whether a good is normal or inferior can change at different income levels.
Cobb-Douglas demand: x₁* = [a/(a+b)] · m/p₁
Cobb-Douglas Engel curve: x₁* = [a/(a+b)] · m/p₁ (linear in m, holding p₁ fixed)
Perfect complements demand: x₁* = bm/(bp₁ + ap₂) for U = min(ax₁, bx₂)
Perfect substitutes demand: x₁* = m/p₁ if a/b > p₁/p₂; x₁* = 0 if a/b < p₁/p₂; any point on the budget line if a/b = p₁/p₂
Necessary condition for an interior optimum: MRS = p₁/p₂ (tangency)
Sufficient condition for a unique interior maximum: MRS = p₁/p₂ and preferences are strictly convex (diminishing MRS)
Engel curves matter for understanding how spending patterns shift as economies grow. As household incomes rise, spending on food as a share of the budget falls (Engel's law), which is why food is often modelled as a necessity rather than a luxury.
Demand curves derived from consumer theory are the microeconomic building blocks behind market demand, which in turn feeds into welfare analysis, tax policy, and antitrust economics.
"The demand curve can slope upward for an inferior good." Only for a Giffen good, which is a very specific subset of inferior goods. Most inferior goods still have downward-sloping demand curves because the substitution effect dominates.
"The PCC and the demand curve are the same thing." They are not. The PCC lives in (x₁, x₂) space and shows how the entire bundle changes. The demand curve lives in (p₁, x₁) space and shows only the quantity of good 1 against its own price.
"Cobb-Douglas goods can be inferior." They cannot. The demand function x₁* = [a/(a+b)] · m/p₁ is strictly increasing in m.
"Necessary conditions are sufficient." Tangency (MRS = MRT) is necessary for an interior optimum but not sufficient. You also need the second-order condition (convexity of preferences) to be sure the tangent point is a maximum rather than a minimum or a saddle point.
⚠️ One of the four long problems asks you to graph either the ICC and Engel curve, or the PCC and demand curve. You need to draw both diagrams side by side: the (x₁, x₂) diagram showing optimal bundles and the derived curve in its own space.
⚠️ Know how to construct the PCC step by step: solve for the optimal bundle at p₁ = 2, then at p₁ = 4, then at p₁ = 6, plot each bundle, and connect.
⚠️ Be ready to explain the difference between necessary and sufficient conditions. A short-answer question on this is likely.
⚠️ Perfect substitutes and perfect complements are fair game for the shorter questions. Be comfortable with corner solutions and the kink condition.
True or false: a Giffen good must be an inferior good.
Fill in the blank: for Cobb-Douglas utility, the price consumption curve when p₁ varies is a ____ line in (x₁, x₂) space.
True or false: the Engel curve for a normal good slopes downward.
Fill in the blank: for perfect complements U = min(x₁, 3x₂), the consumer always chooses bundles where x₁ = ____.
True or false: MRS = p₁/p₂ is a sufficient condition for an interior maximum.
Answers: 1. True. 2. Horizontal. 3. False (it slopes upward). 4. 3x₂. 5. False (it is necessary, not sufficient; you also need convexity).
Q: For U = x₁^(1/3) · x₂^(2/3), m = 90, p₁ = 3, p₂ = 6, find the demand for x₁ and x₂. Then derive the Engel curve for x₁.
A: Spending shares: a/(a+b) = 1/3 on good 1. x₁* = (1/3)(90/3) = 10. x₂* = (2/3)(90/6) = 10. The Engel curve for x₁ is x₁ = (1/3)(m/3) = m/9, a straight line through the origin.
Q: Describe how to derive the demand curve for good 1 from the price consumption curve.
A: Solve for the optimal bundle at several values of p₁ (holding m and p₂ fixed). Each solution gives one point on the PCC in (x₁, x₂) space and one point on the demand curve in (p₁, x₁) space. Connect the demand-curve points to trace out the demand function.
Q: A consumer has U = 4x₁ + x₂, p₁ = 3, p₂ = 2, m = 30. What bundle does the consumer choose?
A: MRS = 4/1 = 4. MRT = 3/2 = 1.5. Since MRS > MRT, the consumer values good 1 more than the market price requires, so they spend everything on good 1. x₁* = 30/3 = 10, x₂* = 0.
Q: Explain why a good can be inferior but not Giffen.
A: When the price of an inferior good rises, the substitution effect reduces quantity demanded (as always) and the income effect increases it (because the consumer is poorer and buys more of the inferior good). If the substitution effect is larger, quantity demanded still falls, and the demand curve still slopes downward. The good is inferior but not Giffen.
The demand curve you derive here is the foundation for the income and substitution effects covered in Doc 3. When you decompose a price change into its two parts, you are splitting a movement along the demand curve into a pure substitution piece and a pure income piece. Compensating and equivalent variation (also Doc 3) are dollar-valued welfare measures built on top of these demand relationships.
price consumption curve, PCC, demand function, individual demand, market demand, income consumption curve, ICC, Engel curve, Engel's law, normal good, inferior good, Giffen good, luxury good, necessity, income elasticity, corner solution, interior solution, necessary condition, sufficient condition, second-order condition, comparative statics