Difficulty: Intermediate | Prerequisites: Basic calculus (partial derivatives), introductory microeconomics.
Consumer theory is the backbone of intermediate microeconomics. It explains how individuals allocate limited income across goods to maximise satisfaction, and how their choices shift when prices or income change. You need a working knowledge of utility functions, budget constraints, and basic calculus (partial derivatives) before diving in. Everything here feeds directly into demand curves, welfare analysis, and general equilibrium later in the course.
Consumers choose bundles where the marginal rate of substitution equals the price ratio, subject to their budget constraint. When prices change, the total effect on quantity demanded splits into a substitution effect (due to relative price changes, always moves opposite the price change) and an income effect (due to purchasing-power changes, direction depends on whether the good is normal or inferior). Special cases such as Giffen goods and perfect complements have distinctive demand behaviour and indifference-curve shapes.
Utility function
A mathematical representation of a consumer's preferences over bundles of goods, assigning higher numbers to more-preferred bundles.
In simple terms, it is a scoring system: the bundle you like best gets the highest score.
Cobb-Douglas utility
A utility function of the form U(x, y) = x^a * y^b. It produces smooth, convex indifference curves and yields demand functions where the consumer spends a fixed share of income on each good.
Think of it as the workhorse utility function in intermediate micro, because the maths is clean and the results are intuitive.
Budget constraint
The equation Pxx + Pyy = M, representing all bundles a consumer can just afford given prices (Px, Py) and income (M). Its slope is -Px/Py.
In simple terms, it is the line showing what you can buy with your pay cheque at current prices.
Marginal Rate of Substitution (MRS)
The maximum amount of good y a consumer is willing to give up for one more unit of good x while remaining on the same indifference curve. Formally, MRS = MUx / MUy, and it equals the negative of the slope of the indifference curve at any point.
Think of it as the consumer's personal exchange rate between two goods.
Indifference curve
A curve in (x, y) space connecting all bundles that yield the same utility level. Indifference curves are typically downward-sloping, convex to the origin, and never cross.
In simple terms, every point on the curve makes the consumer equally happy.
Transitivity
If a consumer prefers bundle A to B, and B to C, then they must prefer A to C. This rules out circular preferences and is required for the consumer to have a well-defined best choice.
Think of it as a no-contradictions rule for preferences.
Income Consumption Curve (ICC)
The locus of optimal bundles as income changes while prices stay fixed. For normal goods the ICC slopes upward; for inferior goods it bends back.
In simple terms, it traces how your shopping basket shifts as you get richer or poorer.
Engel curve
A graph plotting the quantity demanded of one good against income, with prices held constant. Normal goods have upward-sloping Engel curves; inferior goods have downward-sloping (negative slope) Engel curves.
Think of it as the ICC redrawn with income on one axis and quantity on the other.
Substitution effect
The change in quantity demanded due purely to the change in relative prices, holding utility constant (Hicksian decomposition). It always moves in the opposite direction of the price change.
In simple terms, when something gets cheaper relative to alternatives, you tilt toward it.
Income effect
The change in quantity demanded due to the change in real purchasing power caused by a price change, holding relative prices at their new level. For normal goods it reinforces the substitution effect; for inferior goods it works against it.
In simple terms, a price drop makes you feel richer, and what you do with that extra purchasing power is the income effect.
Giffen good
A good for which quantity demanded rises when its price rises, violating the law of demand. This requires the good to be inferior and the income effect to be large enough to outweigh the substitution effect.
Think of it as the rare case where feeling poorer from a price hike forces you to buy more of the cheap staple, not less.
Compensating Variation (CV)
The amount of additional income a consumer would need at new prices to reach their original utility level. CV = E(P_new, U_old) - M.
In simple terms, it is the cash you would need to feel just as well off as before the price change.
Perfect complements
Goods consumed in fixed proportions (e.g. left and right shoes). Their indifference curves are L-shaped, with the kink along the line where the fixed ratio holds.
Think of it as goods that are useless unless paired in the right ratio.
The consumer maximises U(x, y) subject to the budget constraint Pxx + Pyy = M.
At the optimum, the tangency condition holds: MRS = MUx/MUy = Px/Py. The consumer's personal trade-off rate between the goods equals the market's trade-off rate.
The solution also satisfies the budget constraint (the consumer spends all income).
For U = x^a * y^b, the demand functions are:
x* = [a / (a + b)] * (M / Px)
y* = [b / (a + b)] * (M / Py)
The consumer always spends fraction a/(a+b) of income on good x and b/(a+b) on good y, regardless of prices. This constant-expenditure-share property is unique to Cobb-Douglas.
U(x,y) = x^0.5 * y^0.5, so a = b = 0.5. Income M = 120, Px = 2, Py = 4.
x* = (0.5/1) * (120/2) = 30 units.
y* = (0.5/1) * (120/4) = 15 units.
Utility = 30^0.5 * 15^0.5 = sqrt(450) = 21.21 approximately.
CV measures the extra income needed at new prices to restore the consumer's original utility.
Use the expenditure function: for U = x^0.5 * y^0.5, the expenditure function is E(Px, Py, U) = 2 * sqrt(Px * Py) * U.
After Px rises from 2 to 8 (Py stays at 4): E(8, 4, sqrt(450)) = 2 * sqrt(32) * sqrt(450) = 2 * sqrt(14400) = 2 * 120 = 240.
CV = E_new - M_old = 240 - 120 = 120. The consumer would need an extra $120 to be as well off as before.
This framework is how economists measure the welfare cost of price changes such as energy-price shocks or tariffs. The CV gives a dollar figure for how much worse off consumers are, which is more useful for policy than simply noting that quantity demanded fell.
Any price change produces two effects on quantity demanded:
Substitution effect: the change in demand due to the shift in relative prices, holding utility constant. Always moves opposite the price change (price falls, quantity rises).
Income effect: the change in demand due to the shift in real purchasing power. Direction depends on whether the good is normal (same direction as substitution effect) or inferior (opposite direction).
Total effect = substitution effect + income effect.
Both effects reinforce each other. A price decrease raises quantity demanded through both channels. The demand curve slopes downward unambiguously.
The income effect works against the substitution effect. A price decrease makes the consumer feel richer, but for an inferior good, feeling richer means buying less of it.
For most inferior goods the substitution effect still dominates, so the demand curve still slopes downward.
The extreme case of an inferior good where the income effect is so large it overwhelms the substitution effect.
Result: quantity demanded moves in the same direction as price (price rises, quantity rises).
The demand curve slopes upward. Giffen goods are theoretically possible but rarely observed in practice. The classic textbook example is staple foods consumed by very poor households.
The substitution-income decomposition is the tool behind analysing how tax changes, subsidies, or cost-of-living adjustments affect household spending. For instance, a fuel-tax increase has both a substitution effect (switch to public transport) and an income effect (less money left over for everything).
Consumed in fixed proportions (e.g. one left shoe with one right shoe).
Indifference curves are L-shaped, with the kink along the line of the fixed ratio.
No substitution effect exists, because the consumer never trades one good for the other.
The consumer is willing to swap one good for another at a constant rate.
Indifference curves are straight lines with a negative slope.
The consumer buys whichever good is cheaper per unit of utility, or any combination if prices match the MRS.
Traces optimal bundles as income varies, prices held constant.
If the ICC slopes upward in both goods, both are normal.
If the ICC bends back toward one axis, that good is inferior at higher income levels.
Plots quantity of one good against income.
Normal good: positive slope (more income, more of the good).
Inferior good: negative slope (more income, less of the good).
Transitivity (A preferred to B, B preferred to C implies A preferred to C) prevents circular preferences.
Without transitivity, a consumer could be "money-pumped," cycling through trades and losing wealth each round. It is essential for the existence of a well-defined utility function.
Rearranging Pxx + Pyy = M into slope-intercept form: y = -(Px/Py)*x + M/Py.
The slope is -Px/Py, which is the rate at which the market allows the consumer to trade x for y.
This slope is also the Marginal Rate of Transformation (MRT) in a two-good consumer setting.
\text{Budget constraint: } P_x x + P_y y = M\text{Tangency condition: } \frac{MU_x}{MU_y} = \frac{P_x}{P_y}\text{Cobb-Douglas demand: } x^* = \frac{a}{a+b} \cdot \frac{M}{P_x}, \quad y^* = \frac{b}{a+b} \cdot \frac{M}{P_y}\text{Expenditure function (Cobb-Douglas): } E(P_x, P_y, U) = 2\sqrt{P_x P_y} \cdot U\text{Compensating Variation: } CV = E(P_{\text{new}}, U_{\text{old}}) - M\text{Slope of budget constraint: } -\frac{P_x}{P_y}Students often think the MRS is the slope of the budget constraint. The MRS is the slope of the indifference curve. At the optimum they are equal, but they measure different things.
Students often confuse the substitution effect with the total effect. The substitution effect isolates the impact of relative price changes only, holding utility constant. The total change in quantity includes the income effect as well.
Students often think all inferior goods are Giffen goods. A Giffen good is a special, extreme case of an inferior good. Most inferior goods still obey the law of demand because the substitution effect dominates.
Students often set the MRS equal to the price ratio the wrong way round, writing MUy/MUx = Px/Py instead of MUx/MUy = Px/Py. Keep the goods in the same order on both sides.
⚠️ Cobb-Douglas optimisation (tangency condition, demand functions, utility calculation) is near-certain exam material. Practise until you can do it from memory.
⚠️ The Compensating Variation calculation using the expenditure function is a high-value question. Know the expenditure function for Cobb-Douglas cold.
⚠️ Be ready to decompose a price change into substitution and income effects and state clearly which direction each moves.
⚠️ Know the slope of the budget constraint (-Px/Py) and how to distinguish it from MRS. Multiple-choice questions test this routinely.
⚠️ Perfect complements (L-shaped indifference curves) versus perfect substitutes (straight-line indifference curves) is a common one-mark pick-up.
True or False: The MRS is the slope of the budget constraint. (False, it is the slope of the indifference curve.)
Fill in the blank: For Cobb-Douglas utility U = x^a * y^b, the consumer spends fraction ______ of income on good x. (a/(a+b))
True or False: A Giffen good must be an inferior good. (True.)
Fill in the blank: The Engel curve for an inferior good has a ______ slope. (Negative.)
True or False: The substitution effect of a price decrease can be negative (reduce quantity demanded). (False, it always increases quantity demanded of the good whose price fell.)
Q: A consumer has utility U(x,y) = x^0.5 * y^0.5, income M = 120, Px = 2, Py = 4. Find the optimal bundle and utility.
A: x* = (0.5/1)(120/2) = 30, y* = (0.5/1)(120/4) = 15. U = sqrt(30*15) = sqrt(450) = 21.21.
Q: Using the same setup, if Px rises to 8, what is the Compensating Variation?
A: E(8, 4, sqrt(450)) = 2sqrt(84)sqrt(450) = 2sqrt(32)*sqrt(450) = 240. CV = 240 - 120 = $120.
Q: What is the slope of the budget constraint Pxx + Pyy = M?
A: -Px/Py.
Q: Define the substitution effect of a price decrease.
A: The change in quantity demanded due to the change in relative prices, holding utility (real income) constant. It always increases quantity demanded of the good whose price fell.
Q: What is a Giffen good, and what must be true about its income effect?
A: A Giffen good is an inferior good for which the negative income effect of a price decrease outweighs the positive substitution effect, so quantity demanded falls when price falls. The income effect must be larger in magnitude than the substitution effect.
Q: Describe the shape of indifference curves for perfect complements.
A: L-shaped (right-angled), with the kink along the line where the goods are consumed in their fixed ratio.
Consumer theory connects directly to demand curves: the demand function for a good is simply the solution to the optimisation problem as price varies. It also feeds into general equilibrium (Edgeworth box analysis), where two consumers' optimal choices must be consistent with market clearing. The welfare measures here (CV) reappear in public-finance topics like tax incidence and deadweight loss.
Utility maximisation, Cobb-Douglas, optimal bundle, tangency condition, MRS, marginal rate of substitution, budget line, budget constraint slope, indifference curve, income effect, substitution effect, Slutsky decomposition, Hicksian decomposition, Giffen good, inferior good, normal good, Engel curve, Income Consumption Curve, ICC, compensating variation, CV, expenditure function, perfect complements, perfect substitutes, L-shaped indifference curves, transitivity, rational choice, preference axioms, ECON 302, intermediate microeconomics