Difficulty: Intermediate | Prerequisites: comfort with partial derivatives, budget constraint algebra, and the concepts of marginal utility and indifference curves (Documents 1 and 2 of this series).
This set of notes brings the theory of consumer choice to its quantitative conclusion: given a utility function and a budget constraint, how do you find the optimal consumption bundle? You will also learn to classify goods as substitutes or complements from observed behaviour and meet the Giffen good, an infamous theoretical possibility where a price increase actually raises quantity demanded. The Shannon/Great Famine problem ties all three documents together by working through budget constraints, MRS, optimisation, income and substitution effects, and the Giffen classification in a single extended exercise.
At the optimum, the ratio of marginal utilities equals the price ratio (the tangency condition). If a price change causes you to buy less of the other good, the two goods are substitutes; if you buy more of both (or less of both), they are complements. A Giffen good is a special inferior good where the income effect is so strong that raising the price actually increases quantity demanded.
Marginal utility (MU)
The additional satisfaction a consumer gets from one more unit of a good, holding consumption of all other goods constant. Think of it as: how much happier the next slice of pizza makes you.
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. MRS = MU_x / MU_y. In simple terms, this means the number of units of Y you would give up for one more unit of X, without becoming worse off.
Tangency condition (optimality condition)
At the optimal bundle, MRS = P_x / P_y. Equivalently, MU_x / P_x = MU_y / P_y (the marginal utility per dollar is equal across goods). Think of it as: you have squeezed every last drop of satisfaction from your budget when the last pound spent on each good gives you the same happiness.
Substitutes
Two goods are substitutes if a decrease in the price of one leads the consumer to buy less of the other. In simple terms, this means they compete for your attention: cheaper pizza means fewer burgers.
Complements
Two goods are complements if a decrease in the price of one leads the consumer to buy more of the other as well. Think of it as: things that go together, like printers and ink.
Giffen good
A special type of inferior good where the income effect of a price increase is so large that it overwhelms the substitution effect, causing quantity demanded to rise when the price rises. In simple terms, this means a good so central to a poor consumer's budget that when it becomes more expensive, they cannot afford the alternatives and must buy even more of it (the classic textbook example is potatoes during the Irish Famine).
Budget constraint
The equation P_x * X + P_y * Y = I, representing all affordable combinations of goods X and Y given income I and prices P_x, P_y.
Hui consumes pizza (P_pizza = $16) and burgers (P_burger = $8). He chooses 4 pizzas and 8 burgers.
At the optimum, the tangency condition holds:
MU_pizza / MU_burger = P_pizza / P_burger = 16/8 = 2
This means the ratio of marginal utilities is 2: the last pizza gives Hui exactly twice the marginal satisfaction of the last burger, which makes sense because pizza costs exactly twice as much.
When pizza falls to $10, the new tangency condition is:
MU_pizza / MU_burger = 10/8 = 1.25
The ratio of marginal utilities decreases from 2 to 1.25. The consumer re-optimises so that the marginal utility of pizza relative to burgers falls (diminishing MU as he eats more pizza, increasing MU of burgers as he eats fewer).
When pizza drops from $16 to $10, Hui buys 6 pizzas (up from 4) and 7 burgers (down from 8). The decrease in burgers when pizza becomes cheaper tells us pizza and burgers are substitutes: the cheaper pizza draws spending away from burgers.
If burgers had increased to, say, 9, they would be complements (cheaper pizza leads to more of both).
Utility function: U(T, M) = 100T - T^2 + M + M^2/2
where T = potatoes, M = meatloaf.
Budget constraint in 1845:
P_T = 1, P_M = 1, I = 55
So: T + M = 55 (this is l1)
Marginal utilities:
MU_T = dU/dT = 100 - 2T
MU_M = dU/dM = 1 + M
MRS (potatoes for meatloaf):
MRS = MU_T / MU_M = (100 - 2T) / (1 + M)
Solving for the optimal basket in 1845:
Tangency condition: MRS = P_T / P_M = 1/1 = 1
So: (100 - 2T) / (1 + M) = 1
This gives: 100 - 2T = 1 + M, so M = 99 - 2T
Substitute into the budget constraint: T + (99 - 2T) = 55
99 - T = 55, so T = 44, M = 55 - 44 = 11
Optimal basket in 1845: 44 pounds of potatoes, 11 pounds of meatloaf.
Potato price rises to 1.1 shillings. Meatloaf price and income unchanged.
New budget constraint (l2): 1.1T + M = 55
New tangency condition: MRS = P_T / P_M = 1.1/1 = 1.1
(100 - 2T) / (1 + M) = 1.1
100 - 2T = 1.1(1 + M) = 1.1 + 1.1M
From the budget constraint: M = 55 - 1.1T
Substitute: 100 - 2T = 1.1 + 1.1(55 - 1.1T) = 1.1 + 60.5 - 1.21T
100 - 2T = 61.6 - 1.21T
38.4 = 0.79T
T ≈ 48.6 pounds of potatoes
M = 55 - 1.1(48.6) ≈ 55 - 53.46 ≈ 1.54 pounds of meatloaf
Optimal basket in 1848: approximately 48.6 pounds of potatoes, 1.54 pounds of meatloaf.
The government transfer programme (Hicksian compensation) gives Shannon enough income to reach her 1845 utility level at 1848 prices. Under this programme she buys 42 potatoes and 12 meatloaf.
1845 basket: T = 44
Compensated (Hicksian) basket: T = 42
1848 basket: T ≈ 48.6
Substitution effect: 42 - 44 = -2 (she buys fewer potatoes when they become relatively more expensive, holding utility constant).
Income effect: 48.6 - 42 = +6.6 (the loss of purchasing power makes her so much poorer that she buys more potatoes, not fewer).
Total effect: 48.6 - 44 = +4.6 (she buys more potatoes despite the price increase).
The income effect (+6.6) dominates the substitution effect (-2). The price of potatoes rose, yet Shannon bought more of them.
This is the hallmark of a Giffen good: the price goes up, and quantity demanded increases. Potatoes in this model are a Giffen good. They are necessarily inferior (the income effect works in the opposite direction to the substitution effect), and the income effect is large enough to overwhelm the substitution effect.
Historically, the Irish Famine is the classic example: potatoes consumed such a large share of a poor household's budget that a price increase left families too poor to afford meat, forcing them to buy even more potatoes to survive.
The Giffen good concept is largely theoretical, but similar dynamics appear in development economics: staple foods (rice in parts of China, tortillas in Mexico) can behave this way for very low-income households. The tangency condition, by contrast, is the workhorse of every applied consumption model.
Marginal utility:
MU_x = dU/dx (partial derivative of the utility function with respect to good x)
Marginal rate of substitution:
MRS_xy = MU_x / MU_y
Tangency (optimality) condition:
MU_x / MU_y = P_x / P_y
Equivalently: MU_x / P_x = MU_y / P_y (equal marginal utility per dollar)
Budget constraint:
P_x * X + P_y * Y = I
Shannon's utility function:
U(T, M) = 100T - T^2 + M + M^2/2
MU_T = 100 - 2T
MU_M = 1 + M
Income and substitution effect decomposition (numerical):
Substitution effect = Q_compensated - Q_original
Income effect = Q_new - Q_compensated
Total effect = Q_new - Q_original = SE + IE
"If the price of a good rises and a consumer buys more of it, the consumer is irrational." Not necessarily. This is exactly what a Giffen good looks like, and it is fully consistent with rational utility maximisation. The consumer cannot afford the superior alternatives.
"Substitutes and complements are fixed properties of goods." Whether two goods are substitutes or complements depends on the consumer's preferences and the specific price range. Pizza and burgers might be substitutes for one person and complements for someone who always eats both at lunch.
"The MRS is the same as the price ratio at all times." Only at the optimum. Away from the optimum, MRS and the price ratio diverge, and the consumer can improve by reallocating spending.
"Giffen goods are common." They are extremely rare empirically. The conditions are strict: the good must be inferior, it must represent a large share of the consumer's budget, and the income effect must be large enough to dominate the substitution effect.
⚠️ You will almost certainly be asked to set up and solve a tangency condition problem. Practise going from a utility function to MU_x, MU_y, setting MRS = price ratio, then solving simultaneously with the budget constraint.
⚠️ Be ready to determine substitutes vs complements from before/after consumption data. The test is what happens to the quantity of the other good when one good's price changes.
⚠️ The Giffen good question typically asks: compute the income and substitution effects, identify which dominates, and name the type of good. Know the full chain: Giffen implies inferior implies the income effect opposes and overwhelms the substitution effect.
⚠️ The Shannon problem is a full, multi-part optimisation exercise. Expect something similar on an exam: set up the budget constraint, find MRS, solve, then repeat after a parameter change.
True or False: At the optimal bundle, MU_x / P_x = MU_y / P_y. (True)
Fill in the blank: If the price of pizza falls and the consumer buys fewer burgers, pizza and burgers are ______. (substitutes)
True or False: A Giffen good must be an inferior good. (True)
Fill in the blank: The MRS equals the ______ at the consumer's optimal choice. (price ratio)
True or False: If the income effect dominates the substitution effect for an inferior good, quantity demanded rises when price rises. (True)
Q: Hui buys 4 pizzas ($16 each) and 8 burgers ($8 each). What is MU_pizza / MU_burger at his optimum?
A: At the optimum, MU_pizza / MU_burger = P_pizza / P_burger = 16/8 = 2.
Q: Pizza drops to $10. Hui now buys 6 pizzas and 7 burgers. Are pizza and burgers substitutes or complements?
A: Substitutes. The price of pizza fell and the quantity of burgers decreased (from 8 to 7), meaning cheaper pizza drew spending away from burgers.
Q: Given U(T,M) = 100T - T^2 + M + M^2/2, with P_T = P_M = 1 and I = 55, find the optimal bundle.
A: MU_T = 100 - 2T, MU_M = 1 + M. Tangency: (100 - 2T)/(1 + M) = 1, so M = 99 - 2T. Budget: T + M = 55. Substituting: T + 99 - 2T = 55, so T = 44, M = 11.
Q: In 1848, P_T rises to 1.1. Compute the new optimal T.
A: New tangency: (100 - 2T)/(1 + M) = 1.1. Budget: 1.1T + M = 55, so M = 55 - 1.1T. Substituting: 100 - 2T = 1.1 + 1.1(55 - 1.1T) = 61.6 - 1.21T. So 0.79T = 38.4, T ≈ 48.6.
Q: The compensated bundle under the government transfer is T = 42. Compute the substitution and income effects for potatoes. Is this a Giffen good?
A: SE = 42 - 44 = -2 (potatoes fell). IE = 48.6 - 42 = +6.6 (potatoes rose). Total = +4.6. The income effect dominates and runs in the opposite direction to the substitution effect. Potatoes are a Giffen good.
This material directly extends consumer choice theory (Document 1): the income and substitution effects computed here are the same decomposition introduced graphically there, now applied with calculus. The elasticity concepts (Document 2) connect because a Giffen good's demand curve is upward-sloping, giving a positive price elasticity of demand, the opposite of the normal case.
Marginal utility, MU, marginal rate of substitution, MRS, tangency condition, equal marginal utility per dollar, budget constraint, optimal bundle, utility maximisation, substitutes, complements, Giffen good, inferior good, income effect, substitution effect, Hicksian compensation, Great Famine, Irish Famine, potatoes, upward-sloping demand, Slutsky equation, consumer optimisation, microeconomics problem set 3, NYU intro micro