Source: Velez & Guo lecture notes, Pindyck & Rubinfeld Ch. 3-4 | Texas A&M University
Tags: consumer theory, preferences, utility function, indifference curves, marginal utility, MRS, budget constraint, utility maximization, demand, elasticity, normal goods, inferior goods, Giffen goods, income effect, substitution effect, intermediate microeconomics
Consumer theory builds a model of individual decision-making from three ingredients: preferences (what you want), budget constraints (what you can afford), and optimising behaviour (choosing the best affordable bundle). From these, we derive demand functions, measure sensitivity with elasticities, and decompose price changes into substitution and income effects.
Preference (consumer's preference)
The complete collection of an agent's answers to every pairwise "is a at least as good as b?" question over consumption alternatives. Preferences encode all taste information.
Complete preference
A preference where, for any two alternatives a and b, at least one of the following is true: a is at least as good as b, or b is at least as good as a.
Transitive preference
A preference where, for any three alternatives a, b, c: if a is at least as good as b, and b is at least as good as c, then a is at least as good as c. Equivalently, no "cycles" of strict preference exist.
Utility function, U(x, y)
A function assigning a real number to each consumption bundle such that higher numbers correspond to preferred bundles. If U(a) >= U(b), then a is at least as good as b for the agent. The numerical value itself is meaningless; only the ranking matters.
Indifference curve (indifference set)
The set of all bundles (x, y) that yield the same utility level. Found by fixing U(x, y) = k and solving for the relationship between x and y.
More-is-better (monotonicity)
A property where if x' > x and y' > y, then U(x', y') > U(x, y). Implies indifference curves are thin, downward-sloping, and never cross.
Marginal utility (MU)
The partial derivative of the utility function with respect to one good. MUx(x, y) = dU/dx. Measures the rate of change in utility from a small increase in consumption of that good.
Marginal rate of substitution (MRS)
The absolute value of the slope of the indifference curve at a point. MRSxy(x, y) = MUx / MUy. Represents the maximum amount of y the agent would give up for one more unit of x (locally).
Decreasing MRS
The indifference curve becomes flatter as x increases, meaning it becomes harder to substitute x for y the more x you already have. Produces convex indifference curves.
Budget constraint
The set of bundles satisfying px x + py y <= W, where px and py are prices and W is income. The boundary (budget line) is px x + py y = W.
Marginal rate of transformation (MRT)
The absolute value of the slope of the budget line: MRTxy = px / py. Represents the market's trade-off rate between goods.
Demand function, Qx(px, py, W)
The quantity of good x that maximises utility given prices px, py and income W. Demand depends on all three variables.
Demand curve
The graph of quantity demanded against own price, holding other prices and income fixed. Economists plot price on the vertical axis.
Engel curve
The graph of quantity demanded against income, holding prices fixed.
Price elasticity of demand
epx = (dQx/dpx) * (px / Qx). Measures the percentage change in quantity demanded per 1% change in own price.
Income elasticity of demand
eW = (dQx/dW) * (W / Qx). Measures the percentage change in quantity demanded per 1% change in income.
Normal good
A good whose demand increases (or stays constant) as income rises: dQx/dW >= 0, equivalently eW >= 0.
Inferior good
A good whose demand decreases as income rises at some market situation: eW < 0.
Giffen good
A good whose demand increases when its own price rises: dQx/dpx > 0. Requires the good to be inferior, but not all inferior goods are Giffen.
Substitution effect
The change in consumption of x due to a price change, holding utility constant. Always follows the law of demand (price up, quantity down).
Income effect
The remaining change in consumption after accounting for the substitution effect. Can be positive or negative depending on whether the good is normal or inferior.
Why we need utility functions
With infinitely many bundles in continuous space, listing all pairwise preference answers is impractical
Under completeness, transitivity, and a mild continuity assumption, preferences can always be encoded as a utility function
Utility numbers are ordinal, not cardinal: only the ranking matters, not the size of the gap
Common utility function forms
Cobb-Douglas: U(x, y) = x^a * y^b (smooth, curved indifference curves)
Linear / perfect substitutes: U(x, y) = ax + by (straight-line indifference curves, constant MRS)
Perfect complements (Leontief): U(x, y) = min{ax, by} (L-shaped indifference curves, consumed in fixed proportions)
Indifference curves that violate more-is-better
Crossing curves: two bundles in different curves are equal via transitivity, but one is to the north-east of the other
Thick curves: contain two points where one dominates the other
Upward-sloping curves: same logic as thick curves
Marginal utility and MRS
MUx = dU/dx, MUy = dU/dy
MRSxy = MUx / MUy (not MUy / MUx, a common mistake)
"Substitution of x for y" means consuming more x, less y
When MRS is constant (linear utility), the local interpretation extends globally: MRSxy is exactly the amount of y the agent will trade for one unit of x, regardless of bundle size
Decreasing MRS
Indifference curves become flatter as you move right
Reflects diminishing willingness to trade away y for more x
The "first coffee of the day" intuition: the first unit is worth much more than the fifth
Budget line geometry
Equation: y = (W / py) - (px / py) * x
Vertical intercept: W / py (spend all income on y)
Horizontal intercept: W / px (spend all income on x)
Slope: -px / py = -MRTxy
How the budget line shifts
Price of x increases: budget line pivots inward on the vertical intercept (steeper)
Price of y increases: budget line pivots inward on the horizontal intercept (flatter)
Income increases: budget line shifts outward, parallel to itself
Utility maximisation with smooth preferences (interior solution)
Two conditions characterise the optimum (x-hat, y-hat):
Budget line: px x-hat + py y-hat = W
Tangency rule: MRSxy(x-hat, y-hat) = px / py
Solve these two equations for two unknowns. The logic: at the optimum, the indifference curve is tangent to the budget line. If the indifference curve were steeper or flatter than the budget line, the agent could do better by moving along the budget line.
Utility maximisation with linear preferences
Compare MRSxy with px / py
MRSxy > px / py: agent buys only good x (corner solution at horizontal intercept)
MRSxy < px / py: agent buys only good y (corner solution at vertical intercept)
MRSxy = px / py: every point on the budget line is a maximiser
Shortcut: evaluate utility at both intercepts of the budget line. The higher one is the maximiser.
Utility maximisation with non-smooth preferences (e.g. perfect complements)
Cannot use tangency rule; instead, visually locate the indifference curve farthest to the north-east that still touches the budget line
For Leontief preferences, the optimum lies at the kink of the L-shaped curve on the budget line
Application: the food stamps paradox
An agent receiving $135 in food stamps is never worse off, but possibly better off, with $135 in cash
Food stamps restrict spending to food only; cash expands the budget set
The paradox: public support favours stamps over cash, but the policy's goal is improved nutrition (a paternalistic aim), not maximising the agent's own preferences
Deriving demand
Solve the two-equation system (budget line + tangency rule) for general (px, py, W) to get Qx(px, py, W) and Qy(px, py, W)
Example with u = 4x^(1/4) * y^(3/4): Qx = (1/4)(W/px), Qy = (3/4)(W/py)
Demand for x may depend on py through cross-price effects
Price elasticity of demand
Formula: epx = (dQx/dpx) * (px / Qx)
Unit-free measure: a 1% increase in px causes approximately epx % change in Qx
For Cobb-Douglas demand Qx = (1/4)(W/px), the price elasticity is -1 (unit elastic)
Graphically, a flatter demand curve (in the economist's convention of p on the y-axis) indicates more elastic demand at a crossing point
Income elasticity of demand
Formula: eW = (dQx/dW) * (W / Qx)
For Cobb-Douglas demand, income elasticity is 1
eW > 0 means normal good; eW < 0 means inferior good
Normal goods
Demand increases with income (Engel curve is upward-sloping)
On the income-consumption curve, consumption moves north-east as income rises
Aggregate good categories (food, clothing, transport) are typically normal even if individual brands are not
Inferior goods
Demand falls as income rises for some market situation
Engel curve is downward-sloping in that range
Example: instant Ramen noodles; as income rises, the agent switches to better food
Giffen goods
Demand increases with own price (upward-sloping demand curve)
Requires the good to be inferior at the relevant market situation, but inferiority alone is not sufficient
The Irish potato famine is not a genuine example (income fell, not just potato prices rose)
Genuine evidence: rice among poor farmers in certain Chinese provinces (Jensen and Miller, mid-2000s)
Decomposition procedure (Slutsky-Hicks)
Suppose px falls from px to px'. Three steps on a graph:
Plot the initial optimum (x-hat, y-hat) and draw the new (cheaper) budget constraint
Shift the new budget line parallel until it is tangent to the original indifference curve. The tangent point is the "compensated" bundle (x-hat-1, y-hat-1). The substitution effect is x-hat-1 - x-hat.
Find the actual new optimum (x-hat-2, y-hat-2). The income effect is x-hat-2 - x-hat-1.
Total effect = substitution effect + income effect.
Key results
The substitution effect always obeys the law of demand (price down causes quantity up, and vice versa)
The income effect can go either direction
For a normal good, both effects push in the same direction, so demand curves always slope downward
For an inferior good, the income effect works against the substitution effect
A Giffen good is the extreme case where the income effect overwhelms the substitution effect
Aggregate goods are unlikely to be Giffen because aggregate demand is typically normal
MRS formula
MRSxy(x, y) = MUx(x, y) / MUy(x, y)
Tangency condition for utility maximisation
MRSxy = px / py, together with px x + py y = W
Price elasticity of demand
epx(px, py, W) = (dQx/dpx) * (px / Qx)
Income elasticity of demand
eW(px, py, W) = (dQx/dW) * (W / Qx)
Budget line equation
y = (W / py) - (px / py) * x
⚠️ MRS = MUx / MUy, not MUy / MUx. This reversal is the single most common calculation error.
⚠️ "Substitution of x for y" means more x, less y. "Substitution of x with y" reverses the meaning. Be precise with prepositions.
⚠️ Utility values are ordinal. A utility of -100 does not mean "bad"; it only matters relative to other bundles' utility.
⚠️ For linear preferences, do not use the tangency rule blindly. Compare MRS with the price ratio and identify corner solutions.
⚠️ Not every inferior good is Giffen. The income effect must be large enough to overwhelm the substitution effect.
⚠️ Demand is a function of three variables (px, py, W). The demand curve holds two constant.
⚠️ Food stamps vs. cash: cash is weakly better for the agent (larger budget set). The policy rationale for stamps is paternalistic.
Q: An agent has utility U(x, y) = x^(1/3) * y^(2/3), income W = 120, px = 2, py = 4. What bundle maximises utility?
A: MRSxy = (1/3 y^(2/3) x^(-2/3)) / (2/3 x^(1/3) y^(-1/3)) = y / (2x). Set y/(2x) = px/py = 2/4 = 1/2, giving y = x. Substitute into the budget line: 2x + 4x = 120, so x = 20, y = 20.
Q: If Q(p) = 50/p, what is the price elasticity of demand?
A: dQ/dp = -50/p^2. Then ep = (-50/p^2)(p / (50/p)) = (-50/p^2)(p^2/50) = -1. The demand is unit elastic everywhere.
Q: Can a good be inferior but not Giffen?
A: Yes. A good is Giffen only if the (negative) income effect exceeds the (positive) substitution effect in absolute terms. Most inferior goods have a substitution effect that dominates, so their demand still falls when price rises.
Q: An agent has linear utility U = 3x + 5y, with px = 6, py = 8, and W = 48. What does the agent consume?
A: MRSxy = 3/5 = 0.6. Price ratio px/py = 6/8 = 0.75. Since MRSxy < px/py, the indifference curves are flatter than the budget line, so the agent buys only y. Quantity: W/py = 48/8 = 6 units of y.
Q: Graphically, how do you identify the substitution effect of a price decrease in x?
A: Draw a line parallel to the new (flatter) budget line that is tangent to the original indifference curve. The horizontal distance between the original optimum and the tangent point is the substitution effect.
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