Microeconomic Theory: Utility & Optimization
CollegeChapters 6–9 asserted that demand curves slope down and gave the intuition why. Here we build that law from first principles: a rational consumer maximising utility subject to a budget constraint — the formal foundation beneath every downward-sloping curve you've drawn so far.
At a glanceUtility maximisation subject to a budget constraint
A consumer's preferences over two goods are represented by a utility function U(x, y). Marginal utility MUx = ∂U/∂x is the extra satisfaction from one more unit of x; the law of diminishing marginal utility says MUx falls as x rises, holding y fixed. Preferences are drawn as indifference curves — loci of (x, y) bundles giving equal utility — which are convex to the origin and never cross.
The consumer maximises U(x, y) subject to Pxx + Pyy = M. Forming the Lagrangian ℒ = U(x, y) + λ(M − Pxx − Pyy) and setting its partial derivatives to zero yields the tangency condition:
In words: at the optimum, the last dollar spent on either good buys the same extra utility, and the marginal rate of substitution (the slope of the indifference curve) equals the relative price ratio (the slope of the budget line) — the indifference curve is tangent to the budget line. Producer theory is the mirror image: a firm minimises cost subject to an output target, with isoquants tangent to isocost lines where MRTS = MPL/MPK = w/r.
From optimisation to the demand curve — and back to Chapter 7
Solving the tangency condition for every possible Px, holding M and Py fixed, traces out x*(Px) — the individual's Marshallian (ordinary) demand curve. This is the rigorous derivation behind the "law of demand" asserted informally in Chapter 7: it slopes down because a lower Px makes the same bundle affordable at a higher utility, and the tangency point systematically moves outward along the new budget line.
The Slutsky equation formally splits the effect of a price change into two parts foreshadowed in Chapter 6's labour-supply discussion:
The substitution effect (movement along a single indifference curve to the new price ratio) is always negative — cheaper goods are always substituted toward. The income effect (the shift to a new indifference curve caused by the change in real purchasing power) can be positive or negative depending on whether the good is normal or inferior. This is also how consumer surplus (Chapter 9's simple triangle) is refined: the exact welfare measure of a price change is the compensating variation along the Hicksian (utility-constant) demand curve, of which the Marshallian consumer-surplus triangle is only a close approximation.
Worked example: solving a Cobb-Douglas optimisation
Q1Why must indifference curves be convex to the origin, and why can't two cross?
Convexity reflects diminishing MRS: as you have more x and less y, you require ever more x to compensate for giving up one more unit of y — a plausible "variety is valued" assumption. Two indifference curves crossing would imply a single bundle belongs to two different utility levels simultaneously, violating the definition of a well-behaved preference ordering (completeness and transitivity).
Q2What does λ (the Lagrange multiplier) represent economically?
λ is the marginal utility of money — the extra utility obtainable from one more dollar of budget, optimally spent. It is exactly what both MUx/Px and MUy/Py equal at the optimum: the "bang per buck" is equalised across every good you buy, which is precisely why you keep reallocating spending until it is.
Q3For a Cobb-Douglas consumer, why doesn't the expenditure share on x respond to Px?
Because Cobb-Douglas utility has a unit elasticity of substitution built in: the substitution and income effects of a price change exactly offset each other in terms of budget share, though not in terms of quantity. This is a special property of Cobb-Douglas preferences, not a general law — other utility functions (e.g. perfect complements or perfect substitutes) give very different demand responses.
Q4How does this chapter's producer theory connect to Chapter 6's labour market?
Cost-minimising input choice (MRTS = w/r) is the formal machinery behind the MRP = wage hiring rule of Chapter 6. When capital is also variable, a firm chooses both labour and capital so that the marginal product per dollar is equalised across inputs — the same "bang per buck" logic as consumer choice, just applied to production instead of consumption. Microeconomics is one optimisation principle applied twice.
How the ideas connect
Every key idea in this chapter, branching from the core concept — use it to see the whole picture at a glance.
The process, step by step
Worked problems, step by step
Follow each solution line by line, then try to reproduce it on paper before moving on.
Example 1MUx = 20, Px = $4, MUy = 10, Py = $2. Is the consumer optimising?
- Compare MU per dollar: MUx/Px = 20/4 = 5.
- MUy/Py = 10/2 = 5.
- They are equal, so the tangency condition holds.
Example 2MUx/Px = 6 and MUy/Py = 4. Should the consumer buy more x or more y?
- Good x gives 6 utils per dollar; y gives only 4.
- Shift spending toward the higher MU-per-dollar good.
- Buying more x (and less y) raises total utility.
Now you try
Work each one out first, then tap to reveal the worked answer.