ECON5001 · Microeconomic Theory
Consumer Choice, Demand & Comparative Statics
This is the engine room of consumer theory in ECON5001: the consumer picks the affordable bundle that reaches the highest indifference curve, and solving that problem as a function of prices and income is the demand function. The marks live in the method, not the recall — diagnose the preference type first (smooth-convex needs the tangency MRS = p1/p2, while perfect substitutes, perfect complements and concave preferences need corner or proportion logic), then re-solve. Comparative statics pushes one parameter and watches the optimum move: raising income traces the income-offer curve and the Engel curve (normal vs inferior), while changing a price traces the price-offer curve and the ordinary demand curve (ordinary vs Giffen). The Slutsky decomposition then splits any price change into a substitution effect (always opposite the price move) and an income effect (sign set by normal/inferior), which is consumer theory's single most-tested mechanism on the Week-7 mid-semester exam.
What this chapter covers
- 011. The consumer's problem — maximise U(x1,x2) subject to the budget p1x1 + p2x2 = m, with the optimum on the line
- 022. Interior tangency condition — MRS = p1/p2, equivalently equal bang-per-buck MU1/p1 = MU2/p2, plus the binding budget
- 033. Corner solutions — when one good is not bought, the optimum sits on an axis instead of at a tangency
- 044. When tangency fails — perfect substitutes (bang-per-buck), perfect complements (the proportion), concave and quasi-linear
- 055. Ordinary (Marshallian) demand — solving with prices and income as symbols; the Cobb-Douglas constant-share rule
- 066. Income comparative statics — the income-offer curve and the Engel curve; normal vs inferior goods
- 077. Price comparative statics — the price-offer curve builds the demand curve; ordinary vs Giffen goods
- 088. Slutsky decomposition — splitting a price change into a substitution effect plus an income effect, and where Giffen lives
Perfect complements: the optimal bundle via the proportion
- +2Diagnose the preference. Utility is a min{...}, so these are perfect complements with L-shaped indifference curves. The MRS is undefined at the kink, so tangency does not apply — use the proportion instead.
- +2Read the proportion off the utility. The consumer never wastes either good, so the optimum sits on the kink line where 2x1 = x2.
- +2Substitute the proportion into the budget. 4x1 + 6x2 = 160 becomes 4x1 + 6(2x1) = 160, i.e. 4x1 + 12x1 = 16x1 = 160.
- +1Solve. x1* = 10, and x2* = 2(10) = 20. Check the budget: 4(10) + 6(20) = 40 + 120 = 160, exactly m.
- +1Evaluate utility. U = min{2(10), 20} = min{20, 20} = 20 — both arguments are equal, confirming no good is wasted.
Key terms
- Tangency (interior) condition
- The interior optimal-choice rule: the indifference curve is tangent to the budget line, so MRS = p1/p2, equivalently MU1/p1 = MU2/p2 (the last dollar buys the same extra utility on either good). Valid only for smooth, strictly convex preferences.
- Corner solution
- An optimum where one good is not bought at all (xi* = 0), so the chosen bundle sits on an axis rather than at a tangency. It is the rule for perfect substitutes and for concave preferences, where the tangency condition is the wrong tool.
- Ordinary (Marshallian) demand
- The optimal quantities written as functions of prices and income, x*(p1, p2, m). For Cobb-Douglas U = x1^a x2^b each good gets a constant share of income, x1* = a/(a+b) · m/p1, which makes the good normal and unit own-price elastic.
- Engel curve
- The plot of one good's chosen quantity against income, holding prices fixed. It slopes up for a normal good (quantity rises with income) and bends back for an inferior good; a homothetic preference gives a straight ray from the origin.
- Income-offer vs price-offer curve
- The income-offer (income-consumption) curve joins the optima in good space as income rises with prices fixed; the price-offer (price-consumption) curve joins them as one price changes. Re-plotting quantity against income gives the Engel curve, and against own price gives the demand curve.
- Normal vs inferior good
- Normal means quantity rises as income rises (the partial of x with respect to m is positive); inferior means it falls. Many goods are normal when income is low and inferior when income is high, so the Engel curve can bend back.
- Giffen good
- A good whose demand curve slopes up over a range, so quantity demanded rises when its own price rises. It requires a strongly inferior good whose negative income effect outweighs the substitution effect — every Giffen good is inferior, but most inferior goods are not Giffen.
- Slutsky decomposition
- Splitting a price change into a substitution effect (the move along the compensated budget that keeps the old bundle just affordable — always opposite to the price change) plus an income effect (restoring the real budget — positive for normal goods, negative for inferior). The compensated income is m' = p1' x1A + p2 x2A, using new prices and old quantities.
Consumer Choice, Demand & Comparative Statics FAQ
How do I know whether to use the tangency condition or look for a corner?
Read the utility function. A product such as x1^a x2^b is Cobb-Douglas and smooth-convex, so the interior tangency MRS = p1/p2 applies. A sum ax1 + bx2 is perfect substitutes — the indifference curves are straight, so there is no tangency and you compare bang-per-buck MU1/p1 versus MU2/p2 and buy only the winner (a corner). A min{ax1, bx2} is perfect complements — solve the proportion ax1 = bx2 with the budget. Concave preferences also give a corner. Always check at the end that quantities are positive; a negative answer signals the true optimum is a corner.
What is the difference between the income-offer curve and the Engel curve?
They live in different spaces. The income-offer curve plots the optimal bundles (x1 against x2) as income rises with prices fixed. The Engel curve plots one good's quantity against income (x against m). Students lose marks by drawing one when the question asks for the other, so always label which axes you are using.
Why does demand usually slope down, and when can it slope up?
Lowering a good's price does two things: it makes the good relatively cheaper, so you substitute toward it (the substitution effect, always raising quantity), and it frees real income (the income effect). For a normal good both forces push the same way, so demand slopes down. Demand only slopes up for a Giffen good — a strongly inferior good whose negative income effect outweighs the substitution effect — which is rare and only over a range.
How do I compute a Slutsky substitution and income effect?
Find the old bundle A at the old prices, then the new bundle C at the new prices; their difference is the total effect. For the split, compute the compensated income m' = p1' x1A + p2 x2A (new prices, old quantities) so the old bundle is just affordable, and find the bundle B chosen on that compensated budget. The substitution effect is x1B minus x1A and the income effect is x1C minus x1B. The substitution effect always opposes the price change; only the income effect's sign depends on whether the good is normal or inferior.
Is an inferior good the same as a Giffen good?
No. Inferior describes the response to income (quantity falls as income rises). Giffen describes the response to own price (quantity rises as price rises). Every Giffen good must be inferior, but the reverse fails: most inferior goods are not Giffen because the substitution effect still dominates the income effect. Giffen needs the income effect to be both negative and large enough to swamp substitution.
What makes Cobb-Douglas demand so convenient in exams?
For U = x1^a x2^b the demands are x1* = a/(a+b) · m/p1 and x2* = b/(a+b) · m/p2, so each good takes a constant share of income regardless of prices. That implies an income elasticity of exactly 1 (always normal), an own-price elasticity of minus 1 (unit-elastic everywhere) and a zero cross-price effect. Use the share rule as a five-second check on your tangency working, but still show the marginal-utility and budget steps because that is where the marks are.
Exam move
Treat this chapter as a decision tree rather than a list of facts. Step one is always to diagnose the preference type from the functional form (product, sum, min, or quasi-linear), because that dictates the method — tangency, bang-per-buck, or the proportion. Step two is to run that method to a bundle and verify it is genuinely interior (both quantities positive, budget binding). Once you can solve a single optimum cleanly, the comparative statics are just the same optimum re-solved: raise income and you trace the Engel curve, change a price and you trace the demand curve. Invest your heaviest practice in the Slutsky decomposition, since it is the most-tested mechanism and the easiest to fumble — drill the compensated-income line m' = p1' x1A + p2 x2A until it is automatic, and always state the rule that the substitution effect opposes the price change before debating the income effect. Work the unit's problem sets with the source's own numbers first, then re-derive each from a blank page under time pressure, keeping a running list of traps (autopilot tangency on substitutes, calling every inferior good Giffen, mixing up the income-offer and Engel curves) so they stop costing you marks.
