ECOS2001 Chap.1 Budget, Preferences & Utility
Budget, Preferences & Utility
This chapter sets up the consumer's two halves: what they can afford (the budget set) and what they want (preferences). The budget line p₁x₁ + p₂x₂ = m has intercepts m/p₁ and m/p₂ and slope −p₁/p₂, the market rate of exchange; a higher income shifts it out in parallel while a single price change pivots it about the unchanged intercept.
Preferences are described by the axioms (completeness, transitivity, monotonicity, convexity) and drawn as indifference curves whose slope is the marginal rate of substitution. Utility is ordinal, so any positive monotonic transform represents the same preferences and leaves the MRS unchanged.
What this chapter covers
- 01
Budget line: intercepts m/p₁ and m/p₂, slope −p₁/p₂
- 02
How income changes shift, and price changes pivot, the budget line
- 03
Ad valorem and quantity taxes/subsidies; kinked budgets from rationing
- 04
Preference axioms: completeness, transitivity, monotonicity, convexity
- 05
Indifference curves and the MRS = MU₁/MU₂ (a magnitude)
- 06
Ordinal utility: monotonic transforms leave preferences (and the MRS) unchanged
- 07
Special preferences: perfect substitutes, perfect complements, quasi-linear
Budget line and MRS
- 2 marksIntercepts: spend all income on one good. x₁-intercept = m/p₁ = 120/4 = 30; x₂-intercept = m/p₂ = 120/5 = 24.
- 1 markBudget-line slope = −p₁/p₂ = −4/5 = −0.8 (the rate at which the market lets you swap snacks for coffee).
- 1 markMarginal utilities of U = x₁²·x₂: MU₁ = 2x₁·x₂ and MU₂ = x₁².
- 1 markMRS = MU₁/MU₂ = (2x₁x₂)/(x₁²) = 2x₂/x₁ (taken as a positive magnitude).
- 1 markAt (10, 16): MRS = 2·16/10 = 3.2.
Key terms
- Budget line
- The set of bundles costing exactly all income, p₁x₁ + p₂x₂ = m, with intercepts m/p₁ and m/p₂ and slope −p₁/p₂.
- Marginal rate of substitution (MRS)
- The magnitude of the slope of an indifference curve, MRS = MU₁/MU₂ (a positive number); how much of good 2 a consumer will give up for one more unit of good 1 while staying equally happy.
- Ordinal utility
- Utility numbers only rank bundles, not measure happiness; any increasing transform of U represents the same preferences and the same MRS.
- Perfect complements
- Goods consumed in fixed proportions, U = min{ax₁, bx₂}, giving L-shaped indifference curves with a kink on the ray ax₁ = bx₂.
Budget, Preferences & Utility FAQ
What is the difference between the budget-line slope and the MRS?
The budget-line slope −p₁/p₂ is the market's exchange rate, set by prices; the MRS (taken as a positive magnitude, MU₁/MU₂) is the consumer's personal willingness to trade, set by their preferences. Their magnitudes differ everywhere except at the optimal bundle, where |MRS| = p₁/p₂.
Why does a price fall pivot the budget line instead of shifting it?
Spending everything on the good whose price did not change still buys the same amount, so that intercept is fixed; the other intercept moves out because each dollar now buys more of the cheaper good. The line pivots about the unchanged intercept.
Exam move
Be able to draw and label the budget line (both intercepts, slope) in seconds and to compute the MRS from any utility function by taking the two marginal utilities and dividing. Practise the four special-preference shapes so you recognise them instantly in MCQs.
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