ECC1000 Chap.4 Indifference Curves, Preferences and the Consumer Optimum
Indifference Curves, Preferences and the Consumer Optimum
Two exchange rates, and whose they are
The marginal rate of transformation is the rate at which the market lets one good be turned into the other. It lives in the slope of the budget line and equals the ratio of the two prices, so it moves only when a price moves. The marginal rate of substitution is the rate at which the consumer would swap the goods and feel no different.
It lives in the slope of an indifference curve, equals the ratio of the two marginal utilities, and changes as the bundle changes. Almost every Week 3 question is a comparison of those two numbers.
Saying the rate in words before using it
A rate says how many units of the good on the vertical axis must be surrendered to obtain one more unit of the good on the horizontal axis.
Reading it upside down is the commonest error in this block, and attaching units to the sentence catches it: apples per fish is a different number from fish per apple, and only one of them is the slope on the page. Write the sentence, then use the number.
When the budget line bends
Budget constraints are not always straight.
The unit teaches the kinked case through an Australian income-management arrangement in which part of a payment arrives on a card that may be spent only on eligible goods and cannot be withdrawn as cash.
Total spending power is unchanged and the eligible-goods intercept is unchanged, because the whole payment could always have been spent that way; only the other-goods intercept falls, and the constraint runs flat until the restricted balance is exhausted. The kink sits exactly at the size of that balance.
A recipient who would have spent more than the restricted amount on eligible goods anyway is unaffected, since their chosen bundle lies on the segment common to both constraints.
The restriction binds only those who would have spent less, and saying so is the analytical point of the example.
Indifference curves and why they bow
An indifference curve joins the bundles delivering equal total utility, so the consumer is by construction indifferent among them, and a curve further from the origin is preferred.
Moving down a curve, the consumer holds more of the horizontal good and less of the vertical one, so the vertical good becomes scarcer at the margin and worth more while the horizontal good becomes plentiful and worth less. The number of units the consumer will surrender therefore falls: a diminishing marginal rate of substitution, which is what makes a curve bow toward the origin.
Two curves can never cross, because a crossing point would have to sit on two utility levels at once.
Five properties, then one condition
The unit assumes preferences are complete, transitive, continuous, monotonic and convex. The first three make preferences rational; the last two make them well behaved.
Each does a specific job in the diagram: completeness puts every point on some curve, transitivity stops curves crossing, continuity makes them unbroken, monotonicity makes them slope downward with higher curves preferred, and convexity makes them bow so that the slope diminishes. The optimum then spends the whole budget at the point where the highest reachable curve touches the line, which is where the two rates coincide.
That is the equal marginal utility per dollar condition of the previous chapter written a second way, and convexity is what guarantees the tangency is the best point rather than the worst.
What this chapter covers
- 01
The marginal rate of transformation and the price ratio
- 02
The marginal rate of substitution and the utility ratio
- 03
Reading a rate with its units attached
- 04
Kinked budget constraints and where the kink sits
- 05
Who a spending restriction actually binds
- 06
Indifference curves, and why they never cross
- 07
Diminishing marginal rate of substitution
- 08
The five preference properties and their jobs
- 09
The tangency condition and what breaks without convexity
Compare the two rates and give the instruction
- 1Write the market rate from the prices.
- 1Write the consumer's own rate from the marginal utilities.
- 1Compare the two and give the adjustment with its direction.
Key terms
- Marginal rate of transformation
- The rate at which the market converts one good into the other, equal to the price ratio.
- Marginal rate of substitution
- The quantity of one good a consumer will give up for one more unit of another at constant utility.
- Indifference curve
- A line joining every bundle that delivers the same total utility.
- Kinked budget constraint
- A budget boundary whose slope changes at a point, usually because part of the income is restricted.
- Convex preferences
- Preferences under which mixtures are ranked above extremes, so indifference curves bow inward.
- Transitive preferences
- Preferences under which ranking A above B and B above C implies A above C.
Indifference Curves, Preferences and the Consumer Optimum FAQ
Why can two indifference curves never cross?
A crossing point would belong to two curves at once, and each curve represents a single utility level, so the same bundle would deliver two different levels of satisfaction. That contradicts the transitivity assumption the curves are drawn from, which is why crossing curves are treated as a drawing error rather than as an unusual case.
What does a restricted spending card actually change?
It changes the shape of what is affordable without changing the amount paid. Spending on non-eligible goods is capped at the unrestricted portion, so the constraint runs flat until the restricted balance is used and then takes the ordinary slope. Recipients who already spent more than the restricted amount on eligible goods are unaffected.
How do the two rates relate to the rule from the previous chapter?
They are the same condition written differently. Setting the ratio of marginal utilities equal to the ratio of prices rearranges directly into equal marginal utility per dollar, so a tangency on a diagram and an equal-ratio calculation are two presentations of one optimum, and either is acceptable unless the question specifies a method.
What happens to the optimum if preferences are not convex?
The tangency stops being a maximum. Without convexity the highest reachable utility can sit at a corner of the budget line, where the consumer buys only one of the two goods, and the tangency point can even be the worst affordable bundle on the line. Convexity is the assumption doing that work, so a question that removes it is asking for a corner solution.
Does a steeper indifference curve mean the consumer values that good more?
It means the consumer will surrender more of the vertical good for one unit of the horizontal one at that particular bundle. The claim is local rather than general: the same consumer holding a different bundle sits at a different point on the same curve with a different slope, so a rate quoted without the bundle it was read at carries no information.
Exam move
Redraw before you calculate. Sketch the budget line from its intercepts, add two curves, mark the tangency, then state the slope of each line in words with units attached. For the kinked case, practise finding the kink from the restricted amount alone, and say which side of it a given recipient would sit on before drawing anything else.
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