Revise Indifference curves and budget lines for Cambridge 9708 (7.2)

An indifference curve is a graphical representation of all the different combinations of two goods that provide a consumer with an equal level of satisfaction or utility. The consumer is therefore 'indifferent' between any of the combinations on the same curve. A collection of indifference curves for a consumer is called an indifference map, with curves further from the origin representing higher levels of utility. A key assumption is that consumers are rational and their preferences are transitive. Indifference curves are downward sloping from left to right, reflecting the trade-off required to maintain constant utility: to gain a unit of one good, the consumer must give up some of the other. They are also convex to the origin, which illustrates the principle of the diminishing marginal rate of substitution.

The Marginal Rate of Substitution (MRS) is the rate at which a consumer is willing to give up a quantity of one good (Good Y) to obtain one more unit of another good (Good X), while remaining on the same indifference curve. It is the slope of the indifference curve at any given point. The MRS is calculated as the change in Good Y divided by the change in Good X (ΔY/ΔX). A fundamental characteristic of standard indifference curves is the diminishing MRS. This means as a consumer has more of Good X, they are willing to give up progressively less of Good Y to get another unit of Good X. This reflects the idea that the more you have of something, the less you value an additional unit. This principle is what gives the indifference curve its characteristic convex shape.

A budget line, or budget constraint, illustrates all the possible combinations of two goods that a consumer can afford, given their income and the prices of the goods, assuming they spend their entire income. It represents the boundary of the consumer's opportunity set. The line is linear because prices are assumed to be constant. The slope of the budget line is determined by the ratio of the prices of the two goods (-Px/Py). This slope represents the opportunity cost of consuming one good in terms of the other; it is the rate at which the market requires the consumer to trade one good for the other. The intercepts of the budget line show the maximum quantity of each good the consumer could buy if they spent their entire income on only that good.

The position and slope of the budget line are determined by the consumer's income and the prices of the goods. A change in the consumer's income, with prices held constant, will cause a parallel shift in the budget line. An increase in income shifts the line outwards, expanding the consumer's affordable choices, while a decrease in income shifts it inwards, contracting their choices. Conversely, a change in the price of one good, with income and the other price held constant, causes the budget line to pivot. For example, if the price of Good X falls, the budget line will pivot outwards from the Y-intercept, becoming flatter. This indicates that the consumer can now afford more of Good X, and the opportunity cost of Good X has decreased.

A rational consumer aims to maximise their utility, which means reaching the highest possible indifference curve their budget allows. This point of utility maximisation is known as the consumer equilibrium. It occurs at the point where the budget line is tangent to an indifference curve. At this tangency point, the slope of the indifference curve (the MRS) is exactly equal to the slope of the budget line (the price ratio, Px/Py). This equilibrium condition, MRS = Px/Py, means that the rate at which the consumer is willing to trade one good for another is equal to the rate at which the market allows them to trade. Any other affordable combination would lie on a lower indifference curve, providing less satisfaction.

An indifference curve (IC) shows combinations of goods X and Y giving equal utility. Key properties:

1. Downward sloping — to keep utility constant, more X requires less Y.
2. Convex to origin — diminishing marginal rate of substitution (MRS).
3. Higher IC = higher utility — consumer prefers bundles on higher curves.
4. Never cross — each point has one utility level.

The budget line shows all affordable bundles given income (Y) and prices:

Y = P_x·X + P_y·Y

Slope = −P_x/P_y — the opportunity cost of one more unit of X.

X-intercept = Y/P_x; Y-intercept = Y/P_y.

A fall in P_x pivots the line outward on the X-axis (same Y-intercept). A rise in income shifts the entire line outward parallel.

Utility maximisation at tangency: $MRS_{xy} = \frac{P_x}{P_y}$

Equivalently: $\frac{MU_x}{P_x} = \frac{MU_y}{P_y}$

When P_x falls, the budget line pivots outward. The consumer reaches a higher IC at a new tangency — buys more X.

This decomposes into:

• Substitution effect — X relatively cheaper → buy more X, less Y.
• Income effect — real purchasing power rises → buy more of normal goods.

For a Giffen good (rare), the income effect dominates and demand curve slopes upward.