Source: Chapter 4, Microeconomic Theory (Texas A&M University)
Tags: consumer choice, budget constraint, marginal utility, marginal rate of substitution, MRS, marginal rate of transformation, MRT, utility maximisation, indifference curves, perfect substitutes, budget line, ordinal utility, cardinal utility, price ratio, optimal consumption bundle
This chapter covers how consumers make choices under a budget constraint. You need to understand how marginal utility drives trade-offs between goods, how budget lines are constructed and shift, and the conditions for utility maximisation. The key equilibrium condition is that the ratio of marginal utilities must equal the ratio of prices.
Marginal utility (MU)
The additional satisfaction a consumer gains from consuming one more unit of a good, holding consumption of all other goods constant.
Budget constraint (budget line)
The set of all consumption bundles a consumer can afford given their income and the prices of goods. For two goods X and Y: Income = Px · X + Py · Y.
Marginal rate of transformation (MRT)
The rate at which the market allows a consumer to trade one good for another. Equal to the slope of the budget constraint. For good Y in terms of good X: MRT = -Py / Px.
Marginal rate of substitution (MRS)
The rate at which a consumer is willing to give up one good to obtain more of another while remaining on the same indifference curve. For two goods: MRS = MUx / MUy.
Ordinal utility
A ranking of bundles by preference (first, second, third) without implying anything about the magnitude of the difference. A bundle with utility 100 is preferred to one with utility 50, but you cannot say it is "twice as good."
Perfect substitutes
Two goods that a consumer is willing to trade at a constant rate. Indifference curves are straight lines. The consumer spends their entire budget on whichever good gives more utility per pound spent.
Utility maximisation condition
The consumer's optimal bundle occurs where MUx / Px = MUy / Py, i.e. the marginal utility per pound spent is equal across all goods.
Diminishing marginal utility
As a consumer has more of one good (and less of another), the marginal utility of the abundant good falls and the marginal utility of the scarce good rises.
If MU of pizza = 10 and MU of salad = 2, the consumer would give up 5 salads to get one more pizza (MU pizza / MU salad = 10/2 = 5). This is because pizza delivers five times the marginal satisfaction, so the consumer is willing to sacrifice up to 5 units of salad for one pizza.
Be careful with the direction: the consumer gives up the low-MU good to get the high-MU good, not the other way around.
Utility numbers rank bundles but do not measure intensity of preference.
A bundle with total utility 100 is preferred to one with total utility 50, full stop.
You cannot conclude the consumer "likes it twice as much," because utility functions are ordinal. Any monotonic transformation of the utility function represents the same preferences.
The budget constraint for two goods F (food) and S (shelter) with income M, price of food Pf, and price of shelter Ps:
General form: M = Pf · F + Ps · S
All algebraic rearrangements of this equation are equivalent representations. For example, with M = 500, Pf = 2, Ps = 100:
S = 5 - 0.02F
F = 250 - 50S
500 = 2F + 100S
These are all the same budget constraint, just solved for different variables.
The MRT is the slope of the budget line, measuring the market trade-off between two goods.
Formula: MRT of Y for X = -Px / Py
With food on the horizontal axis and shelter on the vertical axis (Pf = 2, Ps = 100):
MRT of food for shelter = -Ps / Pf = -100 / 2 = -50
This means the consumer must give up 50 units of food to obtain 1 more unit of shelter at market prices.
A common error is confusing which price goes in the numerator. The MRT of Y for X uses -Px / Py because you are asking "how much Y must I sacrifice for one more X?"
From F = 250 - 5S:
The F-intercept (250) is income divided by the price of food, so income / Pf = 250
The coefficient on S (5) is the price ratio Ps / Pf, meaning shelter is 5 times the price of food
You cannot determine the actual price levels or income from this equation alone, only the ratios
One price rises, other price and income unchanged: the budget line rotates inward, pivoting from the intercept of the good whose price did not change. The consumer can still buy the same maximum amount of the unchanged good, but less of the good that became more expensive.
Income rises, prices unchanged: the budget line shifts outward in parallel (same slope, higher intercepts).
Both prices double, income unchanged: the budget line shifts inward in parallel.
When goods are perfect substitutes, the consumer compares the MRS to the price ratio:
If MRS > price ratio, spend everything on the good on the X-axis
If MRS < price ratio, spend everything on the good on the Y-axis
If MRS = price ratio, any bundle on the budget line is equally good
Example: coffee and tea are perfect substitutes with MRS of tea for coffee = 2, and both goods have the same price. The price ratio is 1. Since MRS (2) > price ratio (1), coffee delivers more utility per pound, so the consumer spends the entire budget on coffee.
As the consumer moves along an indifference curve, consuming less pizza and more burritos:
MU of pizza increases (it is becoming scarce)
MU of burritos decreases (it is becoming abundant)
The MRS (in absolute value) of burritos for pizza increases, meaning the consumer demands ever more burritos to compensate for each pizza given up
This is the logic behind the convex shape of standard indifference curves.
At the optimum, the consumer equates the marginal utility per pound across all goods:
Condition: MUx / Px = MUy / Py
If MU of pizza = 5, MU of burrito = 10, price of pizza = 20, price of burrito = 15:
MU per pound on pizza: 5/20 = 0.25
MU per pound on burrito: 10/15 = 0.67
Burritos deliver more utility per pound, so the consumer should buy fewer pizzas and more burritos until the ratios equalise.
This condition also lets you solve for unknown prices. If MU pizza = 20, MU burrito = 10, price of pizza = $4, then in equilibrium: 20/4 = 10/Pb, giving Pb = $2.
Budget constraint: M = Px · X + Py · Y
MRT of Y for X: -Px / Py (slope of budget line)
MRS of Y for X: MUx / MUy (slope of indifference curve, absolute value)
Utility maximisation: MUx / Px = MUy / Py (equivalently, MRS = price ratio)
⚠️ Ordinal utility is a ranking only. Never say a consumer "likes bundle A twice as much" just because the utility number is double. This is a classic exam trap.
⚠️ MRT direction matters. The MRT of food for shelter uses -Ps / Pf, not -Pf / Ps. Read the question carefully for which good is being given up.
⚠️ Budget line equation reading: from F = 250 - 5S you know the price ratio (Ps/Pf = 5) but not individual prices or income, unless you have additional information.
⚠️ Perfect substitutes always produce corner solutions unless MRS exactly equals the price ratio. If they are equal, the entire budget line is optimal.
⚠️ The utility maximisation condition MUx / Px = MUy / Py is the workhorse of this chapter. Know it cold, and know how to use it to find an unknown price or determine whether to buy more of good X or good Y.
⚠️ Diminishing marginal utility drives the convexity of indifference curves and the increasing absolute MRS along the curve.
Q: If Fred's marginal utility of pizza is 10 and his marginal utility of salad is 2, what trade-off would he make?
A: He would give up 5 salads to get the next pizza. MU pizza / MU salad = 10/2 = 5, so one pizza is worth 5 salads to him.
Q: Adrian's total utilities of two consumption bundles are 50 and 100. Can we say he likes the second bundle twice as much?
A: No. We can only say he prefers the second bundle. Utility is ordinal, so the magnitudes do not measure intensity of preference.
Q: Joe's income is $500, price of food is $2, price of shelter is $100. What is the MRT of food for shelter?
A: -50. MRT of food for shelter = -Ps / Pf = -100/2 = -50. He must give up 50 units of food to get one more unit of shelter.
Q: From the budget line F = 250 - 5S, what can we determine?
A: The price of shelter is 5 times the price of food. The F-intercept is 250 (income / Pf). We cannot determine the absolute price levels or income without more information.
Q: If coffee and tea are perfect substitutes, MRS of tea for coffee is 2, and both sell for the same price, what does the consumer do?
A: Spends all money on coffee. MRS (2) exceeds the price ratio (1), so coffee delivers more utility per pound than tea.
Q: As a consumer eats less pizza and more burritos, what happens to the MRS of burritos for pizza?
A: It increases in absolute value. MU of pizza rises (scarcer), MU of burritos falls (more abundant), so the consumer demands increasingly more burritos per pizza sacrificed.
Q: MU pizza = 5, MU burrito = 10, price of pizza = $20, price of burrito = $15. Is the consumer maximising utility?
A: No. MU per dollar on pizza (5/20 = 0.25) is less than MU per dollar on burrito (10/15 ≈ 0.67). The consumer should buy fewer pizzas and more burritos.
Q: In equilibrium, MU pizza = 20, MU burrito = 10, price of pizza = $4. What is the price of a burrito?
A: $2. Using the equilibrium condition: 20/4 = 10/Pb, so Pb = 10/(20/4) = 10/5 = $2.
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