Corner Solutions | Econ Academy

The tangency condition works beautifully when the consumer buys positive amounts of both goods. But sometimes the optimum is at a corner — the consumer spends all income on just one good and buys zero of the other. This happens when the indifference curves are too flat or too steep to ever be tangent to the budget line in the interior.

Key Term

Corner Solution

A corner solution occurs when the consumer maximizes utility at one of the intercepts of the budget line, buying only one of the two goods. At a corner, $M R S \neq = P_{X} / P_{Y}$ — instead, the consumer would like to substitute further but can’t because consumption of one good is already at zero.

Interactive — Corner Solution: Buying Only One Good

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Indifference curves and a budget line meeting at a corner of the budget set.

The classic case is perfect substitutes. If goods X and Y are perfect substitutes (straight-line indifference curves), the $M R S$ is constant. Unless $M R S$ happens to equal $P_{X} / P_{Y}$ exactly, the consumer will spend everything on whichever good gives more utility per dollar. For example, if generic aspirin and brand-name aspirin are perfect substitutes, you buy whichever is cheaper — all of one, none of the other.

When MRS always exceeds the price ratio, the consumer values X relatively more than the market price, so they spend everything on X.

Check yourself · no marks

Two brands of bottled water are perfect substitutes. Brand A costs 1 dollar and Brand B costs 1.50 dollars. What does the consumer buy?

Commit to one first. Guessing and being wrong beats reading the answer cold.

Worked Example

A consumer treats goods X and Y as perfect substitutes with $U (X, Y) = 2 X + Y$ . Prices are $P_{X} = 3$ , $P_{Y} = 2$ , income $I = 60$ . Find the optimal bundle.

Practice · 1 / 4

No penalty for a wrong answer

Identify corner solutions:

For $U = X + Y$ with $P_{X} = 4$ and $P_{Y} = 3$ , the consumer will buy:

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