The buyer's problem in three steps, and the rule that marks the best choice

The buyer's problem. A buyer has to decide how to spend her income on different goods; economists call this the problem of choice. She wants the combination of goods that gives her the greatest satisfaction, and which combination that is depends on what she likes and on what she can afford, which in turn depends on the prices of the goods and on her income. Solving the problem therefore takes three steps.

Step 1: what the buyer likes.

• The buyer's tastes are her preferences. A good gives her utility, its capacity to satisfy a want, and this is personal: the same good can satisfy two people very differently.
• The extra utility from one more unit, the marginal utility, usually falls as she consumes more of the good.
• With two goods she can rank bundles, and bundles she likes equally lie on one indifference curve. The amount of the second good she is willing to give up for one more unit of the first, keeping her satisfaction the same, is her marginal rate of substitution (MRS); it usually falls as she has more of the first good.
• Her preferences are assumed to be monotonic: a bundle with more of at least one good and no less of the other is preferred.

Step 2: the prices of the goods.

• The price of a good is the number of rupees she must pay per unit.
• Prices set the terms on which the market lets her swap one good for another. To buy one more unit of good 1 she must spend p1 rupees less on good 2, that is, give up p1/p2 units of it. The price ratio p1/p2 is the rate at which she is able to substitute one good for the other.

Step 3: how much the buyer can spend.

• With income M, she can afford any bundle with p1x1 + p2x2 ≤ M: this is her budget set. The bundles that cost exactly M lie on the budget line, whose slope is −p1/p2.
• A rise in income moves the budget line out parallel to itself, while a change in one price changes its slope.

Putting the steps together. A rational buyer picks, from the bundles she can afford, the one she likes most: she moves to the highest indifference curve her budget allows. With monotonic preferences the chosen bundle is on the budget line, not below it, because a point below always leaves room for more of one good without less of the other.

The decision rule at equilibrium.

• The buyer is in equilibrium where the budget line just touches (is tangent to) an indifference curve. There MRS = p1/p2: the rate at which she is willing to swap the goods equals the rate at which the market lets her swap them, and her whole income is spent.
• Why. If her MRS were above the price ratio, she would value one more unit of good 1 at more of good 2 than the market charges for it; buying more of good 1 and less of good 2 would move her to a preferred bundle. If MRS were below the price ratio, the reverse move would help. Only where the two are equal is no change worth making.
• Single-good form. Looking at one good on its own, she keeps buying while an extra unit is worth at least its price to her, and stops before the unit that is worth less than the price; this is why her demand rises as the price falls.

Example: phuchka and coffee in Kolkata. Our own figures. Riya spends ₹1,200 a month on plates of phuchka (₹30 each) and cups of coffee (₹60 each). The price ratio is 30/60 = 0.5: the market lets her swap one cup of coffee for two plates.

• Suppose she buys 20 plates and 10 cups (₹600 + ₹600 = ₹1,200) and, at that bundle, would give up one cup of coffee for just one extra plate (MRS = 1, above 0.5). Giving up a cup frees ₹60, which buys two plates, so she ends up better off by switching.
• As she buys more phuchka and less coffee, her MRS falls. She stops switching when it reaches 0.5, where the rate at which she is willing to swap equals the market rate: that bundle on her budget line is her equilibrium.

Conclusion. The buyer's problem is solved in three steps: know what she likes (preferences), know the prices, and know how much she can spend (the budget line). The decision rule at equilibrium is that she spends her whole income on the bundle at which her marginal rate of substitution equals the price ratio.