4. Classical Demand Theory

From indifference curves through utility maximization and the Slutsky decomposition to duality with expenditure minimization — the heart of demand theory.

• Indifference Curves and MRS — This scene draws the indifference map of Cobb-Douglas utility. An indifference curve joins bundles that feel equally good. Move the reference bundle and the whole map redraws. The marginal rate of substitution is the tangent slope there, as in the formula below. It says how much of one good replaces one unit of the other. For example, giving up one 🍎 apple feels fine if you get two more 🍌 bananas. Then the MRS is two. The fewer apples you have, the more each apple is worth.
• Utility Maximization — This scene finds the best bundle where the budget line meets an indifference curve. The optimum is where the budget line just touches the curve. Move prices or wealth and the optimum slides along. At that spot the MRS always equals the price ratio, as in the formula below. You can check it numerically at every moment. For example, you have 💰 6,000 won for 🍎 apples and 🍌 bananas, each 1,000 won. If you like both equally, three apples and three bananas are best. If apples get pricier, you buy fewer apples and more bananas.
• The Slutsky Decomposition — This scene splits a price rise into two effects to see why demand falls. A price change brings two effects at once. The substitution effect pushes you away from the pricier good. The income effect comes from feeling poorer than before. You can compensate by the old bundle or by the old utility. The two ways split the formula below differently. For example, if 🍎 apples get pricier, switching to 🍌 bananas is the substitution effect. If you then buy less fruit overall because 💰 money feels tight, that is the income effect.
• Expenditure Minimization and Duality — This scene finds the cheapest way to reach a target utility. Now the problem runs in reverse. Fix the satisfaction level first, then shrink the cost. The cheapest bundle sits where an isocost line touches the curve. Turn on the mirror and it lands exactly on the earlier optimum. Two problems mirroring each other is duality. For example, you want the satisfaction of 🍞 bread, two slices, and 🥛 milk, one glass. You shop for that satisfaction at the lowest cost. To save 💰 money, you shift toward the cheaper good.