Consumer Choice Preferences, Budgets, and Optimization

The Consumer’s Problem Preferences ◦ A consumer prefers one good (or bundle of goods) to another. Income or Budget ◦ Income (Budget) limits a consumer’s buying Consumers are Rational ◦ A consumer seeks the maximum satisfaction from consuming goods and services ◦ Where consumption is limited by income

Preferences Completeness ◦ All pairs of goods (bundles) are ranked Transitivity ◦ A > B and B > C, then A > C Goods are “good” (not “bad”) ◦ More is always preferred to less Described by Indifference Curves ◦ All combinations of goods that yield the same satisfaction (or “utility”)

Shapes of Indifference Curves Convex ◦ Goods are gross substitutes (downward slope) ◦ Diminishing marginal rate of substitution Special cases ◦ Perfect substitutes ◦ Perfect Complements Impossibilities ◦ Intersecting indifference curves ◦ Upward sloping (one “good” is a “bad”) ◦ Circular (Bliss point)

Budget Line All income is spent on goods ◦ P Y Y + P X X = I ◦ All combinations of goods that a consumer can afford to buy Equation for the budget line ◦ Y = (I/P Y ) – (P X /P Y )X Shifts in the budget line ◦ Changes in Prices ◦ Changes in Income

Solutions to the Consumer’s Problem A combination of goods on the highest indifference curve the consumer can afford to reach Budget line tangent to indifference curve ◦ Slope of budget line = slope of indifference curve ◦ -P X /P Y = ΔY/ΔX = -MU X /MU Y = -MRS  -P Y /P X = -MU X /MU Y implies  P Y /P X = MU X /MU Y implies  MU X / P X = MU Y / P Y or  Marginal utility per dollar equal across goods

Special Cases Perfect substitutes ◦ Corner solutions Perfect complements ◦ No response to price changes Non-convexities ◦ Gaps or jumps ◦ Corner solutions

Changes in Income and Prices Income Changes ◦ Budget line shifts out for increase in income ◦ Budget line shifts in for decrease in income Changes in the price of one good ◦ Price decrease  Budget line shifts out along axis for that good ◦ Price increase  Budget line shifts in along axis for that good Substitution and Income effects Revealed Preference

Utility Functions Suppose U = 2FC is a consumer’s utility function for food (F) and clothes (C) Is this a proper utility function? ◦ Increasing in F and C? ◦ Indifference curves downward sloping? ◦ Diminishing MRS? Define an Indifference Curve: 2FC =100 ◦ C = 100/F, ∆C/ ∆F = -100/F 2 < 0 ◦ MRS = 100/F 2

Example: Consumer Choice Max U = 2FC Subject to $500 = $5F + $10C Find MU F /MU C = P F /P C ◦ MU F /MU C = 2C/2F = C/F ◦ C/F = $5/$10 = ½, or C = ½ F Substitute into constraint ◦ $500 = $5F + $10(½ F) = $10F ◦ F = 50, C = ½ F = 25

A Note on Derivatives For any function U = DX a Y b ◦ MU X = ∂U/∂X = aDX a-1 Y b ◦ MU Y = ∂U/∂Y = bDX a Y b-1 MU X /MU Y = (a/b)(Y/X) ◦ (aDX a-1 Y b )/(bDX a Y b-1 ) = (a/b)X -1 Y 1 If a + b = 1, then U = X a Y 1- a ◦ MU X /MU Y = [(a/(1- a)](Y/X) ◦ Known as the Cobb-Douglas functional form Same solutions as lnU = lnD +alnX + blnY

Exercise U = F ¼ C ¾ P F = $2.00 P C = $3.00 Income = $ Find the optimal quantities of food and clothes for this consumer Graph the budget line Illustrate your answer on this graph

Substitution and Income Effects Substitution Effect: ∂X/∂P X | U=U* ◦ The change in the quantity demanded holding utility fixed Income Effect: (∂X/∂I)(∂I/∂P X ) ◦ The change in the quantity demanded when utility changes, holding relative prices fixed Slutsky Equation ◦ dX/dP X = ∂X/∂P X | U=U* + (∂X/∂I)(∂I/∂P X ) ◦ From budget constraint (∂I/∂P X ) = X ◦ dX/dP X = ∂X/∂P X | U=U* - X(∂X/∂I)

Deriving Demand Curves Individual demand ◦ Change prices and record quantites ◦ Graph price quantity combinations Market demand ◦ Add individual demand curves horizontally ◦ Total quantity demanded at each price