Chapter 5 CHOICE

5.1 Optimal Choice Optimal Choice x2 Indifference curves Optimal Choice the bundle of goods that is associated with the highest indifference curve that just touches the budget line. Optimal choice x2* x1* x1

5.1 Optimal Choice The first exception: the indifference curve might not have a tangent line. x1

5.1 Optimal Choice x2 The second exception: the optimal point occurs where the consumption of some good is zero. Indifference curves Budget line x1* x1

5.1 Optimal Choice Indifference curves x2 In general there may more than one optimal bundle that satisfies the tangency condition. Optimal bundles Budget line nonoptimal bundles x1

5.2 Consumer Demand Demanded bundle Demand function The optimal choice of good 1 and 2 at some set of prices and income. Demand function the function that relates the quantities demanded to the values of prices and incomes.

5.3 Some Examples Perfect Substitutes when p1＜p2, x1=m/p1 Indifference curves Perfect Substitutes when p1＜p2, x1=m/p1 when p1=p2, x1 could be any number between 0 and m/p1 when p1＞p2, x1=0 Budget line Optimal choice x1*=m/p1 x1

5.3 Some Examples Perfect complements x1=x2=m/(p1+p2) x2 Indifference curves Optimal choice x2* Budget line x1* x1

5.3 Some Examples Neutrals and Bads The consumer spends all income on the good she likes and doesn’t purchase any of the neutral (bad) good. If commodity 1 is a good and commodity 2 is a bad, then the demand functions will be x1= m/p1 x2=0

5.3 Some Examples Concave Preferences Indifference curves Concave Preferences ---The optimal choice is the boundary point, z, not the interior tangency point, x, because z lies on a higher indifference curve. X nonoptimal choice x1 Z optimal choice

p1x1/m= p1cm/m(c+d)p1= c/(c+d) 5.3 Some Examples Cobb- Douglas Preferences x1=cm/(c+d)p1 x2=dm/(c+d)p2 p1x1/m= p1cm/m(c+d)p1= c/(c+d)

Lagrange multiplier method The standard problem The Lagrangian The F.O.C.

Lagrange multiplier method If the constraints behave in a “regular” manner, any solution must satisfies the F.O.C. If the objective function is concave, and the constraints are convex, then the F.O.C. is also sufficient for optimum.

Lagrange multiplier method Consumer’s problem Can be reformulated as Lagrangian

Lagrange multiplier method F.O.C.

Lagrange multiplier method The budget constraint will be always satisfied for locally nonsatiated preferences. The textbook way:

Lagrange multiplier method What’s the advantage of Kuhn-Tucker F.O.C.? It treats corner solutions. What happens if x1>0 and x2=0? i.e., the indifference curve is steeper than the budget line.

5.5 Implications of the MRS Condition In well-organized markets, it is typical that everyone faces roughly the same prices for goods. People may value their total consumption of the two goods very differently. Everyone who is consuming the two goods must have the same marginal rate of substitution. The fact that prices reflect how people value things on the margin is one of the most fundamental and important ideas in economics.

5.6 Choosing Taxes Quantity tax Income tax a tax on the amount consumed of good. Income tax a tax on income.

5.6 Choosing Taxes quantity tax: p1x1+p2x2=m (p1+t)x1*+p2x2*=m R*=tx1* budget constraint (p1+t)x1*+p2x2*=m revenue R*=tx1* budget constraint for an “equivalent” income tax p1x1+p2x2=m-tx1*

5.6 Choosing Taxes (x1*, x2*) is affordable under income tax. Budget line under income tax cuts the indifference curve at (x1*,x2*). income tax is superior to the quantity tax.