1. Chapter 5 CHOICE

2. 5.1 Optimal Choice Optimal Choicex2
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

3. 5.1 Optimal Choice
The first exception: the indifference curve might not have a tangent line. x1

4. 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. 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

6. 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.

7. 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

8. 5.3 Some Examples Perfect complements x1=x2=m/(p1+p2) x2
Indifference curves
Optimal choice
x2*
Budget line
x1* x1

9. 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

10. 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

11. 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)

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

13. 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.

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

15. Lagrange multiplier method
F.O.C.

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

17. 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.

18. 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.

19. 5.6 Choosing Taxes Quantity tax Income tax
a tax on the amount consumed of good.
Income tax
a tax on income.

20. 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*

21. 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.