1. Choice

2. Economic Rationality
The principal behavioral postulate is that a decisionmaker chooses its most preferred alternative from those available to it.
The available choices constitute the choice set.

3. Economic Rationality
How is the most preferred bundle in the choice set located?
The answer is simple: the consumer chooses the bundle within the choice set that is on the indifference curve that provides the highest level of utility.

4. Rational Constrained Choicex2
x2*
x1* x1

5. Rational Constrained Choicex2
(x1*,x2*) is the most preferred affordable bundle.
x2*
x1* x1

6. Rational Constrained Choice
When x1* > 0 and x2* > 0 the demanded bundle is INTERIOR.
If buying (x1*,x2*) costs €m then the budget is exhausted.

7. Rational Constrained Choicex2
(x1*,x2*) is interior. (a) (x1*,x2*) exhausts the budget: p1x1* + p2x2* = m
x2*
x1* x1

8. Rational Constrained Choicex2
(x1*,x2*) is interior. (b) The slope of the indiff. curve at (x1*,x2*) equals the slope of the budget constraint
x2*
x1* x1

9. Rational Constrained Choice
(x1*,x2*) satisfies two conditions:
(a) the budget is exhausted p1x1* + p2x2* = m
(b) the slope of the budget constraint, -p1/p2, and the slope of the indifference curve containing (x1*,x2*) are equal at (x1*,x2*).

10. Rational Constrained Choice
For well-behaved (monotonic and convex) preferences, tangency is a necessary and sufficient condition for optimality.
At the optimum the consumer substitutes one good for the other at a rate identical to the market’s.

11. Computing Ordinary Demands - a Cobb-Douglas Example
Suppose that the consumer has Cobb-Douglas preferences.

12. Computing Ordinary Demands - a Cobb-Douglas Example
Suppose that the consumer has Cobb-Douglas preferences.
Then

13. Computing Ordinary Demands - a Cobb-Douglas Example
So the MRS is

14. Computing Ordinary Demands - a Cobb-Douglas Example.
So the MRS is
At (x1*,x2*), MRS = -p1/p2 so
(A)

15. Computing Ordinary Demands - a Cobb-Douglas Example.
(x1*,x2*) also exhausts the budget so
(B)

16. Computing Ordinary Demands - a Cobb-Douglas Example.
So now we know that
(A)
(B)

17. Computing Ordinary Demands - a Cobb-Douglas Example.
So now we know that
(A)
Substitute
(B)

18. Computing Ordinary Demands - a Cobb-Douglas Example.
So now we know that
(A)
Substitute
(B)
and get
This simplifies to ….

19. Computing Ordinary Demands - a Cobb-Douglas Example.

20. Computing Ordinary Demands - a Cobb-Douglas Example.
Substituting for x1* in
then gives

21. Computing Ordinary Demands - a Cobb-Douglas Example.
So we have discovered that the most preferred affordable bundle for a consumer with Cobb-Douglas preferences is

22. Computing Ordinary Demands - a Cobb-Douglas Example.

23. Rational Constrained Choice
When x1* > 0 and x2* > 0 and (x1*,x2*) exhausts the budget, and indifference curves have no ‘kinks’, the ordinary demands are obtained by solving:
(a) p1x1* + p2x2* = m
(b) the slopes of the budget constraint, -p1/p2, and of the indifference curve containing (x1*,x2*) are equal at (x1*,x2*).

24. Rational Constrained Choice
But what if x1* = 0?
Or if x2* = 0?
If either x1* = 0 or x2* = 0 then the ordinary demand (x1*,x2*) is at a corner solution to the problem of maximizing utility subject to a budget constraint.

25. Examples of Corner Solutions -- the Perfect Substitutes Case
MRS = -1 x1

26. Examples of Corner Solutions -- the Perfect Substitutes Case
MRS = -1
Slope = -p1/p2 with p1 > p2 x1

27. Examples of Corner Solutions -- the Perfect Substitutes Case
MRS = -1
Slope = -p1/p2 with p1 > p2 x1

28. Examples of Corner Solutions -- the Perfect Substitutes Case
MRS = -1
Slope = -p1/p2 with p1 > p2 x1

29. Examples of Corner Solutions -- the Perfect Substitutes Case
MRS = -1
Slope = -p1/p2 with p1< p2 x1

30. Examples of Corner Solutions -- the Perfect Substitutes Case
So when U(x1,x2) = x1 + x2, the most preferred affordable bundle is (x1*,x2*) where
if p1 < p2
and
if p1 > p2

31. Examples of Corner Solutions -- the Perfect Substitutes Case
MRS = -1
Slope = - p1/p2 with p1= p2 x1

32. Examples of Corner Solutions -- the Perfect Substitutes Case
All the bundles in the constraint are equally the most preferred affordable when p1 = p2 x1

33. Examples of ‘Kinky’ Solutions -- the Perfect Complements Case
U(x1,x2) = min{ax1,x2} x2
x2 = ax1 x1

34. Examples of ‘Kinky’ Solutions -- the Perfect Complements Case
U(x1,x2) = min{ax1,x2} x2
MRS = -
MRS is undefined
x2 = ax1
MRS = 0 x1

35. Examples of ‘Kinky’ Solutions -- the Perfect Complements Case
U(x1,x2) = min{ax1,x2} x2
Which is the most preferred affordable bundle?
x2 = ax1 x1

36. Examples of ‘Kinky’ Solutions -- the Perfect Complements Case
U(x1,x2) = min{ax1,x2} x2
The most preferred
affordable bundle
x2 = ax1 x1

37. Examples of ‘Kinky’ Solutions -- the Perfect Complements Case
(a) p1x1* + p2x2* = m; (b) x2* = ax1*
Substitution from (b) for x2* in (a) gives p1x1* + p2ax1* = m which gives

38. Application: choosing a tax
Which is less harmful, a quantity tax or an income tax, both allowing the same revenue?
1. We can show that an income tax is always better in the sense that given any quantity tax, there is an income tax that leaves the consumer at a higher indifference curve.

39. Application: choosing a tax
2. Outline of the argument:
a) original budget constraint: p1x1 + p2x2 = m
b) budget constraint with quantity tax:
(p1 + t)x1+p2x2 = m
c) optimal choice with tax: (p1+t)x1* + p2x2* = m
d) revenue raised is tx1*

40. Application: choosing a tax
e) income tax (lump-sum) that raises the same amount of revenue leads to budget constraint: p1x1 + p2x2 = m - tx1*
i) this line has the same slope as the original budget line, but lies inwards

41. Application: choosing a tax
ii) also passes through (x1*;x2*)
iii) proof: p1x1*+ p2x2* = m - tx1*
iv) this means that (x1*; x2* ) is affordable under the income tax, so the optimal choice under the income tax must be even better than (x1 *; x2*)