1. Course: Microeconomics Text: Varian’s Intermediate Microeconomics
Chapter 5 Choice
Course: Microeconomics
Text: Varian’s Intermediate Microeconomics

2. Economic Rationality
The principal behavioral postulate is that a decision-maker chooses its most preferred alternative from those available to it.
The available choices constitute the budget set.
How is the most preferred bundle in the budget set located?

3. Rational Constrained Choicex2 x1

4. Rational Constrained Choice
Utility x2 x1

5. Rational Constrained Choice
Utility
Affordable, but not the most preferred affordable bundle. x2 x1

6. Rational Constrained Choice
Utility
The most preferred of the affordable
bundles.
Affordable, but not the most preferred affordable bundle. x2 x1

7. Rational Constrained Choicex2
One will choose the bundle on the outermost indifference curve. Therefore, it is the one that just touch the budget constraint.
Affordable bundles x1

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

9. Rational Constrained Choice
The most preferred affordable bundle is called the consumer’s ORDINARY DEMAND at the given prices and budget.
Ordinary demands will be denoted by x1*(p1,p2,m) and x2*(p1,p2,m).

10. 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.

11. Rational Constrained Choice
Assuming monotonicity and smooth indifference curve, for an interior solution (x1*,x2*), it 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*).

12. Rational Constrained Choice
Condition (b) can be written as:
So, at the optimal choice, the marginal willingness to pay for an extra unit of good 1 in terms of good 2 is the same as the price you actually need to pay.
If the price is lower than your willingness to pay, you would buy more, otherwise buy less. Adjust until they are equal.

13. Mathematical Formulation
The consumer problem can be formulated mathematically as:
One way to solve is to use Lagrangian method:

14. Mathematical Treatment
Assuming the utility function is differentiable, and there exists interior solution
The first order conditions are:

15. Mathematical Treatment
So, we have the same condition as before that MRS equals price ratio.

16. Mathematical Treatment
So, for interior solution, we can solve for the optimal bundle using the two conditions:

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

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

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

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

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

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

23. 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

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

25. Demand for Cobb-Douglas Utility Function
With the optimal bundle:
Note the expenditures on each good:
It is a property of Cobb-Douglas Utility function that expenditure share on a particular good is a constant.

26. Special Example

27. Special Examples
1. Tangency condition is only necessary but not sufficient.
2. There can be more than one optimum.
3. Strict Convexity implies unique solution.

28. Corner Solution 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.
E.g. it happens when the indifference curves are too flat or steep relative to the budget line.

29. Special Example
The indifference curves are too steep in this case to touch the budget line.
The willingness to pay for good 2 is still lower than the price of good 2 even when x2 is zero.

30. Corner Solution
The mathematical treatment for corner solution is more complicated.
We have to consider also the inequality constraints. (x1 ≥0 and x2 ≥0)
We skip the details here.

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

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

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

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

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

36. 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.

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

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

39. Examples of Corner Solutions -- the Non-Convex Preferences Case
Better x1

40. Examples of Corner Solutions -- the Non-Convex Preferences Case

41. Examples of Corner Solutions -- the Non-Convex Preferences Case
Which is the most preferred affordable bundle? x1

42. Examples of Corner Solutions -- the Non-Convex Preferences Case
The most preferred affordable bundle x1

43. Examples of Corner Solutions -- the Non-Convex Preferences Case
Notice that the “tangency solution” is not the most preferred affordable bundle. x2
The most preferred affordable bundle x1

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

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

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

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

48. 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

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

50. Examples of ‘Kinky’ Solutions -- the Perfect Complements Case
U(x1,x2) = min{ax1,x2} x2
(a) p1x1* + p2x2* = m (b) x2* = ax1*
x2 = ax1
x2*
x1* x1

51. Examples of ‘Kinky’ Solutions -- the Perfect Complements Case
(a) p1x1* + p2x2* = m; (b) x2* = ax1*.
-Intuitively, it is clear to have (b) because we don’t want to spend money on something that cannot raise my utility.
-Substitution from (b) for x2* in (a) gives p1x1* + p2ax1* = m which gives

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

53. Implications of MRS condition
Recall
As price is the same for all individuals, every individual has the same MRS at the optimal consumption bundle, even if they have very different preferences.
Thus, it can be interpreted as a social willingness to pay for good 1 in terms of good 2.

54. Summary
In this chapter we put together budget constraint and preference together to analyze consumption decision.
Typically, if we have interior solution, then the optimal condition is MRS=p1 / p2.
It can also have corner solution if one good is not consumed at all.

55. What’s next? Next, we will look at comparative statics.
In particular, how will prices and income changes affect the optimal consumption bundle?