1. Chapter 5 Choice
Key Concept: Optimal choice means a consumer chooses the best he can afford.
Tangency is neither necessary nor sufficient.
The correct necessary and sufficient condition: The set of consumption bundles which is strictly preferred to A by this consumer cannot intersect with the budget set. (月亮形區域為空)

2. Chapter 5 Choice
budget set + preference → choice
Optimal choice: choose the best one can afford. optimal vs. maximal

3. Suppose the consumer chooses bundle A.
A is optimal
A w B for any B in the budget set
↔ The set of consumption bundles which is strictly preferred to A by this consumer cannot intersect with the budget set. (月亮形區域為空)

4. Fig. 5.1

5. A optimal↔月亮形區域為空.
A optimal →月亮形區域為空? (necessary) If not, then月亮形區域不為空, that means there exists a bundle B such that B s A and B is in the budget set. Then A is not optimal.
A optimal ←月亮形區域為空? (sufficient) All B such that B s A is not affordable, so for all B in the budget set, we must have A w B. Hence A is optimal.

6. The indifference curve tangent to the budget line is neither necessary nor sufficient for optimality.
typhoon → rain → wet
sufficient
optimum
necessary

7. necessary: optimal → tangent
not necessary
kinked preferences (perfect complements)
corner solution 角解 (vs. interior solution 內解) (intuition)

8. Fig. 5.2

9. Fig. 5.3

10. sufficient: tangent → optimal
not sufficient: satiation or convexity is violated

11. Fig. 3.7

12. Fig. 5.4

13. Despite all of these, the usual tangent condition MRS1, 2= -p1/ p2 has a nice interpretation. So we still need to get the economic intuition.

14. The MRS is the rate the consumer is willing to pay for an additional unit of good 1 in terms of good 2.
The relative price ratio is the rate the market asks a consumer to pay for an additional unit of good 1 in terms of good 2.
At optimum, these two rates are equal. (主觀相對價格 vs. 客觀相對價格)

15. If they are not equal, we will get:
when |MRS1, 2| > p1/ p2, buy more of 1
when |MRS1, 2| < p1/ p2, buy less of 1

16. We now know what the optimal choice is, let us turn to demand since they are related.
The optimal choice of goods at some price and income is the consumer’s demanded bundle. A demand function gives you the optimal amount of each good as a function of prices and income faced by the consumer.

17. x1 (p1, p2, m): the demand function
At p1, p2, m, the consumer demands x1
Let us work out some examples.

18. Perfect substitutes: (graph)
u(x1, x2) = x1 + x2
p1 > p2: x1 = 0, x2 = m/ p2
p1 = p2: x1 belongs to [0, m/ p1] and x2 = (m- p1 x1)/p2
p1 < p2: x1 = m/ p1, x2 = 0

19. Fig. 5.5

20. Perfect complements: (graph)
u(x1, x2) = min{x1, x2}
x1 = x2 = m/ (p1+ p2)

21. Fig. 5.6

22. Neutrals or bads: why spend money on them?
Discrete goods (just foolhardily compare)
Non convex preferences: probably corner solution

23. Fig. 5.7

24. Fig. 5.8

25. Cobb-Douglas is useful
u(x1, x2) = a lnx1 + (1-a) lnx2
|MRS1, 2| = p1/ p2, so (a/x1)/[(1-a)/x2] = p1/ p2.
This implies that a/(1-a) = p1x1/ p2x2,
so x1 = am/ p1 and x2 = (1-a)m/ p2.
This is useful if when we are estimating utility functions, we find that the expenditure share is fixed.

26. Table 5.1

27. Implication of the MRS condition: at equilibrium, we don’t need to know the preferences of each individual, we can infer that their MRS’ are the same. (This has an useful implication for Pareto efficiency as we will see later.)

28. One small example: butter (price:2) and milk (price: 1)
A new technology that will turn 3 units of milk into 1 unit of butter. Will this be profitable?
Another new tech that will turn 1 unit of butter into 3 units of milk. Will this be profitable?

29. Choosing taxes: quantity tax and income tax
Suppose we impose a quantity tax of t dollars per unit of x1. budget constraint: (p1+t) x1 + p2 x2 = m
optimum: (x1*, x2*) so that (p1+t) x1* + p2 x2* = m, tax revenue is t x1*
to compare, we want income tax R* to raise the same revenue: R* = t x1*

30. optimum at income tax: p1 x1’+ p2 x2’ = m - R*, but we know that
(p1+t) x1* + p2 x2* = m or
p1 x1* + p2 x2* = m - t x1* or
p1 x1* + p2 x2* = m - R*
so (x1*, x2*) is affordable at the case of the income tax.
Hence, (x1’, x2’) w (x1*, x2*). (graph)

31. Fig. 5.9

32. Income tax better than quantity tax?
two caveats:
one consumer, uniform income tax vs. uniform quantity tax (think about the person who does not consume good 1)
tax avoidance or income tax discourages earning
ignore supply side

33. Chapter 5 Choice
Key Concept: Optimal choice means a consumer chooses the best he can afford.
Tangency is neither necessary nor sufficient.
The correct necessary and sufficient condition: The set of consumption bundles which is strictly preferred to A by this consumer cannot intersect with the budget set. (月亮形區域為空)