Chapter 5 Choice Key Concept Optimal choice where

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

• Chapter 5 Choice • budget set + preference → choice • Optimal choice: the best one can afford.

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

A

sufficient optimum necessary

• typhoon → rain → wet sufficient optimum necessary

• 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. • 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.

• The indifference curve tangent to the budget line is neither necessary nor sufficient for optimality. sufficient optimum necessary

• necessary: optimal → tangent • optimal not tangent • kinked preferences • tangent is not defined • (intuition)

• necessary: optimal → tangent • optimal not tangent • corner solution 角解 • vs. interior solution 內解 • (intuition)

• sufficient: tangent → optimal • tangent not optimal • satiation

• sufficient: tangent → optimal • tangent not optimal • convexity is violated

• Despite all of these, the usual tangent condition MRS 1, 2= -p 1/ p 2 has a nice interpretation. • So we still need to get the economic intuition.

• 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)

• If they are not equal, we will get: • when |MRS 1, 2| > p 1/ p 2 • should buy more of 1 • when |MRS 1, 2| < p 1/ p 2 • should buy less of 1

• 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.

• Denote the demand function x 1 (p 1, p 2, m). • At p 1, p 2, m, the consumer demands x 1 • Let us work out some examples.

• Perfect substitutes • u(x 1, x 2) = x 1 + x 2 • p 1 > p 2: x 1 = 0, x 2 = m/ p 2 • p 1 = p 2: x 1 belongs to [0, m/ p 1] and x 2 = (m- p 1 x 1)/p 2 • p 1 < p 2: x 1 = m/ p 1, x 2 = 0

• Perfect complements • u(x 1, x 2) = min{x 1, x 2} • x 1 = x 2 = m/ (p 1+ p 2)

• Neutrals or bads • Why spend money on them?

• Discrete goods • Just compare.

• Non convex preferences • Probably a corner solution

• Cobb-Douglas • u(x 1, x 2) = x 1 ax 21 -a • u(x 1, x 2) = a lnx 1 + (1 -a) lnx 2 • |MRS 1, 2| = p 1/ p 2 • (a/x 1)/[(1 -a)/x 2] = p 1/ p 2

• |MRS 1, 2| = p 1/ p 2 • (a/x 1)/[(1 -a)/x 2] = p 1/ p 2 • a/(1 -a) = p 1 x 1/ p 2 x 2 • x 1 = am/ p 1 and x 2 = (1 -a)m/ p 2 • This is useful if when we are estimating utility functions, we find that the expenditure share is fixed.

• 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 because if their MRS’ are different, they can trade and Pareto improve.

• 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?

• quantity tax and income tax • If the government wants to raise a certain amount of revenue, is it better to raise it via a quantity tax or income tax?

• Suppose gov imposes a quantity tax of t dollars per unit of x 1. • budget constraint: (p 1+t) x 1 + p 2 x 2 = m • at optimum: (x 1*, x 2*) so that • (p 1+t) x 1* + p 2 x 2* = m • tax revenue is t x 1* • to compare, we want income tax R* to raise the same revenue • R* = t x 1*

• (x 1’, x 2’) is the optimum with income tax. • p 1 x 1’+ p 2 x 2’ = m - R* but we know that • (p 1+t) x 1* + p 2 x 2* = m • p 1 x 1* + p 2 x 2* = m - t x 1* • p 1 x 1* + p 2 x 2* = m - R* • so (x 1*, x 2*) is on the income tax budget line • Hence, (x 1’, x 2’) w (x 1*, x 2*).

• Income tax better than quantity tax? • For any consumer, we can find an income tax raising the same tax revenue and the consumer is better off. • Yet the amount of income tax will differ from consumer to consumer. • uniform income tax vs. quantity tax (consider those who do not consume good 1)

• Income tax might discourage earning. • We ignore supply side

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