• Chapter 5 Choice• Budget set + preference → choice• Optimal choice: choose 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 which is strictly preferred to A by this consumer cannot intersect with the budget set. ( 月亮形區域 )

• A optimal↔ 月亮形區域為空 .• A optimal → 月亮形區域為空 ? 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 ? 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.

• Not necessary: kinked preferences (perfect complements), corner solution (vs. interior solution) (!!) (intuition)

• Not sufficient: satiation or convexity is violated

• Not necessary: kinked preferences (perfect complements), corner solution (vs. interior solution) (!!) (intuition)

• Not sufficient: satiation or convexity is violated

optimumsufficient

necessary

• The usual tangent condition MRS1, 2= -p1/ p2 has a nice interpretation. 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 )

• |MRS1, 2| > p1/ p2, buy more of 1• |MRS1, 2| < p1/ p2, 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.

• x1 (p1, p2, m): the demand function• At p1, p2, m, the consumer demands x1

• 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

• Perfect complements: (graph) u(x1, x2) = min{x1, x2} x1 = x2 = m/ (p1+ p2)• Neutrals or bads: why spend money on

them?• Discrete goods (just foolhardily compare)• Non convex preferences: corner solution

• Cobb-Douglas: 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.

Table 5.1

• 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.)

• 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?

• 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

income tax R* to raise the same revenue: R* = t x1*

• optimum at income tax: 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)

• 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