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.

Fig. 5.1

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 optimum sufficient necessary

Fig. 5.2

Fig. 5.3

Fig. 3.7

Fig. 5.4

The usual tangent condition MRS 1, 2 = -p 1 / p 2 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 )

|MRS 1, 2 | > p 1 / p 2, buy more of 1 |MRS 1, 2 | < p 1 / p 2, 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.

x 1 (p 1, p 2, m): the demand function At p 1, p 2, m, the consumer demands x 1 Perfect substitutes: (graph) 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

Fig. 5.5

Perfect complements: (graph) 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 foolhardily compare) Non convex preferences: corner solution

Fig. 5.6

Fig. 5.7

Fig. 5.8

Cobb-Douglas: u(x 1, x 2 ) = a lnx 1 + (1-a) lnx 2 |MRS 1, 2 | = p 1 / p 2, so (a/x 1 )/[(1-a)/x 2 ] = p 1 / p 2. This implies that a/(1-a) = p 1 x 1 / p 2 x 2, so 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.

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 x 1. budget constraint: (p 1 +t) x 1 + p 2 x 2 = m optimum: (x 1 *, x 2 *) so that (p 1 +t) x 1 * + p 2 x 2 * = m income tax R* to raise the same revenue: R* = t x 1 *

optimum at income tax: p 1 x 1 ’+ p 2 x 2 ’ = m - R*, so (x 1 *, x 2 *) is affordable at the case of the income tax. hence, (x 1 ’, x 2 ’) w (x 1 *, x 2 *). (graph)

Fig. 5.9

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