Chapter Five Choice 选择

Structure 5.1 The optimal choice of consumers 5.2 Consumer demand  Interior solution （内解）  Corner solution （角解）  “Kinky” solution 5.3 Example: Choosing taxes

5.1 The optimal choice of consumers The goal of consumers: maximizing utility subject to the budget constraint

The optimal bundle of goods Must be on the budget line  points to the left and below the budget line are no equilibrium. Why?  points to the right and above are no equilibrium either. why? Must on the highest indifference curve that touches the budget line.

Movies CD’s M1M1 C1C1 Highest attainable utility is U 2 U1U1 U2U2 U3U3 The optimal choice

The most preferred affordable bundle x1x1 x2x2 x1*x1* x2*x2* (x 1 *,x 2 *) is the most preferred affordable bundle.

Movies CD’s M1M1 C1C1 Note that slopes are equal here! U1U1 U2U2 U3U3 Equilibrium condition: Geometrically

Rearranging gives Consumer Equilibrium Condition MU C /P C = MU M /P M Movie CD M1M1 C1C1 Equilibrium condition

MU C /P C or MU M /P M : Marginal utiltiy per dollar of expenditure. Equal marginal principle: Utility is maximized when the consumer has equalized the marginal utility per dollar spent on all goods.  Why is this an equilibrium? Equal Marginal Principle

Disequilibrium Point Suppose you are at M 2, C 2. Movies CDs M2M2 C2C2 U1U1 U2U2 C1C1 M1M1 Disequilibrium Equilibrium

5.2 Consumer demand The optimal choice ---the consumer’s ORDINARY DEMAND （一般需求） at the given prices and income. The consumer’s demand functions give the optimal amounts of each of the goods as a function of the prices and the consumer’s income, x 1 *(p 1,p 2,m) and x 2 *(p 1,p 2,m). How to compute the optimal x?

Case1: Interior solution When x 1 * > 0 and x 2 * > 0 the demanded bundle is called INTERIOR solution.

Solve for interior solution (method 1) (x 1 *,x 2 *) satisfies two conditions : (a) p 1 x 1 * + p 2 x 2 * = m (b) tangency

Solve for interior solution (method 2) The conditions may be obtained by using the Lagrangian multiplier method, i.e., constrained optimization in calculus.

Example 1: Cobb-Douglas preference Suppose that the consumer has Cobb- Douglas preferences.

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

Corner solution But what if x 1 * = 0? Or if x 2 * = 0? If either x 1 * = 0 or x 2 * = 0 then maximizing problem has a corner solution ( 角解 ) (x 1 *,x 2 *).

Example 2-- Perfect Substitutes x1x1 x2x2 MRS = 1

Example 3: ‘ Kinky ’ Solutions -- Perfect Complements x1x1 x2x2 U(x 1,x 2 ) = min{ax 1,x 2 } x 2 = ax 1

‘ Kinky ’ Solutions -- the Perfect Complements Case x1x1 x2x2 U(x 1,x 2 ) = min{ax 1,x 2 } x 2 = ax 1

5.3 Choosing Taxes: Various Taxes Quantity tax: on x: (p+t)x Value tax: on p: (1+t)p  Also called ad valorem tax Lump sum tax: T Income tax:  Can be proportional or lump sum

Income Tax vs. Quantity Tax Proposition: Suppose the purpose of taxes is to raise the same revenue, then consumers are better off with income tax than with quantity tax on a certain commodity.

Proof: …