1. Chapter Five Choice 选择

2. 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

3. 5.1 The optimal choice of consumers The goal of consumers: maximizing utility subject to the budget constraint

4. 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.

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

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

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

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

9. 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

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

11. 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?

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

13. 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

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

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

16. 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

17. 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 *).

18. Example 2-- Perfect Substitutes x1x1 x2x2 MRS = 1

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

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

21. 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

22. 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.

23. Proof: …