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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. The optimal choice Movies Highest attainable utility is U2 M1 U2 U1 C1 U3 CD’s The most preferred affordable bundle x2 (x1*,x2*) is the most preferred affordable bundle. x2* x1* x1 Equilibrium condition: Geometrically Movies Note that slopes are equal here! M1 U2 U1 C1 U3 CD’s Equilibrium condition Rearranging gives Consumer Equilibrium Condition MUC/PC= MUM/PM Movie M1 C1 CD Equal Marginal Principle MUC/PC or MUM/PM : 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? Disequilibrium Point Movies Disequilibrium Equilibrium M2 M1 U2 U1 C2 C1 CDs Suppose you are at M2, C2. 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, x1*(p1,p2,m) and x2*(p1,p2,m). How to compute the optimal x? Case1: Interior solution When x1* > 0 and x2* > 0 the demanded bundle is called INTERIOR solution. Solve for interior solution (method 1) (x1*,x2*) satisfies two conditions: (a) p1x1* + p2x2* = 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 CobbDouglas preferences. U( x1 , x 2 ) x1axb2 Computing Ordinary Demands - a Cobb-Douglas Example. So we have discovered that the most preferred affordable bundle for a consumer with Cobb-Douglas preferences U( x1 , x 2 ) x1axb2 is ( x*1 , x*2 ) ( ) am bm , . ( a b)p1 ( a b)p2 Corner solution But what if x1* = 0? Or if x2* = 0? If either x1* = 0 or x2* = 0 then maximizing problem has a corner solution (角解) (x1*,x2*). Example 2-- Perfect Substitutes x2 MRS = 1 x1 Example 3: ‘Kinky’ Solutions -Perfect Complements x2 U(x1,x2) = min{ax1,x2} x2 = ax1 x1 ‘Kinky’ Solutions -- the Perfect Complements Case x2 U(x1,x2) = min{ax1,x2} * x2 x2 = ax1 am p1 ap 2 x*1 m p1 ap 2 x1 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: …