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1、Chapter Five,Choice 消费者最优选择,Where Are We Doing in This Chapter?,After modeling a consumers choice set and his preference (represented by utility functions), we now put them together and model how he/she makes optimal choice. In mathematical terms, this is a constrained maximization problem; In econo

2、mics, this is a rational choice problem.,Rational Constrained Choice,Affordablebundles,x1,x2,More preferredbundles,Rational Constrained Choice,x1,x2,x1*,x2*,Rational Constrained Choice,x1,x2,x1*,x2*,(x1*,x2*) is the mostpreferred affordablebundle.,Rational Constrained Choice,The most preferred affor

3、dable bundle is called the consumers ORDINARY DEMAND at the given prices and budget. Ordinary demands will be denoted byx1*(p1,p2,m) and x2*(p1,p2,m).,Rational Constrained Choice,When x1* 0 and x2* 0 the demanded bundle is INTERIOR. If buying (x1*,x2*) costs $m then the budget is exhausted.,Rational

4、 Constrained Choice,x1,x2,x1*,x2*,(x1*,x2*) is interior.(a) (x1*,x2*) exhausts thebudget; p1x1* + p2x2* = m.,Rational Constrained Choice,x1,x2,x1*,x2*,(x1*,x2*) is interior .(b) The slope of the indiff.curve at (x1*,x2*) equals the slope of the budget constraint.,Rational Constrained Choice,(x1*,x2*

5、) satisfies two conditions: (a) the budget is exhausted; p1x1* + p2x2* = m (b) the slope of the budget constraint, -p1/p2, and the slope of the indifference curve containing (x1*,x2*) are equal at (x1*,x2*).,Computing Ordinary Demands - a Cobb-Douglas Example.,Suppose that the consumer has Cobb-Doug

6、las preferences.,Computing Ordinary Demands - a Cobb-Douglas Example.,Suppose that the consumer has Cobb-Douglas preferences. Then,Computing Ordinary Demands - a Cobb-Douglas Example.,So the MRS is,Computing Ordinary Demands - a Cobb-Douglas Example.,So the MRS is At (x1*,x2*), MRS = -p1/p2 so,Compu

7、ting Ordinary Demands - a Cobb-Douglas Example.,So the MRS is At (x1*,x2*), MRS = -p1/p2 so,(A),Computing Ordinary Demands - a Cobb-Douglas Example.,(x1*,x2*) also exhausts the budget so,(B),Computing Ordinary Demands - a Cobb-Douglas Example.,So we have discovered that the mostpreferred affordable

8、bundle for a consumerwith Cobb-Douglas preferences,is,Computing Ordinary Demands - a Cobb-Douglas Example.,x1,x2,Rational Constrained Choice,When x1* 0 and x2* 0 and (x1*,x2*) exhausts the budget,and indifference curves have no kinks, the ordinary demands are obtained by solving: (a) p1x1* + p2x2* =

9、 y (b) the slopes of the budget constraint, -p1/p2, and of the indifference curve containing (x1*,x2*) are equal at (x1*,x2*).,Rational Constrained Choice,But what if x1* = 0? Or if x2* = 0? If either x1* = 0 or x2* = 0 then the ordinary demand (x1*,x2*) is at a corner solution to the problem of max

10、imizing utility subject to a budget constraint.,Examples of Corner Solutions - the Perfect Substitutes Case,x1,x2,MRS = -1,Slope = -p1/p2 with p1 p2.,Examples of Corner Solutions - the Perfect Substitutes Case,x1,x2,MRS = -1,Slope = -p1/p2 with p1 p2.,Examples of Corner Solutions - the Perfect Subst

11、itutes Case,So when U(x1,x2) = x1 + x2, the mostpreferred affordable bundle is (x1*,x2*)where,and,if p1 p2,if p1 p2.,Examples of Corner Solutions - the Perfect Substitutes Case,x1,x2,MRS = -1,Slope = -p1/p2 with p1 = p2.,Examples of Corner Solutions - the Perfect Substitutes Case,x1,x2,All the bundl

12、es in the constraint are equally the most preferred affordable when p1 = p2.,Examples of Corner Solutions - the Non-Convex Preferences Case,x1,x2,Better,Examples of Corner Solutions - the Non-Convex Preferences Case,x1,x2,Examples of Corner Solutions - the Non-Convex Preferences Case,x1,x2,Which is

13、the most preferredaffordable bundle?,Examples of Corner Solutions - the Non-Convex Preferences Case,x1,x2,The most preferredaffordable bundle,Examples of Corner Solutions - the Non-Convex Preferences Case,x1,x2,The most preferredaffordable bundle,Notice that the “tangency solution”is not the most pr

14、eferred affordablebundle.,Examples of Kinky Solutions - the Perfect Complements Case,x1,x2,U(x1,x2) = minax1,x2,x2 = ax1,Examples of Kinky Solutions - the Perfect Complements Case,x1,x2,MRS = 0,U(x1,x2) = minax1,x2,x2 = ax1,Examples of Kinky Solutions - the Perfect Complements Case,x1,x2,MRS = -,MRS

15、 = 0,U(x1,x2) = minax1,x2,x2 = ax1,Examples of Kinky Solutions - the Perfect Complements Case,x1,x2,MRS = -,MRS = 0,MRS is undefined,U(x1,x2) = minax1,x2,x2 = ax1,Examples of Kinky Solutions - the Perfect Complements Case,x1,x2,U(x1,x2) = minax1,x2,x2 = ax1,Examples of Kinky Solutions - the Perfect Complements Case,x1,x2,U(x1,x2) = minax1,x2,x2 = ax1,The most preferred affordable bundle,Examples of Kinky Solutions - the Perfect Complements Case,x1,x2,U(x1,x2) = minax1,x2,x2 = ax1,