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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 economics,t
2、his is a rational choice problem.,Rational Constrained Choice,Affordablebundles,x1,x2,More preferredbundles,Rational Constrained Choice,The most preferred affordable bundle is called the consumers ORDINARY DEMAND at the given prices and budget.Ordinary demands will be denoted byx1*(p1,p2,m)and x2*(p
3、1,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 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*,
4、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*)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
5、(x1*,x2*)are equal at(x1*,x2*).,Computing Ordinary Demands-a Cobb-Douglas Example.,Suppose that the consumer has Cobb-Douglas preferences.,Computing Ordinary Demands-a Cobb-Douglas Example.,Suppose that the consumer has Cobb-Douglas preferences.Then,Computing Ordinary Demands-a Cobb-Douglas Example.
6、,So the MRS is,Computing Ordinary Demands-a Cobb-Douglas Example.,So the MRS isAt(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 aff
7、ordable 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*=y(b)
8、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 maximizing utility subject
9、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 Substitutes Case,So when U(x1,x2)=x1+x2,the
10、 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 bundles in the constraint are equally the most preferred af
11、fordable 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 the most preferredaffordable bundle?,Examples of Corner Soluti
12、ons-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 preferred affordablebundle.,Examples of Kinky Solutions-the Perfect C
13、omplements 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=0,U(x1,x2)=minax1,x2,x2=ax1,Examples of Kinky Solutions-the Perfect Complements Case
14、,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 preferredaffordable bundle,Examples of Kinky Solutions-the Perfect Complements Case,x1,x2,U(x1,x2)=minax1,x2,x2=ax1,x1*,x2*,(a)p1x1*+p2x2*=m(b)x2*=ax1*,Summary:Three Steps to Find the Optimal Choice of the Consumer,Step 1:Draw the budget set;Step 2:Draw the indifference curves;Step 3:Locate the point of optimal choice and calculate the solution.,
