1、Chapter FiveChoiceEconomic RationalityuThe principal behavioral postulate is that a decisionmaker chooses its most preferred alternative from those available to it.uThe available choices constitute the choice set.uHow is the most preferred bundle in the choice set located?Rational Constrained Choice
2、x1x2Rational Constrained Choicex1x2UtilityRational Constrained ChoiceUtilityx2x1Rational Constrained Choicex1x2UtilityRational Constrained ChoiceUtilityx1x2Rational Constrained ChoiceUtilityx1x2Rational Constrained ChoiceUtilityx1x2Rational Constrained ChoiceUtilityx1x2Rational Constrained ChoiceUti
3、lityx1x2Affordable,but not the most preferred affordable bundle.Rational Constrained Choicex1x2UtilityAffordable,but not the most preferred affordable bundle.The most preferredof the affordablebundles.Rational Constrained Choicex1x2UtilityRational Constrained ChoiceUtilityx1x2Rational Constrained Ch
4、oiceUtilityx1x2Rational Constrained ChoiceUtilityx1x2Rational Constrained Choicex1x2Rational Constrained Choicex1x2AffordablebundlesRational Constrained Choicex1x2AffordablebundlesRational Constrained Choicex1x2AffordablebundlesMore preferredbundlesRational Constrained ChoiceAffordablebundlesx1x2Mor
5、e preferredbundlesRational Constrained Choicex1x2x1*x2*Rational Constrained Choicex1x2x1*x2*(x1*,x2*)is the mostpreferred affordablebundle.Rational Constrained ChoiceuThe most preferred affordable bundle is called the consumers ORDINARY DEMAND at the given prices and budget.uOrdinary demands will be
6、 denoted byx1*(p1,p2,m)and x2*(p1,p2,m).Rational Constrained ChoiceuWhen x1*0 and x2*0 the demanded bundle is INTERIOR.uIf buying(x1*,x2*)costs$m then the budget is exhausted.Rational Constrained Choicex1x2x1*x2*(x1*,x2*)is interior.(x1*,x2*)exhausts thebudget.Rational Constrained Choicex1x2x1*x2*(x
7、1*,x2*)is interior.(a)(x1*,x2*)exhausts thebudget;p1x1*+p2x2*=m.Rational Constrained Choicex1x2x1*x2*(x1*,x2*)is interior.(b)The slope of the indiff.curve at(x1*,x2*)equals the slope of the budget constraint.Rational Constrained Choiceu(x1*,x2*)satisfies two conditions:u(a)the budget is exhausted;p1
8、x1*+p2x2*=mu(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 DemandsuHow can this information be used to locate(x1*,x2*)for given p1,p2 and m?Computing Ordinary Demands-a Cobb-Douglas Example.uSuppose tha
9、t the consumer has Cobb-Douglas preferences.U xxx xa b(,)1212 Computing Ordinary Demands-a Cobb-Douglas Example.uSuppose that the consumer has Cobb-Douglas preferences.uThenU xxx xa b(,)1212 MUUxaxxab11112 MUUxbx xa b22121 Computing Ordinary Demands-a Cobb-Douglas Example.uSo the MRS isMRSdxdxUxUxax
10、xbx xaxbxaba b 211211212121 /.Computing Ordinary Demands-a Cobb-Douglas Example.uSo the MRS isuAt(x1*,x2*),MRS=-p1/p2 soMRSdxdxUxUxaxxbx xaxbxaba b 211211212121 /.Computing Ordinary Demands-a Cobb-Douglas Example.uSo the MRS isuAt(x1*,x2*),MRS=-p1/p2 soMRSdxdxUxUxaxxbx xaxbxaba b 211211212121 /.axbx
11、ppxbpapx21122121*.(A)Computing Ordinary Demands-a Cobb-Douglas Example.u(x1*,x2*)also exhausts the budget sop xp xm1 12 2*.(B)Computing Ordinary Demands-a Cobb-Douglas Example.uSo now we know thatxbpapx2121*(A)p xp xm1 12 2*.(B)Computing Ordinary Demands-a Cobb-Douglas Example.uSo now we know thatxb
12、papx2121*(A)p xp xm1 12 2*.(B)SubstituteComputing Ordinary Demands-a Cobb-Douglas Example.uSo now we know thatxbpapx2121*(A)p xp xm1 12 2*.(B)p xpbpapxm1 12121*.Substituteand getThis simplifies to.Computing Ordinary Demands-a Cobb-Douglas Example.xamab p11*().Computing Ordinary Demands-a Cobb-Dougla
13、s Example.xbmab p22*().Substituting for x1*in p xp xm1 12 2*then givesxamab p11*().Computing Ordinary Demands-a Cobb-Douglas Example.So we have discovered that the mostpreferred affordable bundle for a consumerwith Cobb-Douglas preferencesU xxx xa b(,)1212 is(,)(),().*()xxamab pbmab p1212 Computing
14、Ordinary Demands-a Cobb-Douglas Example.x1x2xamab p11*()xbmab p22*()U xxx xa b(,)1212 Rational Constrained ChoiceuWhen x1*0 and x2*0 and (x1*,x2*)exhausts the budget,and indifference curves have no kinks,the ordinary demands are obtained by solving:u(a)p1x1*+p2x2*=yu(b)the slopes of the budget const
15、raint,-p1/p2,and of the indifference curve containing(x1*,x2*)are equal at(x1*,x2*).Rational Constrained ChoiceuBut what if x1*=0?uOr if x2*=0?uIf either x1*=0 or x2*=0 then the ordinary demand(x1*,x2*)is at a corner solution to the problem of maximizing utility subject to a budget constraint.Exampl
16、es of Corner Solutions-the Perfect Substitutes Casex1x2MRS=-1Examples of Corner Solutions-the Perfect Substitutes Casex1x2MRS=-1Slope=-p1/p2 with p1 p2.Examples of Corner Solutions-the Perfect Substitutes Casex1x2MRS=-1Slope=-p1/p2 with p1 p2.Examples of Corner Solutions-the Perfect Substitutes Case
17、x1x2xyp22*x10*MRS=-1Slope=-p1/p2 with p1 p2.Examples of Corner Solutions-the Perfect Substitutes Casex1x2xyp11*x20*MRS=-1Slope=-p1/p2 with p1 p2.Examples of Corner Solutions-the Perfect Substitutes CaseSo when U(x1,x2)=x1+x2,the mostpreferred affordable bundle is(x1*,x2*)where 0,py)x,x(1*2*1and 2*2*
18、1py,0)x,x(if p1 p2.Examples of Corner Solutions-the Perfect Substitutes Casex1x2MRS=-1Slope=-p1/p2 with p1=p2.yp1yp2Examples of Corner Solutions-the Perfect Substitutes Casex1x2All the bundles in the constraint are equally the most preferred affordable when p1=p2.yp2yp1Examples of Corner Solutions-t
19、he Non-Convex Preferences Casex1x2BetterExamples of Corner Solutions-the Non-Convex Preferences Casex1x2Examples of Corner Solutions-the Non-Convex Preferences Casex1x2Which is the most preferredaffordable bundle?Examples of Corner Solutions-the Non-Convex Preferences Casex1x2The most preferredaffor
20、dable bundleExamples of Corner Solutions-the Non-Convex Preferences Casex1x2The most preferredaffordable bundleNotice that the“tangency solution”is not the most preferred affordablebundle.Examples of Kinky Solutions-the Perfect Complements Casex1x2U(x1,x2)=minax1,x2x2=ax1Examples of Kinky Solutions-
21、the Perfect Complements Casex1x2MRS=0U(x1,x2)=minax1,x2x2=ax1Examples of Kinky Solutions-the Perfect Complements Casex1x2MRS=-MRS=0U(x1,x2)=minax1,x2x2=ax1Examples of Kinky Solutions-the Perfect Complements Casex1x2MRS=-MRS=0MRS is undefinedU(x1,x2)=minax1,x2x2=ax1Examples of Kinky Solutions-the Per
22、fect Complements Casex1x2U(x1,x2)=minax1,x2x2=ax1Examples of Kinky Solutions-the Perfect Complements Casex1x2U(x1,x2)=minax1,x2x2=ax1Which is the mostpreferred affordable bundle?Examples of Kinky Solutions-the Perfect Complements Casex1x2U(x1,x2)=minax1,x2x2=ax1The most preferredaffordable bundleExa
23、mples of Kinky Solutions-the Perfect Complements Casex1x2U(x1,x2)=minax1,x2x2=ax1x1*x2*Examples of Kinky Solutions-the Perfect Complements Casex1x2U(x1,x2)=minax1,x2x2=ax1x1*x2*(a)p1x1*+p2x2*=mExamples of Kinky Solutions-the Perfect Complements Casex1x2U(x1,x2)=minax1,x2x2=ax1x1*x2*(a)p1x1*+p2x2*=m(
24、b)x2*=ax1*Examples of Kinky Solutions-the Perfect Complements Case(a)p1x1*+p2x2*=m;(b)x2*=ax1*.Examples of Kinky Solutions-the Perfect Complements Case(a)p1x1*+p2x2*=m;(b)x2*=ax1*.Substitution from(b)for x2*in(a)gives p1x1*+p2ax1*=mExamples of Kinky Solutions-the Perfect Complements Case(a)p1x1*+p2x
25、2*=m;(b)x2*=ax1*.Substitution from(b)for x2*in(a)gives p1x1*+p2ax1*=mwhich gives21*1appmx Examples of Kinky Solutions-the Perfect Complements Case(a)p1x1*+p2x2*=m;(b)x2*=ax1*.Substitution from(b)for x2*in(a)gives p1x1*+p2ax1*=mwhich gives.appamx;appmx21*221*1 Examples of Kinky Solutions-the Perfect
26、Complements Case(a)p1x1*+p2x2*=m;(b)x2*=ax1*.Substitution from(b)for x2*in(a)gives p1x1*+p2ax1*=mwhich givesA bundle of 1 commodity 1 unit anda commodity 2 units costs p1+ap2;m/(p1+ap2)such bundles are affordable.appamx;appmx21*221*1 Examples of Kinky Solutions-the Perfect Complements Casex1x2U(x1,x
27、2)=minax1,x2x2=ax1xmpap112*xampap212*Choosing Taxes:Various TaxesuQuantity tax:on x:(p+t)xuValue tax:on px:(1+t)pxAlso called ad valorem taxuLump sum tax:TuIncome tax:Can be proportional or lump sumIncome Tax vs.Quantity TaxuOriginal budget:p1x1+p2x2=muAfter quantity tax:(p1+t)x1+p2x2=muAt optimal c
28、hoice(x1*,x2*)(p1+t)x1*+p2x2*=m (5.2)Tax revenue:R*=tx1*uWith an income tax,budget is:p1x1+p2x2=m-tx1*Income vs.Quantity TaxuProposition:(x1*,x2*)is affordable under income taxuEquivalent to:prove that(x1*,x2*)satisfies budget constraint under income tax.uOr,budget constraint holds at point(x1*,x2*).p1x1*+p2x2*=m-tx1*uWhich is true according to(5.2).uIt is not an optimal choice because prices are different.uConclusion:The optimal choice must be more preferred to(x1*,x2*)
