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1、Chapter Five,Choice,Structure,Rational constrained choice Computing ordinary demands Interior solution （内在解） Corner solution （角点解） “Kinky” solution Example: Choosing taxes,Economic Rationality,The principal behavioral postulate is that a decisionmaker chooses its most preferred alternative from thos
2、e available to it. The available choices constitute the choice set. How is the most preferred bundle in the choice set located?,Rational Constrained Choice,x1,x2,Affordable bundles,More preferred bundles,Rational Constrained Choice,Affordable bundles,x1,x2,More preferred bundles,Rational Constrained
3、 Choice,x1,x2,x1*,x2*,Rational Constrained Choice,x1,x2,x1*,x2*,(x1*,x2*) is the most preferred affordable bundle.,Rational Constrained Choice,The most preferred affordable bundle is called the consumers ORDINARY DEMAND （一般需求）at the given prices and budget. Ordinary demands will be denoted by x1*(p1
4、,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 Constrained Choice,x1,x2,x1*,x2*,(x1*,x2*) is interior.,(x1*,x2*) exhausts the budget.,Rational Constrained Choice,x1,x2,x1*,
5、x2*,(x1*,x2*) is interior. (a) (x1*,x2*) exhausts the budget; 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*) satisfies two conditions
6、: (a) the budget is exhausted; p1x1* + p2x2* = m (b) tangency: the slope of the budget constraint, -p1/p2, and the slope of the indifference curve containing (x1*,x2*) are equal at (x1*,x2*).,Meaning of the Tangency Condition,Consumers marginal willingness to pay equals the market exchange rate. Sup
7、pose at a consumption bundle (x1, x2), MRS=-2, -P1/P2=-1 The consumer is willing to give up 2 unit of x2 to exchange for an additional unit of x1 The market allows her to give up only 1 unit of x2 to obtain an additional x1 (x1, x2) is not optimal choice She can be better off increasing her consumpt
8、ion of x1.,x1,x1,x2,Computing Ordinary Demands,Solve for 2 simultaneous equations. Tangency Budget constraint The conditions may be obtained by using the Lagrangian multiplier method, i.e., constrained optimization in calculus.,Computing Ordinary Demands,How can this information be used to locate (x
9、1*,x2*) for given p1, p2 and m?,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 E
10、xample.,So the MRS is,Computing Ordinary Demands - a Cobb-Douglas Example.,So the MRS is At (x1*,x2*), MRS = -p1/p2 so,Computing 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
11、 budget so,(B),Computing Ordinary Demands - a Cobb-Douglas Example.,So now we know that,(A),(B),Computing Ordinary Demands - a Cobb-Douglas Example.,So now we know that,(A),(B),Substitute,Computing Ordinary Demands - a Cobb-Douglas Example.,So now we know that,(A),(B),Substitute,and get,This simplif
12、ies to .,Computing Ordinary Demands - a Cobb-Douglas Example.,Computing Ordinary Demands - a Cobb-Douglas Example.,Substituting for x1* in,then gives,Computing Ordinary Demands - a Cobb-Douglas Example.,So we have discovered that the most preferred affordable bundle for a consumer with Cobb-Douglas
13、preferences,is,Computing Ordinary Demands - a Cobb-Douglas Example.,x1,x2,Rational Constrained Choice: Summary,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) the slopes of the budg
14、et 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 to a
15、budget constraint.,Examples of Corner Solutions - the Perfect Substitutes Case,x1,x2,MRS = -1,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.,Ex
16、amples 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 most preferred affordable bundle is
