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,Affordable bundles,x1,x2,More preferred bundles,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 by x1*(p1,p2,
3、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. (a) (x1*,x2*) exhausts the budget; p1x1* + p2x2* = m.,Rational Constraine
4、d 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:(a) the budget is exhausted; p1x1* + p2x2* = m(b) the slope of the budget constraint, -p1/p2, and t
5、he 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-Douglas preferences.,Computing Ordinary Demands - a Cobb-Douglas Example.,Suppose that the consumer has Cobb-Douglas preferences. Th
6、en,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,(A),Computing Ordinary Demands - a Cobb-Douglas Example.,(x1*,x2*) also exhausts the budget so,(B),Computing Ordinary Demands - a Cob
7、b-Douglas Example.,So we have discovered that the most preferred affordable bundle for a consumer with 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 ha
8、ve no kinks, the ordinary demands are obtained by solving:(a) p1x1* + p2x2* = 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
9、 the ordinary demand (x1*,x2*) is at a corner solution to the problem of maximizing 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
10、,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 (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.,E
11、xamples of Corner Solutions - the Perfect Substitutes Case,x1,x2,All the bundles 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,
12、Examples of Corner Solutions - the Non-Convex Preferences Case,x1,x2,Which is the most preferred affordable bundle?,Examples of Corner Solutions - the Non-Convex Preferences Case,x1,x2,The most preferred affordable bundle,Examples of Corner Solutions - the Non-Convex Preferences Case,x1,x2,The most
13、preferred affordable bundle,Notice that the “tangency solution” is not the most preferred affordable bundle.,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
14、 = 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,x1,x2,MRS = -,MRS = 0,MRS is undefined,U(x1,x2) = minax1,x2,x2 = ax1,Examples of Kinky Solutions - the Perfect Complements
15、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,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.,
