microlower.jpg © 2010 W. W. Norton & Company, Inc. microtitle.jpg microedition.jpg varianname.jpg 5 Choice microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Economic Rationality uThe 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? microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice x1 x2 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice x1 Utility microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice 25% Utility x2 x1 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice x1 x2 Utility microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice 25% Utility x1 x2 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice 25% Utility x1 x2 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice Utility x1 x2 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice Utility x1 x2 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice Utility x1 x2 Affordable, but not the most preferred affordable bundle. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice x1 x2 Utility Affordable, but not the most preferred affordable bundle. The most preferred of the affordable bundles. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice x1 x2 Utility microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice Utility x1 x2 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice Utility x1 x2 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice Utility x1 x2 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice x1 x2 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice x1 x2 Affordable bundles microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice x1 x2 Affordable bundles microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice x1 x2 Affordable bundles More preferred bundles microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice Affordable bundles x1 x2 More preferred bundles microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice x1 x2 x1* x2* microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice x1 x2 x1* x2* (x1*,x2*) is the most preferred affordable bundle. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice uThe most preferred affordable bundle is called the consumer’s ORDINARY DEMAND at the given prices and budget. uOrdinary demands will be denoted by x1*(p1,p2,m) and x2*(p1,p2,m). microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice uWhen x1* > 0 and x2* > 0 the demanded bundle is INTERIOR. uIf buying (x1*,x2*) costs $m then the budget is exhausted. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice x1 x2 x1* x2* (x1*,x2*) is interior. (x1*,x2*) exhausts the budget. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice x1 x2 x1* x2* (x1*,x2*) is interior. (a) (x1*,x2*) exhausts the budget; p1x1* + p2x2* = m. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice u(x1*,x2*) satisfies two conditions: u (a) the budget is exhausted; p1x1* + p2x2* = m u (b) the slope of the budget constraint, -p1/p2, and the slope of the indifference curve containing (x1*,x2*) are equal at (x1*,x2*). microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Computing Ordinary Demands uHow can this information be used to locate (x1*,x2*) for given p1, p2 and m? microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Computing Ordinary Demands - a Cobb-Douglas Example. uSuppose that the consumer has Cobb-Douglas preferences. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Computing Ordinary Demands - a Cobb-Douglas Example. uSuppose that the consumer has Cobb-Douglas preferences. uThen microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Computing Ordinary Demands - a Cobb-Douglas Example. uSo the MRS is microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Computing Ordinary Demands - a Cobb-Douglas Example. uSo the MRS is uAt (x1*,x2*), MRS = -p1/p2 so microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Computing Ordinary Demands - a Cobb-Douglas Example. uSo the MRS is uAt (x1*,x2*), MRS = -p1/p2 so (A) microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Computing Ordinary Demands - a Cobb-Douglas Example. u(x1*,x2*) also exhausts the budget so (B) microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Computing Ordinary Demands - a Cobb-Douglas Example. uSo now we know that (A) (B) microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Computing Ordinary Demands - a Cobb-Douglas Example. uSo now we know that (A) (B) Substitute microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Computing Ordinary Demands - a Cobb-Douglas Example. uSo now we know that (A) (B) Substitute and get This simplifies to …. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Computing Ordinary Demands - a Cobb-Douglas Example. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Computing Ordinary Demands - a Cobb-Douglas Example. Substituting for x1* in then gives microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Computing Ordinary Demands - a Cobb-Douglas Example. So we have discovered that the most preferred affordable bundle for a consumer with Cobb-Douglas preferences is microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Computing Ordinary Demands - a Cobb-Douglas Example. x1 x2 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice uWhen 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* = y u (b) the slopes of the budget constraint, -p1/p2, and of the indifference curve containing (x1*,x2*) are equal at (x1*,x2*). microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Rational Constrained Choice uBut 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of Corner Solutions -- the Perfect Substitutes Case x1 x2 MRS = -1 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of Corner Solutions -- the Perfect Substitutes Case x1 x2 MRS = -1 Slope = -p1/p2 with p1 > p2. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of Corner Solutions -- the Perfect Substitutes Case x1 x2 MRS = -1 Slope = -p1/p2 with p1 > p2. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of Corner Solutions -- the Perfect Substitutes Case x1 x2 MRS = -1 Slope = -p1/p2 with p1 > p2. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of Corner Solutions -- the Perfect Substitutes Case x1 x2 MRS = -1 Slope = -p1/p2 with p1 < p2. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of Corner Solutions -- the Perfect Substitutes Case x1 x2 MRS = -1 Slope = -p1/p2 with p1 = p2. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of Corner Solutions -- the Perfect Substitutes Case x1 x2 All the bundles in the constraint are equally the most preferred affordable when p1 = p2. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of Corner Solutions -- the Non-Convex Preferences Case x1 x2 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of Corner Solutions -- the Non-Convex Preferences Case x1 x2 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of Corner Solutions -- the Non-Convex Preferences Case x1 x2 Which is the most preferred affordable bundle? microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of Corner Solutions -- the Non-Convex Preferences Case x1 x2 The most preferred affordable bundle microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of Corner Solutions -- the Non-Convex Preferences Case x1 x2 The most preferred affordable bundle Notice that the “tangency solution” is not the most preferred affordable bundle. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1 x2 U(x1,x2) = min{ax1,x2} x2 = ax1 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1 x2 MRS = 0 U(x1,x2) = min{ax1,x2} x2 = ax1 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1 x2 MRS = - ¥ MRS = 0 U(x1,x2) = min{ax1,x2} x2 = ax1 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1 x2 MRS = - ¥ MRS = 0 MRS is undefined U(x1,x2) = min{ax1,x2} x2 = ax1 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1 x2 U(x1,x2) = min{ax1,x2} x2 = ax1 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1 x2 U(x1,x2) = min{ax1,x2} x2 = ax1 Which is the most preferred affordable bundle? microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1 x2 U(x1,x2) = min{ax1,x2} x2 = ax1 The most preferred affordable bundle microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1 x2 U(x1,x2) = min{ax1,x2} x2 = ax1 x1* x2* microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1 x2 U(x1,x2) = min{ax1,x2} x2 = ax1 x1* x2* (a) p1x1* + p2x2* = m microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1 x2 U(x1,x2) = min{ax1,x2} x2 = ax1 x1* x2* (a) p1x1* + p2x2* = m (b) x2* = ax1* microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of ‘Kinky’ Solutions -- the Perfect Complements Case (a) p1x1* + p2x2* = m; (b) x2* = ax1*. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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* = m microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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* = m which gives microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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* = m which gives microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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* = m which gives A bundle of 1 commodity 1 unit and a commodity 2 units costs p1 + ap2; m/(p1 + ap2) such bundles are affordable. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1 x2 U(x1,x2) = min{ax1,x2} x2 = ax1