INTMIC9.jpg Chapter 5 Choice 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? Rational Constrained Choice x1 x2 Rational Constrained Choice x1 Utility Rational Constrained Choice 25% Utility x2 x1 Rational Constrained Choice x1 x2 Utility Rational Constrained Choice 25% Utility x1 x2 Rational Constrained Choice 25% Utility x1 x2 Rational Constrained Choice Utility x1 x2 Rational Constrained Choice Utility x1 x2 Rational Constrained Choice Utility x1 x2 Affordable, but not the most preferred affordable bundle. Rational Constrained Choice x1 x2 Utility Affordable, but not the most preferred affordable bundle. The most preferred of the affordable bundles. Rational Constrained Choice x1 x2 Utility Rational Constrained Choice Utility x1 x2 Rational Constrained Choice Utility x1 x2 Rational Constrained Choice Utility x1 x2 Rational Constrained Choice x1 x2 Rational Constrained Choice x1 x2 Affordable bundles Rational Constrained Choice x1 x2 Affordable bundles Rational Constrained Choice x1 x2 Affordable bundles More preferred bundles Rational Constrained Choice Affordable bundles x1 x2 More preferred bundles Rational Constrained Choice x1 x2 x1* x2* Rational Constrained Choice x1 x2 x1* x2* (x1*,x2*) is the most preferred affordable bundle. 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). 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. Rational Constrained Choice x1 x2 x1* x2* (x1*,x2*) is interior. (x1*,x2*) exhausts the budget. Rational Constrained Choice x1 x2 x1* 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 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*). Computing Ordinary Demands uHow can this information be used to locate (x1*,x2*) for given p1, p2 and m? Computing Ordinary Demands - a Cobb-Douglas Example. uSuppose that the consumer has Cobb-Douglas preferences. Computing Ordinary Demands - a Cobb-Douglas Example. uSuppose that the consumer has Cobb-Douglas preferences. uThen Computing Ordinary Demands - a Cobb-Douglas Example. uSo the MRS is Computing Ordinary Demands - a Cobb-Douglas Example. uSo the MRS is uAt (x1*,x2*), MRS = -p1/p2 so Computing Ordinary Demands - a Cobb-Douglas Example. uSo the MRS is uAt (x1*,x2*), MRS = -p1/p2 so (A) Computing Ordinary Demands - a Cobb-Douglas Example. u(x1*,x2*) also exhausts the budget so (B) Computing Ordinary Demands - a Cobb-Douglas Example. uSo now we know that (A) (B) Computing Ordinary Demands - a Cobb-Douglas Example. uSo now we know that (A) (B) Substitute Computing Ordinary Demands - a Cobb-Douglas Example. uSo now we know that (A) (B) Substitute and get This simplifies 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 preferences is Computing Ordinary Demands - a Cobb-Douglas Example. x1 x2 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*). 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. 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. 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 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. 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. Examples of Corner Solutions -- the Non-Convex Preferences Case x1 x2 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 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 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) = min{ax1,x2} x2 = ax1 Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1 x2 MRS = 0 U(x1,x2) = min{ax1,x2} x2 = ax1 Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1 x2 MRS = - ¥ MRS = 0 U(x1,x2) = min{ax1,x2} x2 = ax1 Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1 x2 MRS = - ¥ MRS = 0 MRS is undefined U(x1,x2) = min{ax1,x2} x2 = ax1 Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1 x2 U(x1,x2) = min{ax1,x2} x2 = ax1 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? Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1 x2 U(x1,x2) = min{ax1,x2} x2 = ax1 The most preferred affordable bundle Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1 x2 U(x1,x2) = min{ax1,x2} x2 = ax1 x1* x2* Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1 x2 U(x1,x2) = min{ax1,x2} x2 = ax1 x1* x2* (a) p1x1* + p2x2* = m 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* 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* = m 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 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 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. Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1 x2 U(x1,x2) = min{ax1,x2} x2 = ax1