© 2010 W. W. Norton & Company, Inc. 5 Choice

© 2010 W. W. Norton & Company, Inc. 2 Economic Rationality u The principal behavioral postulate is that a decisionmaker chooses its most preferred alternative from those available to it. u The available choices constitute the choice set. u How is the most preferred bundle in the choice set located?

© 2010 W. W. Norton & Company, Inc. 3 Rational Constrained Choice x1x1 x2x2

© 2010 W. W. Norton & Company, Inc. 4 Rational Constrained Choice x1x1 Utility

© 2010 W. W. Norton & Company, Inc. 5 Rational Constrained Choice Utility x2x2 x1x1

© 2010 W. W. Norton & Company, Inc. 6 Rational Constrained Choice x1x1 x2x2 Utility

© 2010 W. W. Norton & Company, Inc. 7 Rational Constrained Choice Utility x1x1 x2x2

© 2010 W. W. Norton & Company, Inc. 8 Rational Constrained Choice Utility x1x1 x2x2

© 2010 W. W. Norton & Company, Inc. 9 Rational Constrained Choice Utility x1x1 x2x2

© 2010 W. W. Norton & Company, Inc. 10 Rational Constrained Choice Utility x1x1 x2x2

© 2010 W. W. Norton & Company, Inc. 11 Rational Constrained Choice Utility x1x1 x2x2 Affordable, but not the most preferred affordable bundle.

© 2010 W. W. Norton & Company, Inc. 12 Rational Constrained Choice x1x1 x2x2 Utility Affordable, but not the most preferred affordable bundle. The most preferred of the affordable bundles.

© 2010 W. W. Norton & Company, Inc. 13 Rational Constrained Choice x1x1 x2x2 Utility

© 2010 W. W. Norton & Company, Inc. 14 Rational Constrained Choice Utility x1x1 x2x2

© 2010 W. W. Norton & Company, Inc. 15 Rational Constrained Choice Utility x1x1 x2x2

© 2010 W. W. Norton & Company, Inc. 16 Rational Constrained Choice Utility x1x1 x2x2

© 2010 W. W. Norton & Company, Inc. 17 Rational Constrained Choice x1x1 x2x2

© 2010 W. W. Norton & Company, Inc. 18 Rational Constrained Choice x1x1 x2x2 Affordable bundles

© 2010 W. W. Norton & Company, Inc. 19 Rational Constrained Choice x1x1 x2x2 Affordable bundles

© 2010 W. W. Norton & Company, Inc. 20 Rational Constrained Choice x1x1 x2x2 Affordable bundles More preferred bundles

© 2010 W. W. Norton & Company, Inc. 21 Rational Constrained Choice Affordable bundles x1x1 x2x2 More preferred bundles

© 2010 W. W. Norton & Company, Inc. 22 Rational Constrained Choice x1x1 x2x2 x1*x1* x2*x2*

© 2010 W. W. Norton & Company, Inc. 23 Rational Constrained Choice x1x1 x2x2 x1*x1* x2*x2* (x 1 *,x 2 *) is the most preferred affordable bundle.

© 2010 W. W. Norton & Company, Inc. 24 Rational Constrained Choice u The most preferred affordable bundle is called the consumer’s ORDINARY DEMAND at the given prices and budget. u Ordinary demands will be denoted by x 1 *(p 1,p 2,m) and x 2 *(p 1,p 2,m).

© 2010 W. W. Norton & Company, Inc. 25 Rational Constrained Choice u When x 1 * > 0 and x 2 * > 0 the demanded bundle is INTERIOR. u If buying (x 1 *,x 2 *) costs $m then the budget is exhausted.

© 2010 W. W. Norton & Company, Inc. 26 Rational Constrained Choice x1x1 x2x2 x1*x1* x2*x2* (x 1 *,x 2 *) is interior. (x 1 *,x 2 *) exhausts the budget.

© 2010 W. W. Norton & Company, Inc. 27 Rational Constrained Choice x1x1 x2x2 x1*x1* x2*x2* (x 1 *,x 2 *) is interior. (a) (x 1 *,x 2 *) exhausts the budget; p 1 x 1 * + p 2 x 2 * = m.

© 2010 W. W. Norton & Company, Inc. 28 Rational Constrained Choice x1x1 x2x2 x1*x1* x2*x2* (x 1 *,x 2 *) is interior. (b) The slope of the indiff. curve at (x 1 *,x 2 *) equals the slope of the budget constraint.

© 2010 W. W. Norton & Company, Inc. 29 Rational Constrained Choice u (x 1 *,x 2 *) satisfies two conditions: u (a) the budget is exhausted; p 1 x 1 * + p 2 x 2 * = m u (b) the slope of the budget constraint, -p 1 /p 2, and the slope of the indifference curve containing (x 1 *,x 2 *) are equal at (x 1 *,x 2 *).

© 2010 W. W. Norton & Company, Inc. 30 Computing Ordinary Demands u How can this information be used to locate (x 1 *,x 2 *) for given p 1, p 2 and m?

© 2010 W. W. Norton & Company, Inc. 31 Computing Ordinary Demands - a Cobb-Douglas Example. u Suppose that the consumer has Cobb-Douglas preferences.

© 2010 W. W. Norton & Company, Inc. 32 Computing Ordinary Demands - a Cobb-Douglas Example. u Suppose that the consumer has Cobb-Douglas preferences. u Then

© 2010 W. W. Norton & Company, Inc. 33 Computing Ordinary Demands - a Cobb-Douglas Example. u So the MRS is

© 2010 W. W. Norton & Company, Inc. 34 Computing Ordinary Demands - a Cobb-Douglas Example. u So the MRS is u At (x 1 *,x 2 *), MRS = -p 1 /p 2 so

© 2010 W. W. Norton & Company, Inc. 35 Computing Ordinary Demands - a Cobb-Douglas Example. u So the MRS is u At (x 1 *,x 2 *), MRS = -p 1 /p 2 so (A)

© 2010 W. W. Norton & Company, Inc. 36 Computing Ordinary Demands - a Cobb-Douglas Example. u (x 1 *,x 2 *) also exhausts the budget so (B)

© 2010 W. W. Norton & Company, Inc. 37 Computing Ordinary Demands - a Cobb-Douglas Example. u So now we know that (A) (B)

© 2010 W. W. Norton & Company, Inc. 38 Computing Ordinary Demands - a Cobb-Douglas Example. u So now we know that (A) (B) Substitute

© 2010 W. W. Norton & Company, Inc. 39 Computing Ordinary Demands - a Cobb-Douglas Example. u So now we know that (A) (B) Substitute and get This simplifies to ….

© 2010 W. W. Norton & Company, Inc. 40 Computing Ordinary Demands - a Cobb-Douglas Example.

© 2010 W. W. Norton & Company, Inc. 41 Computing Ordinary Demands - a Cobb-Douglas Example. Substituting for x 1 * in then gives

© 2010 W. W. Norton & Company, Inc. 42 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

© 2010 W. W. Norton & Company, Inc. 43 Computing Ordinary Demands - a Cobb-Douglas Example. x1x1 x2x2

© 2010 W. W. Norton & Company, Inc. 44 Rational Constrained Choice u When x 1 * > 0 and x 2 * > 0 and (x 1 *,x 2 *) exhausts the budget, and indifference curves have no ‘kinks’, the ordinary demands are obtained by solving: u (a) p 1 x 1 * + p 2 x 2 * = y u (b) the slopes of the budget constraint, -p 1 /p 2, and of the indifference curve containing (x 1 *,x 2 *) are equal at (x 1 *,x 2 *).

© 2010 W. W. Norton & Company, Inc. 45 Rational Constrained Choice u But what if x 1 * = 0? u Or if x 2 * = 0? u If either x 1 * = 0 or x 2 * = 0 then the ordinary demand (x 1 *,x 2 *) is at a corner solution to the problem of maximizing utility subject to a budget constraint.

© 2010 W. W. Norton & Company, Inc. 46 Examples of Corner Solutions -- the Perfect Substitutes Case x1x1 x2x2 MRS = -1

© 2010 W. W. Norton & Company, Inc. 47 Examples of Corner Solutions -- the Perfect Substitutes Case x1x1 x2x2 MRS = -1 Slope = -p 1 /p 2 with p 1 > p 2.

© 2010 W. W. Norton & Company, Inc. 48 Examples of Corner Solutions -- the Perfect Substitutes Case x1x1 x2x2 MRS = -1 Slope = -p 1 /p 2 with p 1 > p 2.

© 2010 W. W. Norton & Company, Inc. 49 Examples of Corner Solutions -- the Perfect Substitutes Case x1x1 x2x2 MRS = -1 Slope = -p 1 /p 2 with p 1 > p 2.

© 2010 W. W. Norton & Company, Inc. 50 Examples of Corner Solutions -- the Perfect Substitutes Case x1x1 x2x2 MRS = -1 Slope = -p 1 /p 2 with p 1 < p 2.

© 2010 W. W. Norton & Company, Inc. 51 Examples of Corner Solutions -- the Perfect Substitutes Case So when U(x 1,x 2 ) = x 1 + x 2, the most preferred affordable bundle is (x 1 *,x 2 *) where and if p 1 < p 2 if p 1 > p 2.

© 2010 W. W. Norton & Company, Inc. 52 Examples of Corner Solutions -- the Perfect Substitutes Case x1x1 x2x2 MRS = -1 Slope = -p 1 /p 2 with p 1 = p 2.

© 2010 W. W. Norton & Company, Inc. 53 Examples of Corner Solutions -- the Perfect Substitutes Case x1x1 x2x2 All the bundles in the constraint are equally the most preferred affordable when p 1 = p 2.

© 2010 W. W. Norton & Company, Inc. 54 Examples of Corner Solutions -- the Non-Convex Preferences Case x1x1 x2x2 Better

© 2010 W. W. Norton & Company, Inc. 55 Examples of Corner Solutions -- the Non-Convex Preferences Case x1x1 x2x2

© 2010 W. W. Norton & Company, Inc. 56 Examples of Corner Solutions -- the Non-Convex Preferences Case x1x1 x2x2 Which is the most preferred affordable bundle?

© 2010 W. W. Norton & Company, Inc. 57 Examples of Corner Solutions -- the Non-Convex Preferences Case x1x1 x2x2 The most preferred affordable bundle

© 2010 W. W. Norton & Company, Inc. 58 Examples of Corner Solutions -- the Non-Convex Preferences Case x1x1 x2x2 The most preferred affordable bundle Notice that the “tangency solution” is not the most preferred affordable bundle.

© 2010 W. W. Norton & Company, Inc. 59 Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1x1 x2x2 U(x 1,x 2 ) = min{ax 1,x 2 } x 2 = ax 1

© 2010 W. W. Norton & Company, Inc. 60 Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1x1 x2x2 MRS = 0 U(x 1,x 2 ) = min{ax 1,x 2 } x 2 = ax 1

© 2010 W. W. Norton & Company, Inc. 61 Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1x1 x2x2 MRS = -  MRS = 0 U(x 1,x 2 ) = min{ax 1,x 2 } x 2 = ax 1

© 2010 W. W. Norton & Company, Inc. 62 Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1x1 x2x2 MRS = -  MRS = 0 MRS is undefined U(x 1,x 2 ) = min{ax 1,x 2 } x 2 = ax 1

© 2010 W. W. Norton & Company, Inc. 63 Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1x1 x2x2 U(x 1,x 2 ) = min{ax 1,x 2 } x 2 = ax 1

© 2010 W. W. Norton & Company, Inc. 64 Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1x1 x2x2 U(x 1,x 2 ) = min{ax 1,x 2 } x 2 = ax 1 Which is the most preferred affordable bundle?

© 2010 W. W. Norton & Company, Inc. 65 Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1x1 x2x2 U(x 1,x 2 ) = min{ax 1,x 2 } x 2 = ax 1 The most preferred affordable bundle

© 2010 W. W. Norton & Company, Inc. 66 Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1x1 x2x2 U(x 1,x 2 ) = min{ax 1,x 2 } x 2 = ax 1 x1*x1* x2*x2*

© 2010 W. W. Norton & Company, Inc. 67 Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1x1 x2x2 U(x 1,x 2 ) = min{ax 1,x 2 } x 2 = ax 1 x1*x1* x2*x2* (a) p 1 x 1 * + p 2 x 2 * = m

© 2010 W. W. Norton & Company, Inc. 68 Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1x1 x2x2 U(x 1,x 2 ) = min{ax 1,x 2 } x 2 = ax 1 x1*x1* x2*x2* (a) p 1 x 1 * + p 2 x 2 * = m (b) x 2 * = ax 1 *

© 2010 W. W. Norton & Company, Inc. 69 Examples of ‘Kinky’ Solutions -- the Perfect Complements Case (a) p 1 x 1 * + p 2 x 2 * = m; (b) x 2 * = ax 1 *.

© 2010 W. W. Norton & Company, Inc. 70 Examples of ‘Kinky’ Solutions -- the Perfect Complements Case (a) p 1 x 1 * + p 2 x 2 * = m; (b) x 2 * = ax 1 *. Substitution from (b) for x 2 * in (a) gives p 1 x 1 * + p 2 ax 1 * = m

© 2010 W. W. Norton & Company, Inc. 71 Examples of ‘Kinky’ Solutions -- the Perfect Complements Case (a) p 1 x 1 * + p 2 x 2 * = m; (b) x 2 * = ax 1 *. Substitution from (b) for x 2 * in (a) gives p 1 x 1 * + p 2 ax 1 * = m which gives

© 2010 W. W. Norton & Company, Inc. 72 Examples of ‘Kinky’ Solutions -- the Perfect Complements Case (a) p 1 x 1 * + p 2 x 2 * = m; (b) x 2 * = ax 1 *. Substitution from (b) for x 2 * in (a) gives p 1 x 1 * + p 2 ax 1 * = m which gives

© 2010 W. W. Norton & Company, Inc. 73 Examples of ‘Kinky’ Solutions -- the Perfect Complements Case (a) p 1 x 1 * + p 2 x 2 * = m; (b) x 2 * = ax 1 *. Substitution from (b) for x 2 * in (a) gives p 1 x 1 * + p 2 ax 1 * = m which gives A bundle of 1 commodity 1 unit and a commodity 2 units costs p 1 + ap 2 ; m/(p 1 + ap 2 ) such bundles are affordable.

© 2010 W. W. Norton & Company, Inc. 74 Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1x1 x2x2 U(x 1,x 2 ) = min{ax 1,x 2 } x 2 = ax 1