Chapter Five Choice

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?

Rational Constrained Choice x1x1 x2x2

x1x1 x2x2 Utility

Rational Constrained Choice Utility x2x2 x1x1

Rational Constrained Choice x1x1 x2x2 Utility

Rational Constrained Choice Utility x1x1 x2x2

Rational Constrained Choice Utility x1x1 x2x2

Rational Constrained Choice Utility x1x1 x2x2

Rational Constrained Choice Utility x1x1 x2x2

Rational Constrained Choice Utility x1x1 x2x2 Affordable, but not the most preferred affordable bundle.

Rational Constrained Choice x1x1 x2x2 Utility Affordable, but not the most preferred affordable bundle. The most preferred of the affordable bundles.

Rational Constrained Choice x1x1 x2x2 Utility

Rational Constrained Choice Utility x1x1 x2x2

Rational Constrained Choice Utility x1x1 x2x2

Rational Constrained Choice Utility x1x1 x2x2

Rational Constrained Choice x1x1 x2x2

x1x1 x2x2 Affordable bundles

Rational Constrained Choice x1x1 x2x2 Affordable bundles

Rational Constrained Choice x1x1 x2x2 Affordable bundles More preferred bundles

Rational Constrained Choice Affordable bundles x1x1 x2x2 More preferred bundles

Rational Constrained Choice x1x1 x2x2 x1*x1* x2*x2*

x1x1 x2x2 x1*x1* x2*x2* (x 1 *,x 2 *) is the most preferred affordable bundle.

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).

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.

Rational Constrained Choice x1x1 x2x2 x1*x1* x2*x2* (x 1 *,x 2 *) is interior. (x 1 *,x 2 *) exhausts the budget.

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.

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.

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 *).

Computing Ordinary Demands u How can this information be used to locate (x 1 *,x 2 *) for given p 1, p 2 and m?

Computing Ordinary Demands - a Cobb-Douglas Example. u Suppose that the consumer has Cobb-Douglas preferences.

Computing Ordinary Demands - a Cobb-Douglas Example. u Suppose that the consumer has Cobb-Douglas preferences. u Then

Computing Ordinary Demands - a Cobb-Douglas Example. u So the MRS is

Computing Ordinary Demands - a Cobb-Douglas Example. u So the MRS is u At (x 1 *,x 2 *), MRS = -p 1 /p 2 so

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)

Computing Ordinary Demands - a Cobb-Douglas Example. u (x 1 *,x 2 *) also exhausts the budget so (B)

Computing Ordinary Demands - a Cobb-Douglas Example. u So now we know that (A) (B)

Computing Ordinary Demands - a Cobb-Douglas Example. u So now we know that (A) (B) Substitute

Computing Ordinary Demands - a Cobb-Douglas Example. u So now we know that (A) (B) Substitute and get This simplifies to ….

Computing Ordinary Demands - a Cobb-Douglas Example.

Substituting for x 1 * 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. x1x1 x2x2

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 *).

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.

Examples of Corner Solutions -- the Perfect Substitutes Case x1x1 x2x2 MRS = -1

Examples of Corner Solutions -- the Perfect Substitutes Case x1x1 x2x2 MRS = -1 Slope = -p 1 /p 2 with p 1 > p 2.

Examples of Corner Solutions -- the Perfect Substitutes Case x1x1 x2x2 MRS = -1 Slope = -p 1 /p 2 with p 1 > p 2.

Examples of Corner Solutions -- the Perfect Substitutes Case x1x1 x2x2 MRS = -1 Slope = -p 1 /p 2 with p 1 > p 2.

Examples of Corner Solutions -- the Perfect Substitutes Case x1x1 x2x2 MRS = -1 Slope = -p 1 /p 2 with p 1 < p 2.

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.

Examples of Corner Solutions -- the Perfect Substitutes Case x1x1 x2x2 MRS = -1 Slope = -p 1 /p 2 with p 1 = p 2.

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.

Examples of Corner Solutions -- the Non-Convex Preferences Case x1x1 x2x2 Better

Examples of Corner Solutions -- the Non-Convex Preferences Case x1x1 x2x2

x1x1 x2x2 Which is the most preferred affordable bundle?

Examples of Corner Solutions -- the Non-Convex Preferences Case x1x1 x2x2 The most preferred affordable bundle

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.

Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1x1 x2x2 U(x 1,x 2 ) = min{ax 1,x 2 } x 2 = ax 1

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

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

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

Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1x1 x2x2 U(x 1,x 2 ) = min{ax 1,x 2 } x 2 = ax 1

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?

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

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*

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

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 *

Examples of ‘Kinky’ Solutions -- the Perfect Complements Case (a) p 1 x 1 * + p 2 x 2 * = m; (b) x 2 * = ax 1 *.

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

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

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

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.

Examples of ‘Kinky’ Solutions -- the Perfect Complements Case x1x1 x2x2 U(x 1,x 2 ) = min{ax 1,x 2 } x 2 = ax 1