Chapter 5 Choice
Economic Rationality The principal behavioral postulate is that a decisionmaker chooses its most preferred alternative from those available to it. The available choices constitute the choice set. How is the most preferred bundle in the choice set located?
Rational Constrained Choice
Rational Constrained Choice Utility
Rational Constrained Choice Utility
Rational Constrained Choice Utility
Rational Constrained Choice Utility
Rational Constrained Choice Utility
Rational Constrained Choice Utility
Rational Constrained Choice Utility
Rational Constrained Choice Utility Affordable, but not the most preferred affordable bundle.
Rational Constrained Choice Utility The most preferred of the affordable bundles. Affordable, but not the most preferred affordable bundle.
Rational Constrained Choice Utility
Rational Constrained Choice Utility
Rational Constrained Choice Utility
Rational Constrained Choice Utility
Rational Constrained Choice
Rational Constrained Choice Affordable bundles
Rational Constrained Choice Affordable bundles
Rational Constrained Choice More preferred bundles Affordable bundles
Rational Constrained Choice More preferred bundles Affordable bundles
Rational Constrained Choice * *
Rational Constrained Choice ( *, *)isthemost preferred affordable bundle. * *
Rational Constrained Choice The most preferred affordable bundle is called the consumer ss ORDINARY DEMAND at the given prices and budget. Ordinary demands will be denoted by *(p 1,p 2,m) and *(p 1,p 2,m).
Rational Constrained Choice When * > 0 and * > 0 the demanded bundle is INTERIOR. If buying ( *, *) costs $m then the budget is exhausted.
Rational Constrained Choice ( *, *) is interior. ( *, *) exhausts the budget. * *
Rational Constrained Choice ( *, *) is interior. (a) ( *, *) exhausts the budget; p 1 *+p 2 *=m m. * *
Rational Constrained Choice * ( *, *) is interior. (b) The slope of the indiff. curve at ( *x*) *, *) equals the slope of the budget constraint. *
Rational Constrained Choice ( *, *) satisfies two conditions: (a) the budget is exhausted; p 1 *+p 2 *=m (b) the slope of the budget constraint, -p 1 /p 2, and the slope of the indifference curve containing ( *, *) are equal at ( *, *).
Computing Ordinary Demands How can this information be used to locate ( *, *) for given p 1, p 2 and m?
Computing Ordinary Demands - a Cobb-Douglas Example. Suppose that the consumer has Cobb-Douglas preferences. 1 2 1 a b 2 Ux (, x ) x x
Computing Ordinary Demands - a Cobb-Douglas Example. Suppose that the consumer has Cobb-Douglas preferences. Then 1 2 1 a b 2 Ux (, x ) x x MU U ax a x x b 1 1 1 2 1 MU 2 U x 2 1 a b 2 1 bx x
Computing Ordinary Demands - a Cobb-Douglas Example. So the MRS is a b 2 1 1 1 2 2 dx U/ x ax x ax MRS dx U / x a b bx x bx 1 2 1 2 1 1.
Computing Ordinary Demands - a Cobb-Douglas Example. So the MRS is a b 2 1 1 1 2 2 dx U/ x ax x ax MRS dx U / x a b bx x bx 1 2 1 2 1 1. At ( *, *), MRS = -p 1 /p 2 so
Computing Ordinary Demands - a Cobb-Douglas Example. So the MRS is a b 2 1 1 1 2 2 dx U/ x ax x ax MRS dx U / x a b bx x bx 1 2 1 2 1 1. At ( *, *), MRS = -p 1 /p 2 so * ax 2 p 1 * bp x 1 * 2 ap x *. (A) bx p 1 1 2 2
Computing Ordinary Demands - a Cobb-Douglas Example. ( *, *) also exhausts the budget so * * 1 1 2 2 px px m. (B)
Computing Ordinary Demands - a Cobb-Douglas Example. So now we know that bp x 2 1 ap x * * 2 1 * * 1 1 2 2 (A) px px m. (B)
Computing Ordinary Demands - a Cobb-Douglas Example. So now we know that * bp x 1 2 ap x * 2 1 (A) Substitute * * p 1 p 2 m. (B)
Computing Ordinary Demands - a Cobb-Douglas Example. So now we know that * bp x 1 2 ap x * 2 1 (A) Substitute * * p 1 p 2 m. (B) and get px p bp 1 * 1 1 2 x * 1 m. ap 2 This simplifies to.
Computing Ordinary Demands - a Cobb-Douglas Example. x * 1 am ( a b ) p 1.
Computing Ordinary Demands - a Cobb-Douglas Example. x * 1 am ( a b ) p 1. Substituting for * in then gives * * 1 1 2 2 px px m x * 2 bm ( a b) p 2.
Computing Ordinary Demands - a Cobb-Douglas Example. So we have discovered that the most preferred affordable bundle for a consumer with Cobb-Douglas preferences a 1 2 1 b 2 Ux (, x ) x x is * * ( ) ( am bm x 1, x 2 ),. ( a b) p ( a b ) p 1 2
Computing Ordinary Demands - a Cobb-Douglas Example. U ( x, x ) x x 1 2 1 a b 2 * bm ( a b ) p 2 x x * am 1 ( a b) p 1
Rational Constrained Choice When * > 0 and * > 0 and ( *, *) exhausts the budget, and indifference curves have no kinks, the ordinary demands are obtained by solving: (a) p 1 * + p 2 * = y (b) the slopes of the budget constraint, -p 1 /p 2, and of the indifference curve containing ( *, *) are equal at ( *, *).
Rational Constrained Choice But what if * = 0? Or if *=0? If either * = 0 or * = 0 then the ordinary demand ( *, *) is at a corner solution to the problem of maximizing utility subject to a budget constraint.
Examples of Corner Solutions -- the Perfect Substitutes Case MRS =-1
Examples of Corner Solutions -- the Perfect Substitutes Case MRS =-1 Slope = -p 1 /p 2 with p 1 >p 2.
Examples of Corner Solutions -- the Perfect Substitutes Case MRS =-1 Slope = -p 1 /p 2 with p 1 >p 2.
Examples of Corner Solutions -- the Perfect Substitutes Case x y MRS =-1 * 2 p 2 Slope = -p 1 /p 2 with p 1 >p 2. * 0
Examples of Corner Solutions -- the Perfect Substitutes Case MRS =-1 Slope = -p 1 /p 2 with p 1 <p 2. * 0 x * 1 y p 1
Examples of Corner Solutions -- the Perfect Substitutes Case So when U(, )= +x 2, the most preferred affordable bundle is ( *, *) where * * y (x 1,x 2 ),0 if p 1 <p 2 p1 and * * y ( x1,x 2 ) 0, if p 1 >p 2. p2
Examples of Corner Solutions -- the Perfect Substitutes Case MRS =-1 y p 2 Slope = -p 1 /p 2 with p 1 = p 2. y p 1
Examples of Corner Solutions -- the Perfect Substitutes Case y p 2 All the bundles in the constraint are equally the most preferred affordable when p 1 = p 2. y p 1
Examples of Corner Solutions -- the Non-Convex Preferences Case
Examples of Corner Solutions -- the Non-Convex Preferences Case
Examples of Corner Solutions -- the Non-Convex Preferences Case Which is the most preferred affordable bundle?
Examples of Corner Solutions -- the Non-Convex Preferences Case The most preferred affordable bundle
Examples of Corner Solutions -- the Non-Convex Preferences Case Notice that t the tangency solution is not the most preferred affordable bundle. The most preferred affordable bundle
Examples of Kinky Solutions - - the Perfect Complements Case U(, ) = min{a, } = a
Examples of Kinky Solutions - - the Perfect Complements Case U(, ) = min{a, } = a MRS = 0
Examples of Kinky Solutions - - the Perfect Complements Case U(, ) = min{a, } MRS = - = a MRS = 0
Examples of Kinky Solutions - - the Perfect Complements Case U(, ) = min{a, } MRS = - MRS is undefined = a MRS = 0
Examples of Kinky Solutions - - the Perfect Complements Case U(, ) = min{a, } = a
Examples of Kinky Solutions - - the Perfect Complements Case U(, ) = min{a, } Which is the most preferred affordable bundle? = a
Examples of Kinky Solutions - - the Perfect Complements Case U(, ) = min{a, } The most preferred affordable bundle = a
Examples of Kinky Solutions - - the Perfect Complements Case U(, ) = min{a, } * = a *
Examples of Kinky Solutions - - the Perfect Complements Case U(, ) = min{a, } (a) p 1 *+p 2 *=m * = a *
Examples of Kinky Solutions - - the Perfect Complements Case U(, ) = min{a, } (a) p 1 *+p 2 *=m (b) * = a * * = a *
Examples of Kinky Solutions - - the Perfect Complements Case (a) p 1 * + p 2 * = m; (b) * = a *.
Examples of Kinky Solutions - - the Perfect Complements Case (a) p 1 * + p 2 * = m; (b) * = a *. Substitution from (b) for * in (a) gives p 1 *+p 2 a *=m
Examples of Kinky Solutions - - the Perfect Complements Case (a) p 1 * + p 2 * = m; (b) * = a *. Substitution from (b) for * in (a) gives p 1 *+p 2 a *=m which gives * m p ap 1 2
Examples of Kinky Solutions - - the Perfect Complements Case (a) p 1 * + p 2 * = m; (b) * = a *. Substitution from (b) for * in (a) gives p 1 *+p 2 a *=m which gives m x * * am 1 ; x 2 p1 ap2 p1 ap2.
Examples of Kinky Solutions - - the Perfect Complements Case (a) p 1 * + p 2 * = m; (b) * = a *. Substitution from (b) for * in (a) gives p 1 *+p 2 a *=m which gives m x * * am 1 ; x 2 p1 ap2 p1 ap2 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 U(, ) = min{a, } * am p ap 1 2 x m ap = a * 1 1 p1 2