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Chapter Five Choice. Economic Rationality u The principal behavioral postulate is that a decisionmaker chooses its most preferred alternative from those.

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Presentation on theme: "Chapter Five Choice. Economic Rationality u The principal behavioral postulate is that a decisionmaker chooses its most preferred alternative from those."— Presentation transcript:

1 Chapter Five Choice

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?

3 Rational Constrained Choice x1x1 x2x2

4 x1x1 x2x2 Affordable bundles More preferred bundles

5 Rational Constrained Choice Affordable bundles x1x1 x2x2 More preferred bundles

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

7 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).

8 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.

9 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.

10 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.

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

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

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

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

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

16 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)

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

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

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

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

21 Computing Ordinary Demands - a Cobb-Douglas Example.

22 Substituting for x 1 * in then gives

23 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

24 Computing Ordinary Demands - a Cobb-Douglas Example. x1x1 x2x2

25 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 *).

26 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.

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

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

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

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

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

32 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.

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

34 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.

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

36 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

37 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

38 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

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

40 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?

41 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

42 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*

43 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

44 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 *

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

46 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

47 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

48 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

49 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.

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

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