Rational Choice

CHOICE 1. Scarcity (income constraint) 2. Tastes (indifference map/utility function)

ECONOMIC RATIONALITY u The principal behavioral postulate is that a decision-maker 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/found?

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. E

RATIONAL CONSTRAINED CHOICE MRS=  x 2 /  x 1 = p 1 /p 2 Slope of the indifference curve Slope of the budget constraint Individual’s willingness to trade Society’s willingness to trade At Equilibrium E

RATIONAL CONSTRAINED CHOICE u The most preferred affordable bundle is called the consumer’s ORDINARY DEMAND at the given prices and income. 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 x1x1 x2x2 x1*x1* x2*x2* The slope of the indifference curve at (x 1 *,x 2 *) equals the slope of the budget constraint.

RATIONAL CONSTRAINED CHOICE (x 1 *,x 2 *) satisfies two conditions: u (i) the budget is exhausted, i.e. p 1 x 1 * + p 2 x 2 * = m; and u (ii) 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 DEMAND u How can this information be used to locate (x 1 *,x 2 *) for given p 1, p 2 and m? u Two ways to do this 1. Use Lagrange multiplier method 2. Find MRS and substitute into the Budget Constraint

COMPUTING DEMAND Lagrange Multiplier Method Suppose that the consumer has Cobb- Douglas preferences and a budget constraint given by

COMPUTING DEMAND Lagrange Multiplier Method Aim Set up the Lagrangian

COMPUTING DEMAND Lagrange Multiplier Method Differentiate

COMPUTING DEMAND Lagrange Multiplier Method From (1) and (2) Then re-arranging

COMPUTING DEMAND Lagrange Multiplier Method Rearrange Remember

COMPUTING DEMAND Lagrange Multiplier Method Substitute

COMPUTING DEMAND Lagrange Multiplier Method Solve x 1 * and x 2 * and

COMPUTING DEMAND Method 2 Suppose that the consumer has Cobb-Douglas preferences.

COMPUTING DEMAND Method 2 u Suppose that the consumer has Cobb-Douglas preferences.

COMPUTING DEMAND Method 2 u So the MRS is

COMPUTING DEMAND Method 2 u So the MRS is u At (x 1 *,x 2 *), MRS = -p 1 /p 2, i.e. the slope of the budget constraint.

COMPUTING DEMAND Method 2 u So the MRS is u At (x 1 *,x 2 *), MRS = -p 1 /p 2 so (A)

COMPUTING DEMAND Method 2 u (x 1 *,x 2 *) also exhausts the budget so (B)

COMPUTING DEMAND Method 2 u So now we know that (A) (B)

COMPUTING DEMAND Method 2 So now we know that (A) (B) Substitute

COMPUTING DEMAND Method 2 So now we know that (A) (B) Substitute and get This simplifies to ….

COMPUTING DEMAND Method 2

Substituting for x 1 * in then gives

COMPUTING DEMAND Method 2 So we have discovered that the most preferred affordable bundle for a consumer with Cobb-Douglas preferences is

COMPUTING DEMAND Method 2: Cobb-Douglas x1x1 x2x2

Rational Constrained Choice u But what if x 1 * = 0 or 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: Perfect Substitutes x1x1 x2x2 MRS = -1

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

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

Examples of Corner Solutions: Perfect Substitutes x1x1 x2x2 MRS = -1 (This is the indifference curve) Slope = -p 1 /p 2 with p 1 > p 2.

Examples of Corner Solutions: Perfect Substitutes x1x1 x2x2 MRS = -1 Slope = -p 1 /p 2 with p 1 < p 2. ANOTHER EXAMPLE

Examples of Corner Solutions: Perfect Substitutes So when U(x 1,x 2 ) = x 1 + x 2, the most preferred affordable bundle is (x 1 *,x 2 *) where or if p 1 < p 2 if p 1 > p 2.

Examples of Corner Solutions: Perfect Substitutes x1x1 x2x2 MRS = -1 Slope = -p 1 /p 2 with p 1 = p 2. The budget constraint and the utility curve lie on each other

Examples of Corner Solutions: Perfect Substitutes x1x1 x2x2 All the bundles in the constraint are equally the most preferred affordable when p 1 = p 2.

Examples of ‘Kinky’ Solutions: Perfect Complements X 1 (tonic) X 2 (gin) U(x 1,x 2 ) = min(ax 1,x 2 ) x 2 = ax 1 (a =.5)

Examples of ‘Kinky’ Solutions: Perfect Complements x1x1 x2x2 MRS = 0 U(x 1,x 2 ) = min(ax 1,x 2 ) x 2 = ax 1

Examples of ‘Kinky’ Solutions: Perfect Complements x1x1 x2x2 MRS = -  MRS = 0 U(x 1,x 2 ) = min(ax 1,x 2 ) x 2 = ax 1

Examples of ‘Kinky’ Solutions: Perfect Complements 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: Perfect Complements x1x1 x2x2 U(x 1,x 2 ) = min(ax 1,x 2 ) x 2 = ax 1

Examples of ‘Kinky’ Solutions: Perfect Complements 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: Perfect Complements 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: Perfect Complements x1x1 x2x2 U(x 1,x 2 ) = min(ax 1,x 2 ) x 2 = ax 1 x1*x1* x2*x2*

Examples of ‘Kinky’ Solutions: Perfect Complements x1x1 x2x2 U(x 1,x 2 ) = min(ax 1,x 2 ) x 2 = ax 1 x1*x1* x2*x2* and p 1 x 1 * + p 2 x 2 * = m

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

Examples of ‘Kinky’ Solutions: Perfect Complements (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: Perfect Complements (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: Perfect Complements (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: Perfect Complements (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: Perfect Complements x1x1 x2x2 U(x 1,x 2 ) = min(ax 1,x 2 ) x 2 = ax 1