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Choice

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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.

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Economic Rationality u How is the most preferred bundle in the choice set located? u The answer is simple: the consumer chooses the bundle within the choice set that is on the indifference curve that provides the highest level of utility.

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

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

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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.

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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

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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

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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 *).

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Rational Constrained Choice u For well-behaved (monotonic and convex) preferences, tangency is a necessary and sufficient condition for optimality. u At the optimum the consumer substitutes one good for the other at a rate identical to the market’s.

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

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

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

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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)

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

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

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

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

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Computing Ordinary Demands - a Cobb-Douglas Example.

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Substituting for x 1 * in then gives

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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

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

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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 * = m 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 *).

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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.

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

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

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

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

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

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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

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

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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

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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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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

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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?

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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

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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

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Application: choosing a tax u Which is less harmful, a quantity tax or an income tax, both allowing the same revenue? 1. We can show that an income tax is always better in the sense that given any quantity tax, there is an income tax that leaves the consumer at a higher indifference curve.

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Application: choosing a tax 2. Outline of the argument: a) original budget constraint: p 1 x 1 + p 2 x 2 = m b) budget constraint with quantity tax: (p 1 + t)x 1 +p 2 x 2 = m c) optimal choice with tax: (p 1 +t)x 1 * + p 2 x 2 * = m d) revenue raised is tx 1 *

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Application: choosing a tax e) income tax (lump-sum) that raises the same amount of revenue leads to budget constraint: p 1 x 1 + p 2 x 2 = m - tx 1 * i) this line has the same slope as the original budget line, but lies inwards

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Application: choosing a tax ii) also passes through (x1*;x2*) iii) proof: p 1 x 1 *+ p 2 x 2 * = m - tx 1 * iv) this means that (x 1 *; x 2 * ) is affordable under the income tax, so the optimal choice under the income tax must be even better than (x 1 *; x 2 *)

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