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Chapter Two Budgetary and Other Constraints on Choice

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Consumption Choice Sets u A consumption choice set is the collection of all consumption choices available to the consumer. u What constrains consumption choice? –Budgetary, time and other resource limitations.

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Budget Constraints u A consumption bundle containing x 1 units of commodity 1, x 2 units of commodity 2 and so on up to x n units of commodity n is denoted by the vector (x 1, x 2, …, x n ). u Commodity prices are p 1, p 2, …, p n.

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Budget Constraints u Q: When is a consumption bundle (x 1, …, x n ) affordable at given prices p 1, …, p n ?

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Budget Constraints u Q: When is a bundle (x 1, …, x n ) affordable at prices p 1, …, p n ? A: When p 1 x 1 + … + p n x n m where m is the consumers (disposable) income.

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Budget Constraints The bundles that are only just affordable form the consumers budget constraint. This is the set { (x 1,…,x n ) | x 1 0, …, x n and p 1 x 1 + … + p n x n m }.

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Budget Constraints The consumers budget set is the set of all affordable bundles; B(p 1, …, p n, m) = { (x 1, …, x n ) | x 1 0, …, x n 0 and p 1 x 1 + … + p n x n m } u The budget constraint is the upper boundary of the budget set.

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Budget Set and Constraint for Two Commodities x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 1 m /p 2

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Budget Set and Constraint for Two Commodities x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 2 m /p 1

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Budget Set and Constraint for Two Commodities x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 1 Just affordable m /p 2

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Budget Set and Constraint for Two Commodities x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 1 Just affordable Not affordable m /p 2

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Budget Set and Constraint for Two Commodities x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 1 Affordable Just affordable Not affordable m /p 2

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Budget Set and Constraint for Two Commodities x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 1 Budget Set the collection of all affordable bundles. m /p 2

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Budget Set and Constraint for Two Commodities x2x2 x1x1 p 1 x 1 + p 2 x 2 = m is x 2 = -(p 1 /p 2 )x 1 + m/p 2 so slope is -p 1 /p 2. m /p 1 Budget Set m /p 2

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Budget Constraints u For n = 2 and x 1 on the horizontal axis, the constraints slope is -p 1 /p 2. What does it mean?

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Budget Constraints u For n = 2 and x 1 on the horizontal axis, the constraints slope is -p 1 /p 2. What does it mean? u Increasing x 1 by 1 must reduce x 2 by p 1 /p 2.

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Budget Constraints x2x2 x1x1 Slope is -p 1 /p p 1 /p 2

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Budget Constraints x2x2 x1x1 +1 -p 1 /p 2 Opp. cost of an extra unit of commodity 1 is p 1 /p 2 units foregone of commodity 2.

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Budget Constraints x2x2 x1x1 Opp. cost of an extra unit of commodity 1 is p 1 /p 2 units foregone of commodity 2. And the opp. cost of an extra unit of commodity 2 is p 2 /p 1 units foregone of commodity 1. -p 2 /p 1 +1

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Budget Sets & Constraints; Income and Price Changes u The budget constraint and budget set depend upon prices and income. What happens as prices or income change?

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How do the budget set and budget constraint change as income m increases? Original budget set x2x2 x1x1

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Higher income gives more choice Original budget set New affordable consumption choices x2x2 x1x1 Original and new budget constraints are parallel (same slope).

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How do the budget set and budget constraint change as income m decreases? Original budget set x2x2 x1x1

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How do the budget set and budget constraint change as income m decreases? x2x2 x1x1 New, smaller budget set Consumption bundles that are no longer affordable. Old and new constraints are parallel.

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Budget Constraints - Income Changes u Increases in income m shift the constraint outward in a parallel manner, thereby enlarging the budget set and improving choice.

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Budget Constraints - Income Changes u Increases in income m shift the constraint outward in a parallel manner, thereby enlarging the budget set and improving choice. u Decreases in income m shift the constraint inward in a parallel manner, thereby shrinking the budget set and reducing choice.

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Budget Constraints - Income Changes u No original choice is lost and new choices are added when income increases, so higher income cannot make a consumer worse off. u An income decrease may (typically will) make the consumer worse off.

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Budget Constraints - Price Changes u What happens if just one price decreases? u Suppose p 1 decreases.

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How do the budget set and budget constraint change as p 1 decreases from p 1 to p 1? Original budget set x2x2 x1x1 m/p 2 m/p 1 -p 1 /p 2

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How do the budget set and budget constraint change as p 1 decreases from p 1 to p 1? Original budget set x2x2 x1x1 m/p 2 m/p 1 New affordable choices -p 1 /p 2

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How do the budget set and budget constraint change as p 1 decreases from p 1 to p 1? Original budget set x2x2 x1x1 m/p 2 m/p 1 New affordable choices Budget constraint pivots; slope flattens from -p 1 /p 2 to -p 1 /p 2

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Budget Constraints - Price Changes u Reducing the price of one commodity pivots the constraint outward. No old choice is lost and new choices are added, so reducing one price cannot make the consumer worse off.

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Budget Constraints - Price Changes u Similarly, increasing one price pivots the constraint inwards, reduces choice and may (typically will) make the consumer worse off.

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Shapes of Budget Constraints u Q: What makes a budget constraint a straight line? u A: A straight line has a constant slope and the constraint is p 1 x 1 + … + p n x n = m so if prices are constants then a constraint is a straight line.

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