Spending and Income

• Suppose Cathy consumes two goods 1 and 2.

– Quantity of good 1 consumed is x1

– Quantity of good 2 consumed is x2

Budgets –Let p1 and p2 denote the prices of good 1 and good 2, respectively –Let m be Cathy’s (money) income ECON 370: Microeconomic Theory • The amount she spends is p1x1 + p2x2 Summer 2004 – Rice University • Ignoring the possibility of borrowing, she cannot Stanley Gilbert spends more than her income – That is: p1x1 + p2x2 ≤ m

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The Budget Line Budget Constraint: Intercepts

• The line p1x1 + p2x2 = m is often referred to as the budget line. x2 – It shows the maximum possible amounts that can be spent on the two goods. Budget constraint is m /p2 • The Feasible Set is the set of all affordable p1x1 + p2x2 = m consumption bundles Intercepts are m/p1 and m/p2 – That is all bundles (x1, x2) such that p1x1 + p2x2 ≤ m

– And x1 ≥ 0 and x2 ≥ 0.

m /p1 x1

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1 Budget Constraint for Two Goods Budget Set, Constraint for Two Goods

x2 x2 Budget constraint is Budget constraint is m /p2 m /p2 p1x1 + p2x2 = m. p x + p x = m 1 1 2 2 Not affordable Just affordable Affordable w/ cash left

m /p1 x1 m /p1 x1

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Budget Set, Constraint for Two Goods Budget Constraint for Three Goods

x2 Budget constraint is x2 p x + p x + p x = m m / p2 1 1 2 2 3 3 m /p2 p1x1 + p2x2 = m.

Collection of all m / p affordable bundles. 3 x Budget 3 Set m / p x 1 m /p1 1 x1

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2 Slope of the Budget Constraint Changes in Budgets

• What happens if: Since p1x1 + p2x2 = m – Income increases m p1 Then x2 = − x1 – Income decreases p2 p2 – Price of good 1 increases dx p Or 2 = − 1 – Price of good 2 decreases dx1 p2 – All prices and income increase by 10% • We can interpret this as the opportunity cost of a good

• If I want more of good x2, I must give up (p1 / p2) units of good x1 to get it.

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Composite Commodities Composite Commodities (cont)

• We can only conveniently analyze two goods in a budget •Let x1 represent loaves of bread set diagram. •Let x2 represent dollars spent on everything else • In practice people consume a wide variety of goods. • As before, we have p1x1 + p2x2 = m • Often we are interested in describing how some change in • Dividing through by p we get: price or income affects the amount of one good that can be 2 –(p / p )x + x = (m / p ) purchased e.g. loaves of bread. 1 2 1 2 2 • One conclusion of this is that we can define prices based • To consider this case, it is often convenient to treat all on any numeraire we want so long as we are consistent goods other than the good that is of interest as a single about it composite commodity whose quantity is measured in dollars. • Also note that measuring the amount of other goods in dollars is valid only if the relative prices of these other goods is not changing

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3 Non-Linear Budget Constraints The Food Stamp Program

• Not all budget constraints are linear • Popular income support program • People may be prohibited from buying all of a • Coupons given to poor (used to be sold) good they can afford • Can be legally exchanged only for food • Prices may (and often do) vary depending on • Popular with some donors quantity purchased • Popular with agricultural interests • Example: the Food Stamp Program

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The Food Stamp Program The Food Stamp Program

• What is effect on budget constraint? F + G = 100: before stamps. – Suppose m = $100

– Price of food: pF = $1 G

– Price of “all other goods”: pG = $1 – Other goods is “numeraire” good 100 – Budget constraint is F + G = 100 – Key factor: Income available for “other goods” does not change with receipt of food stamps – Suppose receive food stamps for $40 worth of food

100 F

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4 The Food Stamp Program The Food Stamp Program

F + G = 100: before stamps. • If food stamp program is generous, families may be at “kink” of budget set G Budget set after 40 food • What if food stamps can be traded on a black 100 stamps issued. market for $0.50 each? Welfare up since budget set is enlarged

40 100 140 F

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The Food Stamp Program Quantity-Based Prices

• Price may be a function of quantity – Quantity discounts for large buyers G Black market trading – Penalties for buying “too much” expands budget set further 100 • Budget constraints “kinked” where p changes Budget constraint with • Suppose quantity discount: black market trading – p2 constant at $1 and m = 100

– p1 = $2 for 0 ≤ x1 ≤ 20; -p1 / p2 = -2

– p1 = $1 for x1 > 20; -p1 / p2 = -1 – What does the budget set look like? 40 100 140 F

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5 Budget Constraints w/ Quantity Discount Quantity Restrictions

• Suppose as follows: – p = $2 x2 m = $100 1 – p = $1 100 Slope = - 2 20 units @ p=2 2 60 units @ p=1 – m = 100 – But not allowed to buy more than $25 of good 1 • What does the budget set look like? Slope = - 1

20 50 80 x1

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More General Choice Sets Example

• Other constraints on choices • Instead of budgeting Money, let’s budget time – Time constraints (labor choice) – Erika is choosing how much time to work per week – Other resource constraints – If she works, she earns a wage of $25 / hour – In addition, she has $250 in non-wage income • “Available” bundle must meet all relevant constraints simultaneously • What would this budget set look like? • Budget set is intersection of each set formed by each separate constraint

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