Budget Constraints ECO 61 Udayan Roy Fall 2008

Prices, quantities, and expenditures • PX is the price of good X – It is measured in dollars per unit of good X – The consumer pays this price no matter what quantity she buys. – That is, there are no quantity discounts and there is no rationing • X is also the quantity of good X that is purchased by the consumer – It is measured in units of good X per unit of time • PX X = PXX is the consumer’s expenditure on good X

Budget constraint • Assume a world with only two consumer goods, X and Y • Total expenditure = PXX + PYY • M is the consumer’s income or budget • The consumer cannot spend more than her budget allows • PXX + PYY ≤ M is the consumer’s budget constraint

“More-is-better” implies budget exhaustion • A rational consumer will spend every penny available. • PXX + PYY ≤ M becomes PXX + PYY = M • Here’s an example:

Saving • The budget constraint PXX + PYY = M does not imply that saving is being ignored. • We saw earlier that, to an economist, “food delivered now” and “food delivered in the future” are different goods • The former could be our good X and the latter could be good Y • Then PYY would represent saving for the future.

Saving • You may pay today for something that will be delivered at some date in the future. – For example, you may pay Long Island University today for courses you plan to take in 2015 – You may pay today to reserve hotel rooms in London for the 2012 Olympics – You may pay today for the future delivery of the National Geographic magazine • These purchases are the same as saving for the future.

Budget constraint algebra

Budget constraint algebra • If X = 0, then Y = M/PY – This is the maximum amount of good Y that the consumer can buy • Similarly, the maximum amount of good X that the consumer can buy is M/PX • If the consumer’s income (M) increases, both maximums will increase by the same proportion

Budget Constraint: Graph • PSS + PBB = M is the budget constraint • It can be graphed into the budget line:

Budget constraint algebra • If X increases by one unit, then Y must decrease by PX/PY units – This is at the heart of the consumer’s tradeoff – PX/PY is also called the relative price of good X (in units of good Y)

Budget Constraint • Consider Lisa, who buys only burritos (B) and pizza (Z) – If p. Z = $1, p. B = $2, and M = $50, then:

Possible Allocations of Lisa’s Budget Between Burritos and Pizza Lisa’s budget is $50. Burritos are $2 each and pizzas are $1 each.

B, Burritos per semester Budget Constraint: graph 25 = M/p. B 20 a Amount of Burritos From previous slide we have that consumed if all income if: is allocated for Burritos. – p = $1, p = $2, and M = $50, Z B then the budget constraint, L 1, is: b L 1 c 10 Opportunity set Amount of Pizza consumed if all income is allocated for Pizza. d 0 10 30 50 = M/p. Z Z, Pizzas per semester

The Slope of the Budget Constraint • We have seen that the budget constraint for Lisa is given by the following equation: Slope = DB/DZ – The slope of the budget line is the rate at which Lisa can trade burritos for pizza in the marketplace

B, Burritos per semester Changes in the Budget Constraint: An increase in the Price of Pizzas. Slope = -$1/$2 = -0. 5 25 p. Z = $1 M PB B= p. Z = $2 Slope = -$2/$2 = -1 PB Z If the price of Pizza doubles, (increases from $1 to $2) the slope of the budget line increases Loss 0 - PZ = $1 $2 25 50 Z, Pizzas per semester This area represents the bundles she can no longer afford

How taxes affect the budget constraint • A tax of TZ dollars per pizza has the effect of raising the price paid by the buyer from PZ to P Z + T Z. • Therefore, the effect is essentially the same as in the previous slide

Changes in the Budget Constraint: Increase in Income (M) B, Burritos per semester B= 50 M = $100 $50 PB - PZ PB Z If Lisa’s income increases by $50 the budget line shifts to the right (with the same slope!) 25 Gain M = $50 0 50 100 Z, Pizzas per semester This area represents the new consumption bundles she can now afford

Solved Problem • A government rations water, setting a quota on how much a consumer can purchase. • If a consumer can afford to buy 12 thousand gallons a month but the government restricts purchases to no more than 10 thousand gallons a month, how does the consumer’s opportunity set change?

Solved Problem

Income in the budget constraint • We have seen that the consumer’s budget is affected by her income (M) • Therefore, the consumer’s choices (of X and Y) are affected by her income • But it has been implied that income (M) is not affected by the consumer’s choices (of X and Y) • This is not always true: the consumer’s choices (of X and Y) may affect her income (M)

Income in the budget constraint • It is also implicit in my discussion of the budget constraint that income (M) is not affected by prices (of X and Y) • This is not always true: the prices of goods (PX and PY) may affect income (M)

Leisure and consumption • The price of leisure (N) is the wage (w) that is lost Y, Goods per d ay (24 w + M*) /PY Time constraint Slope = -w/PY When w/PY decreases, the budget constraint rotates down Consumption with nonlabor income (M*/PY) 0 N 1 N 2 24 N, Leisure hours per d ay 24 H 1 H 2 0 H, Work hours per d ay

Leisure and consumption II: M* = 0 • The price of leisure (N) is the wage (w) that is lost Y, Goods per d ay 24 w /PY Slope = -w/PY When w/PY decreases, the budget constraint rotates down 24 w/w = 24 0 N 1 N 2 24 N, Leisure hours per d ay 24 H 1 H 2 0 H, Work hours per d ay

Y, Goods per d ay Progressive income tax 24 w (1 -0. 20) /PY Slope = -w (1 -0. 20)/PY Y 0 Slope = -w/PY • Now we have a 20% income tax, but only on income in excess of Y 0. 24 w/w = 24 0 N 1 N 2 24 N, Leisure hours per d ay 24 H 1 H 2 0 H, Work hours per d ay

Income is affected by choices • Other examples where consumers’ choices affect their incomes – How much we save today will affect our future interest income – How much we spend today on which asset (stocks, bonds, college courses) will affect our future incomes