Budget Constraints ECO61 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 consumers expenditure on good X Budget constraint Assume a world with only two consumer goods, X and Y Total expenditure = PXX + PYY M is the consumers income or budget The consumer cannot spend more than her budget allows PXX + PYY M is the consumers budget constraint More-is-better implies budget exhaustion A rational consumer will spend every penny available. PXX + PYY M becomes PXX + PYY = M
Heres 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 PX X PY Y M PY Y M PX X M PX X Y PY M PX Y X PY PY
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 consumers income (M) increases, both maximums will increase by the same proportion M PX Y X PY PY
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 M PX Y X Y must decrease by PX/PY units PY PY This is at the heart of the consumers 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) M PZ Z M PZ B Z PB PB PB If pZ = $1, pB = $2, and M = $50, then: $50 ($1Z ) B 25 0.25Z $2 Possible Allocations of Lisas Budget Between Burritos and Pizza
Lisas budget is $50. Burritos are $2 each and pizzas are $1 each. B, Burritos per semester Budget Constraint: graph 25 = M/pB 20 a From previous slide we have Amount of Burritos consumed if all income that if: is allocated for pZ = $1, pB = $2, and M = $50, Burritos. then the budget constraint, L1,
is: b B L1 c 10 Opportunity set $50 ($1Z ) 25 0.25Z $2 Amount of Pizza consumed if all income is allocated for Pizza. d
0 10 30 50 = M/pZ 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: M PZ B Z PB PB
Slope = B/Z 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 pZ = $1 M PB B=
pZ = $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 PZ + TZ. Therefore, the effect is essentially the same as in the previous slide
Changes in the Budget Constraint: Increase in Income (M) $100 $50 PB B, Burritos per semester B= 50 M = $100 - PZ PB
Z If Lisas 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 consumers opportunity set change? Solved Problem Income in the budget constraint We have seen that the consumers budget is affected by her income (M) Therefore, the consumers choices (of X and Y)
are affected by her income But it has been implied that income (M) is not affected by the consumers choices (of X and Y) This is not always true: the consumers 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 (24w + M*) /PY Time constraint PY Y wH M * PY Y w (24 N ) M * Slope = -w/PY When w/PY decreases, the budget constraint rotates down PY Y 24w wN M * PY Y wN 24 w M * Consumption with nonlabor income (M*/PY)
0 N1 N2 24 N, Leisure hours per d ay 24 H1 H2 0
H, Work hours per d ay Leisure and consumption II: M = 0 * Y, Goods per d ay The price of leisure (N) is the wage (w) that is lost PY Y wH 24w /PY PY Y w (24 N ) PY Y 24w wN PY Y wN 24w
Slope = -w/PY When w/PY decreases, the budget constraint rotates down 24w/w = 24 0 N1 N2 24 N, Leisure hours per d ay 24
H1 H2 0 H, Work hours per d ay Y, Goods per d ay Progressive income tax 24w (1-0.20) /PY Slope = -w (1-0.20)/PY Y0 Slope = -w/PY Now we have a 20%
income tax, but only on income in excess of Y0. 24w/w = 24 0 N1 N2 24 N, Leisure hours per d ay 24
H1 H2 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
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