Definition
A budget constraint is an accounting identity that describes the consumption options available to an agent with a limited income (or wealth) to allocate among various goods.(Econmodel, 2007) It is important to understand that the budget constraint is an accounting identity, not a behavioral relationship. An agent may well determine his behavior by considering his budget constraint, but his budget constraint is a given element of the problem he faces.
A budget constraint forces consumers to make choices. A budget constraint requires the individual to select a group or bundle of goods that is afordable. (Baye 123). The total amount a consumer wishes to purchase may not exceed the total amount of money the consumer has available to spend. The consumer must make choices on what is important to them and purchase accordingly. A combination of x and y goods may be purchased, but they may not exceed the individuals budget.

1. For an individual or household, the condition that income equals expenditure (in a static model), or that income minus expenditure equals the value of increased asset holdings (in a dynamic model). http://www-personal.umich.edu/~alandear/glossary/b.html

Wikipedia states that "a Budget Constraint represents the combinations of goods and services that a consumer can purchase given current prices and his income. Consumer theory uses the concepts of a budget constraint and a preference ordering to analyze consumer choices. Both concepts have a ready graphical representation in the two-good case."
Budget constraints can be altered when someone experiences a change in income or the product price changes. Increases in income give the consumer more purchasing power and they are able to buy more of a product, shifting the demand curve to the right for normal goods and the reverse for inferior goods. Decreases in income will shift the demand curve left for normal goods and shift the line to the right for inferior goods.

Part of budget constraints includes budget sets. A budget set has been defined as the bundles of goods a consumer can afford. Px and Py represent the prices of goods X and Y and M represents the consumer's income, which can be any amount. The mathematical expression is as follows:

(Px)(X) +(Py)(Y) is less than or equal to M.

This set defines the different combinations of goods X and Y that the consumer is able to purchase. If the entire income is spent on the two goods, equality occurs. This is called the budget line. The budget line is formally defined as the bundles of goods that exhaus a conusmer's income.

(Px)(X) + (Py)(Y) = M

This line defines any possible combination of X and Y used to exhaust the consumer's income.

The following graph was found on www.wikipedia.org and depicts the budget constraint line. labeled BC and three indifference curves for goods X and Y. Bayes defines an indifference curve as a curve that defines the combinations of two goods that give a consumer the same level of satisfaction. Using the budget constraint and indifference curves of a consumer, you can identify the bundle of goods that will satisfy the consumer without overextending the income available.
Income effect and price effect deal with how the change in price of a commodity changes the consumption of the good. The theory of consumer choice examines the trade-offs and decisions people make in their role as consumers as prices and their income changes.
You can find additional information on indifference curves from the following link: indifference curves

external image Consumer_constraint_choice.png


The slope of the budget line represents the Market Rate of Substitution, which is the rate at which one good may be traded for another in the market; slope of the budget line. (Bayes, p.125)

A change in the price of good X or Y or a change in consumer income will both affect the budget line.
A positive change in income will result in the consumer having greater purchasing power and the budget line will shift to the right. Conversly, a decrease in income will shift the budget line to the left.

A change in the price of good X or Y decreases, the budget line will rotate counterclockwise as more of the good can be purchased at the lower price, holding income constant. The reverse holds true when the price of good X or Y increases.
Opportunity Set and Budget Line
A consumer’s budget constraint depends on the prices of the goods she (he) wants to consume and the amount of money she (he) has to spend. Suppose;
PY=$2, PX=$1, and Y=$50


Source: Washington University

A person’s schedule of preferences, or family of indifference curves for various goods, is assumed to be independent of the opportunities he or she has foe expressing those preferences. But actual purchases depend upon the level of the level of in come and on relative prices, with the prices affecting the manner in which one allocates the income among different goods. Possible patterns of expenditures of a given sum of money on two goods may be illustrated on an indifference graph by using a budget constraint line. Such a line shows the various combinations of the two goods which can be purchased with a certain expenditure, given the prices of the two goods. To put in another way, the budget line separates what you can afford (opportunity set) from what you can’t. The budget line consists of all values of B and Z satisfying.
In mathematical form:
Y=PB*B+PZ*Z

Y is income in $, B quantity of good on vertical axis, Z is quantity of good on horizontal axis, and PB, PZ are their money prices. In previous example, Y=$50, PB=$2, and PZ=$1. The opportunity set consists of B and Z satisfying:

Y≥PB*B+PZ*Z

We assume consumers spend all of their income (saving is just another good). As known, more is better, so consumers will want to be on the budget line (to not throw away money).The slope of the budget line indicates how much more of the vertical axis good (B) the consumer must give up in order to purchase a unit of the horizontal axis good (Z).The absolute slope (i.e., ignoring the negative sign) of the budget line is the ratio PZ / PB or relative price. It is sometimes called the marginal rate of transformation

What happens to a consumers budget constraint when his (her) income changes or when the prices of the goods change can be explained now.
An increase (decrease) in the price of the good on the horizontal axis (Z) shifts the budget line in (out) while making it steeper (flatter). However, an increase (decrease) in money income shifts the budget line out (in) without changing its slope. To learn more about slope, the meaning of elasticity and how that can affect budget constraints, click on the following link: Slope and elasticity



Source: Washington University

Budget constraints show that “relative price” is what matters. Suppose: Income increases by 100%, the price of the good on the vertical axis (B) increases by 100%, the price of the good on the horizontal axis (Z) stays constant. The effect on the budget line is the same as if income and vertical axis good price unchanged while price of horizontal axis good falls 50%


Burritos
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Pizza
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2Y/PB
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Questions

1) The budget line when Y=$ 500, PX=$10 and PY=$20 is;

a) 500=$20X+$15Y
b) 500=$10X+$20Y
c) 500=$40X+$5Y
d) 500=$20X+$20Y

Answer: The consumer’s budget line is 500=$10X+$20Y (b)

2) The equation for consumer’s budget line when Y=$300, PX=6 and PY= 30 equals to;
a)Y=50-0.3X
b)Y=100-0.2X
c)Y=10-0.5X
d)Y=10-0.2X

Answer: The consumer’s budget line is $300 = $6X + $30Y. Rearranging terms and
solving for Y results in Y=10-0.2X (d)

3) Jack spends all his income on beef and pasta. He has $40 per week to spend on the two goods. The price of beef is $4 per pound and the price of pasta is $2 per pound. One possible graph of the budget constraint would have _ on the x axis and on the y axis.
    1. income; price of beef
    2. quantity of beef; price of beef
    3. price of beef; quantity of beef
    4. quantity of beef; income
    5. quantity of beef; quantity of pasta


Answer: Quantities of goods takes place on the axis of budget constraint. Hence,the answer is E.


The slope of the budget line is:

a.) the marginal rate of substitution
b.) the elasticity of demand
c.) the market rate of substitution
d.) none of the above.



References: