#### Budget Constraint

Budget Constraint-Classical demand theory:

We’ve used the term individual and the technical term “choice set”. This was done in order to emphasize the very general nature of the theories of choice we are studying. However, microeconomics deals predominantly with choices made on markets for goods and services and it is therefore more convenient to use the words: “consumer” and “budget set” or “budget constraint”. The latter word emphasizes that the typical consumer is constrained in his/her choices of goods to consume by the available budget. By “budget” we may mean the monthly flow of money income that the individual receives from selling goods and services (usually mostly labor services), but more generally it may stand for the market value of all goods and services which the consumer has property rights to (i.e., has the legal right to dispose of, for example by exchanging them for other goods and services). In the latter case one often use the word “wealth”, which is a stock concept, instead of income (a “flow” concept). In most applications of the theory we will think about the flow of money income over a given time period as determining the consumer’s budget over that same period. If we now define Y as the money income over the observation period and if we look at the consumer’s choice of the quantity demanded of two goods (1 and 2), given prices p1 and p2, we write the consumer’s budget constraint as,

p1Q1 + p2Q2 ≤ Y

The left-hand side of this inequality shows the total expenditure on both goods during the time period. The inequality sign just says that our consumer cannot spend more than his/hers available money income. This rule out the (very real) possibility that the consumer can borrow. In Figure below the consumer’s budget set is drawn. This is the set of points, or commodity bundles (Q1, Q2) which satisfy the budget constraint, and there only non-negative quantities of the two goods are considered (Q1 ≥ 0, Q2 ≥ 0).The point A in the figure (the intercept on the vertical axis) is found by setting Q1 = 0 and solving the budget equation for Q2:

Q2max = Y/p2,

i.e., this shows the maximum quantity that the consumer can buy during the time period, given the constraint of not spending more than Y. The point B is then,

Q1max = Y/p1,

with an analogous interpretation. We can now view Q2 as a function of Q1, and derive an equation for the budget line:

Q2 = (Y/p2) – (p1/p2) . Q1

Note that the ratio p1/p2 is the relative price of good Q1 in terms of good Q2. We can interpret this ratio as the (external) exchange ratio, i.e., it shows at what rate the consumer can “buy” more units of good Q1, measured in units of good Q2. Another interpretation of this ratio is that it is the opportunity cost of increasing consumption by one unit more of good Q1. This can be expressed by considering a given increase in purchase of good 1, (ΔQ1) together with a decrease in the purchase of good 2, (ΔQ2), such that the budget equation still holds,

p1 (Q1 + ΔQ1) + p2 (Q2 + ΔQ2) = Y

Since, p1Q1 + p2Q2 = Y, it must be the case that,

p1ΔQ1 + p2ΔQ2 = 0,
and that,

(ΔQ2/ΔQ1) = - (p1/p2)

We should note a couple of things about our budget set. For one thing it may seem very restrictive to look only at two goods, even if it makes it easier to draw pictures. Well, in some situations it is restrictive, but in other situations it is not. If we’re primarily interested in studying how the quantity demanded of good 1 change as its own price changes, we may interpret Q2 as the quantity demanded of all other goods. The price of good 2 would then be a price index of all other goods, and the relative price ratio p1/p2, would be the price of good 1 in terms of all other goods. Another thing we should note is that if we multiply all prices and income by the same factor, the budget set will be unchanged. This is the same thing as to say that only relative price changes will affect the opportunities open to the consumer, and therefore their actual choices. E.g. (perfectly anticipated) inflation will not have any impact on real choice variables. A third thing to note is that our budget set often does not have the neat triangular shape, as it has in shown figure. For example, if the consumer gets a discount (i.e., a lower price) for quantities bought over a certain threshold, the relative price ratio from that point on will be lower than for quantities below the threshold, and the budget set will have a kink. The reverse case would be if consumption over a certain threshold is taxed at a higher rate than below the threshold. The budget line will then be steeper beyond the threshold. Yet another case occurs than there is rationing, and it is not possible (legal) to consume more than a certain number of units of a good per time period.

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