The Budget Constraint given Limited Income
The budget constraint is representative of one of the founding principles in economics i.e., that of scarcity. I mentioned earlier that some people are billionaires, and clearly fortunate enough to have seemingly unlimited budget constraints. This may be true, but the problem of scarcity is not entirely overcome by more or less unlimited purchasing power. Even billionaires have other constraints to deal with - most notably time.
Consumption of one good may negate consumption of another once time is added to the picture e.g., we may like to purchase an all expenses paid trip to Milan, but we cannot simultaneously enjoy a trip to Madrid.
Constraints abound in the real world, but in economics we focus on primarily on the budget constraint and limited income, because it is the one constraint that best helps us to explain consumer behavior.
Budget Line Formula
A useful way to remember the budget line is to regard as the line that separates that which is affordable and that which is not. With this in mind it is easy to remember that all points on the line infer that all income is being spent. This can be represented in a formulaic expression that includes all goods consumed rather than being restricted to the two-good model above:
I = A(Pa) + B(Pb) + C(Pc) + D(Pd) ...
This simple formula just adds up all of the spending on individual goods in order to derive the budget line formula. There can be as many or as little goods included, but it becomes impossible to represent this in a graphs. Three goods would require a three dimensional plane rather than a two dimensional line, and I've no clue what a graph with more than three dimensions would look like.
Budget Line Example (changing prices & income)
In the examples below I've illustrated what happens to the budget line when the price of a good decreases, and when the disposable income of a consumer increases.
Consumer spending and price changes
With a price change, the effect leads to a pivot of the budget line around the point at which it touches one axis or the other. In the example below I've illustrated what happens when the price of good x on the horizontal axis decreases from x to x'. At the lower price, the consumer finds that a greater quantity of this good can be purchased (up to the limit I/Px').
The effect of this is to pivot the budget line around its intersection point on the vertical axis to a new higher line (from BL to BL'). With a higher budget line, the consumer is able to afford baskets of goods on a higher indifference curve, and moves from a basket of x,y to x'y'.
Notice here that the lower cost of good x has actually led the consumer to increase his/her consumption of it so much that consumption of good y is reduced compared to the original basket. This is not forced to happen, but it is certainly possible that such things can happen depending on the types of goods involved. For details on this, see my article about the price consumption curve.
Income changes shift the Budget Line
In the next example I've shown how an increase in income leads to a higher budget line at all points, not just a pivot around one good. If prices are help constant, the increase in income will lead to a higher budget line that is parallel to the original line, as illustrated by the shift from BL to BL' in the graph below.
An increase in income will tend to lead to an increase in demand for both goods, as illustrated by the move from basket x,y on the lower indifference curve, to x'y' on the new higher indifference curve. Again, it is not guaranteed that demand for both goods will increase with higher income, it all depends on the types of goods under analysis. For details on this, see my article about the income consumption curve.
A Corner Solution
I mentioned above that there is a scenario in which the preferred basket of goods at the highest affordable indifference curve can, in extreme circumstances, occur where the MRS has a different slope to the budget line. One such scenario is illustrated below, it is called a corner solution.
With a corner solution, the consumer's preference is to spend all of his/her income one good only, and would like to consume even more of that good if it were affordable. This leads to a situation in which the indifference curve passes through the highest achievable quantity of the preferred good, rather than just touching that point.
This happens in the graph above for good x, and the slope of the MRS line is greater than the slope of the budget line in this situation. If you recall that the slope of the budget line is given by -Px/Py, this corner solution arises where MRS > -Px/Py.
It is also possible that a corner solution could occur on the other axis, with an indifference curve that is flatter than the budget line. In that case MRS < -Px/Py.