Using indifference curves and an assumption of constant prices and a fixed income in a two good world will give the following diagram. The consumer can choose any point on or below the budget constraint line BC. This line is diagonal since it comes from the equation <math>X \times px + Y \times py <= Income</math>. In other words, the amount spent on both goods together is less than or equal to the income of the consumer. The consumer will choose the indifference curve with the highest utility that is within the budget constraint. I3 has all the points outside of their budget constraint so the best that they can do is I2. This will result in them purchasing X* of good X and Y* of good Y.
More usefully, this can now be used to predict the effect of various shifts in the constraint. The below graphic shows the effect of a price shift for good y. If the price of Y increases from where it is at BC2, the budget constraint will shift to BC1. Notice that since the price of X does not change, the consumer can still buy the same amount of X if they only choose to buy good X. On the other hand, if they choose to buy only good Y, they will be able to buy less of good Y since it's price increased. This causes the amount of good Y bought to shift from Y2 to Y3, and the amount of good X bought to shift from X2 to Y3. Similar things happen with the shift from BC2 to BC3.
If this shifts are repeated with many different prices for good Y, a demand curve for good Y can be constructed. If the price for good Y is fixed and the price for good X is varied, a demand curve for good X can be constructed. The below diagram shows this for good y.
Another important item that can change is the income of the consumer. As long as the prices remain constant, changing the income will create a parellel shift of the budget constraint. Increasing the income will shift the budget constraint right since more of both can be bought, and decreasing income will shift it left.
Depending on the indifference curves the amount of a good bought can either increase, decrease or stay the same when income increases. In the diagram below, good Y is a normal good since the amount purchased increased as the budget constraint shifted from BC1 to the higher income BC2. Good X is an inferior good since the amount bought decreased as the income increases.
Every price change can be converted into a income effect and a substition effect. The substition effect is basically a price change that changes the slope of the budget constraint, but leaves the consumer on the same indifference curve. This effect will always cause the consumer to substitute away from the good that is becoming comparatively more expensive. If the good in question is a normal good, than the income effect will re-enforce the substition effect. If the good is inferior, then the income effect will lessen the substition effect. If the income effect is opposite and stronger than the substition effect, the consumer will buy more of the good when it becomes more expensive. There is no generally agreed upon example of this happening, known as a Giffen good[?].
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