### 8.4 Budget optimum

So, which bundle of goods will the consumer choose? On the one hand, it should lie on the highest possible indifference curve, but on the other hand it must be ensured that the consumer can afford this bundle of goods. Therefore, it should be on or below the budget line. The highest achievable indifference curve is the one that just touches, or mathematically speaking, is tangent to the budget line. Let us illustrate this with the help of the graph: If we move point A from the origin towards the budget line, we cross several indifference curves (red lines), but as long as we are still below the budget line, there are still affordable points on higher lying indifference curves. The green shining indifference curve is the one maximally attainable. The optimum point is highlighted green. If we move from this point to the right or left along the budget line, we get to lower lying indifference curves, whereas if we move further away from the origin to higher lying indifference curves, these are no longer affordable with the given income.
The second switch shows or hides the optimal bundle of consumption. The first switch displays both the indifference curve in point A and the marginal rate of substitution in the form of the tangent.
In the optimum case, the slope of the budget line corresponds exactly to the slope of the indifference curve, i.e. it is equal to the marginal rate of substitution (the slope of a curve at a certain point is defined by the tangent touching there). This means that the price ratio of the two goods on the market (= relative price) corresponds exactly to the ratio at which the consumer is willing to exchange Good 2 for Good 1 (=MRS). Hence, in the optimum case, the market and the consumer household value the two goods equally in relation to each other. The relative value that the good has for the consumer is exactly reflected by the relative price of this good.
The budget line, and thus the relative price ratio, can be modified by shifting the axes intersection points.
To illustrate this with an example, we show the marginal rate of substitution and move point A on the budget line to the upper left, so that the MRS is approximately 4. In this case, the marginal rate of substitution is greater than the price ratio given by the market, which is 1 in the initial scenario (slope of the above budget line is -1).
How can we illustrate the fact that in the optimum case, MRS must be equal to the price ratio? At the chosen consumption point A, the household would be prepared to exchange 4 units of Good 2 for 1 unit of Good 1 without any loss of benefit, and the resulting new combination of goods would be just as satisfying. On the market, however, he would even get 4 units of Good 1 for 4 units of Good 2 . Consequently, he has an incentive to actually improve his position by exchanging 4 units of Good 2 for 4 of Good 1 and thus to choose a different combination of Good 2 and Good 1 , i.e. to move point A. Therefore, the consumer will not choose this combination of goods where his MRS is 4. If one continues this thought experiment, one inevitably comes to the conclusion that the household would never choose a consumption bundle where its MRS is unequal to the price ratio. Otherwise, the market would always give him an incentive to swap one good for another. Only when MRS and price ratio are equal there is no reason for the household to change the ratio of Good 1 to Good 2 by exchanging them on the market. So, we now know that the household will only choose bundles of goods where its marginal rate of substitution is equal to the price ratio. In the graph, this is where the tangent to the indifference curve at point A is parallel to the budget line. Since we have assumed that the budget will be completely consumed, the household will not choose a bundle A that is below the budget constraint. These conditions thus lead to the same optimum as our first approach did.

(c) by Christian Bauer
Prof. Dr. Christian Bauer
Chair of monetary economics
Trier University
D-54296 Trier
Tel.: +49 (0)651/201-2743
E-mail: Bauer@uni-trier.de
URL: https://www.cbauer.de