### 8.2 The budget line

The assumption that all possible bundles of consumption are on a straight line when the budget is fully consumed can be proven by looking at the mathematical relationship between income and expenditure. If we name the budget $\mathit{Budget}$, the price of Good 1 $P1$, the amount of Good 1 $x$, the price of Good 2 $P2$ and the amount of units of Good 2 $y$, we get: $\mathit{Budget}=x\ast P1+y\ast P2$ or, if we solve this equation (budget equation) for $x$:

 $\text{y}=\frac{\text{Budget}}{P2}-\text{x}\frac{P1}{P2},$

thus, a straight line equation with the unknowns $x$ and $y$ with the slope $\frac{P1}{P2}$. This straight line is called budget line or budget constraint. It contains all consumption bundles that the household can afford when its $\mathit{Budget}$ is completely consumed. Bundles of consumption below the budget line, represented here by the green area, can as well be afforded by the consumer, but there would still be income left. Bundles of consumption above the budget line cost more than the household has at its disposal. When showing test point T, the appearing text explains the situation: if we move T into the green area, the costs of the consumption bundle are lower than the income, on the budget line they correspond exactly to the income and if T is above the red budget line, the costs are higher than the income. The point C in turn represents a bundle of goods where the budget is actually used up. If you move this point, it becomes clear once again that additional consumption of one good reduces the consumption of the other good. The extent to which this exchange takes place is reflected by the slope of the budget line (the negative sign of the slope is ignored here for reasons of simplification). For example, if the slope is 2, for each unit more of Good 1 , we have to give up two units of Good 2 . This can be illustrated by shifting point C. The slope of the budget line is the relative price of the two goods. For example, if a unit of Good 1 costs 6  and one unit of Good 2 3 , the ratio of the price of Good 1 to the price of Good 2 is 6 : 3 = 2. One unit of Good 1 is twice as expensive as one unit of Good 2 . In other words, for the price of one unit of Good 1 , you can get 2 units of Good 2 . Here, we could read this ratio in the light blue box: a relative price of two units of Good 2 per unit of Good 1 corresponds to a slope of the budget line of 2.

(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