# Slope of the Budget Line

When we try to maximise utility constraint to a budget. We find the utility function or indifference curve which is having the budget line as tangent. So for solving a question we can equate the slope of budget line with the slope of a indifference curve and we will get the maximising utility coordinates.In many of the text books they equate ratio of prices with the tangent slope, but the slope of the budget line is negative where as the relative prices are positive. How does this work ? Why are they not considering the - sign of budget line ?

They do consider it. The precise mathematical formulation is usually $$-\frac{p_1}{p_2} = \text{MRS}(x_1,x_2)$$ or $$\frac{p_1}{p_2} = \left| \text{MRS}(x_1,x_2) \right|$$ in the basic cases, when MRS is a negative number (the slope of the indifference curve at basket $$(x_1,x_2)$$).

The +1 answer of Giskard is correct but I wanted to expand on it as to provide more intuition.

They do consider the negative sign and it is because they both have negative slopes so the negative numbers cancel each other out. This is the result of plotting consumption of one good on $$x$$-axis and another on $$y$$ axis. Consider the following picture from the Mankiw Principles of Economics textbook:

the picture in Mankiw textbook plots a situation that is similar to the following:

$$U(x,y) = x^{\alpha}y^{\beta} \text{ s.t. } m = p_x x + p_y y$$

where $$x$$ would be quantity of pizza and $$y$$ quantity of pepsi, $$p_x$$ and $$p_y$$ their respective prices and $$m$$ budget constraint. Of course, we cannot verify that Mankiw used exactly this specification for utility but I needed to put some concrete utility here and using different one would not fundamentally change the answer. You see even though the budget constraint is given as:

$$m = p_x x + p_y y$$

in order to plot it on a cartesian coordinate system we need to isolate $$y$$ (Pepsi). So actually the budget line is plot of this:

$$y = \frac{m}{p_y} - \frac{p_x}{p_y}x$$

Furthermore, in order to plot indifference curve we have to do the same to utility while holding utility constant so we will have:

$$U = x^{\alpha}y^{\beta} \implies y = \left( \frac{U}{x^{\alpha}}\right)^\frac{1}{\beta}$$

Now in the first case you can easily see that the slope is negative because it is linear function. However, in the second case the slope is negative as well. You can verify that by taking the derivative with respect to $$x$$ which will give you the slope of the function and $$dy/dx <0$$. To be more specific the slope of indifference curve will be:

$$-\frac{\alpha}{\beta} \left( \frac{U}{x^{\alpha}}\right)^{\frac{1}{\beta}-1} \left( \frac{U}{x^{\alpha-1}}\right)$$

Hence what you are comparing is the slope of budget constraint to slope of indifference curve as at the tangent point their slopes have to match:

$$-\frac{p_x}{p_y} = -\frac{\alpha}{\beta} \left( \frac{U}{x^{\alpha}}\right)^{\frac{1}{\beta}-1} \left( \frac{U}{x^{\alpha-1}}\right) \Leftrightarrow \frac{p_x}{p_y} = \frac{\alpha}{\beta} \left( \frac{U}{x^{\alpha}}\right)^{\frac{1}{\beta}-1} \left( \frac{U}{x^{\alpha-1}}\right)$$

Hence here the two negative values would cancel. Of course, in classic (undergraduate) textbook you wont do it this way as that would be too difficult, but it is done by comparing it to $$MRS$$ (which is equal to the slope of indifference curve) where Giskards answer applies.

However, a fundamental intuitive reason behind your answer is that simply both budget constraint and indifference curve are having negative slopes. You can see it just by looking at the picture - they are both monotonically decreasing so their slope will be negative. You might not always see it in a calculations because depending on how exactly you derive the two slopes you might already cancel the negative signs. For example, typical textbook will just tell you to compare ratio of prices $$p_x/p_y$$ to marginal rate of substitution which will be equivalent to the slope of indifference curve but in its absolute value.