DERIVATION OF THE SLOPES OF AN INDIFFERENCE CURVE AND A BUDGET CONSTRAINT


Consider one indifference curve. Let the individual consume a small additional amount of x. The effect of this on total utility is given by the rate of change of utility with respect to a change in x, U/Qx , multiplied by the amount of the change in x, Qx. Next suppose that the individual reduces his or her consumption of y in such a way that the loss of utility caused by having less y exactly compensates for the gain in utility from having more x. The loss in utility from having less y is given by the rate of change of utility with respect to a change in y, U/Qy , multiplied by the amount of the change in y, Qy.

If we constrain the changes in x and y to be along one indifference curve, as illustrated in Panel a in Figure 3.9, then the overall change in utility must be zero. Therefore we have

Rearranging this we obtain

and so

But the final term in this expression is the negative of the ratio of the marginal utility of x to the marginal utility of y. [See panel b in Figure 3.9]. So we conclude that

Slope of indifference curve =

The slope of the budget constraint

The equation of a budget constraint is

Rearrange this to give an equation for Qy in terms of Qx as follows:

Therefore, the slope of the budget line is given by – (Px/Py), as shown in panel (c).