Graphical Derivation of Cross-Price Demand Schedule
As before, we
start with the indifference mapping in X1-X2 space,
where each indifference curve traces out all combinations of
the two goods X1 and X2 which yield the same level
of utility. Given a fixed budget M and goods prices P1
and P2 for the goods X1 and X2,
we can superimpose the budget line, which shows all just-affordable
combinations of X1 and X2.
We assume a rational consumer chooses the X1-X2 combination which maximizes his/her utility subject to this budget constraint. This optimal combination occurs at the unique tangency point between the budget line and the indifference curve representing the highest affordable level of utility. The slope of the budget line is the price ratio P2/P1. The slope of the indifference curve is the ratio of marginal utilities MU2/MU1.
We derive the cross-price demand schedule for X1 by varying P2, holding P1 and M constant, and tracing out the utility-maximizing level of X1 consumed at each level of P2. As P2 is reduced, the budget line shifts outward to new tangency points on successively higher indifference curves. In this example, X1 and X2 are substitutes, so that declines in P2 cause consumers to substitute X2 for X1. This animation shows successively lower optimal consumption levels of X1 (and higher optimal consumption levels of X2) as P2 declines.
If X1 and X2 were complementary goods, the cross-price demand schedule would be negatively-sloped, and a decline in P2 would stimulate increased consumption of both X1 and X2.