# Budget lines and Indifference curves: How do you figure where exactly you should draw the IC tangent to a budget line?

In Consumer Choice, we frequently encounter budget lines and indifference curves. For a given budget line, the optimization point is at the point where the indifference curve is just tangent to the budget line. For example, here is a graph used to illustrate how inferior goods behave with increases in income:

Here, as the budget line shifts to $$I_3$$, we're shown that consumption of $$z$$, an inferior good, decreases. My question is: what's stopping me from drawing the indifference curve elsewhere? Why can't I draw the indifference curve tangent to $$I_3$$, just right of $$z_1$$ so that $$z_3 > z_1$$?

Sure, the MRS should be equal to the ratio of the price of the goods, but we can show that anywhere along the budget line, right?

Also, I understand that we can't draw the IC in a way that they intersect. So, I can't draw the IC for $$I_3$$ towards the end of $$I_3$$ (near the x-axis) because it'll intersect other indifference curves -- but how do we know where to draw the IC?

In other words, I can only reproduce this diagram for inferior goods because I've seen it. If I had to graphically represent this without having seen it earlier, there is no way I'd have figured that $$z_3$$ is lesser than $$z_1$$.

• Please let me know if I can improve my question. – WorldGov Jan 20 '19 at 15:27
• You have answered the question. The ratio of the prices is equal to the MRS. – superhulk Jan 20 '19 at 22:52

In order to know the exact bundle for that maximizes utility, you need to know what the consumer's preferences are. Nevertheless, if you knew that good $$z$$ is an inferior good for this particular consumer at her current income level, you should still be able to draw this diagram. The only thing you need is to recall the definition of an inferior good: a good is said to be inferior if as income increases its consumption decreases. This alone implies that $$z_3 Observe that in the diagram the budget constraint is drawn for three income levels. As income increases from $$m_1$$ to $$m_2,$$ consumption of goes from $$z_1$$ to $$z_2,$$ implying that $$z$$ is an inferior good. If income increases further to $$m_3,$$ consumption needs to reduce relative to $$z_2.$$
Note that it could have been the case that as income grew from $$m_1$$ to $$m_2$$ consumtion of $$z$$ increased, but from $$m_2$$ to $$m_3$$ it decreased. This is why I emphasized the role of consumer income: a given good may be normal at some income levels and then inferior for other, presumably larger, income levels.