The income consumption curve and homotheticity

Income consumption curve

The dotted orange line was obtained by imposing the tangency condition. You can convince yourself that it describes the income consumption curve by:

  1. varying income (shift red slider)

and:

  1. adjusting the basket to reach the highest utility level, subject to the budget constraint (move the gray slider along the budget constraint segment)

Budget shares at different income levels

Once you've convinced yourself above, you can just look for the intersection between the budget line and income consumption curve to see in what ratio the two goods are consumed at different levels of income.

For instance, in the second case, you can keep prices at 1, and look for I=6, I=12, I=22, etc.