Question: If you have studied consumer indifference curves, try the following analysis of a family's decision about choosing the "optimum" number of children which can help you to understand better why income and fertility are inversely related from a neoclassical economic perspective. Draw axes where you measure the "number of children" on the horizontal axis and "family income" on the vertical axis. Draw in a few normally shaped, convex-to-theorigin indifference curves. The shapes of the curves indicate that income (a proxy for "all other goods") and children are substitutes for a family in providing satisfaction or utility. Draw and label a "current income" budget line and find the "optimum" number of children, given that income level. You will remember that the optimum will be at the tangency point of that linear budget line and the highest attainable indifference curve.
Show on the horizontal axis the number of children "chosen"; label that quantity . Now, let income increase, but at the same time let the "price" of children also increase (in terms of the opportunity cost of lost income for time spent in caring for children by, for example, women with greater opportunities for employment). Thus the new budget line will both shift out and be steeper than the original line, indicating an increase in the relative "price" of children compared to all other goods as income rises. You should be able to show that it is quite possible that the new optimum number of children, , that a family now chooses is fewer because of the higher relative price of children (as a result of their higher opportunity cost) at the higher income level.