Difference between Engel curve and income expansion path

Question

I was scrolling through many questions on this site about income offer path and it appeared to me that income offer path, income expansion path, engel curve all are different terms with same meaning, i.e. graph of the engel function.

Can somebody explain what is the difference between these terms, if any?

Answer 1

Both of these curves describe the same phenomenon, the change in consumer choice as income changes. The difference is in the displayed variables. The Engel curve is a relationship between the consumption of a good $x$ and income $I$, whereas the income expansion path (IEP) shows the relationship between the consumption of a good $x$ and a good $y$ as income increases. These are the coordinate systems the curves are usually displayed in, so Engel in $(x,I)$, IEP in $(x,y)$.

The Engel curve would show you if good $x$ is inferior.

One can also learn this from IEP (it would have a negative slope somewhere), and in case both goods are normal IEP would show you if the consumption of one good would increase at a faster rate than that of the other.

Answer 2

The Income Offer Curve (which is the same as the Income Expansion Path) shows us the effect of a change in nominal money income on the consumption of both goods (in a 2 good model) in the real 2 good indifference curve space. Thus, to derive the income offer curve, one shifts the budget constraint by varying money income, and joins all the points of optimality. If both goods are normal goods, the income offer curve is upward sloping; if one of them is inferior, it is downward sloping.

The Engel Curves can be derived by looking at the Income Offer Curve. The Engel Curve shows the relationship between ONE particular good and money income in a graph i.e. in the X-I space. The Engel Curve for both goods can be figured out by looking at the Income Offer Curve.

So, in a 2 good model, the income offer curve summarises the effect of a change in money income on both goods.