# Vertical income expansion path: explanation?

On the horizontal axis we have good x, on the vertical axis good y. The income expansion path is vertical. Which of the following assumptions is wrong?

• no homotheticity
• no strong monoticity
• represantative consumer
• The assumption is "no homotheticity", as in we assume the utility function is not homothetic? Can you flesh out the question a bit? Also, seems like a homework problem. Where do you get stuck? Commented Jan 5 at 21:29
• I'm not sure of my answer. With no homotheticity, it is assumed that the preferences of the consumer are not homothetic. Since in homotheticity there is an income expansion path that goes through the origin, I think that in this case ther is no homotheticity. But im not sure about monoticity. I think there is monoticity since the IC are negativly sloped, but don't know wheter is is a correct assumption, since the concept op strong monoticity remains vague for me ("monoticity = IC's are decreasing since more is better?) Commented Jan 5 at 21:35

If a consumer has the utility function $$U(x,y) = x + y$$, and $$p_x > p_y$$, then the optimal bundle is $$(x,y) = (0,m/p_y)$$. This seems to result in a vertical income expansion path.
Note that the same is true for $$\hat{U}(x,y) = y - x,$$ and $$\tilde{U}(x,y) = \ln (x+1) + y.$$