\(U(X ; Y) = XY = 90\)

\(Y=\frac{90}{X}\)

\(MRS_{xy} = \frac{\Delta Y}{\Delta X} = - \frac{90}{X^2}\)

With X = 6, we have:

\(MRS_{xy}=-\frac{90}{36}=-\frac{5}{2}\)

\(I = P_xX + P_yY\)

\(Y = \frac{I}{P_y} - \frac{P_x}{P_y}X\)

So the slope is equal to \(- \frac{P_x}{P_y}\) So an increase in income does not change the slope.

If \(P_x = 8\) , \(P_y = 4\) and \(I = 45\) then we have:

\(Y = \frac{45}{4} - 2X\)

So the slope is equal to \(-2\). If income increases from 45 to 60, the slope of the budget constraint does not change and the equation is the following:

\(Y = 15 - 2X\)