Microeconomics, deriving demand equations from utility unction

Question

Utility function: U=X+XY and I=XPx+YPy Good Y is a composite good, so Py= 1

What would be the demand function for good X?

I got Mux=1+y and Muy=x and since Mux/Px=Muy/Py

1+y/Px=x/py

so that I=XPx+(Xpx-1)Py---> X=(I+Py)/(Px(1+Py)

Is this correct? Or am I supposed to set Py= to 1 in the calculations so that it would instead be X= (I+1)/2Px?

Also, would it be correct to say that these are substitutes? And good X is a normal good?

Answer 1

You're right in that $\frac{1+y}{p_x}=\frac{x}{p_y}$. This implies that $yp_y=xp_x-p_y$, and using this in the budget constraint $I=xp_x+yp_y$ yields $ I =xp_x+xp_x-p_y=2xp_x-p_y$ or

$$x=\frac{I+p_y}{2p_x}.$$

In order to determine whether these goods are gross substitutes (or complements), you need to compute $\frac{\partial x}{\partial p_y}$ and observe that this is

$$\frac{\partial x}{\partial p_y}=\frac{1}{2p_x}>0.$$

This means that increases in the price of $y$ are associated with increases in the consumption of $x$, a behaviour that corresponds to gross substitutes.

As for whether $x$ is a normal good, you need to determine if increases in income are associated with increases or decreases in its consumption, so we calculate

$$\frac{\partial x}{\partial I}=\frac{1}{2p_x}>0.$$

As consumption of $x$ increases with $I$, we say that $x$ is a normal good (has positive income elasticity).