Question $2$:

Suppose a consumer has a utility function $U(x_1,x_2)=ln{x}_1+ln{x}_2.$ The consumer takes prices $(p_1$ and $p_2)$ and income $($I$)$ as given.

Find the demand functions for $x_1$ and $x_2$ assuming $\frac{\text{I}}{p_2}>1.$ What is special about these demand functions? Are both goods normal? Are these tastes homothetic?

Now find the demand functions for $x_1$ and $x_2$ assuming You probably assumed the opposite above, so now will you find something different. Explain.

Graph the income consumption curve for good $2$ as a function of income for given prices.

Find the consumer's indirect utility function when $\frac{\text{I}}{p_2}>1.$ Verify Roy's identity.