the indirect utility function states that higher indirect indifference curves carry lower utility. however it also assumes homogenity of degree zero in price and income. how?
please explain it to me in terms of simple language as i am not familiar with linear programming techniques.
The indirect utility function is homogeneous of degree zero in prices and income because if you multiply all prices and income by the same positive constant, then this has no effect on the budget constraint, and so must have no effect on the solution to the utility maximization problem. Since multiplying all prices and income by a positive constant does not affect the maximized utility, the indirect utility function is homogeneous of degree zero.
For example, with two goods $1$ and $2$ with prices $p_1$ and $p_2$ the budget constraint is
$$p_1x_1+p_2x_2\leq m \tag{BC1}$$
where $x_1$ and $x_2$ are the quantities of goods $1$ and $2$, and where $m$ is income. Let $V(p_1,p_2,m)$ denote the indirect utility (the maximized utility given prices and income). To show $V$ is homogeneous of degree zero, we need to show that for any $\lambda>0$ we have
$$V(\lambda p_1,\lambda p_2, \lambda m)=V(p_1,p_2,m) \tag{1}$$
If we multiply all prices and income by $\lambda>0$, then the budget constraint for becomes
$$(\lambda p_1)x_1+(\lambda p_2)x_2\leq (\lambda m) \tag{BC2}$$
But dividing by $\lambda$ shows that BC2 is equivalent to the BC1, and so the utility maximization problem is unchanged. Hence (1) holds.
A level curve of the indirect utility function is the set of points $(p_1,p_2,m)$ satisfying the equation $V(p_1,p_2,m)=U$ for some level of utility $U$.
If we hold $m$ constant, then we can plot level curves in in $p_1p_2$-space. Since $V$ is decreasing in prices (assuming monotonic preferences), the level curves will be downward sloping and level curves to the northeast will represent lower values of utility.
