Without knowing the Slutsky equation and income/substitution effect, how can I show a certain good is inferior or Giffen? Say I've got a function $x_1(p_1,p_2,m)$ where $p_1, p_2$ are the prices for good 1, good 2 respectively and m is the income.
Now, I haven't heard of the Slutsky equation yet nor the income/substitution effect.
How do I show for this function that it is an inferior/normal and ordinary/Giffen good?
My initial thought was to simply use the partial derivate and to check whether it's greater or less than zero, since
$\frac{\partial x_1(...)}{\partial m} \gt 0$ should be true for normal goods, $\frac{\partial x_1(...)}{\partial m} \lt 0$ for inferior goods, $\frac{\partial x_1(...)}{\partial p_1} \gt 0$ for Giffen goods and $\frac{\partial x_1(...)}{\partial p_1} \lt 0$ for ordinary goods.
Is this correct?
Is this all I can do to formally show that the good is normal/ordinary? Is this sufficient?
Also, if I can show that the same function for $x_1(m)$ is homogeneous of degree 1 (i.e. homothetic), i.e. $tx_1(m)=x_1(tm)$, this means that the partial derivative $\frac{\partial x_1(...)}{\partial m}$ is homogeneous of degree 0, so it would be a constant. Could I thus conclude that the good is strictly normal/inferior (is that a term?), i.e. it doesn't change if the income changes?
yes. Notice that a good is normal, inferior Giffen for a certain value of prices/income. For example, a good might be inferior at some prices/income but normal for some other values of the prices/income.
In some cases, however, you can make stronger statments. For example, homothetic preferences always lead to normal demands as
$$
x_1(p_1, p_2,m) = x_1(p_1, p_2,1) m 
$$
where $x_1(p_1, p_2,1)$ is the unit income demand function which is always greater or equal to zero.
From this, it follows that homothetic preference can never lead to Giffen goods (but to show this, you do need the Slutsky decomposition).