# 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. – tdm May 11 at 15:31
• You're right; however if I can show that the partial derivative $\partial x_1(...)/\partial m$ is not dependent on $m$, that means it's a constant and the function $x_1$ is homothetic. So for homothetic functions this shouldn't matter? @tdm – j3141592653589793238 May 11 at 15:34
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.