9.9 The derived demand: Engel curves and demand curve

The budget problem

provides a solution for the optimal consumption quantity $x$, which depends on the model parameters $p_x$, $p_y$ and $B$. So you could write

for an applicable function $F$. If we consider a Cobb-Douglas utility function $U\left(x,y\right)=x^{\alpha }{y}^{\beta }$, we get

In this case, the function $F$ is independent of $p_y$. $F$ can be regarded as a function of a parameter and the influence of this parameter on the quantity $x$ demanded can be analyzed. This is called derived demand. If one considers $x$ as a function of the price of $x$, one analyzes the individual or the ??, as we have already examined in detail in the chapter Market. If one considers $x$ as a function of the budget of $x$, one analyses the so-called Engelkurven, since the budget can be seen as equivalent to income.
In the case of the Cobb-Douglas utility function we obtain for the market demand

where $\tilde{c}$ is a suitable constant. The demand curve is therefore monotonically falling, as usual.
As Engel curve we obtain

where $\tilde{c}$ is again a suitable constant. Here, the Engel curve is monotonically rising. The good is normal and not inferior.