### 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

 $x=f\left({p}_{x},{p}_{y},B\right)$

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

 $x=\frac{\alpha }{\alpha +\beta }B\frac{1}{{p}_{x}}.$

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

 $x\left({p}_{x}\right)=\stackrel{̃}{c}\cdot \frac{1}{{p}_{x}},$

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

 $x\left(B\right)=\stackrel{̃}{c}\cdot B,$

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

(c) by Christian Bauer
Prof. Dr. Christian Bauer
Chair of monetary economics
Trier University
D-54296 Trier
Tel.: +49 (0)651/201-2743
E-mail: Bauer@uni-trier.de
URL: https://www.cbauer.de