### 9.9 The derived demand: Engel curves and demand curve

The budget problem

$$\underset{x,y}{\mathrm{max}}U\left(x,y\right)\text{undertheconditionthat}x{p}_{x}+y{p}_{y}=B$$ |

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

$$x\left({p}_{x}\right)=\stackrel{\u0303}{c}\cdot \frac{1}{{p}_{x}},$$ |

where $\stackrel{\u0303}{c}$
is a suitable constant. The demand curve is therefore monotonically falling, as
usual.

As Engel curve we obtain

$$x\left(B\right)=\stackrel{\u0303}{c}\cdot B,$$ |

where $\stackrel{\u0303}{c}$
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