9.9 The derived demand: Engel curves and demand curve

The budget problem

max x,yU x,y under the condition that xpx + ypy = B

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

x = f px,py,B

for an applicable function F. If we consider a Cobb-Douglas utility function U x,y = xαyβ, we get

x = α α + βB 1 px.

In this case, the function F is independent of py. 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 px = c̃ 1 px,

where c̃ is a suitable constant. The demand curve is therefore monotonically falling, as usual.
As Engel curve we obtain

x B = c̃ B,

where c̃ 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