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July 9th, 2011, 11:05 AM   #1
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Economics - Consumer expenditure/demand curve problem

Hello everyone!

Having trouble with this math-exercise, and I hoped someone could be of assistance:

Suppose the demand curve is given by x(p) = 1 – p. The consumer’s expenditure is px(p) = p(1 – p). Graph the expenditure. What price maximizes the consumer’s expenditure?

Thanks in advance.

Christian Hallas
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July 9th, 2011, 12:22 PM   #2
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Re: Economics - Consumer expenditure/demand curve problem

Let E(p) = px(p) = p(1 - p) represent the expenditure. We see E(p) is a downward opening parabola, thus it's maximum will occur where p is on the axis of symmetry. Setting E(p) = 0, we see it has roots at p = 0 and p = 1. The axis of symmetry will then be the line:



thus a price of p = 1/2 will maximize the expenditure. E(1/2) = 1/4. A sketch of the graph can then be drawn by passing a parabolic arc through the points:

(0,0), (1/2, 1/4), (1,0)

Another way is to write:



Complete the square:





Thus, we find the vertex is at the point

Since you posted this in the calculus section, we could find dE/dp, and equate to zero to find the p that maximizes E.

=\frac{1}{2}" />

We see meaning E is concave down for all p, meaning the extremum is indeed a maximum.
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