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July 9th, 2011, 11:05 AM  #1 
Newbie Joined: Jul 2011 Posts: 1 Thanks: 0  Economics  Consumer expenditure/demand curve problem
Hello everyone! Having trouble with this mathexercise, 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 
July 9th, 2011, 12:22 PM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  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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