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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 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
 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: $p=\frac{0+1}{2}=\frac{1}{2}$ 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: $E(p)=p(1-p)=-p^2+p$ Complete the square: $E(p)=-$$p^2-p+\frac{1}{4}$$+\frac{1}{4}$ $E(p)=-$$p-\frac{1}{2}$$^2+\frac{1}{4}$ Thus, we find the vertex is at the point $$$\frac{1}{2},\frac{1}{4}$$$ 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{dE}{dp}=-2p+1=0\:\therefore\=\frac{1}{2}" /> We see $\frac{d^2E}{dp^2}=-2$ meaning E is concave down for all p, meaning the extremum is indeed a maximum.

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# if |e|=1 then show that the demand curve is parabola

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