Çözüldü Price, Revenue and Marginal Revenue Functions - Analytic Geometry - Differentiation - Microeconomics

A store sells 250 pairs of Brand X running shoes each month when priced at $100 per pair. It has been determined that for every $5 decrease in price, an additional 20 pairs will be sold.

a) Determine the demand or price function.
b) Determine the revenue function.
c) Find the revenue when sales are 250 pairs and 290 pairs of shoes per month.
d) Determine the marginal revenue when sales are 290 pairs of shoes per month.

a) Price Function as a straight line (linear function) versus the number of pairs of shoes: p(x) = f(x)
The line passing through (250, 100) and (270, 95) is [ p(x) - 100 ] / (100 - 95) = (x - 250) / (250 - 270)
p(x) = -x / 4 + 162.5

b) Revenue Function: R(x) = x·p(x) = x·(-x / 4 + 162.5) = -x^2 / 4 + 162.5·x

c)
R(250)= -250^2 / 4 + 162.5·250 = $25,000
R(290)= $26,100

d) R'(x) = dR / dx = -x / 2 + 162.5
R'(290) = -290 / 2 + 162.5 = $17.50 / pair of shoes sold.

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(Question 10)