Question

Based on market research data, you have derived the following price-volume function:

                                x (p) = 23.600 e^-0,5p

1. Please compute the price elasticities at the following price points:

p1 = $1.50 USD

p2 = $2.80 USD

p3 = $3.60 USD

p4 = $5.00 USD

2. Please compute the price at which revenues are optimal

Answer 1

Ans. Demand function,

x = 23.6e-0.5p

Differentiating with respect to p, we get,

dx/dp = -0.5*23.6e-0.5p

=> Price elasticity of demand, E = dx/dp * p/x

E = -11.8e-0.5p * p/x

a) At p1 = $1.5, x = 11.15

=> E = -11.8e-0.5*1.5*1.5/11.15 = -0.75

At p2 = $2.80 , x = 5.82

=> E = -11.8e-0.5*2.8*2.80/5.82 = -1.4

At p3 = $3.60, x = 3.9

=> E = -11.8e-0.5*3.6*3.6/3.9 = -1.8

At p4 = $5, x = 1.94

=> E = -11.8e-0.5*5*5/1.94 = -2.5

b) Total Revenue, TR = P*x = 23.6e-0.5p*p

For optimal revenue,

Differentiate TR with respect to p

=> dTR/dp = 23.6e-0.5p - 0.5*p*23.6e-0.5p = 0

=> p = $2

Thus, revenue is optimal at p = $2

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