Question

You are the product manager for the shower gel. The market research unit has estimated the following price-volume function:

                                 
                                           x(p) = 500 ∙p^-4

1. Please compute the price elasticities and interpret them. In which price area will price increases result in higher revenues?
2. Next, please compute the price that maximizes profits. For doing so, please assume that 1) the variable unit costs are $0.90 USD and 2) fixed costs are negligible.

Answer 1

Ans. Demand function,

x = 500p - 4 or p = (x+4)/500

=> Total Revenue, TR = p*x = (x2 + 4x)/500

=> Marginal Revenue, MR = dTR/dx = (2x + 4)/500

As fixed cost is negligible, so, total cost equals variable cost,

TC = 0.90x

=> Marginal cost, MC = dTC/dx = 0.90

At equilibrium,

MR = MC

=> (2x+4)/500 = 0.90

=> 2x + 4 = 450

=> x = 223 units

=> p = $0.454

For price elasticity,

Price elasticity of supply is infinite because MC is constant.

For price elasticity of demand,

Differentiate demand function with respect to p

=> dx/dp = 500

Elasticity = dx/dp * p/x = 500 * 0.454/223 = 1.018

Thus, these elasticities represents that a small change in price will lead to a very large change in quantity supplied and a 1% change in price will lead to a 1.018% change in quantity demanded.

And because demand for shower gel is positively related to price, so, increase in price will increase revenue and hence profits because marginal cost is constant

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