Consider a market with constant elasticity of demand (ε) and constant marginal cost equal to c. The gross increase in profit from increasing price by dp is dp(q + dq). The gross decrease in profit from increasing price by dp is -(m)( p)(dq) where m is the prevailing price-cost margin ( p - c)/ p. [Hint: dq is negative.]
(a) Show that the gross increase in profits from raising price by dp equals pqt(1 - tε). Recall that t = (dp/ p).
(b) Show that the gross decrease in profits from raising price by dp equals mptqε.
(c) Use (a) and (b) to determine the maximum margin that makes a price increase of 100t % unprofitable.