Consider a monopolist with constant marginal cost c(c1) which faces the following demand...

a) The profit function of the monopolist is:

{eq}\begin{align*} \pi &= TR - TC\\ &= pQ - cQ\\ &= p + p\sqrt A - {p^2} - c - c\sqrt A - cp \end{align*} {/eq}

The first order conditions are:

{eq}\begin{align*} \dfrac{{\partial \pi }}{{\partial p}} &= 1 + \sqrt A - 2p - c = 0\\ \dfrac{{\partial \pi }}{{\partial A}} &= 0.5p{\left( A \right)^{ - 0.5}} - 0.5c{\left( A \right)^{ - 05}} = 0 \end{align*} {/eq}

The optimal price and expenditure can be computed using the equations derived above:

{eq}\begin{align*} 0.5p{\left( A \right)^{ - 0.5}} &= 0.5c{\left( A \right)^{ - 05}}\\ {p^*} &= c\\ 1 + \sqrt A - 2p &= c\\ 1 + \sqrt A - 2c &= c\\ \sqrt A &= 3c - 1\\ {A^*} &= 9{c^2} + 1 - 6c \end{align*} {/eq}

b) The price elasticity of demand is:

{eq}\begin{align*} {e_p} &= \dfrac{{\partial Q}}{{\partial p}} \times \dfrac{p}{Q}\\ &= - 1 \times \dfrac{p}{{1 + \sqrt A - p}}\\ &= - 1 \times \dfrac{c}{{1 + \sqrt {{{\left( {3c - 1} \right)}^2}} - c}}\\ &= \dfrac{{ - c}}{{2c}}\\ &= - 0.5 \end{align*} {/eq}

The advertising elasticity of demand is:

{eq}\begin{align*} {e_A} &= \dfrac{{\partial Q}}{{\partial A}} \times \dfrac{A}{Q}\\ &= 0.5{A^{ - 0.5}} \times \dfrac{{{{\left( {3c - 1} \right)}^2}}}{{1 + \sqrt A - p}}\\ &= \dfrac{{0.5}}{{\left( {3c - 1} \right)}} \times \dfrac{{{{\left( {3c - 1} \right)}^2}}}{{1 + \sqrt {{{\left( {3c - 1} \right)}^2}} - c}}\\ &= \dfrac{{3c - 1}}{{4c}} \end{align*} {/eq}