91影视

\(\begin{aligned}TR &= P \times Q\\ &= (100 - 3Q + 4{A^{\frac{1}{2}}})Q\\ &= 100Q - 3{Q^2} + 4{A^{\frac{1}{2}}}Q\end{aligned}\)

Profit refers to the difference between the total revenue and cost incurred on fixed and variable factors.

Profit = TR 鈥� C

\({\rm{Pr = (100Q - 3}}{{\rm{Q}}^{\rm{2}}}{\rm{ + 4}}{{\rm{A}}^{\frac{{\rm{1}}}{{\rm{2}}}}}{\rm{Q) - (4}}{{\rm{Q}}^{\rm{2}}}{\rm{ + 10Q + A)}}\)

First-order condition:

\(\begin{aligned}\frac{{\partial \Pr }}{{\partial Q}} &= 0\\100 - 6Q + 4{A^{\frac{1}{2}}} - 8Q - 10 &= 0\\90 - 14Q + 4{A^{\frac{1}{2}}} &= 0{\rm{ - - - - - - - - - - - - - - - - - - equation (i)}}\end{aligned}\)

\(\begin{aligned}\frac{{\partial \Pr }}{{\partial A}} &= 0\\\left( {\left( {\frac{1}{2}} \right)\frac{4}{{{A^{\frac{1}{2}}}}}} \right)Q - 1 &= 0\\{A^{\frac{1}{2}}} &= 2Q{\rm{ - - - - - - - - - - - - - - - - - equation (ii)}}\end{aligned}\)

From equation (i) and (ii)

\(\begin{aligned}90 - 14Q + 4{A^{\frac{1}{2}}} &= 0\\90 - 14Q + 4\left( {2Q} \right) &= 0\\6Q &= 90\\Q &= 15\end{aligned}\)

Putting the value of Q in equation (ii)

\(\begin{aligned}{A^{\frac{1}{2}}} &= 2(15)\\A &= {\left( {2 \times 15} \right)^2}\\A &= \left( {4 \times 225} \right)\\A &= 900\end{aligned}\)

Putting the value of A and Q in the demand curve, the value of P is calculated:

\(\begin{aligned}P &= 100 - 3\left( {15} \right) + 4{\left( {900} \right)^{\frac{1}{2}}}\\ &= 100 - 45 + {\rm{120}}\\ &= 220 - 45\\ &= 175\\\end{aligned}\)

The value of A is 900, the value of Q is 15, and the value of P is 175.

The firm's marginal cost is calculated by differentiating the total cost of a firm. It is calculated below:

\(\begin{aligned}MC &= \frac{{d\left( {4{Q^2} + 10Q + A} \right)}}{{dQ}}\\ &= 8Q + 10\\ &= 8 \times 15 + 10\\ &= 130\\\end{aligned}\)

The formula for Lerner鈥檚 index is:

\(\begin{aligned}L &= \frac{{\left( {P - MC} \right)}}{P}\\ &= \frac{{\left( {175 - 130} \right)}}{{175}}\\ &= 0.2571\end{aligned}\)