Blog: The XYZ firm's total cost function is TC=5Q^2 + 20Q+100. If the firm's product could sell in a competitive market at a price of $120 per unit, what...

Sunday, June 28, 2015

The XYZ firm's total cost function is TC=5Q^2 + 20Q+100. If the firm's product could sell in a competitive market at a price of $120 per unit, what...

You may evaluate the profit function, such that:

$\displaystyle{p}{\left({Q}\right)}={T}{R}{\left({Q}\right)}-{T}{C}{\left({Q}\right)}$

$\displaystyle{p}{\left({Q}\right)}={120}{Q}-{\left({5}{{Q}}^{{2}}+{20}{Q}+{100}\right)}$

$\displaystyle{p}{\left({Q}\right)}={120}{Q}-{5}{{Q}}^{{2}}-{20}{Q}-{100}$

$\displaystyle{p}{\left({Q}\right)}=-{5}{{Q}}^{{2}}-{100}{Q}-{100}$

The first order condition for profit maximization is $\displaystyle\frac{{{d}{p}}}{{{d}{Q}}}={0}$ , such that:

$\displaystyle\frac{{{d}{p}}}{{{d}{Q}}}=\frac{{{d}{\left(-{5}{{Q}}^{{2}}-{100}{Q}-{100}\right)}}}{{{d}{Q}}}$

$\displaystyle\frac{{{d}{p}}}{{{d}{Q}}}=-{10}{Q}-{100}\Rightarrow-{10}{Q}-{100}={0}\Rightarrow-{10}{Q}={100}\Rightarrow{Q}=-{10}$

The total maximized profit is obtained by substituting -10 into the equation of $\displaystyle{p}{\left({Q}\right)}=-{5}{{Q}}^{{2}}-{100}{Q}-{100}$ , such that:

$\displaystyle{p}{\left({Q}\right)}=-{5}{{\left(-{10}\right)}}^{{2}}-{100}{\left(-{10}\right)}-{100}$

$\displaystyle{p}{\left({Q}\right)}=-{500}+{1000}-{100}$

$\displaystyle{p}{\left({Q}\right)}={400}$

Hence, the total maximized profit is $400.

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Sunday, June 28, 2015

The XYZ firm's total cost function is TC=5Q^2 + 20Q+100. If the firm's product could sell in a competitive market at a price of $120 per unit, what...

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