Consolidated Salt Company Sells Table Salt to Both Retail Grocery$$\begin{array} { l }
\text {Retail grocery chains:}&P _ { 1 } = 180 - 8 Q _ { 1 } \
\text {Commercial users:}&P _ { 2 } = 100 - 4 Q _ { 2 }
\end{array}$$

Question

Consolidated Salt Company Sells Table Salt to Both Retail Grocery$$\begin{array} { l } 
\text {Retail grocery chains:}&P _ { 1 } = 180 - 8 Q _ { 1 } \
\text {Commercial users:}&P _ { 2 } = 100 - 4 Q _ { 2 }
\end{array}$$

Examlex · Accepted Answer

The Answer is (a)? = TR ? TC = P₁Q₁ + P₂Q₂ ? TC = (180 ? 8Q₁)Q₁ + (100 ?4Q₂)Q₂ ? [50 + 20(Q₁ + Q₂)] = 180Q₁ ? 8Q₁² + 100Q₂ ? 4Q₂² ? 50 ? 20Q₁ ? 20Q₂ = ?50 + 160Q₁ + 80Q₂ ? 8Q₁² ? 4Q₂² (b) $(a)? = TR ? TC = P1Q1 + P2Q2 ? TC = (180 ? 8Q1)Q1 + (100 ?4Q2)Q2 ? [50 + 20(Q1 + Q2)] = 180Q1 ? 8Q12 + 100Q2 ? 4Q22 ? 50 ? 20Q1 ? 20Q2 = ?50 + 160Q1 + 80Q2 ? 8Q12 ? 4Q22 (b) \begin{aligned} \frac { \partial \pi } { \partial Q _ { 1 } } & = 0 \ \frac { \partial \pi } { \partial Q _ { 1 } } & = 160 - 16 \mathrm { Q } _ { 1 } \ 0 & = 160 - 16 \mathrm { Q } _ { 1 } ^ { * } \ \mathrm { Q } _ { 1 } ^ { * } & = 10 ext { units } \ \mathrm { P } _ { 1 } ^ { * } & = 180 - 8 ( 10 ) \ & = \$ 100 / ext { unit } \end{aligned} \begin{array} { l } \frac { \partial \pi } { \partial Q _ { 2 } } = 0 ext { (conditions for maximizing } \pi ext { ) } \\ \frac { \partial \pi } { \partial Q _ { 2 } } = 80 - 8 Q _ { 2 } \\ 0 = 80 - 8 Q _ { 2 } ^ { * } \\ Q _ { 2 } ^ { * } = 10 ext { units } \\ P _ { 2 } ^ { * } = 100 - 4 ( 10 ) \\ = \$ 60 / ext { unit } \end{array} \begin{aligned} \pi ^ { * } & = - 50 + 160 ( 10 ) + 80 ( 10 ) - 80 ( 10 ) ^ { 2 } - 4 ( 10 ) ^ { 2 } \ & = \$ 1150 \end{aligned} (c)180 ? 8Q1 = 100 ? 4Q2 \begin{aligned} Q _ { 2 } & = \frac { - 80 + 8 Q _ { 1 } } { 4 } \ & = - 20 + 2 Q _ { 1 } \end{aligned} ? = ?50 + 160Q1 + 80(?20 + 2Q1)? 8Q12 ? 4(?20 + 2Q1)2 = ?50 + 160Q1 ? 1600 + 160Q1 ? 8Q12 ? 1600 + 320Q1 ? 16Q12 = ?3250 + 640Q1 ? 24Q12 \frac { \mathrm { d } \pi } { \mathrm { dQ } } = 0 (condition for maximizing \left. \pi ight) \frac { \mathrm { d } \pi } { \mathrm { dQ } _ { 1 } } = 640 - 48 \mathrm { Q } _ { 1 } 0 = 640 - 480 Q _ { 1 } ^ { * } Q _ { 1 } ^ { * } = 13.33 ext { units } \begin{aligned} Q _ { 2 } ^ { * } & = - 20 + 2 ( 13.33 ) \ & = 6.67 ext { units } \end{aligned} \begin{aligned} \mathrm { Q } ^ { * } & = \mathrm { Q }^ { * } _ { 1 } + \mathrm { Q }^ { * } _ { 2 } \ & = 13.33 + 6.67 \ & = 20 ext { units } \end{aligned} \begin{aligned} P ^ { * } & = P _ { 2 } ^ { * } = 180 - 8 ( 13.33 ) \ & = \$ 73.36 / ext { unit } \end{aligned} \begin{aligned} \pi & = - 3250 + 640 ( 13.33 ) - 24 ( 13.33 ) ^ { 2 } \ & = \$ 1,016.67 \end{aligned}$

\begin{aligned}\frac { \partial \pi } { \partial Q _ { 1 } } & = 0 \\frac { \partial \pi } { \partial Q _ { 1 } } & = 160 - 16 \mathrm { Q } _ { 1 } \0 & = 160 - 16 \mathrm { Q } _ { 1 } ^ { * } \\mathrm { Q } _ { 1 } ^ { * } & = 10 ext { units } \\mathrm { P } _ { 1 } ^ { * } & = 180 - 8 ( 10 ) \& = \$ 100 / ext { unit }\end{aligned}

\begin{array} { l } \frac { \partial \pi } { \partial Q _ { 2 } } = 0 ext { (conditions for maximizing } \pi ext { ) } \\\frac { \partial \pi } { \partial Q _ { 2 } } = 80 - 8 Q _ { 2 } \\0 = 80 - 8 Q _ { 2 } ^ { * } \\Q _ { 2 } ^ { * } = 10 ext { units } \\P _ { 2 } ^ { * } = 100 - 4 ( 10 ) \\= \$ 60 / ext { unit }\end{array}

\begin{aligned}\pi ^ { * } & = - 50 + 160 ( 10 ) + 80 ( 10 ) - 80 ( 10 ) ^ { 2 } - 4 ( 10 ) ^ { 2 } \& = \$ 1150\end{aligned}

(c)180 ? 8Q₁ = 100 ? 4Q₂

\begin{aligned}Q _ { 2 } & = \frac { - 80 + 8 Q _ { 1 } } { 4 } \& = - 20 + 2 Q _ { 1 }\end{aligned}

? = ?50 + 160Q₁ + 80(?20 + 2Q₁)? 8Q₁² ? 4(?20 + 2Q₁)² = ?50 + 160Q₁ ? 1600 + 160Q₁ ? 8Q₁² ? 1600 + 320Q₁ ? 16Q₁² = ?3250 + 640Q₁ ? 24Q₁²

\frac { \mathrm { d } \pi } { \mathrm { dQ } } = 0

(condition for maximizing

\left. \pi ight)

\frac { \mathrm { d } \pi } { \mathrm { dQ } _ { 1 } } = 640 - 48 \mathrm { Q } _ { 1 }

0 = 640 - 480 Q _ { 1 } ^ { * }

Q _ { 1 } ^ { * } = 13.33 ext { units }

\begin{aligned}Q _ { 2 } ^ { * } & = - 20 + 2 ( 13.33 ) \& = 6.67 ext { units }\end{aligned}

\begin{aligned}\mathrm { Q } ^ { * } & = \mathrm { Q }^ { * } _ { 1 } + \mathrm { Q }^ { * } _ { 2 } \& = 13.33 + 6.67 \& = 20 ext { units }\end{aligned}

\begin{aligned}P ^ { * } & = P _ { 2 } ^ { * } = 180 - 8 ( 13.33 ) \& = \$ 73.36 / ext { unit }\end{aligned}

\begin{aligned}\pi & = - 3250 + 640 ( 13.33 ) - 24 ( 13.33 ) ^ { 2 } \& = \$ 1,016.67\end{aligned}

Consolidated Salt Company Sells Table Salt to Both Retail Grocery

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