Question

A monopolistic producer of two goods, 1 and 2, has a joint total cost function

where and denote the quantity of items of goods 1 and 2, respectively that are produced. If P1 and P2 denote the corresponding prices then the demand equations are

Using the Lagrange multiplier approach, find the maximum profit if the firm is contracted to produce a total of 15 goods of either type. Estimate the new optimal profit if the production quota rises by 1 unit.

Answer 1

$TC=10Q_1+Q_1Q_2+10Q_2$

Then $max\pi ~~~~~~~ Q_1+Q_2\le 15$

Then $L=P_1Q_1+P_2Q_2-TC +\lambda [15-Q_1-Q_2]$

$L=40Q_1-Q_1^2-Q_2^2+30Q_2-10Q_2+2Q_1Q_2 +\lambda [15-Q_1-Q_2] \\[9pt] L=40Q_1-Q_1^2-Q_2^2+20Q_2+2Q_1Q_2+\lambda[15-Q_1-Q_2]$

Now,

$\dfrac{dL}{dQ_1}=40-2Q_1+2Q_2-\lambda=0$

and $\dfrac{dL}{dQ_2}=-2Q_2+20+2Q_1-\lambda=0$

So From FOC:

$40-2Q_1+2Q_2=20+2Q_1-2Q_2 \\[9pt] \Rightarrow 20=4Q_1-4Q_2 \\[9pt] \Rightarrow Q_1-Q_2=5$

and Q_1+Q_2=15

Solving above equation we get-

$Q_1=10,Q_2=5$

Max $\pi =P_!Q_1+P_2Q_2-TC$

$P_1=50-10+5=45, P_2=30+20-5=45$

Then,

$\pi=45(15)-10(10)-50-50 \\ =675-100-100=475$

Now $\lambda=40-2(Q_1-Q_2)$

      $=40-2(5) =30$

So if quota is incresed by 1 unit, then optimal output is increased by $30.