Question

Consider a firm is having monopoly in production of two goods \(X\) and \(Y\). The markets for the two goods do not interact. The profit function is given by

Hint:

For maximizing a two-variable profit function, set both first partial derivatives equal to zero and solve the resulting simultaneous equations.

Answer 1

Correct Answer: 142.5

Step 1: Write the profit.
\[ \pi=25x+13y-2.5x^2-y^2+xy-10 \]

Step 2: First order condition in x.
\[ \frac{\partial\pi}{\partial x}=25-5x+y=0\;\Rightarrow\;y=5x-25 \]

Step 3: First order condition in y.
\[ \frac{\partial\pi}{\partial y}=13-2y+x=0\;\Rightarrow\;x=2y-13 \]

Step 4: Solve the pair.
Substituting $y=5x-25$ gives $x=2(5x-25)-13$, so $9x=63$ and $x=7$. Then $y=5(7)-25=10$.

Step 5: Put back to get profit.
\[ \pi=175+130-122.5-100+70-10=142.5 \]
\[ \boxed{142.5} \]