Question

A competitive firm’s production function is f(x₁, x₂) = 6x^1/2₁ + 8x^1/2₂. The price of factor 1 is $1 and the price of factor 2 is $4. The price of output is $8. What is the profit-maximizing quantity of output?

a.	416
b.	208
c.	204
d.	419
e.	196

Answer 1

The answer is ' \(\mathbf{b}\).' Find the profit function in terms of the two factor of production.

$$ \begin{aligned} &P=T R-T C \\ &P=8\left(6 x_{1}^{1 / 2}+8 x_{2}^{1 / 2}\right)-\left(1 x_{1}+4 x_{2}\right) \\ &P=48 x_{1}^{1 / 2}+64 x_{2}^{1 / 2}-x_{1}-4 x_{2} \end{aligned} $$

Find the optimal quantity of \(x_{1}\) as follows.

$$ \begin{aligned} \frac{\partial P}{\partial x_{1}} &=0 \\ \frac{48}{2 x_{1}^{1 / 2}}-1 &=0 \\ x_{1} &=576 \end{aligned} $$

Find the optimal quantity of \(x_{2}\) as follows.

$$ \frac{\partial P}{\partial x_{2}}=0 $$

$$ \begin{aligned} \frac{64}{2 x_{2}^{1 / 2}}-4 &=0 \\ x_{2} &=64 \end{aligned} $$

Find the optimal level foutput.

$$ \begin{aligned} &q=6 x_{1}^{1 / 2}+8 x_{2}^{1 / 2} \\ &q=6(576)^{1 / 2}+8(64)^{1 / 2} \\ &q=144+64 \\ &q=208 \end{aligned} $$