Solution to Part 1:

The production function is:

$y = x_1^{7/10} x_2^{3/10}$

The firm is leasing $x_2 = 3$ cubicles. Plugging $x_2 = 3$ into the production function:

$y = x_1^{7/10} \cdot 3^{3/10}$

Let $C = 3^{3/10}$, which is a constant. The simplified production function is:

$y = C \cdot x_1^{7/10}$

The firm operates in a competitive market, so its goal is to maximize profits. The profit function is:

$\pi = P \cdot y - \omega_1 \cdot x_1$

Substituting $y = C \cdot x_1^{7/10}$:

$\pi = P \cdot C \cdot x_1^{7/10} - \omega_1 \cdot x_1$

To maximize profit, take the derivative of $\pi$ with respect to $x_1$, set it equal to zero, and solve for $x_1$:

$\frac{d\pi}{dx_1} = P \cdot C \cdot \frac{7}{10} \cdot x_1^{-3/10} - \omega_1 = 0$

Rearranging:

$P \cdot C \cdot \frac{7}{10} \cdot x_1^{-3/10} = \omega_1$

Simplify for $x_1$:

$x_1^{3/10} = \frac{P \cdot C \cdot 7}{10 \cdot \omega_1}$

$x_1 = \left( \frac{P \cdot C \cdot 7}{10 \cdot \omega_1} \right)^{10/3}$

Recall $C = 3^{3/10}$. Substituting $C$:

$x_1 = \left( \frac{P \cdot 3^{3/10} \cdot 7}{10 \cdot \omega_1} \right)^{10/3}$

The firm's demand function for employees depends on the relationship between $P$ and $\omega_1$. Thus, the correct answer for Part 1 is:

$\boxed{B. \ x^*_1 = \left(\frac{P \cdot 0.97}{\omega_1}\right)^{-3.33}}$

---

Solution to Part 2:

Given:

• $P = 40$
• $\omega_1 = 20$

Substitute these values into the demand function derived:

$x_1 = \left( \frac{P \cdot 3^{3/10} \cdot 7}{10 \cdot \omega_1} \right)^{10/3}$

First, compute $3^{3/10}$:

$3^{3/10} \approx 1.231$

Substitute into the equation:

$x_1 = \left( \frac{40 \cdot 1.231 \cdot 7}{10 \cdot 20} \right)^{10/3}$

Simplify the numerator and denominator:

$x_1 = \left( \frac{344.68}{200} \right)^{10/3}$

$x_1 = \left( 1.7234 \right)^{10/3}$

Calculate $1.7234^{10/3}$:

$x_1 \approx 3.42$

Thus, the firm will hire approximately 3.42 employees.

---

Final Answers:

1. Part 1: $\boxed{B}$
2. Part 2: $\boxed{3.42 \ \text{employees}}$

---

Would you like further explanation on any steps or related concepts? Here are 5 related questions you might explore:

1. How does the elasticity of substitution affect input demand functions?
2. What happens to the firm's labor demand if the price of output changes?
3. How would the firm adjust its production function if it could lease more cubicles?
4. Can we derive a similar demand function for cubicles if labor is fixed?
5. How do changes in cost parameters ($\omega_1$) influence optimal hiring?

Tip: Understanding how marginal productivity interacts with costs is crucial in analyzing profit-maximizing behavior in firms.