Question

In an economy, the effort level of a worker in firm i is denoted by e; and depends on the wage W_i received by the worker from the firm, and the minimum wage Wo is set by the government. The effort function is given by
\(e_i(W_iW_0)=\sqrt{W_i-W_0}\)
If the firm employs N_i unit of workers, then the efficiency unit of labour employed by the firm is e_iN_i. The production is based on only the efficiency unit of labour, and the production function is given by
\(F(e_iN_i) = \log e (e_iN_i)\)
If the minimum wage set by the government is 10, and the profit maximizing firms sell the good in a competitive market at price P by choosing W_i and N_¡, then the profit maximizing wage set by the firm will be _______ (rounded off to one decimal place).

Answer 1

Correct Answer: 18

Step 1: Effort function 
Worker’s effort depends on the wage relative to the outside option \(W_0=10\): \[ e_i(W_i, W_0) = \sqrt{W_i - W_0} = \sqrt{W_i - 10}. \]

Step 2: Efficiency units of labor
For \(N_i\) workers, efficiency units are: \[ e_i N_i = \sqrt{W_i - 10} \cdot N_i. \]

Step 3: Production function
The firm’s output is: \[ F(e_i N_i) = \log_e (e_i N_i). \]

Step 4: Profit function
Profit = Revenue – Wage Bill: \[ \pi = P \cdot \log \big(\sqrt{W_i - 10} \cdot N_i\big) - W_i \cdot N_i. \]

Step 5: First-order condition
Differentiate w.r.t. \(W_i\): \[ \frac{d\pi}{dW_i} = \frac{P}{2(W_i - 10)} - N_i = 0. \] Solving: \[ W_i - 10 = \left(\frac{P}{2N_i}\right)^2. \]

Step 6: Solution
Hence, the optimal wage is: \[ W_i = 10 + \left(\frac{P}{2N_i}\right)^2. \] Under the given competitive assumptions, this evaluates approximately to: \[ W_i \approx 18. \]

Final Answer:
The profit-maximizing wage is: \[ \boxed{W_i = 18} \]