Question

A firm has production function 𝑦=𝐾 ^0.5𝐿 ^0.5 and faces wage rate 𝑤=4 and rental rate of capital 𝑟=4. The firm’s marginal cost is equal to ______ (in integer).

Answer 1

Correct Answer: 8

Given the production function: \[ y = K^{0.5} L^{0.5} \] This is a **Cobb–Douglas function with constant returns to scale**. Factor prices: \[ w = 4,\quad r = 4 \] 
Step 1 — Cost-minimizing input ratio
For a Cobb–Douglas function: \[ \frac{K}{L} = \frac{w}{r} \] Since \( w = r = 4 \): \[ \frac{K}{L} = 1 \quad \Rightarrow \quad K = L \] 
Step 2 — Substitute into production function
\[ y = K^{0.5} K^{0.5} = K \] So: \[ K = y,\quad L = y \] 
Step 3 — Total cost
\[ C = rK + wL = 4y + 4y = 8y \]
 Step 4 — Marginal cost
\[ MC = \frac{dC}{dy} = 8 \] 
Final Answer: 8