Question

A competitive firm can sell any output at price 𝑃 = 1. Production depends on capital alone, and the production function 𝑦 = 𝑓(𝐾) is twice continuously differentiable, with
𝑓(0) = 0, 𝑓 ′ > 0, 𝑓 ′′ < 0, \(lim\\_{ 𝐾→0 }\) 𝑓 ′ (𝐾) = ∞ , \(lim\\_{ 𝐾→∞}\) 𝑓 ′ (𝐾) = 0.
The firm has positive capital stock 𝐾̅  to start with, and can buy and sell capital at price 𝑟 per unit of capital. If the firm is maximizing profit then which of the following statements is NOT CORRECT?

• A. If 𝐾̅ is large enough, profit maximizing 𝑦 = 0 and the profit is 𝑟𝐾̅
• B. If 𝑓′(𝐾̅) > 𝑟, the firm will buy additional capital
• C. If 𝑓′(𝐾̅) < 𝑟, the firm will sell some of its capital
• D. If 𝑓 ′(𝐾̅) = 𝑟, the firm will neither buy nor sell any capital

Answer 1

The Correct Option is A

The given problem involves analyzing a firm's decision-making based on its production function and capital market conditions. Let's break down the problem to find which statement is NOT correct.

We have: 

• Price of output \(P = 1\)
• Production function \(y = f(K)\), where \(f'(K) > 0, f''(K) < 0\)
• \(\lim_{K \to 0} f'(K) = \infty\) and \(\lim_{K \to \infty} f'(K) = 0\)
• Capital can be bought or sold at price \(r\) per unit
• Positive initial capital stock \(\bar{K}\)

The firm's profit \(\pi\) is given by:

\(\pi = P \cdot f(K) - r \cdot K\)

Since \(P = 1\), we have:

\(\pi = f(K) - r \cdot K\)

To maximize profit, the firm uses its marginal product of capital condition, \(f'(K) = r\), to decide whether to buy, sell, or hold capital.

Let's consider each option:

1. If \(\bar{K}\) is large enough, profit maximizing \(y = 0\) and the profit is \(r \bar{K}\):
   • If \(\bar{K}\) is very large, \(f'(\bar{K})\) approaches 0, making the capital almost unproductive. Under this situation, the firm would rather sell off all capital and simply gain a profit of \(r \bar{K}\), contradicting the idea that production output \(y=0\) always maximizes profit.
2. If \(f'(\bar{K}) > r\), the firm will buy additional capital:
   • True, when the marginal product of capital is greater than the cost of capital, acquiring more capital increases the firm's profit.
3. If \(f'(\bar{K}) < r\), the firm will sell some of its capital:
   • True, selling capital reduces costs, improving profit when the marginal product is less than the cost of obtaining capital.
4. If \(f'(\bar{K}) = r\), the firm will neither buy nor sell any capital:
   • True, as the firm is already earning the maximum profit per the initial conditions.

Thus, the statement that is NOT correct is: 
If \(\bar{K}\) is large enough, profit maximizing \(y = 0\) and the profit is \(r \bar{K}\) as the firm will always prefer a positive \(y\) if it adds to profits.