“The conventional analysis of profit maximisation breaks down if the entrepreneur sells his output and possesses a production function which is homogenous of degree one.” Explain.

Understanding the Core Concepts

Profit Maximization: In traditional economic theory, profit maximization occurs where Marginal Revenue (MR) equals Marginal Cost (MC). This implies firms have some degree of control over price, particularly in imperfectly competitive markets. The firm actively chooses the output level that yields the highest difference between total revenue and total cost.

Homogenous Production Function of Degree One: A production function is said to be homogenous of degree one if, when all inputs are increased by a certain proportion, the output increases by the same proportion. Mathematically, this is represented as: Q = f(L, K), where if L is multiplied by 't' and K is multiplied by 't', then Q is also multiplied by 't'. This implies constant returns to scale. A common example is the Cobb-Douglas production function: Q = A * L^α * K^β, where α + β = 1.

The Breakdown of Conventional Analysis

When a firm operates with a production function homogenous of degree one and sells its output in a perfectly competitive market, several key changes occur:

• Price Taker: In a perfectly competitive market, the firm is a price taker. It cannot influence the market price and must accept the prevailing price.
• Marginal Revenue Equals Price: Because the firm is a price taker, its Marginal Revenue (MR) is equal to the market price (P). MR = P.
• Profit Maximization Condition: The profit maximization condition becomes MR = MC, which simplifies to P = MC.
• The Critical Issue: The production function being homogenous of degree one means that the firm’s cost structure is directly linked to its output level. As output increases, costs increase proportionally. This means the firm’s MC curve is perfectly elastic (horizontal) at the market price.

Implications for Supply Curve and Firm Behavior

The combination of P=MC and a perfectly elastic MC curve has significant implications:

• Supply Curve: The firm’s supply curve becomes perfectly elastic at the market price. It will supply any quantity demanded at that price.
• No Active Decision-Making: The firm doesn’t actively *choose* its output level to maximize profit in the traditional sense. It simply produces the quantity demanded at the given market price. The price is determined by market forces, not by the firm’s individual decisions.
• Long-Run Equilibrium: In the long run, economic profits are driven to zero due to free entry and exit. The firm operates at the minimum point of its Average Total Cost (ATC) curve, but this is a result of market forces, not active profit maximization.

Mathematical Illustration

Let's consider a simple Cobb-Douglas production function: Q = L^0.5K^0.5. If the firm doubles both labor (L) and capital (K), output (Q) also doubles. If the market price is P, and the firm’s cost of labor and capital are wL and rK respectively, the firm will produce where P = MC. Because of constant returns to scale, the MC curve will be horizontal at the market price, meaning the firm will supply whatever quantity the market demands at that price.

Limitations and Real-World Considerations

While this model provides a theoretical understanding, real-world firms rarely operate under perfectly competitive conditions or with perfectly homogenous production functions. However, it serves as a useful benchmark for understanding the impact of market structure and production characteristics on firm behavior. Factors like product differentiation, barriers to entry, and technological change can all alter the assumptions of this model.