Unit 2: Cost and Production Function

1. Homogenous and Homothetic Production Functions

Production functions describe the technical relationship between inputs and outputs. For ECODSC 251, we focus on specific mathematical properties of these functions.

Homogenous Production Function

A production function is said to be homogenous of degree 'n' if multiplying all inputs by a constant factor 'k' results in the output being multiplied by 'k' raised to the power of 'n'.

Formula: f(kL, kK) = k^n . f(L, K)

• If n = 1: Constant Returns to Scale (CRS).
• If n > 1: Increasing Returns to Scale (IRS).
• If n < 1: Decreasing Returns to Scale (DRS).

Homothetic Production Function

A function is homothetic if it is a monotonic transformation of a homogenous function. In these functions, the slope of the isoquant (Marginal Rate of Technical Substitution - MRTS) remains constant along any ray from the origin.

2. Cobb-Douglas Production Function

The Cobb-Douglas function is widely used to represent the relationship between inputs, typically Labour and Capital, and the resulting output.

Standard Formula: Q = A . L^a . K^b

Where:

• Q: Total Output.
• L, K: Quantities of Labour and Capital.
• A: Efficiency parameter.
• a, b: Output elasticities of inputs.

Key Properties:

• Returns to Scale are determined by the sum of (a + b).
• The elasticity of substitution for a Cobb-Douglas function is always constant and equal to 1.

3. C.E.S. Production Function

The Constant Elasticity of Substitution (C.E.S.) production function allows the elasticity of substitution to be a constant value other than one.

Formula: Q = A [ alpha . K^(-rho) + (1 - alpha) . L^(-rho) ] ^ (-1/rho)

In this function, the elasticity of substitution (sigma) is defined as sigma = 1 / (1 + rho).

4. Elasticity of Factor Substitution

The Elasticity of Factor Substitution measures the ease with which a firm can substitute one factor for another (e.g., Labour for Capital) when relative factor prices change.

• It is defined as the percentage change in the capital-labour ratio divided by the percentage change in the MRTS.
• It helps determine the curvature of the isoquant.

5. Expansion Path

The Expansion Path is a line representing the cost-minimizing combinations of inputs for every possible level of output when input prices (wages and rental rates) are held constant.

• It shows how the firm expands its production in the long run.
• For homogenous functions, the expansion path is a straight line through the origin.

6. Derivation of Cost Function from Production Function

The cost function describes the relationship between the minimum cost of production and the level of output.

Mathematical Derivation Process:

1. State the Production Function: Q = f(L, K).
2. Define the Cost Equation: C = wL + rK.
3. Optimize using the Lagrangian method or by equating MRTS to the factor price ratio (w/r).
4. Solve for optimal L and K in terms of Q, w, and r.
5. Substitute L* and K* back into the cost equation to obtain C(Q).