Define the term ‘Shepard’s lemma’. Assume that the production function of a producer is given by Q=5L0.5 K0.3, where Q,L and K denote output, labour and capital respectively. If labour cost ₹ 1 per unit and capital ₹2, find the least cost combination of inputs (L&K)

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Introduction

In microeconomics, cost minimization is an essential concept where a firm aims to produce a given level of output at the lowest possible cost. Shepard’s Lemma is a theoretical tool that helps in deriving input demand functions from the cost function. In this answer, we will define Shepard’s Lemma and apply it to the given production function to find the least-cost combination of labour and capital when input prices are provided.

Definition: Shepard’s Lemma

Shepard’s Lemma states that the partial derivative of the cost function with respect to the price of an input gives the conditional demand for that input. In other words, if C(w, r, Q) is the cost function, then:

∂C/∂w = L(w, r, Q): The conditional demand for labour
∂C/∂r = K(w, r, Q): The conditional demand for capital

This lemma is crucial in duality theory, where we analyze production using both the primal (production) and dual (cost) functions.

Given Data

Production Function: Q = 5L^0.5 × K^0.3
Price of Labour (w) = ₹1 per unit
Price of Capital (r) = ₹2 per unit

We are asked to find the least-cost combination of L and K to produce a given level of output using cost minimization.

Step 1: Cost Function and Lagrangian Setup

The total cost function is:

C = wL + rK = 1×L + 2×K = L + 2K

We minimize cost C subject to the production function constraint Q = 5L^0.5 K^0.3

Using the method of Lagrange multipliers:

Lagrangian: 𝓛 = L + 2K + λ(Q – 5L^0.5 K^0.3)

Step 2: First Order Conditions

∂𝓛/∂L = 1 – λ × 5 × 0.5 × L^-0.5 × K^0.3 = 0

∂𝓛/∂K = 2 – λ × 5 × 0.3 × L^0.5 × K^-0.7 = 0

∂𝓛/∂λ = Q – 5L^0.5 × K^0.3 = 0

Step 3: Solve the Equations

From the first two equations, eliminate λ:

(1) 1 = λ × 2.5 × L^-0.5 × K^0.3
(2) 2 = λ × 1.5 × L^0.5 × K^-0.7

Divide (1) by (2):

(1)/(2): 1/2 = (2.5 / 1.5) × (L^-1) × (K¹)

Simplify:

1/2 = (5/3) × (K / L)

⇒ K / L = 3 / 10

⇒ K = (3/10) × L

Step 4: Plug Back into Production Function

Q = 5L^0.5 × K^0.3

Substitute K = 0.3L:

Q = 5L^0.5 × (0.3L)^0.3

Q = 5 × L^0.5 × 0.3^0.3 × L^0.3 = 5 × 0.3^0.3 × L^0.8

Let’s assume Q = 10 (for calculation purposes):

10 = 5 × 0.3^0.3 × L^0.8

0.3^0.3 ≈ 0.697

So, 10 = 5 × 0.697 × L^0.8 = 3.485 × L^0.8

⇒ L^0.8 = 10 / 3.485 ≈ 2.87

⇒ L = (2.87)^1.25 ≈ 4.64

⇒ K = 0.3 × 4.64 ≈ 1.39

Step 5: Least Cost

Total Cost = L + 2K = 4.64 + 2 × 1.39 = 4.64 + 2.78 = ₹7.42

Least-cost combination: L ≈ 4.64 units, K ≈ 1.39 units, Total Cost ≈ ₹7.42

Conclusion

Shepard’s Lemma helps in deriving conditional input demands from the cost function. By applying this concept, we were able to find that for the given production function Q = 5L^0.5K^0.3 and input prices (Labour ₹1, Capital ₹2), the least cost combination to produce 10 units of output is approximately 4.64 units of labour and 1.39 units of capital. The total cost incurred is approximately ₹7.42. This analysis is a useful tool in microeconomic decision-making for firms focused on cost-efficiency.