Profit Maximisation in Perfect Competiti

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Profit Maximisation in Perfect Competition Objective: Maximise profit ($\pi$), which is the difference between Total Revenue (TR) and Total Cost (TC). $\pi = TR - TC$ 1. Total Revenue (TR) Calculation Given Price $P = £200$ Quantity $Q = 3.1K^{0.3}L^{0.25}$ $TR = P \times Q = 200 \times (3.1K^{0.3}L^{0.25}) = 620K^{0.3}L^{0.25}$ 2. Total Cost (TC) Calculation Cost of Capital $P_K = £42$ Cost of Labour $P_L = £5$ $TC = (P_K \times K) + (P_L \times L) = 42K + 5L$ 3. Profit Equation Formulation $\pi = TR - TC = 620K^{0.3}L^{0.25} - 42K - 5L$ 4. First-Order Conditions for Profit Maximisation Set partial derivatives of $\pi$ with respect to each input to zero. For Capital (K): $\frac{\partial \pi}{\partial K} = 0.3 \times 620K^{(0.3-1)}L^{0.25} - 42 = 0$ $186K^{-0.7}L^{0.25} = 42$ (Equation 1) For Labour (L): $\frac{\partial \pi}{\partial L} = 0.25 \times 620K^{0.3}L^{(0.25-1)} - 5 = 0$ $155K^{0.3}L^{-0.75} = 5$ (Equation 2) 5. Determine Optimal Input Ratio Divide Equation 1 by Equation 2: $\frac{186K^{-0.7}L^{0.25}}{155K^{0.3}L^{-0.75}} = \frac{42}{5}$ $1.2 \times \frac{L^{0.25}L^{0.75}}{K^{0.3}K^{0.7}} = 8.4$ $1.2 \times \frac{L}{K} = 8.4$ $\frac{L}{K} = \frac{8.4}{1.2} = 7$ $\mathbf{L = 7K}$ 6. Solve for Specific Values of K and L Substitute $L = 7K$ into Equation 1: $186K^{-0.7}(7K)^{0.25} = 42$ $186K^{-0.7} \times 7^{0.25} \times K^{0.25} = 42$ $186 \times 7^{0.25} \times K^{(-0.7 + 0.25)} = 42$ $186 \times 1.6266 \times K^{-0.45} = 42$ $302.55 \times K^{-0.45} = 42$ $K^{-0.45} = \frac{42}{302.55} \approx 0.1388$ $K = (0.1388)^{1/(-0.45)} = (0.1388)^{-2.222...} \approx \mathbf{80.5}$ units Now, calculate L using $L = 7K$: $L = 7 \times 80.5 \approx \mathbf{563.5}$ units Conclusion To maximise profit, the firm should use approximately 80.5 units of Capital (K) and 563.5 units of Labour (L) .