Question

Given: Q = 20K0.3 L0.7, C* = $1,000, r = $30 and w = $10. Determine the amount of labor and capital that maximizes output and determine the optimal amount of output. Calculate and interpret the Lagrangian multiplier.

Answer 1

$$30K+10L=1000$$

$$Q=20K^{0.3}L^{0.7}+\lambda (1000-30K-10L)$$

$$\frac{\delta Q}{\delta K}=6(\frac{L}{K})^{0.7}-30 \lambda$$

$$\frac {\delta Q}{\delta L}=14(\frac{K}{L})^{0.3}-10\lambda$$

$$L=7K$$

$$K=10$$

$$L=70$$

$$\lambda=0.78$$