Answer to Question #209422 in Financial Math for Zozo

Question

Kusho Industries produces and sells computer chips. Its (hourly) production function is 𝒒 = 𝟒𝑲𝟎.𝟒𝑳𝟎.𝟔, while its (hourly) cost function is 𝒄 = 𝟐𝟎𝑳 + 𝟖𝟎𝑲. Furthermore, Kusho must produce 𝒒𝟎 = 𝟒𝟎𝟎 computer chips per hour.

a. Which levels of 𝑳 and 𝑲 satisfy the first-order conditions for the constrained minimisation of Kusho’s

cost? Use the Lagrange Multiplier (LM) method. Also, find and interpret the value of the Lagrange

multiplier (𝝀). [8]

b. Show that 𝑴𝑹𝑻𝑺 = 𝒘 at the constrained cost minimising levels of 𝑳 and 𝑲 obtained above. [2]

Answer 1

As per Lagrange method, the cost is minimized subject to production function.

 Minimize the$C = 20L + 80K$ subject to $Q = 4 K^{0.4}L^{0.6}$

Lagrange Function:

$R= 20L + 80K - λ(4 K^{0.4}L^{0.6} – Q)$

Maximization of R with respect to L and K,

$R_L = 20 – λ (4 \times 0.6 L{-0.4}K^{0.4})$

$R_L = 0$

$λ = (\frac{25}{3}) (\frac{L}{K})^{0.4} ............(1)$

$R_K = 80 – λ (4 \times 0.4 L^{0.6} K{-0.6})$

$R_K = 0\\ λ = 50 (\frac{K}{L})^{0.6}..................(2)\\ R λ = - λ (4 K^{0.4}L^{0.6} – Q)\\ R λ = 0\\ Q = 4 K^{0.4}L^{0.6} ....................(3)\\$

Equating both value of λ as follows:

$L = 6K ..................(4)$

Putting equation (4) in equation (3) as follows:

$Q = 4 K^{0.4}L^{0.6}\\ 400 = 4 K^{0.4}L^{0.6}\\ 400 = 4 K^{0.4}(6K)^{0.6}\\ K = 34.12\\ L = 6 \times 34.12\\ = 204.76$

The value of Lagrange multiplier is calculated as follows:

$λ = 50 (\frac{K}{L})^{0.6}\\ = 50 \times 0.3412\\ = 17.06$

It shows the change in output due to one unit change in labor or capital.

b.

The MRTS is calculated as follows:

$MRTS = \frac{MP_L}{MP_K}\\ = \frac{3K}{2L}\\ = 3 \times \frac{34.12}{2}\times204.76\\ = 0.25$

The value of MRTS will be equal to ratio of wage rate and rental rate.

$\frac{w}{r} = \frac{20}{80}\\ = 0.25$

At equilibrium, the MRTS is equal to ratio of wage rate and rental rate.