Question: Suppose ZiLog Incorporated develops 8-bit micrologic semiconductor devices. The company has now developed a new 12-bit micrologic device. On of its customers, GSUTex wishes to ascertain the technical efficiency of the new device and plans to order three 8-bit devices and two 10-bit devices. They will utilize the two micrologic semiconductor devices to test the productivity gains for is cost minimizing strategy. The Cost Analysis Staff at GSUTex, a group of newly minted MBA’s have determined that the firm’s cost function is given as

\[C\left( x,y \right)=400+{{x}^{\frac{1}{4}}}{{x}^{\frac{1}{4}}}{{y}^{\frac{1}{2}}}=400{{x}^{\frac{1}{2}}}{{y}^{\frac{1}{2}}}\]

where \(C\left( x,y \right)\) is the multi-product cost function for ZiLog; x, the 8-bit device, and y, the 10-bit device. The company plan to purchase 10 devices.

What is the firms Lagrangian specification?