(See Solution) 4. Suppose ZiLog Incorporated develops 8-bit micrologic - #80071
PART II
4. 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.
4.1 What is the firm’s objective function? Also describe the two terms in the firm’s objective function
4.2 What is the firm’s constraint?
4.3 What is the firms Lagrangian specification?
4.4 Determine the quantity of x and y that minimizes the firm’s total cost.
4.5 Define and interpret the Lagrange multiplier.
4.6 Why would one want to use this method and what advantages does it offer as an applied tool?
6. Suppose that Zilog finds that its marginal cost for an upcoming new 16-bit micrologic device under construction is $10 and the firm’s Customer Demand is
\[{{Q}^{D}}=100{{P}^{-1.5}}P{{R}^{-0.5}}{{I}^{2}}\]
where Q is the quantity demanded of the micrologic devices; P is the price; PR is the price of a related good, and I is the per capita disposable income.
6.1 What type of function is the firm’s demand curve and why is it of practical value to demand applications?
6.2 Determine the firm’s optimal price
6.3 Is PR a complement or substitute?
6.4 Suppose that per capita disposable income is supposed to increase by 10% and inflation by 3.5%. How would you incorporate such results in your analysis?
Part III.
7. Show that the cross partial derivatives for the following function are equal and explain why
\[Q\left( x,y \right)={{x}^{1/2}}\left( 1+\frac{x}{y} \right)={{x}^{1/2}}+\frac{{{x}^{3/2}}}{y}\]
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