You work for a specialty precision parts firm that has a 30% market share in the industry. According to your market research, the total demand for the product you manufacture will be around 200,000. You are tasked to determine the best allocation of resources given the following cost function: 

TC = 40 K + 10 L

You have also estimated the production function and came up with the following:

Q = 10 K ^0.5 L ^0.5

a) What is the optimal level of labor (L) and capital (K) which will allow you to minimize cost at the production level at which you can sell your output? What is your optimal total cost? Make sure you solve for lambda (Lagrangian multiplier).

b) Your main competitor's plant experiences some major problems forcing it to shut down. This competitor had a 20% market share in the industry. The management of your company sees this as an opportunity to grab more market share for your firm and increase it from 30% to 50%. Assuming that you have the capacity to increase production, how does this increase affect your total cost? Explain how you got to your answer. Please be as thorough as possible in order for you to get the most points.

c) Based on new research, that has recently become available, the president of the company believes that the market will expand by 50% (use the situation in part a) above as an initial reference point for this question) and there will be a scramble among all firms in this market to grab as much market share for the additional demand. The president has instructed the plant manager to ramp up production and produce as much output as possible since he believes that the company will be able to sell whatever it can produce. The president sets a production budget equal to the calculated total cost you calculated in a) above. How much output will your firm be able to produce? Compare your results with what you got above and explain your results.

d) Based on your answer in c) above, use graphical analysis and the concept of isoquants and isocosts to illustrate why you got the answer you arrived at. Draw out your answer and explain in a detailed fashion why your answer makes sense.