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Question 1. Costs and Productivity The world of cauldron manufacturing is as murky and obscure as the brews they're used for. A typical cauldron manufacturing rm requires workers and heavy machinary to manipulate the steel. Their production flmction can be understood according to the following q=LaKb where L the labor input with a price of w and K is the capital input with a price of 7'. Suppose in the short run K = 16 is xed and L is variable. Part a) Compute the marginal product of labor (M PL) and the average product of labor (APL). Is the M PL increasing or decreasing as L increases? Is the M PL greater than, less than, or equal to the APL? Part b) Find expressions for the xed cost (F0), variable cost (VC), average cost (AC), and marginal cost (M C). These should all be functions of some combination of q, w, and 1'. Part c) Find the level of output (q) as a function of w and 7' at the point where AC is minimized. Part d) Suppose a $10 subsidy per-unit is imposed, what are the new cost mctions (FC, VC', AC, and MC)? Hint: Think about how a per-unit tax affects the supply curve, but think about it in terms of how it may aect cost curves. Question 2. Firm Optimization