### Determination of production function

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You have been hired as a plant manager for a firm that produces widgets (Q) in Angola, Indiana. Widget production requires machine time (K) and labor time (L). The factory works three eight-hour shifts each day. Below is input and output data from the last twenty-one shifts:

 Shift Widgets Machine Hours Labor Hours 1 4067 2540 1029 2 4313 2786 1042 3 4493 2757 1165 4 4503 2397 1421 5 4066 2754 920 6 4641 2364 1556 7 3929 2604 916 8 4222 2530 1131 9 4272 2852 986 10 4728 2637 1399 11 4310 2724 1073 12 3980 2430 1039 13 5011 2773 1499 14 4495 2492 1341 15 3971 2433 1032 16 4597 2652 1298 17 4662 2392 1548 18 4533 2296 1532 19 4400 2722 1128 20 4392 2829 1065 21 4715 2511 1487

1. Your engineers tell you that the marginal product of both labor and machine time diminishes as the volume of production expands during any one shift. What is your production function? You should round all parameters to exactly two decimal places.The chief operating officer of the company asks you to meet a production goal of 4800 widgets during the next shift.

2. In a table, calculate how many machine hours are required to meet the Q = 4800 production goal if L=300, L=400, L=500, L=600, L=700, L=800, L=900, and L=1000.  Graph the isoquant that these calculations imply.  Explain in very clear and complete terms why the isoquant has the shape that you observe.

3. The unit depreciation expense of one machine hour is \$130 and the expense of one labor hour is \$65.  What combination of machine and labor hours minimizes the cost of producing 4800 widgets?  Do you believe that the firm should make production at Angola more or less capital intensive?  Explain your reasoning in full detail.

4. The firm is considering a movement of the plant to Shenzen, China where labor is cheaper.  The same mathematical relationship between inputs and outputs will hold.  A new plant in Shenzen would enjoy a one hour expense for labor of \$40.  Because of a scarcity of modern equipment, the expense of one machine hour would jump to \$320.  Can the move to China be financially justified?  Theoretically, how will the mix of inputs in Shenzen compare to the mix of inputs in Angola?

For political reasons, the plant is not moved to China.  You receive news that certain machinery is unavailable for the foreseeable future.  Due to unscheduled changeover work, use of machinery is constrained to 1500 hours each shift.  This news comes at a bad time.  The COO tells you that market demand for output has dramatically grown.  The equilibrium price for a widget has risen to \$160.

5. You are told to produce a quantity that maximizes profit. How many units do you produce and what is your profit?  How many machine and labor hours are used in production.

6. At this level of profit-maximizing output, what is your marginal cost, average variable cost, and average fixed cost of production?

7. The COO notifies all plant managers in the company of a simplified bonus program.  A plant manager's bonus will be tied to one measurement - average production cost.  The lower a plant's average production cost, the higher the bonus.  Is this type of bonus structure in the interest of the company?  Use theoretical and graphical insights from chapter five of the textbook to explain your reasoning.

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