How much shoplifting should we allow? An example of production and cost analysis

If you are looking for some practice you should check out the following example. 

Production relationships

Let's look at the following simple example.  . A local retailer is experiencing theft problems at its new super store and has decided to hire security guards. The firm wants to minimize the total cost of thefts. The following table shows how the number of security guards affects the number of jewelry pieces per week.  Use these data to calculate the average and marginal productivity of the guards. [Hint: the product of the jewelry guards is the product 'saved' (not stolen)]

Cost relationships

Let's return to our local retailer who is experiencing theft problems at its new super store and who has decided to hire security guards. The firm wants to calculate the costs of saving the jewelry from shoplifting to be able to answer questions such as: how much does it cost me to 'save' 20 items from being stolen?   The following table shows how the number of security guards affects the number of jewelry pieces per week.  You are to fill in the table with the cost variables ( Total, Variable and Fixed Costs and Average), the average cost variables (Average total cost, Average variable cost, and Average fixed cost), and Marginal Cost.  It will cost you $200 for a camera which will be needed and $200 per guard.

Once you have filled in the table you should construct a graph of the total relationships (total cost, variable cost, fixed cost on the vertical axis and output (pieces saved) on the horizontal axis).  Also construct a graph of the average and marginal relationships (average cost, average variable cost, average fixed cost, marginal cost against pieces of jewelry saved).

Revenue and profit relationships and profit maximization: choice of output

The table below contains the data that you calculated above in the cost part of the example.  What is new is the final columns that provides you with data on the price at which you could sell the 'saved' items.  For example, if 30 items were saved, the 30 items could be sold at a price of $23.  If 36 items were saved, the price that they could be sold at would drop to $22. With these data you are to fill in the information in the remaining columns.  This would include total revenue (TR), marginal revenue (MR), and profit. Once you have filled in these columns you should be able to determine the profit maximizing level of output (units saved).  Once you have filled in the table you might also want to construct a graph of profit and a graph of marginal cost and marginal revenue. 

Revenue and profit relationships and profit maximization: choice of input

to see how things change when we look at the input market let's return to our local retailer who is experiencing theft problems.  The following table shows how the number of security guards affects the number of jewelry pieces per week.

To move any further with the problem we need some additional information - the price of the security guards (inputs0 and the price of the saved jewelry (output).  For this problem assume each security guard is paid $200 a week, the fixed cost of the security guards (camera) is $200, and the cost of a stolen piece of jewelry is $25, how many security guards should the firm hire? Also assume that the cost of a stolen piece of jewelry is $25.  based on these data you should be able to fill in the table below and determine the profit maximizing level of guards. And you have done that you might want to look at how things change when each security guard is paid $200 a week and the cost of a stolen piece of jewelry is $50, how many security guards should the firm hire.    

Revenue and profit relationships and profit maximization: perfect competition

What is new is the final column which provides data on the price you could sell the 'saved' items at.  For example, if 30 items were sold, the 30 items could be sold at a price of $25.  If 36 items were saved, the price they could be sold at would be $25. What is new about this is that the price does not change as the output (savings) changes - the essence of perfect competition.  With these data, what would be the optimal level of output?  Is this a long run equilibrium, and if not, what is likely to happen?