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Now assume that instead of a monopoly x is produced in a perfectly competitive market. Write the profit maximization problem for a competitive firm in this market. Explain how this problem differs from 2a. 

Write down the Lagrangian. (Use L to represent the Lagrangian and m for the multiplier; if you need more multipliers label them as m1, m2, m3, … etc.) 

Write down the complementary slackness conditions. 

Write down the complementary slackness conditions. 

Assume there is perfect competition with a tax of $2 per unit of x. What is the deadweight loss of this tax?

Provide a formula for J and K. Note that these are values on the graph’s axis. Hint: The solution must be in terms of the problem’s parameters. To solve this problem, you will use the production level (y), scalars (a, b ,c, d, and e), and input prices (p1 and p2). You may use one or a combination of such parameters. The final formulas must not show x1 nor x2.

How does the production level y1 compare to y2? Is it great, smaller, or equal?  How about y4 and y3? Justify your answer based on the isoquant graphs.

The following description will be used in questions 1a to 1d and for bonus question I. Consider two distinct production functions (f1 and f2) that use two inputs (x1 and x2) to produce one output (y). Such functions have the form f1 = min (a*x1 , b*x2) and f2= c*x1+d*x2, where a,b,c and d are positive scalars, p1 is the cost of one unit of input x1, p2 is the cost of one unit of input x2, and y represents the production level.     

How does the production level y1 compare to y2? Is it great, smaller, or equal?  How about y4 and y3? Justify your answer based on the isoquant graphs.

After solving 2d, you can find that the market will offer 40 units of x at a price of 20. As expected, the competitive market can offer x at a lower price than the monopoly. Calculate the dead weight loss from the monopoly.