Mathematic-econ

Problem Set 3 (Ch. 3 & 4)

1. (4 points) Let () = 3.
   a. Is  an increasing function everywhere?
   b. Discuss the curvature of . Specifically, for what values of  that 
   is concave and for what values of  that  is convex?
2. (4 points) Let  = ()=( − 27) 1
   3 + 3 be a firm’s short-run production
   function where  denotes the output level and  denotes the labor input.
   a. Derive the function of marginal product of labor ().
   b. Does this production technology display diminishing marginal product everywhere? Explain.
3. (Follow the previous question.) Assume the cost of labor is $1 per unit
   and the firm has a fixed cost of $100.
   a. (3 points) Derive this firm’s short-run cost function ()
   b. (2 points) Is () increasing everywhere?
   c. (2 points) Discuss the curvature of () just like in 1.b.
4. (4 points) Suppose () = 3 + 3
   22 − 6 + 9 Use F.O.C. and S.O.C. to
   find the local maximum and local minimum.
5. (6 points) A perfectly competitive firm has a cost function () = 1
   3 3 −
   122 + 150 + 600 and the market price of its output is 70.
   a. Formulate this firm’s profit maximization problem.
   b. Solve the firm’s optimal quantity.
   c. Calculate the resulting profit level.
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Sample Solution