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b) What is MP1, MP2, and the Technical Rate of Substitution?
c) Does this production demonstrate diminishing marginal product for x1? Why?
d) x1 costs the firm $2 per unit, x2 costs the firm $5 per unit. Factor x1is fixed in the short run at 30 units. Write down the profit maximization problem of the firm (no solving required).
e) What is the profit-maximizing (optimal) amount of x2employed by the firm from problem in part (d)?
f) What happens to the answer to part (e) if unit cost of x2decreases to $3? g) What is the firms profit for the problem in part (d)?
h) Now suppose factor x1is no longer fixed, and the firm can control both
x1 and x2in the long run. Write down the profit maximization problem and the optimality conditions.
i) Find the optimal usage of x1 and x2for part (h)
j) Find the profit of the firm for the problem in part (h) k) compare the answer to part (g) to that of part (j)
2 Question 2
Consider the cost minimization problem: MinC = w1x1 + w2x2
s.t. f(x ,x ) = 50×0.3×0.7 = y
a) Suppose the firm must produce 100 outputs. Write down the Lagrangian for the problem