Since you know all about perfect competition, monopoly, and oligopoly, we can find out how various types of firms might feel about uncertainty concerning the prices of its factors of production and output. Consider a profit-maximizing firm that produces a single good from several factors. The firm is characterized by a production function y = f(x1, ... , Xn), where y is the level of output obtainable from factor inputs x1, ... , Xn. We will use p to denote the price of the output good, and Wi to denote the price of factor input i. When there is uncertainty a priori about these prices, the firm is allowed to choose its production plan after any uncertainty in prices resolves. (c) Finally, consider this question in the context of a von Stackelberg duopoly. Two firms produce an undifferentiated commodity for which demand is given by P = A - X, where P is price and X is total supply. Demand-is unchanging. Each firm has production technology with a fixed cost F for producing anything at all, and a (constant) variable cost c for producing every unit. Note well that the fixed cost F can be avoided by producing zero. Firm 1 moves first, deciding whether to produce anything and, if it does produce, the amount xi that it will supply. Firm 2 sees the decisions of firm 1, and then it decides whether to produce anything at all and, if so, how much. (In cases where firm 2 is indifferent between two courses of action, you may assume that it acts in a way that maximizes the profits of firm 1.) The firms regard the "factor costs" F and c as given to them; their actions do not affect those costs. Is the result analogous to the result in part (a) of this problem (and in part (b), if it is correct there) correct in this case for firm 1? That is, for all (F, c) and (F', d), is the firm's average of profits given those two levels of costs at least as large as its level of profits when costs are ((F + F') /2, (c+ c') /2)? If so, can you conjecture how general the general proposition at work here is? If not, can you succinctly state what went wrong? (One way to do this is to give the "general proposition" in a fashion that applies to those cases [(a) and, perhaps, (b)] where the result is true, and fails to apply in this case.)
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(c) Finally, consider this question in the context of a von Stackelberg duopoly. Two firms produce an undifferentiated commodity for which demand is given by P = A - X, where P is price and X is total supply. Demand-is unchanging. Each firm has production technology with a fixed cost F for producing anything at all, and a (constant) variable cost c for producing every unit. Note well that the fixed cost F can be avoided by producing zero. Firm 1 moves first, deciding whether to produce anything and, if it does produce, the amount xi that it will supply. Firm 2 sees the decisions of firm 1, and then it decides whether to produce anything at all and, if so, how much. (In cases where firm 2 is indifferent between two courses of action, you may assume that it acts in a way that maximizes the profits of firm 1.) The firms regard the "factor costs" F and c as given to them; their actions do not affect those costs.
Is the result analogous to the result in part (a) of this problem (and in part (b), if it is correct there) correct in this case for firm 1? That is, for all (F, c) and (F', d), is the firm's average of profits given those two levels of costs at least as large as its level of profits when costs are ((F + F') /2, (c+ c') /2)?
If so, can you conjecture how general the general proposition at work here is? If not, can you succinctly state what went wrong? (One way to do this is to give the "general proposition" in a fashion that applies to those cases [(a) and, perhaps, (b)] where the result is true, and fails to apply in this case.)
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