Consider a firm in a perfectly competitive market that produces a quantity y of a single goodat known cost c(y). Profit when the price is p is py − c(y). We compare two cases: (i) whenthe price is uncertain, equal to the random variable p˜ (the “uncertainty case”), and (ii) whenthe price is certain, equal to p = E[˜p] for sure (the “certainty case”).a) The firm chooses the level of output before observing the price of output. Theowner of the firm is risk neutral with respect to profits. State the maximization problems forthe two cases (i) and (ii), and determine whether or not the two problems are equivalent. Howdo the solutions and values of the two maximization problems compare?(b) The firm chooses the level of output before observing the price of output. Theowner of the firm is a risk-averse expected utility maximizer with respect to profits, for astrictly increasing and strictly concave utility function u. That is, utility when output is y andthe price is p is u(py −c(y)). State the utility maximization problem for case (i) and case (ii).For which case is the maximization problem equivalent to when the owner is risk neutral?(c) The firm chooses output after observing the price. The firm is risk neutral withrespect to profit. Viewed from before observing the price, the decision problem of the firmis to choose a plan y : R+ → R+ which states the level of output y(p) as a function of theobserved price p. Assuming only that each problem has a solution, show that the (ex-ante)value of the problem in the uncertainty case is higher than that of the problem in the certaintycase. Show that the values are the same only if, in the uncertainty case, there is a solution inwhich the output level does not depend on the price.
Consider a firm in a
at known cost c(y). Profit when the
the price is uncertain, equal to the random variable p˜ (the “uncertainty case”), and (ii) when
the price is certain, equal to p = E[˜p] for sure (the “certainty case”).
a) The firm chooses the level of output before observing the price of output. The
owner of the firm is risk neutral with respect to profits. State the maximization problems for
the two cases (i) and (ii), and determine whether or not the two problems are equivalent. How
do the solutions and values of the two maximization problems compare?
(b) The firm chooses the level of output before observing the price of output. The
owner of the firm is a risk-averse expected utility maximizer with respect to profits, for a
strictly increasing and strictly concave utility function u. That is, utility when output is y and
the price is p is u(py −c(y)). State the utility maximization problem for case (i) and case (ii).
For which case is the maximization problem equivalent to when the owner is risk neutral?
(c) The firm chooses output after observing the price. The firm is risk neutral with
respect to profit. Viewed from before observing the price, the decision problem of the firm
is to choose a plan y : R+ → R+ which states the level of output y(p) as a function of the
observed price p. Assuming only that each problem has a solution, show that the (ex-ante)
value of the problem in the uncertainty case is higher than that of the problem in the certainty
case. Show that the values are the same only if, in the uncertainty case, there is a solution in
which the output level does not depend on the price.
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