This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Northwestern University Marciano Siniscalchi Fall 2009 Econ 3310 HOMEWORK 2 DUE MONDAY, OCTOBER 19, IN CLASS. 1. Forecasting and Random Variables Consider a finite state space S , a probability P on S , and a random variable X : S R . Suppose that the following penalty scheme is offered to a decisionmaker: she is asked to report a number , and then she will be subject to the penalty [ X ( s ) ] 2 in each state s S . Show that, if the individual minimizes her expected loss from this penalty scheme, using the probability P , then her optimal forecast is = E[ X ] = s X ( s ) P ( { s } ), the expectation of X with respect to the probability P . Observe that the simple penalty scheme for a single event we talked about in class as a prelude to our theorem about admissible forecasts is actually a special case of this result: just take X = 1 E , the indicator function of E . 2. Forecasting and Expected Utility This question asks you to verify something we claimed to be true in class: expectedutility decisionmakers provide admissible forecasts, even though the forecast systems they provide will not, in general, correspond to the subjective probability they use to evaluate bets. Consider a finite state space S , and consider an individual who must provide a forecast system : P ( S ) R , under the penalty scheme we described in class: that is, if the realized state is s , the individual must pay E P ( S ) [1 E ( s )...
View
Full
Document
 Spring '09
 Marciano
 Economics

Click to edit the document details