averse. It can be shown that the partial derivative of v with respect to μw is positive, and the partial derivative of v with respect to σw is negative; thus more expected wealth is always liked, and more risk (as measured by the standard deviation of wealth) is always disliked. A mean-standard deviation indifference curve is defined as the locus of points (σw, μw) with σw plotted horizontally, such that Eu(w) has the same value at all points on the locus. Then the derivatives of v imply that every indifference curve is upward sloped: that is, along any indifference curve dμw / dσw > 0. Moreover, it can be shown that all such indifference curves are convex: along any indifference curve, d2μw / d(σw)2 > 0. Example 2: The portfolio analysis in example 1 can be generalized. If there are n risky assets instead of just one, and if their returns are jointly elliptically distributed, then all portfolios can be characterized completely by their mean and variance—that is, any two portfolios with identical mean and variance of portfolio return have identical distributions of portfolio return—and all possible portfolios have return distributions that are location-scale-related to each other. Thus portfolio optimization can be implemented using a two-moment decision model. Example 3: Suppose that a price-taking, risk-averse firm must commit to producing a quantity of output q before observing the market realization p of the product's price. Its decision problem is to choose q so as to maximize the expected utility of profit: Maximize Eu(pq – c(q) – g), where E is the expected value operator, u is the firm's utility function, c is its variable cost function, and g is its fixed cost. All possible distributions of the firm's random revenue pq, based on all possible choices of q, are location-scale related; so the decision problem can be framed in terms of the expected value and variance of revenue. Non-expected-utility decision making If the decision-maker is not an expected utility maximizer, decision-making can still be framed in terms of the mean and variance of a random variable if all alternative distributions for an unpredictable outcome are location-scale transformations of each other. See also Decision theory Intertemporal portfolio choice Microeconomics References Expected utility Financial risk modeling Portfolio theories