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Non‑convex sets have been incorporated in the theories of general economic equilibria, [2] of market failures, [3] and of public economics. [4] These results are described in graduate-level textbooks in microeconomics , [ 5 ] general equilibrium theory, [ 6 ] game theory , [ 7 ] mathematical economics , [ 8 ] and applied mathematics (for ...
A set of convex-shaped indifference curves displays convex preferences: Given a convex indifference curve containing the set of all bundles (of two or more goods) that are all viewed as equally desired, the set of all goods bundles that are viewed as being at least as desired as those on the indifference curve is a convex set.
Convex preferences imply that the indifference curves cannot be concave to the origin, i.e. they will either be straight lines or bulge toward the origin of the indifference curve. If the latter is the case, then as a consumer decreases consumption of one good in successive units, successively larger doses of the other good are required to keep ...
Right graph: With fixed probabilities of two alternative states 1 and 2, risk averse indifference curves over pairs of state-contingent outcomes are convex. In economics and finance, risk aversion is the tendency of people to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter ...
The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
A function f is concave over a convex set if and only if the function −f is a convex function over the set. The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield.
Convex demand curve. The demand is called convex (with respect to the origin [7]) if the (generally down-sloping) curve bends upwards, concave otherwise. [8] The demand curvature is fundamentally hard to estimate from the empirical data, with some researchers suggesting that demand with high convexity is practically improbable.
The following are among the properties of log-concave distributions: If a density is log-concave, so is its cumulative distribution function (CDF). If a multivariate density is log-concave, so is the marginal density over any subset of variables. The sum of two independent log-concave random variables is log-concave. This follows from the fact ...