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The first-derivative test depends on the "increasing–decreasing test", which is itself ultimately a consequence of the mean value theorem. It is a direct consequence of the way the derivative is defined and its connection to decrease and increase of a function locally, combined with the previous section.
A differentiable function f is (strictly) concave on an interval if and only if its derivative function f ′ is (strictly) monotonically decreasing on that interval, that is, a concave function has a non-increasing (decreasing) slope. [3] [4] Points where concavity changes (between concave and convex) are inflection points. [5]
When an increase in output by one firm raises the marginal revenues of the other firms, production decisions are strategic complements. When an increase in output by one firm lowers the marginal revenues of the other firms, production decisions are strategic substitutes. A supermodular utility function is often related to complementary goods ...
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.
Let and let (;): be a family of functions parameterized by , where (,) is a partially ordered set (or poset, for short). How does the correspondence (;) vary with ?. Standard comparative statics approach: Assume that set is a compact interval and (;) is a continuously differentiable, strictly quasiconcave function of .
[5] [6] If the person experiences an increase in wealth, he/she will choose to increase (or keep unchanged, or decrease) the number of dollars of the risky asset held in the portfolio if absolute risk aversion is decreasing (or constant, or increasing). Thus economists avoid using utility functions such as the quadratic, which exhibit ...
Assuming that the factor costs are constant (that is, that the firm is a perfect competitor in all input markets) and the production function is homothetic, a firm experiencing constant returns will have constant long-run average costs, a firm experiencing decreasing returns will have increasing long-run average costs, and a firm experiencing ...
The term convex is often referred to as convex down or concave upward, and the term concave is often referred as concave down or convex upward. [3] [4] [5] If the term "convex" is used without an "up" or "down" keyword, then it refers strictly to a cup shaped graph .