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Mathematical methods for economic theory: 3.4 Quasiconcavity and quasiconvexity

mjo.osborne.economics.utoronto.ca · 4,147 words · saved by 1 readers

Now forget about the cows and the snow. Ask yourself whether the function defining the surface of the mountain is concave. It is if every straight line connecting two points on the surface lies everywhere on or under the surface. If, for example, the mountain is a perfect dome (half of a sphere), then this condition is satisfied, so that the function defined by its surface is concave. The condition is satisfied also if the mountain is a perfect cone. In this case, every straight line connecting the peak to another point on the surface lies exactly on the surface. Now suppose that the mountain is a deformation of a cone that gets progressively steeper at higher altitudes—call it a “pinched cone”. (Many mountains seem to have this characteristic when you try to climb them.) That is, suppose that when viewed from far away, the mountain looks like this: In this case, a straight line from the top of the mountain to any other point on the surface does not lie on or under the surface, but rat

Definitions and basic properties Think of a mountain in the Swiss Alps: cows grazing on the verdant lower slopes, snow capping the majestic peak. Now forget about the cows and the snow. Ask yourself whether the function defining the surface of the mountain is concave . It is if every straight line connecting two points on the surface lies everywhere on or under the surface. If, for example, the mountain is a perfect dome (half of a sphere), then this condition is satisfied, so that the function defined by its surface is concave. The condition is satisfied also if the mountain is a perfect cone

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