This preview shows pages 1–2. Sign up to view the full content.

Answers to Econ 210A Midterm, October 2010 Question 1. f ( x 1 ,x 2 ) = p max { x 1 ,x 2 } A. The function f is homogeneous of degree 1/2. To see this, note that for all t > 0 and all ( x 1 ,x 2 ) f ( tx 1 ,x 2 ) = p max { tx 1 ,tx 2 } = p t max { x 1 ,x 2 } = t 1 / 2 p max { x 1 ,x 2 } (1) = t 1 / 2 f ( x 1 ,x 2 ) B. The function f is neither concave, nor quasi-concave. One should be able to see this quickly by drawing an indiﬀerence curve and seeing that the at-least- as-good set is not convex. Therefore the function is not quasi-concave. Since a concave function must be quasi-concave, f cannot be concave either. One can show that the function is not quasi-concave by means of a single example. Let x = (1 , 0) and x 0 = (0 , 1). Then f ( x ) = 1 f ( x 0 ) = 1. Now consider the convex combination tx + (1 - t ) x 0 = ( t,t ) where 0 < t < 1. If f is quasi-concave, then it must be that f ( tx + (1 - t ) x 0 ) f ( x 0 ). But f ( tx + (1 - t ) x 0 ) = t < f ( x 0 ) = 1. So f cannot be quasi-concave. We also note from this same example that tf ( x ) + (1 - t ) f ( x 0 ) = 1 > f ( tx + (1 - t ) x 0 ) = t . But if f is a concave function tf ( x )+(1 - t ) f ( x 0 ) f ( tx +(1 - t ) x 0 ), so x is not a concave function. C The function f is not a convex function. The square root sign should be a tipoﬀ. Square root is a strictly concave function. To show that f is not a convex function, consider the two points x = (1 , 0) and x 0 = (4 , 0). Then f ( x ) = 1 and f ( x 0 ) = 2. Therefore 1 2 f ( x ) + 1 2 f ( x 0 ) = 2 . 5 . But f ( 1 2 x + 1 2 x 0 ) = f (2 . 5 , 0) = 2 . 5 < 1 2 f ( x ) + 1 2 f ( x 0 ) The function f is a quasi-convex function. A quick way to see that this must be true is to look at an indiﬀerence curve and see that it looks like the “worse-than” sets must be convex. To show this formally, suppose

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}