10. Vanna Boogie likes to have large parties. She also has a strong preference for having exactly as many men as women at her parties. In fact, Vanna's preferences among parties can be represented by the utility function U(x,y) = min{2x-y, 2y-x}, where x is the number of women and y is the number of men. The utility is 10 along the curve where x = y.
a) Use a pencil to draw the locus of points at which x = y. What point on this curve gives Vanna a utility of 10? Use blue ink to draw the line along which 2y-x = 10. When min{2x-y, 2y-x} = 2y-x, there are (more men than women, more women than men)? Draw a squiggly red line over the part of the blue line for which U(x,y) = min{2x-y, 2y-x} = 2y-x. This shows all the combinations that Vanna thinks are just as good as (10,10), but where there are (more men than women, more women than men)? Now draw a blue line along which 2x-y = 10. Draw a squiggly red line over the part of this new blue line for which min{2x-y, 2y-x} = 2x-y. Use a pencil to shade in the area on the graph that represents all combinations that Vanna likes at least as well as (10,10).
b) Suppose that there are 9 men and 10 women at Vanna's party. Would Vanna think it was a better party or a worse party if 5 more men came to her party?
c) If Vanna has 16 women at her party and more men than women, and if she thinks the party is exactly as good as having 10 men and 10 women, how many men does she have at the party?
d) If Vanna has 16 women at her party and more women than men, and if she thinks the party is exactly as good as having 10 men and 10 women, how many men does she have at her party?
