Question

1. Prove mathematically, using Jensen’s inequality, that the budget set for a consumer choice problem with 2 goods, fixed prices and consumer wealth, and no ability to consume negative quantities, is a convex set.

Answer 1

Let us consider some weights and variables tf(x1) + (1-t) f(x2) graph of the function is the convex function of weighted means: f (tx1 + (1-t)x2) jensen's inequality is: t f(x1) + (1-t) f(x2) >= f (tx1 + (1-t)x2) The basic idea of Jensen's inequality is as follows: F[g (x)] <= [F(x)] Now if we take two quantities of consumption goods x1 and x2, all positive quantities and fixed prices p1 and p2 instead of tand (1-t) We can write the budget constraint as p1x1+p2x2, where p1 and p2 are constants. F[g (x1, x2)] = p1x1+p2x2; this should be the convex function g [F(X)] stands for the means of all points on the curve (p1x1i+p2x2i); I stands for all points along the curve or F(g(x1,x2)] Now for any positive numbers of x1 and x2, at any point on the convex function or the curved line, we find p1x1+p2x2 >= (p1x1+p2x2)/ (p1+p2); this holds for any positive numbers of x and p | x1 A Budget line = g [F(x)] Utility U = F(x1,x2) X2 In the above diagram the curved line represents the choice set of the consumers (due to diminishing marginal utility of a good, as a consumer consumes more of x1 his consume less of x2 and vice versa. Any point on the secant line or the average of the points on the convex function at each point is greater than any corresponding point on the curve. The secant line is the budget line p1x1+p2x2 while the curved line is the utility function, best represented as U (x1, x2)