, Consider a consumer with Bernoulli utility function u(x1,x2), with a level of income m. Consider two possible set of prices, p = (P1,02) or p'' = (pi,p2). The consumer must decide whether they prefer a risky situation in which prices will be either p or p', each occurring with probability 1/2, or a safe situation in which prices are not random and equal to (p+p')/2 V (a) [10 points] First, assume that u(x1,22) = f(x1 + x2 where f is a strictly increasing function. Suppose that p= (1,3) and that p' = (3,1). Show that, irrespective of what the function f is, the consumer prefers the risky prices over the safe one. Let us now assume until the end of the problem that u(x1, x2) (minke where f is again(strictly increasing and also continuous /Prices can now be either (7:27) or p' = (1/7.1/7), where y>1. (b) (10 points) Show that when y is large enough, the consumer prefers the risky prices over the safe one. Jllint: look at what happens in the limit when y+ 20. We now assume that f, on top of being strictly increasing and continuous, is also strictly concave. Again, prices can either be p = (y) or p = (1/7,1/y) with equal probabilities, where y > 1. (c) [5 points) Show that my 27 (d) (5 points) Illustrate graphically that when fils sufficiently concave, the consumer can prefer the safe price over the risk ones. m 2 , Consider a consumer with Bernoulli utility function u(x1,x2), with a level of income m. Consider two possible set of prices, p = (P1,02) or p'' = (pi,p2). The consumer must decide whether they prefer a risky situation in which prices will be either p or p', each occurring with probability 1/2, or a safe situation in which prices are not random and equal to (p+p')/2 V (a) [10 points] First, assume that u(x1,22) = f(x1 + x2 where f is a strictly increasing function. Suppose that p= (1,3) and that p' = (3,1). Show that, irrespective of what the function f is, the consumer prefers the risky prices over the safe one. Let us now assume until the end of the problem that u(x1, x2) (minke where f is again(strictly increasing and also continuous /Prices can now be either (7:27) or p' = (1/7.1/7), where y>1. (b) (10 points) Show that when y is large enough, the consumer prefers the risky prices over the safe one. Jllint: look at what happens in the limit when y+ 20. We now assume that f, on top of being strictly increasing and continuous, is also strictly concave. Again, prices can either be p = (y) or p = (1/7,1/y) with equal probabilities, where y > 1. (c) [5 points) Show that my 27 (d) (5 points) Illustrate graphically that when fils sufficiently concave, the consumer can prefer the safe price over the risk ones. m 2