Provide the solutions.

Question 2 When we studied consumer theory in ECN200A, we introduced various functions like utility functions, indirect utility functions, expenditure functions, Walrasian demand functions, Hicksian demand functions etc. There are other functions that are sometimes useful in the context of consumer theory. Let us use our knowledge of consumer theory and the techniques we learned to study one such function that is not discussed in Mas- Colell, Whinston, and Green (1995). Fix a consumption bundle g E X C R4 with g * 0. We will use this consumption bundle as a reference point. We want to define a function that measures how many units of this reference consumption bundle g a consumer is willing to give up in order to move from some utility level u to some consumption bundle r E X. Such a function may be useful in the context of development economics of societies in which one commodity (e.g., rice) is a natural reference commodity already. It is also of conceptual significance as it helps us to understand the consumer problem as a problem of maximizing the difference between benefits and costs. To this end, for reference consumption bundle ge X, g # 0, and utility level u, define the benefit function by b(x, u) = max {BER : u(x - Bg) > u,x - BgE X} ifx -Bg E X, u(x - Bg) 2 u for some B otherwise a. Let's try first to understand this function graphically by assuming L = 2. Consider first Figure 1 (a). It depicts an indifference curve representing a utility level u and a reference consumption bundle g. Further, it depicts b, the number of units of g the consumer is willing to give up to move from the indifference curve representing u to the consumption bundle r. Explain now what happens in Figure 1 (b). b. Explain what happens in Figure 1 (c).a. Let's try first to understand this function graphically by assuming L = 2. Consider first Figure 1 (a). It depicts an indifference curve representing a utility level u and a reference consumption bundle g. Further, it depicts b, the number of units of g the consumer is willing to give up to move from the indifference curve representing u to the consumption bundle z. Explain now what happens in Figure 1 (b). b. Explain what happens in Figure 1 (c). c. Let's derive the benefit function for the case of a Cobb-Douglas utility function u(x) = [12, x,' for at > 0, ( = 1,..., L, I E R. Set g = (1,0, ...,0). Then b(r, u) = max B s.t. (x1 - B) "1 x,' Zu. (>2 Derive b(x, u) (I.c., solve for the / that corresponds to b(I, u).) d. Consider now again the general definition of the benefit function defined above. Argue that b(r, u) is nonincreasing in u. e. Argue that if r e R4 and a + age R4, then b(r + ag, u) = a + b(r, u). f. Show that if the utility function u is quasiconcave with respect to z, then b(x, u) is concave with respect to the r.