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Fix a consumption bundle 9 X CR with 90. 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 y to some consumption bundle z E X.

Such 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 g E X, g 0, and utility level u, define the benefit function by b(x, y) = max{ ER: u(1-8g) u,x-Bg EX} if x-Bg E X, u(x - Bg) u for some 6 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 z.

Explain now what happens in Figure 1 (b).

QUESTION 1 Having seen the success of Uber, you have decided to start a similar company in Davis, which you will call Unter. Unter will only provide rides within the city of Davis. The new and revolutionary feature of Unter is that, instead of charging per ride, it is based on a yearly membership. If you become a member of Unter then, within the year covered by your membership, you can take as many rides as you like for free. Each Davis resident is characterized by a pair (r, x), where / is the reservation price for membership (he/she is willing to become a member as long as the membership charge is less than or equal to r) and K is the number of rides that he/she would take per year (note that * is a constant, independent of the cost of becoming a member: riding a car around Davis is not particularly enjoyable, thus residents would make use of Unter only if needed; note also that each resident, if indifferent between applying and not applying for membership, he/she will apply). Unter's cost of providing each ride is constant and equal to c > 0. You find yourself in a situation of asymmetric information: each Davis resident knows his/her own (r, x), but you (the owner of Unter) only know the distribution P over the set of possible pairs (r,x) [ P(r, K) is the fraction of Davis residents who are characterized by the pair (r, x)]. You are risk neutral. Let / be the number of Davis residents. Because of tax reasons, you need to run a small company by enrolling as members not more than K residents. From now on by "price" we will mean the membership fee. Let us start with a simple example, where /= 24,000 and K = 50; furthermore, the possible pairs (r, A) and, for each pair, the number of residents who are characterized by that pair, are given in the following table. (r,K): (100,8) (120, 12) (100,16) (100, 24) (140,24) (120,36) (140,36) (120,40) (140,72) sidents: 2,000 1, 000 3,000 . 2, 000 1, 000 2, 000 4, 000 4,000 5,000 What price will you charge? [You preferences reflect the capitalist society in which you grew up: you want to make as much money as possible! Clearly, your answer will have to be conditional on the value of c, which is the cost of providing each ride.] (b) For this question assume that K > n, = N). Assume a uniform distribution, that is, 1=1 j=1 ny = ny, for every i, j, s, te (1, 2,...,10}QUESTION 1 Having seen the success of Uber, you have decided to start a similar company in Davis, which you will call Unter. Unter will only provide rides within the city of Davis. The new and revolutionary feature of Unter is that, instead of charging per ride, it is based on a yearly membership. If you become a member of Unter then, within the year covered by your membership, you can take as many rides as you like for free. Each Davis resident is characterized by a pair (r, x), where / is the reservation price for membership (he/she is willing to become a member as long as the membership charge is less than or equal to r) and K is the number of rides that he/she would take per year (note that * is a constant, independent of the cost of becoming a member: riding a car around Davis is not particularly enjoyable, thus residents would make use of Unter only if needed; note also that each resident, if indifferent between applying and not applying for membership, he/she will apply). Unter's cost of providing each ride is constant and equal to c > 0. You find yourself in a situation of asymmetric information: each Davis resident knows his/her own (r, x), but you (the owner of Unter) only know the distribution P over the set of possible pairs (r,x) [ P(r, K) is the fraction of Davis residents who are characterized by the pair (r, x)]. You are risk neutral. Let / be the number of Davis residents. Because of tax reasons, you need to run a small company by enrolling as members not more than K residents. From now on by "price" we will mean the membership fee. Let us start with a simple example, where /= 24,000 and K = 50; furthermore, the possible pairs (r, A) and, for each pair, the number of residents who are characterized by that pair, are given in the following table. (r,K): (100,8) (120, 12) (100,16) (100, 24) (140,24) (120,36) (140,36) (120,40) (140,72) sidents: 2,000 1, 000 3,000 . 2, 000 1, 000 2, 000 4, 000 4,000 5,000 What price will you charge? [You preferences reflect the capitalist society in which you grew up: you want to make as much money as possible! Clearly, your answer will have to be conditional on the value of c, which is the cost of providing each ride.] (b) For this question assume that K > n, = N). Assume a uniform distribution, that is, 1=1 j=1 ny = ny, for every i, j, s, te (1, 2,...,10}