Question

Consider two-person households, which run on the principle each should experience the same utility from consumption. Let C1 = Person 1’s consumption, and C2 = Person 2’s consumption. They have $90/day to spend on consumption for the whole household.a) Assume the individual’s utility equals how much they consume. Determine how much each will consume, and what their utility levels will beb) In this case the first person feels competitive with the second, but the second feels compersion for the first. Their respective utility functions are as follows: U1 (C1,C2) = C1 * ( C1/C2), U2(C1,C2) = C1 * C2 Determine how much each will consume, and what their utility levels will be.c) Let the household be as if in part b. However, person 1 can increase the second person’s sense of compersion by giving a small gift that costs k. When they do utility functions become:U1 (C1,C2) = (C1 – k) * [ (C1 - k) / (C2 + k) ] , U2(C1,C2) = (C1 - k) 2 * (C2 + k) Determine the formulas for C1 and C2. If k = .5, determine how much each consumes and their utility levels.d) Describe how the gift changed their relationship and shares of consumption. e) Assume that a person must consume at least $5/day to sustain the household. Which of the above scenarios are sustainable as they are. Are there any where the gift giving becomes problematic ? What endogenous constraints would be required to make it sustainable (determine the new consumptions levels and utilities be, can the household still run by the equal utility principle?)

Answer 1

Answer and Explanation:

(A) In the event that U1 = C1 and U2 = C2,

at that point two get both same utility C1 must equal to C2

C1 + C2 = 90

substituting C1 in the place of C2 in the above equation we get

2C1 = 90

C1 = 90/2

C1 = 45

C2 = 45

Each one has utility = 45

(B) U1 = U2

C1(c1/c2) = c1*c2

c1 = c2^2

c1 + c2 = 90

substituting c1 = c2^2 in c1 + c2 = 90

c2^2 + c2 = 90

c2 = 9

c1 = 81

U1 = U2 = 981 = 729

(C)  U1 = u2

(C1 - 5)({c1 - 5}/{c2 + 5}) = (c1 - 5)(c2 + 5)

C1 - 5 = (c2 + 5)^2

C1 - 5 = c2^2 + 25 + 10c2

C1 = c2^2 + 30 + 10c2

90 = c2^2 + 30 + 10c2 + c2

0 = c2^2 - 60 + 11c2

C2 = 4,-15

-15 not possible C2 = 4

C1 = 90 - 4 = 86

U1 = U2 = 729

(D)  Due to satisfy their principle, gift decrease the consumption of individual 2 by same amount and Increase consumption of individual 1 by same amount.