A consumer has preferences for two goods. Her preferences satisfy Axioms 1 through 4 as discussed in class.

. A consumer has preferences for two goods. Her preferences satisfy Axioms 1 through 4 as discussed in class.

1. Plot and label the following bundles. (6 points)
   • ????  ????5,5????
   • ????  ????1,9????
   • ????  ????9,8????
   • ????  ????5,6????
   • ????  ????0,2????
2. Assume ????A is indifferent to B ????. On a single line, list all the bundles in descending order of preference using (≻) to denote strict preference and (~) to denote indifference between adjacent pairs as, e.g., in the form ???? ≻ B ≻ C ~ D ≻ E. (6 points)

2. Consider an agent with preferences represented by the utility function:

????????????u(x,y)= 2(xy)+5
a. For each pair of bundles (i.e. ???? and ????), indicate whether ???? ≻ K, J~K, or J ≺ K. (4 points)

????A: ????7, 10????                                           C: ????2, 8????                                             E: ????7, 3????                                             G: ????10, 10????

B: ????4, 7???? _________________________ D: ????8, 4???? _________________________ F: ????6, 9???? _________________________ H: ????9, 6???? _________________________

b. Using the bundles in (a), make a list that orders the bundles according to the agent’s preferences. Start the descending list with the most preferred bundle and end with the least preferred bundle. (For example, as before ???? ≻ B ≻ C ~ D ≻ E ≻ ⋯ ) (4 points)

c. Consider a bundle K = (1,y) which contains one unit of good ???? and some amount of good y ????. If the consumer is indifferent between bundle ????K  and the bundle ????(3,3)????, how much of good ???? y does the bundle ????K contain? (4 points)

3. Consider the following four consumers (c1 ????????c2???? c3 ????c4)???? ???????????? with the following utility functions: (5 points each)

c1-  u(x,y)=4x+y

c2- u(x,y)= x^1/3 y^2/3

c3- u(x.y)= min(2x,2y)

c4- u(x,y)= min (x,3y)

On the appropriate graph below, draw each consumer’s indifference curves through the following points: ????(2,2????), (????4,4)????, (????6,6)???? and (????8,8)????. For consumer 2, your indifference curves will be a rough sketch and at best an approximate, but it should still “make sense”.

4. For the following exercise you need to draw indifference curves for people with preferences that exhibit different axioms. Everybody has preferences defined over two goods: ???? (spam) and ???? (cheesecake). Please draw an arrow to indicate the direction of preferences. Note that there may be multiple ways to draw each case, but you only need to show one.

a. Spamantha has preferences such that ????????????MUx >0, ???????????? ???? MUy >0, satisfy strong monotonicity, but do not satisfy strict convexity. (4 points)

b. Spamson has preferences such that ????????????MUx< 0, ???????????? ???? MUy< 0, and do not satisfy strong monotonicity or strict convexity. (4 points)

c. Spammy has preferences such that ???????????? ???? MUx>0, ???????????? ???? MUy< 0, and do not satisfy strong monotonicity or strict convexity. (4 points)

d. Spamuel has preferences such that ???????????? ???? MUx>0, ???????????? ???? MUy< 0, do not satisfy strong monotonicity, but do satisfy strict convexity. (4 points)

5. Determine whether each of the following utility functions are monotonic transformations of u(x,y)????????????. One way you can do this is by taking partial derivative of each utility function with respect to both x???? and y????. If we assume that ????????????u(x,y) ???????? is strictly increasing in ???? x and y????, then the function is a monotonic transformation of ????????????u(x,y)???????? if it is also strictly increasing in ???? x and ????y , for all x and y. This means the partial derivative of a monotonic transformation with respect to ???? x and ???? y must be both greater than 0. Show work, and make sure you state clearly whether the function is a monotonic transformation of ????????????u(x,y)???????? or not. (4 points each)

a(x,y)=2/3u(x,y)

b(x,y)= u(x,y)^3 -93

c(x,y) = 2/u(x,y)

d(x,y)= -2ln(u(x,y))

e(x,y)= 3u(x,y)+2u(x,y)^3

6. Solve for the amounts of goods x ???? and ????y that a consumer with the following preferences and budget constraint demands. For full credit you must show your work and state any assumption(s) you make. (20 points)

max ( 3+x )  (5+y)

s.to. 5x+2y less than or equal to 100

 

Looking for a Similar Assignment? Order now and Get 10% Discount! Use Coupon Code "Newclient"