Problem set

Paper , Order, or Assignment Requirements

  1. (20 points) As part of your answer to the following questions, make sure to specify

what the universe of possible alternatives is.

(a) (10 points) Give an example of a relation that is complete but not transitive and

show it is complete and not transitive.

(b) (10 points) Give an example of a relation that is transitive but not complete and

show it is transitive and not complete.

  1. (40 points) Suppose X = fx; y; zg, and we have the following information about

an individual’s choice function:

c (B1) = fxg, c (B2) = fyg, and c (B3) = fzg,

where B1 = fx; yg, B2 = fy; zg and B3 = fx; zg.

(a) (10 points) Show that c satis…es …nite nonemptiness and choice coherence.

(b) (20 points) Show that there does not exist a utility function U : fx; y; zg ! R,

that can produce these choices via the usual formula (discussed in class):

c (Bn) = fx 2 Bn : U (x)  U (y) for all y 2 Bng , for n = 1; 2; 3.

(c) (10 points) Explain why your answers to parts (a) and (b) do not contradict the

Fundamental Theorem of Mindless Economics from lectures.

  1. (40 points) There are two goods in the economy, meat and potatoes. The price

of meat is p1 dollars per kilogram of meat. The price of potatoes is p2 per kilogram

of potatoes. The consumer has wealth W dollars to spend on meat and potatoes. Let

x1 denote the quantity of meat in kilograms she chooses to buy and let x2 denote the

quantity of potatoes in kilograms she chooses to buy. Suppose throughout that she can

only purchase non-negative quantities of either good.

Solve the following utility maximisation problems when the preferences of a consumer

are characterised by the following utility functions. For each part clearly specify what is

the quantity of each good the consumer wishes to purchase given her budget constraint.

(a) (10 points) U (x1; x2) = x21

x2

(Hint: MU1 (x1; x2) = 2x1x2 and MU2 (x1; x2) = x21

).

(b) (10 points) U (x1; x2) = ln x1 + ln x2

(Hint: MU1 (x1; x2) = 1

x1

and MU2 (x1; x2) = 1

x2

).

(c) (10 points) U (x1; x2) = x1 + ln x2

(Hint: MU1 (x1; x2) = 1 and MU2 (x1; x2) = 1

x2

).

(d) (10 points) U (x1; x2) = maxfx1; x2g That is,

U (x1; x2) =

 

x1 if x1  x2

x2 if x1 < x2

.

Warning: In this case MU1 (x1; x2) and MU2 (x1; x2) do not exist! But even if

they did, you would not want to …nd the bundle that equated the marginal rate

of substitution with the price ratio.