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Intermediate Microeconomics - Consumer Theory Problem Set 1

1. Problem associated with rst video: Increasing transformation

Consider the following utility function u(x 1 , x 2 ) = x1

/2

1 x1

/2

2 .

(a) Calculate the MRS for this utility function.

(b) Consider the bundles ( x 1 , x 2 ) = (4 , 4) and ( x 1 , x 2 ) = (1 , 9). Which bundle is the mostpreferred (i.e. yield the highest utility)?

(c) Consider the following function f (x ) = x 4 and the dene a new utility function v(x 1 , x 2 ) =f (u (x 1 , x 2 )).

i. Show that v(x 1 , x 2 ) is an increasing transformation of u(x 1 , x 2 ).

ii. Answer, without calculating, which of the bundles above will be the most preferredaccording to v(x 1 , x 2 ).

iii. Calculate the MRS for v(x 1 , x 2 ). How it compares with the MRS for u (x 1 , x 2 ). Why?

2. Problem associated with second video: Demand with Cobb-Douglas utility func-tion

All the workers in a factory have the same utility function u(x 1 , x 2 ) = x 1 x 2 , earn the incomeof 10 and face the same prices p1 = 1 and p2 = 1.

(a) What is the optimal consumption bundle for the workers? What utility do they obtain?.

(b) Graph the indifference curve that goes through the optimal consumption bundle. In thesame graph draw the budget set.

(c) Unfortunately for the workers there is an increase in p1 of 100% . What is the new p1 ?What is the new optimal consumption bundle and what utility do they obtain?

(d) In the same graph as in point b, draw the indifference curve that goes through the newoptimal consumption bundle and also draw the new budget set.

3. Problem associated with third video: Demand with Perfect Complements utility

function

Mara lives in the beautiful country of Inationlandia. She earns 30 million pesos a month.She only consumes hamburguers and buns. The price of a dozen hamburgers is 2 million pesosand the prices of a dozen buns is 1 million pesos.

(a) Graph her budget set with dozens of hamburguers in the x axis and dozens of buns inthe y axis.

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(b) What is the slope of the budget line? Give an interpretation for this number.

(c) Tiring of having to deal with too many zeros in the peso bills the government decided toerase six zeros of every price and income. Graph her new budget set.

(d) Mara likes to eat each hamburguer with one bun. She doesnt like to eat hamburguersor buns alone. Graph her indifference curves that go through the following points (5,5),(10,10) and (15,15).

(e) Find a utility function that represents her preferences.

(f) Given her income and the prices of hamburguers and buns, what will she buy?

4. Problem associated with fourth video: demand with perfect substitutes

Consider the following utility function u(x 1 , x 2 ) = 2 x 1 + x 2 .

(a) Calculate the MRS for this utility function.(b) Given an income m and prices p1 and p2 derive the demand function for good 1. Represent

it graphically.

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